Non-commutative Hodge structures: Towards matching categorical and geometric examples

NON-CO MMUT A TIVE HODGE STR UCTURES: TO W ARDS MA TCHING CA TEGORIC AL AND GEOMETRIC EXAMPLES D. SHKL Y A RO V Abstra ct. T he sub ject of the presen t w ork is the de Rh am part o f non-commutativ e Hodge structures on the p eriodic cyclic homology of diﬀerential graded algebras and categories. W e dis- cuss explicit formula s for the corresp onding connection on the p erio dic cyclic homology viewed as a bundle o ver the punctured formal disk. Our main result sa ys that for the category of ma- trix facto rizations of a p olynomial the f ormula s repro duce, up to a certain shift, a well-kno wn connection on the associated twisted d e Rham cohomology which plays a central role in the geometric app roac h to the Ho dge theory of isolated singularities. Contents 1. In tro du ction 1 2. F ormalism of mixed complexes and u -connections 8 3. u -Connections on th e cyclic complexes of dg algebras 11 4. Mixed complexes and u -connections asso ciated with p olynomials 20 5. Applications of Theorem 1.1 23 App end ix A. Pro ofs for Section 2 26 App end ix B. Useful formulas 30 App end ix C . Pro ofs for S ection 3 32 App end ix D. Pro ofs for Section 4 44 References 56 1. Introduction The study of v arious generalizations of the classica l Ho dge theory has ev olv ed into a v ast branc h of mathematics; a compr ehensiv e o v erview of relev an t n otions and references can b e found in [22 ]. The present w ork is in the framew ork of the Ho dge theory of categories, as outlined in [1, 2, 12, 15, 16]. W e refer th e reader to these sources for an in tro duction to the circle of id eas surr ounding the su b ject, as well as motiv ation and references. Our main goal is to present a p iece of evidence (see Theorem 1.1 b elo w) in fa v or of the idea that the categorical Ho dge th eory s h ould include some kno wn geometric examples of generalized Ho dge structures as sp ecial cases. 2010 Mathematics Subje ct Classiﬁc ation. 16E45,16E40. This researc h was sup p orted by the ERC Starting Indep endent Researc her Grant StG N o. 2047 57-TQFT (K. W endland PI). 1 2 D. SHKL Y AR O V The p erio dic cyclic h omology of a diﬀerentia l graded (dg) category is well- kno wn to b e a direct generalization of the de Rham cohomolog y of a space, and it is natural to exp ect it to carry a Ho dge-lik e structure. This exp ectation has b een con v erted in to a p recise conjecture in [12, Section 2.2.2], at least in the case of the so-called homologically smo oth and prop er d g catego ries which are to b e thought of as analogs of s mo oth and pr op er v arieties. According to the conjecture, the cyclic homology of s uc h a category can b e end ow ed with a non-c ommutative Ho dge structur e . Despite the terminology , non-comm utativ e Ho d ge stru ctur es are deﬁned with- out referring to non-comm utativ e mathematics; they are a sub ject of their o wn, with a num b er of geometric applications [8, 10, 12]. The deﬁ n ition of a non -commutati v e Ho dge stru cture inv olv es tw o sets of data called the “de Rham data ” and the “Betti d ata” in [12]. Roughly , the former generalizes th e Ho dge ﬁ ltration of a classical Ho d ge stru cture (cf. section 3.5) while the latter is an an alog of the Q -structur e. Th e main obs tacle to pro ving the aforemen tioned conjecture seems to ha v e b een the fact that the p erio d ic cyclic h omology of, sa y , a C -linear dg category do es not carry an ob vious Q -stru ctur e in general. More on this can b e found in [12, S ection 2.2.6] and [11, Section 8]. The p resen t work concerns th e de Rh am part of the s ou ght-for non -commutati v e Ho d ge structure on th e cyclic homology whose origin has b een understo o d for some time n o w. The de Rham data of a non-comm utativ e Ho d ge structure can b e deﬁn ed as a pair ( H , ∇ ) where H is a Z / 2-graded free C { u } -mo du le of ﬁn ite ran k and ∇ is a meromorphic connection on H with a p ole of ord er at m ost 2 at the origin. There is a formal counte rpart of su c h data in whic h the r ing C { u } of conv ergent series is replaced with the rin g C [[ u ]] of f ormal series. W e will b e considerin g only the formal analog of the de Rh am d ata du e to the very nature of our main example, the cyclic h omology . In wh at follo ws, th e term “bundle on the formal (resp. punctured formal) disk” is u sed as a synonym of “free C [[ u ]]-mod ule of ﬁ nite rank (resp. ﬁ nite-dimensional C (( u ))-v ecto r sp ace)”. Accordingly , we will sp eak of “bu ndles with conn ection (on the formal disk)” instead of “the de Rham data of non-comm utativ e Ho dge stru ctures”. A general w a y of pro d ucing such bu n dles with connections was outlined in [12, S ection 4.2.4]. Let us repr o duce this idea here u sing three examples. Let X b e a compact K¨ ahler manifold and A ( X ) th e Z / 2-graded sp ace of complex C ∞ -forms on X , with the Z / 2-grading giv en by th e parit y of the degree of diﬀerent ial form s . Consider the Z / 2-graded C [[ u ]]-linear complex ( A ( X )[[ u ]] , ¯ ∂ + u∂ ) where A ( X )[[ u ]] stand s for the space of form al p o w er series with co eﬃcient s in A ( X ). By th e classical Ho d ge theory , its cohomology H ∗ ( A ( X )[[ u ]] , ¯ ∂ + u∂ ) is a v ector b undle on the form al d isk of rank dim C H D R ( X ). It carries a NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 3 meromorphic connection d eﬁned as f ollo ws . First, deﬁ n e ∇ X : A ( X )(( u )) → A ( X )(( u )) b y ∇ X = d du + Γ ′ 2 u , Γ ′ ∈ End( A ( X )) , Γ ′ | A p,q ( X ) = q − p and then observe th at (1.1) [ ∇ X , ¯ ∂ + u∂ ] = 1 2 u ( ¯ ∂ + u∂ ) Therefore, ∇ X induces a connection on the cohomology H ∗ ( A ( X )(( u )) , ¯ ∂ + u∂ ), the latter b eing the restriction of the vect or bun dle H ∗ ( A ( X )[[ u ]] , ¯ ∂ + u∂ ) to the p unctured formal d isk. F or our second example, let us tak e a p olynomial w = w ( y 1 , . . . , y k ) on Y = C k suc h that (1.2) w (0 , . . . , 0) = ∂ 1 w (0 , . . . , 0) = . . . = ∂ k w (0 , . . . , 0) = 0 W e will assume that the origin is the only critical p oin t of w . T o pro d uce a bund le with connec- tion asso ciated with w w e will mimic the ab ov e construction for K¨ ahler manifolds. Namely , let Ω( Y ) b e th e Z / 2-graded space of (holomorphic) diﬀeren tial forms with p olynomial co eﬃcien ts on Y . Consider the so-called twiste d de R ham c omplex (Ω( Y )[[ u ]] , − dw + ud ) where d is the (holomorphic) d e Rham diﬀerentia l and dw is the op erator of wedge multiplic ation with dw (as b efore, we v iew it as a Z / 2-graded C [[ u ]]-linear complex). I t is a classical fact that H ∗ (Ω( Y )[[ u ]] , − dw + ud ) = ( Ω k ( Y )[[ u ]] / ( − dw + ud )Ω k − 1 ( Y )[[ u ]] ∗ = k m o d 2 0 otherwise and that the cohomology is a free C [[ u ]]-mod ule, i.e. a v ector bu n dle on the formal d isk . It carries a connection b y the same argument as b efore: the op erator ∇ w : Ω( Y )(( u )) → Ω( Y )(( u )) giv en by (1.3) ∇ w = d du + w u 2 + Γ u , Γ ∈ End(Ω( Y )) , Γ | Ω p ( Y ) = − p 2 satisﬁes (cf. (1.1)) (1.4) [ ∇ w , − dw + ud ] = 1 2 u ( − dw + ud ) thereb y giving r ise to a connection on th e cohomology (cf. [12, Lemma 3.10]). The latter example is closely r elated to the sub ject of th e Gauss-Manin sys tems and the Briesk orn lattices (see section 5) and th rough that to the stud y of generalized Ho dge struc- tures and F rob enius m anifolds asso ciated with isolated singularities (see [9, 23] for review and references). Our last example is algebraic. Let A b e a diﬀerential Z / 2-graded algebra. Th e pr evious tw o examples suggest that we sh ould replace the complexes ( A ( X )[[ u ]] , ¯ ∂ + u∂ ) , (Ω( Y )[[ u ]] , − dw + ud ) 4 D. SHKL Y ARO V with the complex ( C ( A )[[ u ]] , b + uB ) (here ( C ( A ) , b ) is the Ho chsc hild c hain complex of A and B is the C on n es diﬀerenti al) and try to rep eat the construction. This do es not qu ite work sin ce, in general, the cohomology of the ab ov e complex has lots of torsion C [[ u ]]-submo du les meaning it is n ot a vec tor bund le. There are tw o p ossible options at th is p oint : either we work with those d g algebras for which the torsion su bmo du les d o not o ccur (conjecturally , th is is so for an y h omologica lly smooth and prop er dg algebra [16]), or w e replace the cohomolo gy of ( C ( A )[[ u ]] , b + uB ) by its image in the cohomology of ( C ( A )(( u )) , b + uB ), the p erio dic cyclic homology of A . Cho osing the ﬁrst option is analogous to working with compact K ¨ ahler manifolds only . Th en, as we discussed previously , there is a c hance that the resu lting bu ndles with connections can b e p romoted to full- ﬂedged n on-comm utativ e Hod ge structures. W e will kee p consid ering arbitrary dg algebras or, more pr ecisely , dg algebras with ﬁn ite-dimensional p erio dic cyclic homology; the latter condition guaran tees that we are still dealing with ﬁnite r ank v ector bu n dles on the formal disk. The p erio dic cyclic h omology , viewed as a bund le on the pun ctured form al disk, should carry a connection by a general argum en t ([16, Section 11.5],[1 2 , Section 2.2 .5]) in v olving the s o- called non-comm utativ e Gauss -Manin connection [7] (along the parameter of a one-parameter deformation of A ). In order to get a conn ection at the lev el of the p erio dic cyclic complex one should, p erhaps, rep eat th e same argum ent but for a r eﬁ ned v ersion of the non-comm utativ e Gauss-Manin connection obtained in [2 8] (see also [5 ]). W e hop e to return to th is idea on another o ccasion. A t the momen t, b orro wing an idea from [16, S ection 11.5], we can w rite out a connectio n ∇ A on the p erio d ic cyclic complex of A satisfying the same prop erty as in our geometric examples ab o v e: (1.5) [ ∇ A , b + uB ] = 1 2 u ( b + uB ) An explicit formula for ∇ A can b e f ound in s ection 3.3 (Prop osition 3.4). The origin of the form ula is easy to explain. The reader familiar with the non-comm utativ e diﬀerentia l calculus will notice that ∇ A is built by com bining a grading op erator on C ( A ) with th e non-comm utativ e Cartan homotop y op erator (cf. [7 , S ection 2]) corresp onding to the diﬀerent ial d of A . Then th e relation (1.5 ) for this connection is a sligh t v ariation on the non-commutat iv e Cartan homotop y form ula [7, (2.1)] (or, r ather, a very sp ecial case of the latter whic h go es bac k to [19]). Since ∇ A is written in terms of some b asic op erations on the Ho chsc h ild complex, it can b e easily generalized to deﬁn e a connection ∇ D on the cyclic complex of an arb itrary diﬀeren tial Z / 2-graded c ate gory D . NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 5 Notice that wh en the Z / 2-grading on A can b e lifted to a Z -grading, there is an other ob vious connection on the cyclic complex satisfying (1.5), namely ∇ A g r = d du + Γ ′ 2 u where Γ ′ is the corresp onding Z -grading op erator on C ( A ). It turns out that ∇ A and ∇ A g r coincide at the cohomologica l lev el (see 1 section 3.4). This observ atio n shows that the d e Rham part of non-comm utativ e Ho dge stru ctur es on the p erio d ic cyclic homology is not ve ry interesting in the case of Z -graded catego ries (in other w ords, the connections are not essen tial f or deve loping the Ho d ge theory in the Z -graded setting; for more on this, see s ection 3.5). On the other h and, this same observ atio n implies that our ﬁrst geometric example is a sp ecial case of th e algebraic one, at least, wh en X is a smo oth pr o jectiv e v ariet y . Let us presen t the argum ent in a v ery sk etc hy m anner since this is kn o wn, and is not the main p oin t of the pap er an yw a y . The claim is that th er e is an isomorp hism of b undles on the formal d isk H ∗ ( A ( X )[[ u ]] , ¯ ∂ + u∂ ) ∼ = H ∗ ( C (par dg X )[[ u ]] , b + uB ) (here par dg X stands for the dg category of p erfect complexes 2 on X ) which in d uces an isomor- phism of connections 3 ( H ∗ ( A ( X )(( u )) , ¯ ∂ + u∂ ) , ∇ X ) ∼ = ( H ∗ ( C (par dg X )(( u )) , b + uB ) , ∇ par dg X ) T o see this, note ﬁ rst that the category par dg X is Z -graded and therefore we can replace ∇ par dg X with ∇ par dg X g r . F urthermore, H ∗ ( A ( X )[[ u ]] , ¯ ∂ + u∂ ) ∼ = H ∗ ( A ( X ) , ¯ ∂ )[[ u ]] H ∗ ( C (par dg X )[[ u ]] , b + uB ) ∼ = H ∗ ( C (par dg X ) , b )[[ u ]] since the left-hand sides are known to be f ree C [[ u ]]-mo dules. T he right -hand sides can b e endo w ed w ith meromorph ic connections giv en by th e same formulas that d eﬁne ∇ X and ∇ par dg X g r and the isomorphisms ab ov e can b e easily chosen so that they p reserv e the connections. T o conclude the argumen t, we observ e th at the t w o Γ ′ s - in the deﬁnitions of ∇ X and ∇ par dg X g r , resp ectiv ely - matc h un der a well -kno wn isomorphism H ∗ ( A ( X ) , ¯ ∂ ) ∼ = H ∗ ( C (par dg X ) , b ) The goal of the present pap er is to explain why our se c ond geometric example is a sp ecial case of th e algebraic one. Namely , we w ill pr o v e 1 If we knew t hat ∇ A w as the connection that the aforementioned argumen t p ro du ced then this statemen t w ould also follo w from th e argument. 2 See [14, Section 5.3] and references th erein for a d iscussion of the cyclic homology of this dg category . 3 Let us use the same notation for connections at the level of complexes and at the level of cohomology . 6 D. SHKL Y ARO V Theorem 1.1. Ther e is an isomorphism of bund les on the formal disk H ∗ (Ω( Y )[[ u ]] , − dw + ud ) ∼ = H ∗ ( C (MF( w ))[[ u ]] , b + uB ) which induc es an isomorphism of c onne c tions ( H ∗ (Ω( Y )(( u )) , − dw + ud ) , ∇ w ) ∼ =  H ∗ ( C (MF( w ))(( u )) , b + uB ) , ∇ MF( w )  Her e MF( w ) stands for the dg c ate gory of matrix factorizations of w . Let u s omit the d eﬁnition of MF( w ) sin ce w e will not n eed it. W e will us e as a b lac k b o x a result of [6] w hic h sa ys that this d g category is (quasi-)equiv alent to the category of dg mo dules o v er a certain diﬀeren tial Z / 2-graded algebra, A w . This r esult allo ws us to r eplace the cyclic complex of the catego ry MF( w ) w ith the m uc h smaller cyclic complex of A w . The d g algebra is actually quite simp le, so let us r epro du ce its d eﬁnition here. As a Z / 2-graded algebra, it is the tensor p ro duct C [ Y ] ⊗ End C V where V is the Z / 2-graded space of p olynomials in k o dd v ariables θ 1 , . . . , θ k . The only part of A w that dep end s on w is the diﬀerentia l. Th e d iﬀerential dep ends, in fact, on more than j ust w itself 4 : one also n eeds to pic k a decomp osition of w of the form w = y 1 w 1 + . . . + y k w k Then the diﬀerent ial on A w is simply the commutato r with D ( w ) := P i  y i ∂ ∂ θ i + w i θ i  . Clearly , D ( w ) 2 = w and so d w = [ D ( w ) , − ] squares to 0. The second k ey ingred ien t used in the p ro of of Theorem 1.1 is an explicit quasi-isomorphism from the Ho c hsc hild complex of MF( w ) to the complex (Ω( Y ) , − dw ) constructed recen tly in [26] b y com bining results of lo c.cit. with some facts obtained earlier in [4, 6]. W e will n eed only the comp osition of th is quasi-isomorphism w ith the embed ding of the Ho c hsc hild complexes ( C ( A w ) , b ) ֒ → ( C (MF( w )) , b ) The comp osition is still a quasi-isomorphism, by the aforement ioned result of [6 ]. It is worth w h ile to emphasize that th e results from [6, 26] we need (and, as a consequence, Theorem 1.1 itself ) hold true if w e replace the pair ( C [ Y ] , w ) with m ore general pairs ( R, w ) ha ving isolate d critical lo ci (cf. deﬁn itions in [6 , Section 3] and a discussion and citations at the end of [26, Section 3]). F or example, we could hav e started with the pair ( C [ Y ] (0) , w ) w here Y = C k , C [ Y ] (0) is the lo cal rin g at the origin, and w satisﬁes (1.2) and has an isolated singularit y at the origin. F or th e sak e of simplicit y we will stic k to the p olynomial algebra and will not attempt to pr esen t the r esu lts in their “most natur al” generalit y . 4 Thus, our notation A w is a b it misleading. NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 7 T o conclud e, let us p oint out that T h eorem 1.1 has s ome imp lications for the classical singu- larit y theory (not related to n on-comm utativ e Ho dge s tructures). Th ese w ill b e d iscussed brieﬂy in section 5. Organization of the pap er. Th e main b o dy of the p ap er comprises four sections. In section 2 w e devel op some simple con v enien t f ormalism of mixed complexes with conn ec- tions. It can b e view ed as a step to w ard d eﬁning the derive d c ate gory of m ixed complexes with connections but w e do n ot pursue this id ea due to lac k of motiv ation. In section 3 w e recall th e deﬁn ition of the mixed cyclic complexes of dg algebras, b oth unital and non-u nital, and then mo v e on to describing connections on th ese complexes. W e pr esen t sev eral equiv alen t sets of formulas for the connections and discuss separately the case of d iﬀer- en tial Z -graded algebras. The section is concluded with an explanation of why the language of connections is su itable for deve loping the Ho dge theory in the categorical f r amew ork. In section 4 w e explain the pr o of of Theorem 1.1 . Section 5 is devot ed to some app lications of Th eorem 1.1. In ord er to k eep the main b o dy of the text as short as p ossible, we ha v e collecte d all the pro ofs in several app en dices whic h o ccupy most of the pap er. Con v en tions. All complexes and dg algebras in this p ap er are Z / 2-graded, unless w e ex- plicitly say otherwise. Th at is, our complexes are pairs ( C , d ), where C is a Z / 2-graded v ector space C = C ev en ⊕ C odd and d is an o dd op erator suc h that d 2 = 0, and our dg algebras are su c h complexes equipp ed with a compatible m ultiplication. In sections 2.1–3.4 we are wo rking o v er a grou n d ﬁeld k w hose c haracteristic is n ot equal to 2. Starting f rom section 3.5 the ground ﬁeld is C . W e will b e also considering complexes o v er the ﬁeld of form al Laurent series b ut it will b e clear from the con text (or an explicit comment) whic h of the ﬁelds is imp ortant at a giv en m omen t. Ac kno w ledgemen ts. I w ould lik e to th an k Y. Soib elman for exp laining some b asics of the catego rical Ho dge theory to me sev eral yea rs ago. I am also grateful to M. Herb s t, C. Hertling, D. Mu r fet, A. P olishch uk , H. Ruddat, E. Sc h eidegger, C. Sev enhec k, and K. W endland for inspiring d iscussions on matrix facto rizations and Ho dge theory . Sp ecial thanks are due to C. Seve nhec k for a n umber of in teresting commen ts on the su b ject matter of th is p ap er, to the anon ymous referee for v ery v aluable su ggestions on how the exp osition could b e improv ed, and to b oth of them for sp otting several t yp os. 8 D. SHKL Y ARO V 2. F ormalism of mixed compl exes and u -co nnections 2.1. Mixed complexes. A mixe d c omplex is a triple ( C , b, B ) where ( C , b ) is a ( k -linear) com- plex and B is an o dd op erator on C that an ti-co mmute s with b and squares to 0. In the Z / 2 setting, the mixed complexes are merely double complexes but the use of the term “mixed” suggests the analogy with the mixed complexes in the con v en tional Z -graded setting [17], and also emphasizes the fact that b and B p la y diﬀerent roles. Recall that we are using u to den ote a formal (eve n) v ariable, playing the r ole of a co ordin ate on the formal disk, and V (( u )) to denote the k (( u ))-v ector sp ace of formal Laurent s eries with co eﬃcien ts in V where V is any k -vect or space. Giv en a mixed complex ( C , b, B ), w e w ill call the k (( u ))-linear complex ( C (( u )) , b + uB ) the u -totalization of ( C , b, B ). As it should b e clear from the Introd uction, the mixed complexes and their u -totali zations are used as means to constru ct vec tor bu ndles o v er the formal d isk. With this in mind , w e will restrict our selv es to mixed complexes of ﬁnite typ e , i.e. those complexes whose u -totalizations ha v e ﬁnite-dimensional cohomology ov er k (( u )). A morphism of mixed complexes is deﬁned as a m orphism of the un d erlying ordinary com- plexes that commutes w ith the B -op erators. W e will use also the follo w ing w eak er n otion 5 : a u -morphism of mixed complexes is a k (( u ))-linear morphism of the corresp onding u -totaliz ations (as complexes). An y u -morphism f ( u ) : ( C , b, B ) → ( C ′ , b ′ , B ′ ) can b e written in the form f ( u ) = ∞ X i = k u i f i for some k ∈ Z wh ere the co eﬃcien ts f i : C → C ′ are ev en op erators. The reader, we hop e, understands ho w to extend this terminology to other notions used in the stud y of complexes, su c h as quasi-isomorphisms, homotop y equiv alences, etc. F or in stance, t w o u -morphism s f ( u ) and g ( u ), with the s ame domain and range, w ill b e called u -homotop ic if there is an o dd k (( u ))-linear op erator H ( u ) : C (( u )) → C ′ (( u )) (a u -homotop y ) such th at f ( u ) − g ( u ) = H ( u )( b + uB ) + ( b ′ + uB ′ ) H ( u ) Again, eac h u -homotop y has the form H ( u ) = ∞ X i = k u i H i 5 This is a version of the S -morph isms [17]. NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 9 where k ∈ Z and H i : C → C ′ are o dd op erators. Acco rdingly , t w o mixed complexes ( C , b, B ) and ( C ′ , b ′ , B ′ ) will b e called u -homotop y e q u ivalent if th ere are u -morph isms ι ( u ) : ( C , b, B ) ⇄ ( C ′ , b ′ , B ′ ) : p ( u ) (2.1) suc h that ι ( u ) p ( u ) and p ( u ) ι ( u ) are u -h omotopic to id C ′ and id C , resp ectiv ely . It is a standard fact th at for complexes of v ector spaces th ere is not m uc h diﬀerence b et w een homotop y equiv alences and qu asi-isomorphisms. T his fact applies to our setting since we are w orking with complexes of vec tor spaces ov er k (( u )): Prop osition 2.1. Any u -quasi- i somorph ism f ( u ) : ( C , b, B ) → ( C ′ , b ′ , B ′ ) c an b e c omplete d to a u -homotopy e quivalenc e f ( u ) : ( C , b, B ) ⇄ ( C ′ , b ′ , B ′ ) : g ( u ) . Giv en a m ixed complex ( C , b, B ), the image of the cohomolog y of ( C [[ u ]] , b + uB ) in that of ( C (( u )) , b + uB ) under the canonical ( k [[ u ]]-linear) embedd ing of the complexes will b e called the c anonic al u -lattic e . The canonical u -lattic e p la ys the role of the bun dle on the formal disk asso ciated with the mixed co mplex. W e will say th at a u -morphism of mixed complexes is r e gu lar if the image of the canonical u -lattice u nder the induced op erator on the cohomology of the u -totalizat ions b elongs to the canonical u -lattice. In particular, a u -homotop y equiv alence will b e called regular if b oth u -morph isms in v olv ed are regular. 2.2. u -Connections. The f ollo wing deﬁn ition h as b een inspired by the examples w e d iscu ssed in the Introd uction (see (1.1), (1.4 ), (1.5)): A u -c onne ction on a mixed complex ( C , b, B ) is a k -linear op erator ∇ : C (( u )) → C (( u )) satisfying the p rop erties: [ ∇ , u ] = id C (meaning ∇ = d du + A ( u ) where A ( u ) is a k (( u ))-linear op erator from C (( u )) to itself ) and [ ∇ , b + uB ] = 1 2 u ( b + uB ) ( ⇔ [ A ( u ) , b + uB ] = 1 2 u ( b − uB ) ) (2.2) The equalit y (2.2 ) implies that ∇ descends to the cohomolog y of the u -total ization, thereby giving rise to a connection on the corresp onding bu ndle on the punctured formal d isk. Remark 2.2. Notice that sin ce ∇ do es not commute with the diﬀerent ial, a u -connection on a mixed complex is not a Z / 2-graded complex of D -mo d u les on the punctur ed formal disk, in th e con v en tional sense. Ho w ev er, w ith any mixed complex with a u -connection one can asso ciate 10 D. SHKL Y ARO V a Z -graded (unb ounded) complex of D -mo dules as f ollo ws . Let ( C , b, B , ∇ ) b e as ab o v e and D n = D n ( C , b, B , ∇ ) b e th e sequence of D -mo du les given by D n = (  C ev en (( u )) , ∇| C ev en (( u ))  h n 2 i , n ev en  C odd (( u )) , ∇| C odd (( u ))  h n 2 i , n o dd (2.3) where ( V (( u )) , ∇ ) h n 2 i := ( V (( u )) , ∇ − n 2 · 1 u ), a T ate t wist of ( V (( u )) , ∇ ) (cf. [12, Section 2.1.7]). Then b y (2.2) the diﬀerenti al b + uB : D n → D n +1 is a morp hism of D -mo dules for all n . W e will return to this construction in s ection 3.5; apart from that s ection, it will not b e u sed in what follo ws. T o deﬁne m orp hisms of mixed complexes with u -connections, we need Lemma 2.3. L et ( C , b, B , ∇ = d du + A ( u )) and ( C ′ , b ′ , B ′ , ∇ ′ = d du + A ′ ( u )) b e mixe d c omplexes with u -c onne ctions. Then for any u -morphism f ( u ) of the mixe d c omplexes d f ( u ) du + A ′ ( u ) f ( u ) − f ( u ) A ( u ) (2.4) is a u -morphism. Pro of is given in s ection A.1.  W e will say that f ( u ) deﬁ nes a morphism of mixe d c omplexes with u - c onne ctions , or simply a morphism of u - c onne ctions , if (2.4 ) is u -homotopic to 0. As the next claim sho ws, b eing a morph ism of u -connections is stable un der passing to u - homotopic u -morphisms . Prop osition 2.4. L et ( C , b, B , ∇ ) , ( C ′ , b ′ , B ′ , ∇ ′ ) b e mixe d c omplexes with u -c onne ctions and f ( u ) , g ( u ) : ( C , b, B ) → ( C ′ , b ′ , B ′ ) two u -morphisms that ar e u -homotopic to e ach other. If f ( u ) is a morphism of u -c onne ctions then so i s g ( u ) . Pro of is given in s ection A.2.  The follo wing statemen t follo ws immediately fr om the d eﬁnitions: Prop osition 2.5. The c omp osition of morphisms of u - c onne c tions is a morphism of u -c onne ctions. W e w ill sa y that t w o mixed complexes w ith u -connections ( C , b, B , ∇ ) and ( C ′ , b ′ , B ′ , ∇ ′ ) (or simply t w o u -connections ∇ and ∇ ′ ) are homotopy gauge e qui valent if there is a u -homotop y equiv alence (2.1) such that p ( u ) and ι ( u ) are morp h isms of u -connections. By Prop osition 2.5, homotopy gauge equiv alence is an equ iv alence relation. NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 11 Prop osition 2.6. If f ( u ) : ( C , b, B , ∇ ) → ( C ′ , b ′ , B ′ , ∇ ′ ) is a u -quasi-isomorphism and a mor- phism of u -c onne ctions then ∇ and ∇ ′ ar e homotopy gauge e qu ivalent. Pro of is given in s ection A.3.  Homotop y gauge equiv alence can b e reﬁned as f ollo w s : W e ma y r equire it to b e fur nished b y regular u -homotop y equiv alences, in the sense of the previous section. This is similar to the notion of holomorphic equiv alence in the classical theory of connections. In all the examples w e will consider in th e remaining sections, the u -homotop y equiv alences will b e regular, and so in all the statemen ts we will formulate homotopy gauge equ iv alence can b e un dersto o d in the ab o v e stronger sense. A v ery sp ecial instance of homotop y gauge equiv alence is the case when one u -connection is obtained from another one, living on the same mixed complex, b y adding a u -endomorphism that is u -homotopic to 0 (in this case the tw o u -connections are homotop y gauge equiv alent with resp ect to the iden tit y endomorphism of th e mixed complex). Since there is not an y actual “gauging” in this cases, we will simply sa y that tw o suc h u -connections are e qual up to a u -homotop y . W e will conclud e this section with the follo wing result which allo ws one to transfer u -connections using u -homotop y equ iv alences. Prop osition 2.7. Given a u -homotopy e quivalenc e (2.1) and a u -c onne c tion ∇ ′ = d du + A ′ ( u ) on ( C ′ , b ′ , B ′ ) , ther e e xi sts a u -c onne ction ∇ on ( C , b, B ) that is homo topy gauge e quivalent to ∇ ′ with r e sp e ct to p ( u ) and ι ( u ) . E xplicitly, one c an set ∇ = d du + A ( u ) , A ( u ) := p ( u ) dι ( u ) du + p ( u ) A ′ ( u ) ι ( u ) + 1 2 u H ( u )( b − uB ) (2.5) wher e H ( u ) is any u -homotopy suc h that p ( u ) ι ( u ) = id C + ( b + uB ) H ( u ) + H ( u )( b + uB ) . Pro of is given in s ection A.4.  3. u -Connections on the cyclic comple xes of dg algeb ras 3.1. The cyclic complexes of unit a l dg algebras. Let A b e a (not necessarily u nital) dg algebra and sA stand f or A with the rev ersed Z / 2-grading. Giv en a ∈ A , th e corresp ond ing elemen t of sA will b e denoted b y sa . The parit y of a will b e denoted b y | a | ; thus, | sa | = | a | − 1. Let C ( A ) = A ⊗ T ( sA ) = ∞ M n =0 A ⊗ s A ⊗ n 12 D. SHKL Y ARO V equipp ed with the indu ced Z / 2-grading. W e will write the elemen ts of A ⊗ sA ⊗ n as a 0 [ a 1 | a 2 | . . . | a n ] (i.e. a 0 [ a 1 | a 2 | . . . | a n ] = a 0 ⊗ sa 1 ⊗ sa 2 ⊗ . . . ⊗ sa n ), or simp ly a 0 , if n = 0. Throughout the pap er, w e will use the follo wing con v en tion: for an op er ator T from C ( A ) to an ywhere we w ill write T n +1 for the restriction of T on to A ⊗ sA ⊗ n . C ( A ) is the u nderlying Z / 2-graded space of the Ho c hsc hild c hain complex of A . Let us recall the deﬁnition of the Ho c hsc hild diﬀerent ial, b . Let τ d enote the cyclic p erm utation on C ( A ): τ n +1 ( a 0 [ a 1 | a 2 | . . . | a n ]) = ( − 1) | sa 0 | P n i =1 | sa i | a 1 [ a 2 | . . . | a n | a 0 ] One can easily see that τ n +1 n +1 = 1. Set δ (0) n +1 ( a 0 [ a 1 | a 2 | . . . | a n ]) = da 0 [ a 1 | a 2 | . . . | a n ] µ (0) n +1 ( a 0 [ a 1 | a 2 | . . . | a n ]) = ( 0 n = 0 ( − 1) | a 0 | a 0 a 1 [ a 2 | . . . | a n ] n ≥ 1 and δ ( i ) n +1 := τ − i n +1 δ (0) n +1 τ i n +1 , µ ( i ) n +1 := τ − i n µ (0) n +1 τ i n +1 , i = 1 , . . . , n Explicitly , δ ( i ) n +1 ( a 0 [ a 1 | a 2 | . . . | a n ]) = ( − 1) P i − 1 k =0 | sa k | a 0 [ a 1 | . . . | da i | . . . | a n ] (3.1) µ ( i ) n +1 ( a 0 [ a 1 | a 2 | . . . | a n ]) = ( ( − 1) P i k =0 | sa k | +1 a 0 [ a 1 | . . . | a i a i +1 | . . . | a n ] i < n − ( − 1) | sa n | ( | a 0 | + P n − 1 k =1 | sa k | ) a n a 0 [ a 1 | . . . | a n − 1 ] i = n (3.2) Then the Ho c hsc hild diﬀerential b is deﬁ ned as follo ws 6 b = b ( δ ) + b ( µ ) , b ( δ ) n +1 = n X i =0 δ ( i ) n +1 , b ( µ ) n +1 = n X i =0 µ ( i ) n +1 (3.3) That b squares to 0 can b e deduced f orm the formulas w e w r ite out in section B. The cohomology of ( C ( A ) , b ) is called the Ho c hsc hild h omology of A . Let u s recall no w the deﬁnition of the m ixed complex that computes the c yc lic homolo gy of (unital) dg algebras. The under lyin g ordinary complex is ( C ( A ) , b ), and B is deﬁned as follo ws. Assume A is unital. S et N n +1 = n X i =0 τ i n +1 , h n +1 ( a 0 [ a 1 | a 2 | . . . | a n ]) = 1[ a 0 | a 1 | a 2 | . . . | a n ] 6 This deﬁnition of the Ho chsc hild diﬀerential is easily seen to b e equiv alen t to th e one given in [7, Section 1]. NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 13 Then B n +1 = (1 − τ − 1 n +2 ) h n +1 N n +1 Clearly , B is an o d d op erator that squares to 0. Again, the formulas from section B imp ly that B anti-c omm utes with b oth b ( δ ) and b ( µ ). The cohomology of the u -totalizat ion of this mixed complex is called the p erio dic cyclic h omology of A . 3.2. The cyclic complexes of non-unital dg algebras. Notice that when A is non-un ital, the deﬁnition of B from the previous section do es not mak e sens e. T o ﬁ x th e pr oblem, one replaces the Ho chsc h ild c hain complex with a qu asi-isomorphic one, whic h con tains a replacement for the unit. Th e aim of the present section is to recall the deﬁnition of this new complex. Let A b e a dg algebra, with or without unit. Denote by A + the dg algebra obtained from A b y adjoining a unit: A e = A ⊕ k e, | e | = 0 de = 0 , ae = ea = a ∀ a ∈ A Consider the f ollo w in g Z / 2-graded ve ctor space: C e ( A ) = A ⊕ ∞ M n =1 ( A e ⊗ sA ⊗ n ) The op erators δ ( i ) n +1 and µ ( i ) n +1 on C ( A ) extend to C e ( A ) via the formulas (3.1 ) and (3.2), and w e get a complex ( C e ( A ) , b e = b e ( δ ) + b e ( µ )) wh ere the diﬀerenti als b e ( δ ), b e ( µ ) are deﬁn ed as b efore. It is nothing b ut the so-called norm alizatio n of the u sual Ho c hsc hild complex of A e [17]. The ve ctor space C e ( A ) admits the follo wing d ecomp osition: C e ( A ) = C ( A ) ⊕ C + ( A ) (3.4) where C + ( A ) = ∞ M n =1 ( k e ⊗ sA ⊗ n ) Note, how eve r, that C + ( A ) is not stable un der b e ( µ ), so it is not a sub complex. With (3.4) in m in d, w e can represen t the elemen ts of C e ( A ) by column -vect ors with t wo comp onent s, the ﬁrs t comp onen t b eing an element of C ( A ) and the second one an elemen t of C + ( A ). Accordingly , op erators on C e ( A ) can b e represen ted by 2 × 2 matrices. F or example, the diﬀerent ial b e ( δ ) preserve s the decomp osition (3.4) and therefore can b e repr esen ted by the matrix  b ( δ ) 0 0 b ( δ )  14 D. SHKL Y ARO V The op erator µ (0) , on th e other han d , preserves C ( A ) but maps C + ( A ) to C ( A ); it is then represent ed by the matrix  µ (0) µ (0) 0 0  Note that, formally , the t w o copies of b ( δ ) in the ﬁr st matrix (or µ (0) in the second matrix) represent diﬀer ent op erators, since they h a v e diﬀerent domains/ranges. In this notation, the diﬀerential b e ( µ ) is easily seen to corresp ond to the follo w ing up p er- triangular matrix  b ( µ ) b v ( µ ) 0 b h ( µ )  , where 7 b v ( µ ) n +1 = µ (0) n +1 + µ ( n ) n +1 : k e ⊗ sA ⊗ n → A ⊗ s A ⊗ ( n − 1) b h ( µ ) n +1 = n − 1 X i =1 µ ( i ) n +1 : k e ⊗ sA ⊗ n → k e ⊗ sA ⊗ ( n − 1) Let us describ e the analog of B for the new complex ( C e ( A ) , b e ). Deﬁne h e : C ( A ) → C + ( A ) b y h e n +1 ( a 0 [ a 1 | a 2 | . . . | a n ]) = e [ a 0 | a 1 | a 2 | . . . | a n ] Then, in our matrix notation, B e : C e ( A ) → C e ( A ) is giv en by B e n +1 =  0 0 h e n +1 N n +1 0  B e is an o dd op erator that squares to 0 and an ti-comm utes with b e : b e ( δ ) B e = − B e b e ( δ ) , b e ( µ ) B e = − B e b e ( µ ) W e ha v e n o w t w o a p riori diﬀerent deﬁn itions of the cyclic h omology for unital dg algebras, namely , via ( C ( A ) , b, B ) and ( C e ( A ) , b e , B e ). Let u s explain why the t w o deﬁnitions are equiv a- len t. Let A b e a u n ital dg algebra. Consider the follo wing u -morp hisms ι ( u ) : ( C ( A ) , b, B ) → ( C e ( A ) , b e , B e ) , ι ( u ) =  id C ( A ) 0  + u  0 h e hN  (3.5) p ( u ) : ( C e ( A ) , b e , B e ) → ( C ( A ) , b, B ) , p ( u ) =  id C ( A ) (1 − τ − 1 ) hµ (0)  (3.6) (the op er ator hµ (0) in the last line m aps e [ a 0 | a 1 | a 2 | . . . | a n ] to 1[ a 0 | a 1 | a 2 | . . . | a n ]). 7 W e extend our prev ious conv ention regarding subscripts as follo ws: F or an op erator T from C e ( A ) to anywhere w e will write T n +1 for the restriction of T on to ( A ⊗ sA ⊗ n ) ⊕ ( k e ⊗ sA ⊗ n ). NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 15 Prop osition 3.1. ι ( u ) and p ( u ) establish a u -homotop y e quivalenc e b etwe en ( C ( A ) , b, B ) and ( C e ( A ) , b e , B e ) . Pro of is given in s ection C.1.  The last result in this section relates th e cyclic complexes of a dg algebra A and its opp osite dg algebra A ◦ . Recall that A ◦ is ju st A with a new p r o duct: a ′ ⊗ a ′′ 7→ ( − 1) | a ′ || a ′′ | a ′′ a ′ Consider the f ollo w in g isomorphism of Z / 2-graded v ector s paces: Φ = Φ A : C e ( A ) → C e ( A ◦ ) , a 0 [ a 1 | a 2 | . . . | a n ] 7→ ( − 1) n + P 1 ≤ i 0 s. t. P a l R l l ! < ∞} . Th e dimension of G is µ , the Milnor num b er of the s in gularit y . The Briesk orn lattice is deﬁned as f ollo ws : G 0 := Ω k ( Y an ) (0) /dw ∧ d Ω k − 2 ( Y an ) (0) The canonical map Ω k ( Y an ) (0) → G is kno wn to indu ce an em b edd ing G 0 ֒ → G w h ose image we will also denote by G 0 . G 0 is a free C { { u } } -submo d u le in G of rank µ and so G = G 0 [ u − 1 ] (this is one r eason to call G 0 lattic e ). Observe that [ t, u ] = u 2 , so it is n atural to set ∂ u := u − 2 t . S ince the resulting C { { u } } [ u − 1 ][ ∂ u ]- mo dule structure on G enco d es the original D -mo dule structure (it is called th e F our ier-Laplace transform of th e latter), we will f r om no w on f orget the v ariable t and w ork exclusiv ely with u . W e don’t know whether or n ot the analytic versions of the Gauss-Manin sys tem and the Briesk orn lattice we just d escrib ed admit an y categ orical interpretation. F ortunately , d ue to the regularit y of the D -mo dule G it suﬃces for a num b er of questions (e.g. compu ting the NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 25 aforemen tioned inv arian ts of the singularit y) to w ork w ith the f ollo w in g formal v ersions: b G 0 := C [[ u ]] ⊗ C { { u } } G 0 , b G := b G 0 [ u − 1 ] These formal v ersions are precisely what we are able to reconstru ct using the categ orical ap- proac h. Namely , it w as prov ed in [25, Section 2] that b G 0 ≃ H k ( b Ω • ( Y ) (0) [[ u ]] , − dw + ud ) , b G ≃ H k ( b Ω • ( Y ) (0) (( u )) , − dw + ud ) where b Ω • ( Y ) (0) are the spaces of formal diﬀeren tial forms. Moreo ver, the argument in [25] sho ws that w e w ould ha v e got the s ame b G 0 and b G if we started with the sp aces Ω • ( Y alg ) (0) of diﬀerent ial forms with co eﬃcient s in the lo cal algebra C [ Y ] (0) of rational functions r egular at the origin. This, together w ith the remark at the v ery end of the Introdu ction r egarding the generalit y of our results, implies th at the C [[ u ]]-mo dule b G 0 and the C (( u ))-linear space b G can b e extracted from the category of matrix factorizati ons f or the pair ( C [ Y ] (0) , w ). What ab out the D -mo dule stru ctur e? It follo ws from the ab o v e deﬁn itions that the action of ∂ u on b G ≃ H k ( b Ω • ( Y ) (0) (( u )) , − dw + ud ) is giv en by the formula (5.1) ∂ u = d du + w u 2 The connection on th e twiste d de Rham cohomology indu ced b y (1.3), wh ic h we are able to pro du ce starting f r om the category of matrix factorizations, diﬀers fr om (5.1 ) by the term k 2 · 1 u , a kind of T ate twist (cf. the deﬁnition right after the equation (2.3)). Note that k , the num b er of v ariables w d ep ends on, c annot b e extracted fr om the catego ry MF( w ) du e to the celebrated Kn¨ orrer p erio dicit y: the categ ories MF( w ) and MF( w + xy ) are equiv alen t. T o su mmarize the previous discussion, Theorem 1.1 allo w s us to repro d uce, u p to the ab ov e T ate t wist, the formal Gauss-Manin system and the formal Briesk orn lattice starting from the catego rical data. As a consequence, the category of matrix factorizations do es rememb er ab out the inv arian ts that are enco d ed in the Gauss-Manin system, but only mo du lo information in - v olving the n um b er k . Let us explain this using t wo examples. Our ﬁr st example is the sp e ctrum of w . I t is d eﬁned as follo ws. Th e D -mo d ule b G is regular and, thus, it carries the canonical decreasing ﬁ ltration V α b G (see section 3.5). This ﬁ ltration induces a ﬁltration on b G 0 /u b G 0 ≃ b Ω k ( Y ) (0) /dw ∧ b Ω k − 1 ( Y ) (0) ( ≃ Jac( w )) b y the images of V α b G ∩ b G 0 . Th en th e sp ectrum is the function d : Q → Z , d ( α ) := dim Gr α V ( b G 0 /u b G 0 ) 26 D. SHKL Y ARO V Its su pp ort is known to b elong to the in terv al (0 , k ); moreo v er, d ( α ) = d ( k − α ) (cf. [21, Example 1.8]). W e can rep eat this construction starting from the connection in d uced by (1.3) instead of (5.1 ). The r esulting V -ﬁltration will diﬀer from the previous one by a shif t: V α new b G = V α + k 2 old b G . T hus, w e will obtain a sh ifted v ersion of the sp ectrum d new ( α ) = d old ( α + k 2 ), an even function d new ( − α ) = d new ( α ) with sup p ort in ( − k 2 , k 2 ). Our second example is the Ste enb rink Ho dge ﬁltr ation on the v anishing cohomology of w . The latter is nothing bu t HP k (MF( w )) with its n on-comm utativ e Ho dge ﬁltration (see section 3.5), as one can infer by ins p ecting the formulas presen ted in [24]. Since k is not a categ orical in v ariant , we reco v er the Steen brink Ho dge ﬁ ltration only up to an un determined shift. Appendix A. Proofs for S ection 2 A.1. Proof of Lemma 2.3. By (2.2 )  d f ( u ) du + A ′ ( u ) f ( u ) − f ( u ) A ( u )  ( b + uB ) = d f ( u ) du ( b + uB ) + ( b ′ + uB ′ ) A ′ ( u ) f ( u ) + 1 2 u ( b ′ − uB ′ ) f ( u ) − ( b ′ + uB ′ ) f ( u ) A ( u ) − 1 2 u f ( u )( b − uB ) Th us,  d f ( u ) du + A ′ ( u ) f ( u ) − f ( u ) A ( u )  ( b + uB ) − ( b ′ + uB ′ )  d f ( u ) du + A ′ ( u ) f ( u ) − f ( u ) A ( u )  = d f ( u ) du ( b + uB ) − ( b ′ + uB ′ ) d f ( u ) du + 1 2 u ( b ′ − uB ′ ) f ( u ) − 1 2 u f ( u )( b − uB ) T o see that the latter equals 0, one can s im p lify the ﬁrst part of the expr ession usin g the Leibniz rule as follo ws: d f ( u ) du ( b + uB ) − ( b ′ + uB ′ ) d f ( u ) du = d du  f ( u )( b + uB ) − ( b ′ + uB ′ ) f ( u )  −  f ( u ) d du ( b + uB ) − d du ( b ′ + uB ′ ) f ( u )  = − f ( u ) B + B ′ f ( u ) NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 27 A.2. Proof of Prop osition 2.4. I t suﬃces to show th at any u -morphism, u -h omotopic to 0, is a morp hism of u -connections. Let f ( u ) = H ( u )( b + uB ) + ( b ′ + uB ′ ) H ( u ) Then, computing mo dulo terms u -homotopic to 0 d f ( u ) du + A ′ ( u ) f ( u ) − f ( u ) A ( u ) ∼ H ( u ) B + B ′ H ( u ) + A ′ ( u ) H ( u )( b + uB ) + A ′ ( u )( b ′ + uB ′ ) H ( u ) − H ( u )( b + uB ) A ( u ) − ( b ′ + uB ′ ) H ( u ) A ( u ) = H ( u ) B + B ′ H ( u ) + A ′ ( u ) H ( u )( b + uB ) + (( b ′ + uB ′ ) A ′ ( u ) + 1 2 u ( b ′ − uB ′ )) H ( u ) − H ( u )( A ( u )( b + uB ) − 1 2 u ( b − uB )) − ( b ′ + uB ′ ) H ( u ) A ( u ) ∼ H ( u ) B + B ′ H ( u ) + 1 2 u ( b ′ − uB ′ ) H ( u ) + H ( u ) 1 2 u ( b − uB ) = 1 2 u ( b ′ + uB ′ ) H ( u ) + H ( u ) 1 2 u ( b + uB ) ( ∼ ab ov e stands for ‘ u -homotopic’). A.3. Proof of Prop osition 2.6. Due to Pr op osition 2.1, we only n eed to pr o v e Lemma A.1. If one of the u - morphisms p ( u ) , ι ( u ) is a morph ism of u -c onne ctions then the other one is also a morphisms of u -c onne ctions. Pro of. Let us assume for instance that ι ( u ) is a morphism of u -connections: dι ( u ) du + A ′ ( u ) ι ( u ) − ι ( u ) A ( u ) ∼ 0 Then p ( u ) dι ( u ) du p ( u ) + p ( u ) A ′ ( u ) ι ( u ) p ( u ) − p ( u ) ι ( u ) A ( u ) p ( u ) ∼ 0 or, using the Leibniz ru le d du ( p ( u ) ι ( u )) p ( u ) − dp ( u ) du ι ( u ) p ( u ) + p ( u ) A ′ ( u ) ι ( u ) p ( u ) − p ( u ) ι ( u ) A ( u ) p ( u ) ∼ 0 (A.1) Using p ( u ) ι ( u ) = id C + ( b + uB ) H ( u ) + H ( u )( b + uB ) (A.2) ι ( u ) p ( u ) = id C ′ + ( b ′ + uB ′ ) H ′ ( u ) + H ′ ( u )( b ′ + uB ′ ) (A.3) 28 D. SHKL Y ARO V (A.1) is equiv alen t to dp ( u ) du + A ( u ) p ( u ) − p ( u ) A ′ ( u ) ∼ d du (( b + uB ) H ( u ) + H ( u )( b + uB )) p ( u ) − dp ( u ) du  ( b ′ + uB ′ ) H ′ ( u ) + H ′ ( u )( b ′ + uB ′ )  + p ( u ) A ′ ( u )  ( b ′ + uB ′ ) H ′ ( u ) + H ′ ( u )( b ′ + uB ′ )  − (( b + uB ) H ( u ) + H ( u )( b + uB )) A ( u ) p ( u ) Th us, it suﬃces to sho w that th e latter expr ession is u -homotopic to 0. By (2.2) and the Leibniz rule, it is u -homotopic to ( B H ( u ) + H ( u ) B ) p ( u ) − dp ( u ) du  ( b ′ + uB ′ ) H ′ ( u ) + H ′ ( u )( b ′ + uB ′ )  + 1 2 u p ( u )( b ′ − uB ′ ) H ′ ( u ) + 1 2 u H ( u )( b − uB ) p ( u ) ∼ B H ( u ) p ( u ) + H ( u ) B p ( u ) − B p ( u ) H ′ ( u ) + p ( u ) B ′ H ′ ( u ) + 1 2 u p ( u )( b ′ − uB ′ ) H ′ ( u ) + 1 2 u H ( u )( b − uB ) p ( u ) = B H ( u ) p ( u ) − B p ( u ) H ′ ( u ) + 1 2 u p ( u )( b ′ + uB ′ ) H ′ ( u ) + 1 2 u H ( u )( b + uB ) p ( u ) = B H ( u ) p ( u ) − B p ( u ) H ′ ( u ) + 1 2 u ( b + uB ) p ( u ) H ′ ( u ) + 1 2 u H ( u ) p ( u )( b ′ + uB ′ ) ∼ B H ( u ) p ( u ) + 1 2 u ( b − uB ) p ( u ) H ′ ( u ) − 1 2 u ( b + uB ) H ( u ) p ( u ) = 1 2 u ( b − uB ) p ( u ) H ′ ( u ) − 1 2 u ( b − uB ) H ( u ) p ( u ) = 1 2 u ( b − uB )( p ( u ) H ′ ( u ) − H ( u ) p ( u )) The latter coincides with ( b + uB )  A ( u )( H ( u ) p ( u ) − p ( u ) H ′ ( u ))  +  A ( u )( H ( u ) p ( u ) − p ( u ) H ′ ( u ))  ( b ′ + uB ′ ) Indeed, this follo ws fr om (2.2) and th e equalit y ( H ( u ) p ( u ) − p ( u ) H ′ ( u ))( b ′ + uB ′ ) = − ( b + uB )( H ( u ) p ( u ) − p ( u ) H ′ ( u )) whic h, in its turn, is due to (A.2) and (A.3).  NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 29 A.4. Proof of Prop osition 2.7 . First of all, we need to s h o w that (2.5) is a u -connection, i.e. w e need to v erify the prop er ty (2.2):  d du + p ( u ) dι ( u ) du + p ( u ) A ′ ( u ) ι ( u ) + 1 2 u H ( u )( b − uB ) , b + uB  = B + p ( u )  dι ( u ) du , b + uB  + p ( u )[ A ′ ( u ) , b + uB ] ι ( u ) +  1 2 u H ( u )( b − uB ) , b + uB  = B − p ( u )  ι ( u ) , d du ( b + uB )  + 1 2 u p ( u )( b − uB ) ι ( u ) +  1 2 u H ( u )( b − uB ) , b + uB  = B − p ( u ) [ ι ( u ) , B ] + 1 2 u p ( u )( b + uB − 2 uB ) ι ( u ) +  1 2 u H ( u )( b − uB ) , b + uB  = B − p ( u ) ι ( u ) B + p ( u ) B ι ( u ) + 1 2 u p ( u ) ι ( u )( b + uB ) − p ( u ) B ι ( u ) +  1 2 u H ( u )( b − uB ) , b + uB  = 1 2 u ( b + uB ) − (( b + uB ) H ( u ) + H ( u )( b + uB )) B + 1 2 u (( b + uB ) H ( u ) + H ( u )( b + uB ))( b + uB ) +  1 2 u H ( u )( b − uB ) , b + uB  = 1 2 u ( b + uB ) By Lemm a A.1 to complete the pro of it is enough to s ho w that ι ( u ) is a m orp hism from ∇ to ∇ ′ , i.e. dι ( u ) du + A ′ ( u ) ι ( u ) − ( ι ( u ) p ( u ) dι ( u ) du + ι ( u ) p ( u ) A ′ ( u ) ι ( u ) + 1 2 u ι ( u ) H ( u )( b − uB )) ∼ 0 This can b e rewritten u sing (A.3) as f ollo ws : −  ( b ′ + uB ′ ) H ′ ( u ) + H ′ ( u )( b ′ + uB ′ )  dι ( u ) du −  ( b ′ + uB ′ ) H ′ ( u ) + H ′ ( u )( b ′ + uB ′ )  A ′ ( u ) ι ( u ) − 1 2 u ι ( u ) H ( u )( b − uB ) ∼ 0 Using the Leibn iz rule and (2.2), the left-hand side is u -homotopic to − d du  ( b ′ + uB ′ ) H ′ ( u ) ι ( u ) + H ′ ( u ) ι ( u )( b + uB )  + d du  ( b ′ + uB ′ ) H ′ ( u ) + H ′ ( u )( b ′ + uB ′ )  ι ( u ) + 1 2 u H ′ ( u )( b ′ − uB ′ ) ι ( u ) − 1 2 u ι ( u ) H ( u )( b − uB ) ∼ − B ′ H ′ ( u ) ι ( u ) − H ′ ( u ) ι ( u ) B + B ′ H ′ ( u ) ι ( u ) + H ′ ( u ) B ′ ι ( u ) + 1 2 u H ′ ( u )( b ′ − uB ′ ) ι ( u ) − 1 2 u ι ( u ) H ( u )( b − uB ) = − H ′ ( u ) ι ( u ) B + H ′ ( u ) B ′ ι ( u ) + 1 2 u H ′ ( u )( b ′ − uB ′ ) ι ( u ) − 1 2 u ι ( u ) H ( u )( b − uB ) 30 D. SHKL Y ARO V = − H ′ ( u ) ι ( u ) B + 1 2 u H ′ ( u )( b ′ + uB ′ ) ι ( u ) − 1 2 u ι ( u ) H ( u )( b − uB ) = − H ′ ( u ) ι ( u ) B + 1 2 u H ′ ( u ) ι ( u )( b + uB ) − 1 2 u ι ( u ) H ( u )( b − uB ) = 1 2 u H ′ ( u ) ι ( u )( b − uB ) − 1 2 u ι ( u ) H ( u )( b − uB ) = ( H ′ ( u ) ι ( u ) − ι ( u ) H ( u )) 1 2 u ( b − uB ) Rep eating the argum en t at th e end of the p ro of of Lemma A.1 , the latter is u -homotopic to 0. Appendix B. Use ful formulas In this app endix we list v arious “commutat ion relations” b et w een the op erators int ro du ced in section 3. T o a v oid dup licating results, w e will view δ ’s, µ ’s, τ etc. as op erators on C ( A e ) where A e is the u n italizati on of A . Then most of the formulas b elo w ma y b e inte rpreted in t w o wa ys, namely , as equalities b et w een op erators whose domain is either C ( A ) or C + ( A ). Similarly , we will use th e sym b ols lik e b ( µ ) , b v ( µ ) or b h ( µ ) to d enote the op erators on C ( A e ) giv en by the same formulas as in the main b o dy of the text. Let us also d enote b y µ ( ∗ ) the op erator on C ( A e ) deﬁned by µ ( ∗ ) n +1 := µ ( n ) n +1 Using this notation, the op erators b h ( µ ), b v ( µ ) and b ( µ ) are related as follo ws b ( µ ) = b v ( µ ) + b h ( µ ) = µ (0) + µ ( ∗ ) + b h ( µ ) W e will also omit sub scripts in symb ols lik e µ ( i ) n +1 , when it d o es not lead to confu sion. Lemma B.1. a ) δ ( k ) τ j = τ j δ ( k + j ) b ) µ ( k ) n τ j n = ( τ j n − 1 µ ( k + j ) n k + j ≤ n − 1 τ j − 1 n − 1 µ ( k + j − n ) n k + j ≥ n c ) b ( δ ) τ = τ b ( δ ) d ) ( b ( µ ) − µ ( ∗ ) ) τ = τ ( b ( µ ) − µ (0) ) e ) b ( µ )(1 − τ ) = (1 − τ )( b ( µ ) − µ (0) ) f ) ( b ( µ ) − µ ( ∗ ) ) N = N b ( µ ) NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 31 Pro of. a) and b) follo w fr om deﬁ nitions; c) follo ws from a); d) and e) follo w from b). Finally , b oth hand sides of f ) are equal to N µ (0) N .  Lemma B.2. a ) δ ( i ) δ ( j ) = − δ ( j ) δ ( i ) ( ∀ i, j ) b ) µ ( i ) µ ( j ) = − µ ( j ) µ ( i +1) ( j ≤ i ) c ) δ ( i ) n µ ( j ) n +1 =            − µ ( j ) n +1 δ ( i +1) n +1 0 ≤ j < i ≤ n − 1 − µ ( j ) n +1 δ ( i ) n +1 0 ≤ i < j ≤ n − 1 − µ ( n ) n +1 δ ( i ) n +1 1 ≤ i ≤ n − 1 , j = n − µ ( i ) n +1 δ ( i +1) n +1 − µ ( i ) n +1 δ ( i ) n +1 0 ≤ i = j ≤ n (When i = n in the last line, δ ( i ) n and δ ( i +1) n +1 stand for δ (0) n and δ (0) n +1 , r e sp e ctively.) Pro of. a) Using the deﬁnitions of the δ ’s, δ (0) n +1 δ ( l ) n +1 = − δ ( l ) n +1 δ (0) n +1 when l 6 = 0. This holds tr u e for l = 0 as w ell (b oth hand sides v anish). Th e general case reduces to this sp ecial one by means of Lemma B.1 a). b) Again, by th e deﬁnition of the µ ’s, µ ( l ) n µ (0) n +1 = − µ (0) n µ ( l +1) n +1 when 1 ≤ l ≤ n − 2. This holds true for l = 0 and n − 1, as in these cases it is equ iv alen t to th e asso ciativit y of the m ultiplication. The general case reduces to this sp ecial one by m eans of Lemma B.1 b). P art c) is pr ov ed similarly (the last case is just the Leibniz ru le).  The follo wing lemma is a corollary of the pr evious one: Lemma B.3. a ) b ( δ ) δ ( i ) = − δ ( i ) b ( δ ) b ) b ( δ ) µ ( i ) = − µ ( i ) b ( δ ) The remaining form ulas inv olve the op erator h e . Lemma B.4. a ) δ (0) h e = 0 , δ ( i ) h e = − h e δ ( i − 1) ( i ≥ 1) b ) µ (0) h e = id, µ ( i ) n +2 h e n +1 = − h e n µ ( i − 1) n +1 (1 ≤ i ≤ n ) , µ ( ∗ ) h e = − τ − 1 32 D. SHKL Y ARO V Pro of. Th e equalities are straigh tforw ard except, p erhaps, the very last one. W e need to show that τ n +1 µ ( n +1) n +2 h e n +1 = − id n +1 . By (3.2) τ n +1 µ ( n +1) n +2 h e n +1 ( a 0 [ a 1 | a 2 | . . . | a n ]) = τ n +1 µ ( n +1) n +2 ( e [ a 0 | a 1 | a 2 | . . . | a n ]) = − ( − 1) | sa n | P n − 1 k =0 | sa k | τ n +1 ( a n [ a 0 | . . . | a n − 1 ]) = − ( − 1) | sa n | P n − 1 k =0 | sa k | ( − 1) | sa n | P n − 1 k =0 | sa k | ) a 0 [ a 1 | . . . | a n ] = − a 0 [ a 1 | . . . | a n ]  F rom this lemma one easily d educes Lemma B.5. a ) b ( δ ) h e = − h e b ( δ ) b ) b v ( µ ) h e = 1 − τ − 1 c ) b h ( µ ) h e = − h e ( b ( µ ) − µ ( ∗ ) ) = − h e ( µ (0) + b h ( µ )) The formulas ab o v e hold true if w e replace C ( A e ) by C ( A ) and h e b y h . Appendix C. Proofs for Section 3 C.1. Pro of of Proposition 3.1. W e will start b y proving Lemma C.1. ι ( u ) and p ( u ) ar e u - morphisms of mixe d c ompl exes. Pro of. Let u s sho w that ι ( u ) is a u -morph ism. The only p art that is not immediate is the equalit y B e  id C ( A ) 0  −  id C ( A ) 0  B =  0 h e hN  b − b e  0 h e hN  By Lemmas B.1 c), B.5 a)  0 h e hN  b ( δ ) − b e ( δ )  0 h e hN  = 0 so w e need to show that B e  id C ( A ) 0  −  id C ( A ) 0  B =  0 h e hN  b ( µ ) − b e ( µ )  0 h e hN  or, equiv alent ly  − (1 − τ − 1 ) hN h e N  =  − b v ( µ ) h e hN h e hN b ( µ ) − b h ( µ ) h e hN  The ﬁrst comp onents of the t w o vec tors coincide b y Lemma B.5 b ), so it remains to prov e that h e N = h e hN b ( µ ) − b h ( µ ) h e hN . NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 33 The latter follo ws from L emm as B.1 f ), B.5 c), B.4 b ). Let us sho w no w that p ( u ) is a u -morphism. The only n on -trivial part is the equalit y b ( µ )(1 − τ − 1 ) hµ (0) = b v ( µ ) + (1 − τ − 1 ) hµ (0) b h ( µ ) of op erators from C + ( A ) to C ( A ). Since h e : C ( A ) → C + ( A ) is bijectiv e, it is enough to prov e that b ( µ )(1 − τ − 1 ) hµ (0) h e = b v ( µ ) h e + (1 − τ − 1 ) hµ (0) b h ( µ ) h e view ed as op erators on C ( A ). Since µ (0) h e = 1 and b v ( µ ) h e = (1 − τ − 1 ) (Lemmas B.4, B.5), the latter is equ iv alen t to b ( µ )(1 − τ − 1 ) h = (1 − τ − 1 ) + (1 − τ − 1 ) hµ (0) b h ( µ ) h e whic h, by Lemmas B.1 e), d ), redu ces to ( µ (0) + b h ( µ )) h = 1 + hµ (0) b h ( µ ) h e What remains is to use Lemmas B.4 b), B.5 c).  T o complete the pro of of Prop osition 3.1, w e w ill p ro v e Lemma C.2. p ( u ) ι ( u ) = id C ( A ) + ( b + uB ) H ( u ) + H ( u )( b + uB ) , H ( u ) := u (1 − τ − 1 ) hhhN (C.1) ι ( u ) p ( u ) = id C e ( A ) + ( b e + uB e ) H e ( u ) + H e ( u )( b e + uB e ) , H e ( u ) :=  0 0 0 h e hµ (0)  (C.2) Pro of. (C.1): Ob s erv e that p ( u ) ι ( u ) = id C ( A ) + u (1 − τ − 1 ) hhN W e will show that bH ( u ) + H ( u ) b = u (1 − τ − 1 ) hhN , B H ( u ) + H ( u ) B = 0 The second equalit y is obvio us. Let us pro v e the ﬁrst one. By Lemmas B.1 c), B.5 a) b ( δ ) H ( u ) + H ( u ) b ( δ ) = 0 so w e only need to sho w that b ( µ )(1 − τ − 1 ) hhhN + (1 − τ − 1 ) hhhN b ( µ ) = (1 − τ − 1 ) hhN This follo ws from Lemmas B.1 e),d),f ), B.4 b), B.5 c). 34 D. SHKL Y ARO V (C.2): Ob s erv e that ι ( u ) p ( u ) = id C e ( A ) +  0 (1 − τ − 1 ) hµ (0) 0 − id C + ( A )  + u  0 0 h e hN 0  W e will show that b e H e ( u ) + H e ( u ) b e =  0 (1 − τ − 1 ) hµ (0) 0 − id C + ( A )  , B e H e ( u ) + H e ( u ) B e =  0 0 h e hN 0  Again, th e latter equalit y is ob vious and we w ill pr o v e only the f ormer one. By Lemm as B.1 c), B.5 a), B.3 b) b e ( δ ) H e ( u ) + H e ( u ) b e ( δ ) = 0 so it remains to sho w that  b ( µ ) b v ( µ ) 0 b h ( µ )   0 0 0 h e hµ (0)  +  0 0 0 h e hµ (0)   b ( µ ) b v ( µ ) 0 b h ( µ )  =  0 (1 − τ − 1 ) hµ (0) 0 − id C + ( A )  or, equiv alent ly b v ( µ ) h e hµ (0) = (1 − τ − 1 ) hµ (0) , b h ( µ ) h e hµ (0) + h e hµ (0) b h ( µ ) = − id C + ( A ) The ﬁrst equalit y is du e to Lemma B.5 b). The second equalit y is du e to Lemmas B.5 c), B.4 b), B.2 b).  C.2. Pro of of Proposition 3.4. That the op erators ∇ and ∇ ◦ are du al to eac h other follo ws from the form ulas we wrote out wh ile pr o ving Prop osition 3.2. Thus, it suﬃces to sho w that only one of the op erators ∇ , ∇ ◦ is a u -connection. Let u s sho w that, say , ∇ ◦ is a u -connection. Lemma C.3. a ) [ U ◦ ( δ ) , b e ( δ )] = [ U ◦ ( δ ) , b e ( µ )] = 0 b ) [ V ◦ ( δ ) , B e ] = [ V ◦ ( δ ) , b e ( δ )] = 0 c ) [ V ◦ ( δ ) , b e ( µ )] + [ U ◦ ( δ ) , B e ] = 1 2 b e ( δ ) d ) [Γ , b e ( δ )] = 0 , [Γ , b e ( µ )] = 1 2 b e ( µ ) , [Γ , B e ] = − 1 2 B e Pro of. P art a): Th e ﬁrst comm utator equals 0 b y Lemma B.3 . Let us pro v e th e second equ alit y . W riting b oth b e ( µ ) and U ◦ ( δ ) in the matrix notation, the s econd equalit y is equiv alen t to µ ( n − 1) n δ ( n − 1) n · n X i =0 µ ( i ) n +1 = n − 1 X i =0 µ ( i ) n · µ ( n ) n +1 δ ( n ) n +1 (C.3) where b oth h and sides are view ed as op erators either on C ( A ) or on C + ( A ). NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 35 By Lemma B.2 c) µ ( n − 1) n δ ( n − 1) n n X i =0 µ ( i ) n +1 = µ ( n − 1) n δ ( n − 1) n n − 2 X i =0 µ ( i ) n +1 + δ ( n − 1) n µ ( n − 1) n +1 + δ ( n − 1) n µ ( n ) n +1 ! = − µ ( n − 1) n n − 2 X i =0 µ ( i ) n +1 δ ( n ) n +1 + µ ( n − 1) n +1 δ ( n − 1) n +1 + µ ( n − 1) n +1 δ ( n ) n +1 + µ ( n ) n +1 δ ( n − 1) n +1 ! = − µ ( n − 1) n n − 1 X i =0 µ ( i ) n +1 δ ( n ) n +1 + µ ( n − 1) n +1 δ ( n − 1) n +1 + µ ( n ) n +1 δ ( n − 1) n +1 ! = − µ ( n − 1) n n − 1 X i =0 µ ( i ) n +1 δ ( n ) n +1 −  µ ( n − 1) n µ ( n − 1) n +1 + µ ( n − 1) n µ ( n ) n +1  δ ( n − 1) n +1 By Lemma B.2 b), the latter expression equ als the right-hand side of (C .3 ). Part b): The only non-trivial statemen t is th e second equalit y . It is a consequence of Lemmas B.1 c), B.3 a), B.5 a). Part c): Ob serv e that 2[ V ◦ ( δ ) , b e ( µ )] n + 2[ U ◦ ( δ ) , B e ] n =  ∗ 11 0 ∗ 21 ∗ 22  where ∗ 11 = − b v ( µ ) n +1 h e n n − 1 X i =1 i X j =1 τ j n δ ( i ) n + µ ( n ) n +1 δ ( n ) n +1 h e n N n ∗ 21 = h e n − 1 n − 2 X i =1 i X j =1 τ j n − 1 δ ( i ) n − 1 b ( µ ) n − b h ( µ ) n +1 h e n n − 1 X i =1 i X j =1 τ j n δ ( i ) n − h e n − 1 N n − 1 µ ( n − 1) n δ ( n − 1) n ∗ 22 = h e n − 1 n − 2 X i =1 i X j =1 τ j n − 1 δ ( i ) n − 1 b v ( µ ) n − h e n − 1 N n − 1 µ ( n − 1) n δ ( n − 1) n By Lemma B.4, ∗ 11 = − (1 − τ − 1 n ) n − 1 X i =1 i X j =1 τ j n δ ( i ) n − µ ( n ) n +1 h e n δ ( n − 1) n N n = − n − 1 X i =1 ( τ i n − 1) δ ( i ) n + τ − 1 n δ ( n − 1) n N n = − n − 1 X i =1 τ i n δ ( i ) n + n − 1 X i =1 δ ( i ) n + δ (0) n N n = − n − 1 X i =1 δ (0) n τ i n + n − 1 X i =1 δ ( i ) n + δ (0) n N n = − n − 1 X i =0 δ (0) n τ i n + n − 1 X i =0 δ ( i ) n + δ (0) n N n = b ( δ ) n 36 D. SHKL Y ARO V T o simplify ∗ 22 , let u s use the fact that h e : C ( A ) → C + ( A ) is bijectiv e. F or α ∈ A ⊗ sA ⊗ ( n − 2) h e n − 1 n − 2 X i =1 i X j =1 τ j n − 1 δ ( i ) n − 1 b v ( µ ) n h e n − 1 ( α ) − h e n − 1 N n − 1 µ ( n − 1) n δ ( n − 1) n h e n − 1 ( α ) = h e n − 1 n − 2 X i =1 i X j =1 τ j n − 1 δ ( i ) n − 1 (1 − τ − 1 n − 1 )( α ) + h e n − 1 N n − 1 µ ( n − 1) n h e n − 1 δ ( n − 2) n − 1 ( α ) = h e n − 1 n − 2 X i =1 i X j =1 τ j n − 1 δ ( i ) n − 1 ( α ) − h e n − 1 n − 2 X i =1 i X j =1 τ j n − 1 δ ( i ) n − 1 τ − 1 n − 1 ( α ) − h e n − 1 N n − 1 τ − 1 n − 1 δ ( n − 2) n − 1 ( α ) = h e n − 1 n − 2 X i =1 i X j =1 τ j n − 1 δ ( i ) n − 1 ( α ) − h e n − 1 n − 2 X i =1 i X j =1 τ j − 1 n − 1 δ ( i − 1) n − 1 ( α ) − h e n − 1 N n − 1 δ ( n − 2) n − 1 ( α ) = h e n − 1 n − 2 X j =1 τ j − 1 n − 1 δ ( n − 2) n − 1 ( α ) − h e n − 1 n − 3 X i =0 δ ( i ) n − 1 ( α ) − h e n − 1 N n − 1 δ ( n − 2) n − 1 ( α ) = h e n − 1 n − 2 X j =0 τ j − 1 n − 1 δ ( n − 2) n − 1 ( α ) − h e n − 1 n − 2 X i =0 δ ( i ) n − 1 ( α ) − h e n − 1 N n − 1 δ ( n − 2) n − 1 ( α ) = − h e n − 1 n − 2 X i =0 δ ( i ) n − 1 ( α ) By Lemma B.5 a) the latter equals b ( δ ) n ( h e n − 1 ( α )). Thus, ∗ 22 = b ( δ ) n . It remains to show that ∗ 21 = 0. By Lemma B.5 c) it is en ou gh to prov e that n − 2 X i =1 i X j =1 τ j n − 1 δ ( i ) n − 1 b ( µ ) n + ( b ( µ ) n − µ ( n − 1) n ) n − 1 X i =1 i X j =1 τ j n δ ( i ) n − N n − 1 µ ( n − 1) n δ ( n − 1) n = 0 (C.4) By Lemma B.2 c) δ ( i ) n − 1 b ( µ ) n = − i X k =0 µ ( k ) n ! δ ( i +1) n − n − 1 X k = i µ ( k ) n ! δ ( i ) n for i ≥ 1. Th erefore n − 2 X i =1 i X j =1 τ j n − 1 δ ( i ) n − 1 b ( µ ) n = − n − 2 X i =1 i X j =1 i X k =0 τ j n − 1 µ ( k ) n δ ( i +1) n − n − 2 X i =1 i X j =1 n − 1 X k = i τ j n − 1 µ ( k ) n δ ( i ) n = − n − 1 X i =2 i − 1 X j =1 i − 1 X k =0 τ j n − 1 µ ( k ) n δ ( i ) n − n − 2 X i =1 i X j =1 n − 1 X k = i τ j n − 1 µ ( k ) n δ ( i ) n Th us, the coeﬃcient ”in front of ” δ ( i ) n on the left-hand side of (C.4) equals − i − 1 X j =1 i − 1 X k =0 τ j n − 1 µ ( k ) n − i X j =1 n − 1 X k = i τ j n − 1 µ ( k ) n + ( b ( µ ) n − µ ( n − 1) n ) i X j =1 τ j n , (C.5) if 2 ≤ i ≤ n − 2, − n − 1 X k =1 τ n − 1 µ ( k ) n + ( b ( µ ) n − µ ( n − 1) n ) τ n , NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 37 if i = 1, and − n − 2 X j =1 n − 2 X k =0 τ j n − 1 µ ( k ) n + ( b ( µ ) n − µ ( n − 1) n ) n − 1 X j =1 τ j n − N n − 1 µ ( n − 1) n , if i = n − 1. W e will p ro v e that (C.5) v anishes and lea v e the second and th e th ir d cases to the reader: − i − 1 X j =1 i − 1 X k =0 τ j n − 1 µ ( k ) n − i X j =1 n − 1 X k = i τ j n − 1 µ ( k ) n + ( b ( µ ) n − µ ( n − 1) n ) i X j =1 τ j n = − i − 1 X j =1 i − 1 X k =0 τ j − k n − 1 µ (0) n τ k n − i X j =1 n − 1 X k = i τ j − k n − 1 µ (0) n τ k n + n − 2 X k =0 i X j =1 τ − k n − 1 µ (0) n τ k + j n = − i − 1 X j =1 i − 1 X k =0 τ j − k n − 1 µ (0) n τ k n − i X j =1 n − 1 X k = i τ j − k n − 1 µ (0) n τ k n + i X j =1 j + n − 2 X k = j τ j − k n − 1 µ (0) n τ k n The in terv al for k in the th ir d expression con tains the in terv al for k in the second one, so − i − 1 X j =1 i − 1 X k =0 τ j − k n − 1 µ (0) n τ k n − i X j =1 n − 1 X k = i τ j − k n − 1 µ (0) n τ k n + i X j =1 j + n − 2 X k = j τ j − k n − 1 µ (0) n τ k n = − i − 1 X j =1 i − 1 X k =0 τ j − k n − 1 µ (0) n τ k n + i X j =1 i − 1 X k = j τ j − k n − 1 µ (0) n τ k n + i X j =1 j + n − 2 X k = n τ j − k n − 1 µ (0) n τ k n = − i − 1 X j =1 i − 1 X k =0 τ j − k n − 1 µ (0) n τ k n + i X j =1 i − 1 X k = j τ j − k n − 1 µ (0) n τ k n + i X j =1 j − 2 X k =0 τ j − k − n n − 1 µ (0) n τ k + n n = − i − 1 X j =1 i − 1 X k =0 τ j − k n − 1 µ (0) n τ k n + i X j =1 i − 1 X k = j τ j − k n − 1 µ (0) n τ k n + i X j =1 j − 2 X k =0 τ j − k − 1 n − 1 µ (0) n τ k n = − i − 1 X j =1 i − 1 X k =0 τ j − k n − 1 µ (0) n τ k n + i X j =1 i − 1 X k = j τ j − k n − 1 µ (0) n τ k n + i − 1 X j =0 j − 1 X k =0 τ j − k n − 1 µ (0) n τ k n = − i − 1 X j =1 i − 1 X k =0 τ j − k n − 1 µ (0) n τ k n + i − 1 X j =1 i − 1 X k = j τ j − k n − 1 µ (0) n τ k n + i − 1 X j =1 j − 1 X k =0 τ j − k n − 1 µ (0) n τ k n = − i − 1 X j =1 j − 1 X k =0 τ j − k n − 1 µ (0) n τ k n + i − 1 X j =1 j − 1 X k =0 τ j − k n − 1 µ (0) n τ k n = 0 Part d) is s traigh tforw ard.  Lemma C.3 imp lies [ U ◦ ( δ ) , b e ] = 0 , [ V ◦ ( δ ) + Γ , b e ] + [ U ◦ ( δ ) , B e ] = 1 2 b e , [ V ◦ ( δ ) + Γ , B e ] = − 1 2 B e , 38 D. SHKL Y ARO V whic h pr ov es (2.2) for ∇ ◦ . C.3. Pro of of Proposition 3.5. Let us show that 2 u 2 ( ∇ − ∇ ◦ ) = ( b e + uB e )  δ (0) − δ (0) N δ (0) 0 0  +  δ (0) − δ (0) N δ (0) 0 0  ( b e + uB e ) Using the matrix expressions for b e and B e , the statemen t can b e s een to follo w from the follo wing equalities: b ( µ ) n +1 δ (0) n +1 + δ (0) n b ( µ ) n +1 = − µ (0) n +1 δ (1) n +1 − µ ( n ) n +1 δ ( n ) n +1 , b ( µ ) n +1 δ (0) n +1 N n +1 + δ (0) n N n b ( µ ) n +1 = 0 , δ (0) n N n b v ( µ ) n +1 h e n = 0 , h e n +1 N n +1 δ (0) n +1 − h e n +1 N n +1 δ (0) n +1 N n +1 = − h e n +1 n X i =1 n +1 X j =1 τ j n +1 δ ( i ) n +1 Since the last t w o equalities are qu ite s traigh tforw ard, we will giv e pro ofs of the ﬁrst t w o only . The ﬁrst equalit y follo ws from Lemma B.2 c): δ (0) n b ( µ ) n +1 = δ (0) n µ (0) n +1 + δ (0) n µ ( n ) n +1 + δ (0) n n − 1 X i =1 µ ( i ) n +1 = − µ (0) n +1 δ (1) n +1 − µ (0) n +1 δ (0) n +1 − µ ( n ) n +1 δ (0) n +1 − µ ( n ) n +1 δ ( n ) n +1 − n − 1 X i =1 µ ( i ) n +1 δ (0) n +1 = − µ (0) n +1 δ (1) n +1 − µ ( n ) n +1 δ ( n ) n +1 − n X i =0 µ ( i ) n +1 δ (0) n +1 = − µ (0) n +1 δ (1) n +1 − µ ( n ) n +1 δ ( n ) n +1 − b ( µ ) n +1 δ (0) n +1 The pro of of the second equalit y u ses Lemmas B.1 f ), B.2 c) and the compu tation w e ju st did: b ( µ ) n +1 δ (0) n +1 N n +1 + δ (0) n N n b ( µ ) n +1 = b ( µ ) n +1 δ (0) n +1 N n +1 + δ (0) n ( b ( µ ) n +1 − µ ( n ) n +1 ) N n +1 = ( b ( µ ) n +1 δ (0) n +1 + δ (0) n b ( µ ) n +1 ) N n +1 − δ (0) n µ ( n ) n +1 N n +1 = ( − µ (0) n +1 δ (1) n +1 − µ ( n ) n +1 δ ( n ) n +1 ) N n +1 − δ (0) n µ ( n ) n +1 N n +1 = ( − µ (0) n +1 δ (1) n +1 − µ ( n ) n +1 δ ( n ) n +1 + µ ( n ) n +1 δ (0) n +1 + µ ( n ) n +1 δ ( n ) n +1 ) N n +1 = ( − µ (0) n +1 δ (1) n +1 + µ ( n ) n +1 δ (0) n +1 ) N n +1 = ( − µ (0) n +1 τ − 1 n +1 δ (0) n +1 τ n +1 + µ (0) n +1 τ − 1 n +1 δ (0) n +1 ) N n +1 = 0 NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 39 C.4. Pro of of Corollary 3.6. Let us show that 2 u 2 ( e ∇ − ∇ ) = ( b e + uB e ) H ( u ) + H ( u )( b e + uB e ) where H ( u ) = H 0 + uH 1 : C e ( A ) → C e ( A ) and H 0 =  µ (0) µ (0) 0 0  , H 1 =  0 0 h e N ′ 0  , N ′ n +1 = n +1 X j =1 j τ j n +1 W e will use the follo wing prop ert y of N ′ : (1 − τ − 1 ) N ′ = − N − 2Γ + 1 Let us compute ( b e ( δ ) + b e ( µ ) + uB e ) H 0 + H 0 ( b e ( δ ) + b e ( µ ) + uB e ) ﬁr st. By Lemma B.3 b), b e ( δ ) H 0 + H 0 b e ( δ ) = 0. By Lemma B.2 b ) b e ( µ ) H 0 + H 0 b e ( µ ) =  b ( µ ) µ (0) + µ (0) b ( µ ) b ( µ ) µ (0) + µ (0) b ( µ ) 0 0  = 2 U ( µ ) Finally , B e H 0 + H 0 B e =  N 0 h e N µ (0) h e N µ (0)  =: T Th us, ( b e ( δ ) + b e ( µ ) + uB e ) H 0 + H 0 ( b e ( δ ) + b e ( µ ) + uB e ) = 2 U ( µ ) + uT (C.6) Next, let us compute ( b e ( δ ) + b e ( µ ) + uB e ) H 1 + H 1 ( b e ( δ ) + b e ( µ ) + uB e ). Note that b e ( δ ) H 1 + H 1 b e ( δ ) = 0 by Lemmas B.1 c) and B.5 a). Al so, B e H 1 + H 1 B e = 0 f or obvious reasons. F ur thermore, by Lemma B.5 b), c) b e ( µ ) H 1 + H 1 b e ( µ ) =  b v ( µ ) h e N ′ 0 b h ( µ ) h e N ′ + h e N ′ b ( µ ) h e N ′ b v ( µ )  =  (1 − τ − 1 ) N ′ 0 − h e ( µ (0) + b h ( µ )) N ′ + h e N ′ b ( µ ) h e N ′ b v ( µ )  =  − N − 2Γ + id 0 − h e ( µ (0) + b h ( µ )) N ′ + h e N ′ b ( µ ) h e N ′ b v ( µ )  =  − N − 2Γ + id 0 − h e µ (0) N ′ − h e b h ( µ ) N ′ + h e N ′ b ( µ ) − h e N µ (0) − 2Γ  (C.7) T o exp lain the last transition ab o v e, we’ll take α ∈ C ( A ) and notice that h e N ′ b v ( µ ) h e ( α ) = h e N ′ (1 − τ − 1 )( α ) = − h e ( N + 2Γ − 1)( α ) = − h e N µ (0) ( h e ( α )) − 2Γ( h e ( α )) 40 D. SHKL Y ARO V Let us simp lify the remainin g en try of th e matrix (C.7): N ′ n b ( µ ) n +1 − µ (0) n +1 N ′ n +1 − b h ( µ ) n +1 N ′ n +1 = n X i =0 n X j =1 j τ j n µ ( i ) n +1 − n +1 X j =1 n − 1 X i =0 j µ ( i ) n +1 τ j n +1 = n X i =0 n X j =1 j τ j n µ ( i ) n +1 − n +1 X j =1 n − 1 X i =0 j τ − i n µ (0) n +1 τ i + j n +1 = n X i =0 n X j =1 j τ j n µ ( i ) n +1 − n X j =1 n − j X i =0 j τ j n µ ( i + j ) n +1 − n +1 X j =2 n − 1 X i = n − j +1 j τ j − 1 n µ ( i + j − n − 1) n +1 By reindexing and resumming one gets n X i =0 n X j =1 j τ j n µ ( i ) n +1 − n X i =1 i X j =1 j τ j n µ ( i ) n +1 − n − 1 X i =0 n X j = i +1 ( j + 1) τ j n µ ( i ) n +1 = − n − 1 X i =0 n X j = i +1 τ j n µ ( i ) n +1 = − N n µ (0) n +1 − n − 1 X i =1 n X j = i +1 τ j n µ ( i ) n +1 Inserting this f orm ula into (C.7 ) giv es b e ( µ ) H 1 + H 1 b e ( µ ) = − T − 2Γ + 2 V ( µ ) (C.8) Th us, by (C.6) and (C.8 ) ( b e ( δ ) + b e ( µ ) + uB e ) H ( u ) 2 u 2 + H ( u ) 2 u 2 ( b e ( δ ) + b e ( µ ) + uB e ) = U ( µ ) u 2 + V ( µ ) − Γ u whic h ﬁn ishes the pro of. C.5. Pro of of C orollary 3.7. Th e pr o of is based on formula (2.5) app lied to (3.9 ), with ι ( u ), p ( u ), and H ( u ) give n by (3.5),(3.6), and (C.1 ), resp ectiv ely . Let us compute all the summands in (2.5). First, observe that p ( u ) dι ( u ) du =  1 (1 − τ − 1 ) hµ (0)   0 h e hN  = (1 − τ − 1 ) hhN ∼ 0 since (1 − τ − 1 ) hhN = ( b + uB ) H ( u ) u + H ( u ) u ( b + uB ) where H ( u ) = u (1 − τ − 1 ) hhhN (see the p ro of of Part a) of Lemma C .2). F ur thermore, by Lemmas B.4 a),b) p ( u ) U ( δ ) ι ( u ) = 1 2  1 (1 − τ − 1 ) hµ (0)   − µ (0) δ (1) − µ (0) δ (1) 0 0   1 uh e hN  = − 1 2 µ (0) δ (1) = U un ( δ ) NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 41 and similarly p ( u ) V ( δ ) ι ( u ) = V un ( δ ). It remains to compute p ( u )Γ ι ( u ): p ( u )Γ ι ( u ) = Γ + u (1 − τ − 1 ) hµ (0) Γ h e hN = Γ + u (1 − τ − 1 ) hµ (0) ( h e hN Γ − h e hN ) = Γ + u (1 − τ − 1 ) hhN Γ − u (1 − τ − 1 ) hhN ∼ Γ + (( b + uB ) H ( u ) + H ( u )( b + uB ))Γ By P art d) of Lemma C.3, the latter is u -homotopic to Γ − 1 2 H ( u )( b ( µ ) − uB ) T o su mmarize, p ( u ) dι ( u ) du + p ( u ) U ( δ ) u 2 ι ( u ) + p ( u ) V ( δ ) u ι ( u ) + p ( u ) Γ u ι ( u ) + 1 2 u H ( u )( b − uB ) ∼ U un ( δ ) u 2 + V un ( δ ) u + Γ u − 1 2 u H ( u )( b ( µ ) − uB ) + 1 2 u H ( u )( b − uB ) = U un ( δ ) u 2 + V un ( δ ) u + Γ u + 1 2 (1 − τ − 1 ) hhhN b ( δ ) C.6. Pro of of Prop osition 3.8. Let u s start by in tro ducing the follo wing op erators on C e ( A ): γ ( i ) n +1 ( a 0 [ a 1 | . . . | a n ]) = deg ( a i ) a 0 [ a 1 | . . . | a n ] They extend to C ( A e ) and the extended op erators satisfy the relations γ ( i ) n +1 = τ − i n +1 γ (0) n +1 τ i n +1 Ob viously , Γ ′ n +1 = n X i =0 γ ( i ) n +1 + 2Γ n +1 Lemma C.4. a ) γ ( i ) δ ( j ) = ( δ ( i ) γ ( i ) + δ ( i ) i = j δ ( j ) γ ( i ) i 6 = j b ) γ ( i ) n µ ( j ) n +1 =      µ ( i ) n +1 γ ( i ) n +1 + µ ( i ) n +1 γ ( i +1) n +1 i = j µ ( j ) n +1 γ ( i ) n +1 i < j µ ( j ) n +1 γ ( i +1) n +1 i > j c ) γ ( i ) h e = ( 0 i = 0 h e γ ( i − 1) i ≥ 1 Pro of. Part c) is obvio us. The rest is very similar to the pr o of of Lemma B.2: the case i = 0 is easy; Lemma B.1 a),b) r ed uces the general case to this sp ecial one.  Let us sho w that 2 u 2 ( ∇ g r − ∇ ) = ( b e + uB e ) b H ( u ) + b H ( u )( b e + uB e ) 42 D. SHKL Y ARO V where b H ( u ) = b H 0 + u b H 1 : C e ( A ) → C e ( A ) and ( b H 0 ) n +1 =  µ (0) n +1 γ (1) n +1 µ (0) n +1 γ (1) n +1 0 0  , ( b H 1 ) n +1 = 0 0 h e n +1 P n i =1 P n +1 j = i +1 τ j n +1 γ ( i ) n +1 0 ! Let us compu te ( b e ( δ ) + b e ( µ ) + uB e ) b H 0 + b H 0 ( b e ( δ ) + b e ( µ ) + uB e ) ﬁ r st. By L emmas B.3 b), C.4 a) b e ( δ ) b H 0 + b H 0 b e ( δ ) = − 2 U ( δ ) By Lemma C.4 b ) b e ( µ ) b H 0 + b H 0 b e ( µ ) =  ∗ ∗ 0 0  where ∗ = b ( µ ) µ (0) γ (1) + µ (0) ( b ( µ ) γ (1) + µ (0) γ (2) + µ (1) γ (2) − µ (0) γ (1) ) Using Lemma B.2 b) b ( µ ) µ (0) γ (1) + µ (0) ( b ( µ ) γ (1) + µ (0) γ (2) + µ (1) γ (2) − µ (0) γ (1) ) = ( b ( µ ) µ (0) + µ (0) b ( µ )) γ (1) + ( µ (0) µ (0) + µ (0) µ (1) ) γ (2) − µ (0) µ (0) γ (1) = µ (0) µ (0) γ (1) − µ (0) µ (0) γ (1) = 0 i.e. b e ( µ ) b H 0 + b H 0 b e ( µ ) = 0. Also, b y Lemmas C.4 c), B.4 b) B e b H 0 + b H 0 B e =  γ (0) N 0 h e N µ (0) γ (1) h e N µ (0) γ (1)  =: T ′ Th us, ( b e ( δ ) + b e ( µ ) + uB e ) b H 0 + b H 0 ( b e ( δ ) + b e ( µ ) + uB e ) = − 2 U ( δ ) + uT ′ (C.9) Let us compute ( b e ( δ ) + b e ( µ ) + uB e ) b H 1 + b H 1 ( b e ( δ ) + b e ( µ ) + uB e ) now. First, by Lemmas B.1 c), B.5 a), C.4 a) b e ( δ ) b H 1 + b H 1 b e ( δ ) = − 2 V ( δ ) (C.10) Also, B e b H 1 + b H 1 B e = 0. It remains to sim p lify b e ( µ ) b H 1 + b H 1 b e ( µ ). b e ( µ ) n +1 ( b H 1 ) n + ( b H 1 ) n − 1 b e ( µ ) n =  ∗ 11 0 ∗ 21 ∗ 22  where ∗ 11 = b v ( µ ) n +1 h e n n − 1 X i =1 n X j = i +1 τ j n γ ( i ) n , NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 43 ∗ 21 = b h ( µ ) n +1 h e n n − 1 X i =1 n X j = i +1 τ j n γ ( i ) n + h e n − 1 n − 2 X i =1 n − 1 X j = i +1 τ j n − 1 γ ( i ) n − 1 b ( µ ) n , ∗ 22 = h e n − 1 n − 2 X i =1 n − 1 X j = i +1 τ j n − 1 γ ( i ) n − 1 b v ( µ ) n By Lemma B.5, ∗ 11 = (1 − τ − 1 n ) n − 1 X i =1 n X j = i +1 τ j n γ ( i ) n = − n − 1 X i =1 ( τ i n − 1) γ ( i ) n = − n − 1 X i =1 τ i n γ ( i ) n + n − 1 X i =1 γ ( i ) n = − n − 1 X i =0 τ i n γ ( i ) n + n − 1 X i =0 γ ( i ) n = − n − 1 X i =0 γ (0) n τ i n + Γ ′ n − 2Γ n = − γ (0) n N n + Γ ′ n − 2Γ n (C.11) T o simplify ∗ 22 , tak e α ∈ A ⊗ sA ⊗ ( n − 2) and apply ∗ 22 to h e n − 1 ( α ): h e n − 1 n − 2 X i =1 n − 1 X j = i +1 τ j n − 1 γ ( i ) n − 1 b v ( µ ) n h e n − 1 ( α ) = h e n − 1 n − 2 X i =1 n − 1 X j = i +1 τ j n − 1 γ ( i ) n − 1 (1 − τ − 1 n − 1 )( α ) = h e n − 1 n − 2 X i =1 n − 1 X j = i +1 τ j n − 1 γ ( i ) n − 1 ( α ) − h e n − 1 n − 2 X i =1 n − 1 X j = i +1 τ j − 1 n − 1 γ ( i − 1) n − 1 ( α ) = h e n − 1 n − 2 X i =1 n − 1 X j = i +1 τ j n − 1 γ ( i ) n − 1 ( α ) − h e n − 1 n − 3 X i =0 n − 2 X j = i +1 τ j n − 1 γ ( i ) n − 1 ( α ) = h e n − 1 γ ( n − 2) n − 1 ( α ) + h e n − 1 n − 3 X i =1 n − 1 X j = i +1 τ j n − 1 γ ( i ) n − 1 ( α ) − h e n − 1 n − 3 X i =1 n − 2 X j = i +1 τ j n − 1 γ ( i ) n − 1 ( α ) − h e n − 1 n − 2 X j =1 τ j n − 1 γ (0) n − 1 ( α ) = h e n − 1 γ ( n − 2) n − 1 ( α ) + h e n − 1 n − 3 X i =1 γ ( i ) n − 1 ( α ) − h e n − 1 n − 2 X j =1 τ j n − 1 γ (0) n − 1 ( α ) = h e n − 1 n − 2 X i =0 γ ( i ) n − 1 ( α ) − h e n − 1 N n − 1 γ (0) n − 1 ( α ) = n − 1 X i =0 γ ( i ) n h e n − 1 ( α ) − h e n − 1 N n − 1 γ (0) n − 1 µ (0) n h e n − 1 ( α ) = (Γ ′ n − 2Γ n ) h e n − 1 ( α ) − h e n − 1 N n − 1 µ (0) n γ (1) n − 1 h e n − 1 ( α ) Th us, ∗ 22 = Γ ′ n − 2Γ n − h e n − 1 N n − 1 µ (0) n γ (1) n − 1 (C.12) 44 D. SHKL Y ARO V F ur thermore, by L emmas B.5 c), B.1 b), C.4 b) ∗ 21 = − h e n − 1 n − 1 X i =1 n X j = i +1 n − 2 X k =0 µ ( k ) n τ j n γ ( i ) n + h e n − 1 n − 2 X i =1 n − 1 X j = i +1 n − 1 X k =0 τ j n − 1 γ ( i ) n − 1 µ ( k ) n = − h e n − 1 n − 2 X i =1 n − 1 X j = i +1 n − j − 1 X k =0 τ j n − 1 µ ( k + j ) n γ ( i ) n − h e n − 1 n − 1 X i =1 n X j = i +1 n − 2 X k = n − j τ j − 1 n − 1 µ ( k + j − n ) n γ ( i ) n + h e n − 1 n − 2 X i =1 n − 1 X j = i +1 τ j n − 1 ( µ ( i ) n γ ( i ) n + µ ( i ) n γ ( i +1) n ) + h e n − 1 n − 2 X i =1 n − 1 X j = i +1 n − 1 X k = i +1 τ j n − 1 µ ( k ) n γ ( i ) n + h e n − 1 n − 2 X i =1 n − 1 X j = i +1 i − 1 X k =0 τ j n − 1 µ ( k ) n γ ( i +1) n or, after reind exing − h e n − 1 n − 2 X i =1 n − 1 X j = i +1 n − 1 X k = j τ j n − 1 µ ( k ) n γ ( i ) n − h e n − 1 n − 1 X i =1 n − 1 X j = i j − 1 X k =0 τ j n − 1 µ ( k ) n γ ( i ) n + h e n − 1 n − 2 X i =1 n − 1 X j = i +1 τ j n − 1 µ ( i ) n γ ( i ) n + h e n − 1 n − 1 X i =2 n − 1 X j = i τ j n − 1 µ ( i − 1) n γ ( i ) n + h e n − 1 n − 2 X i =1 n − 1 X j = i +1 n − 1 X k = i +1 τ j n − 1 µ ( k ) n γ ( i ) n + h e n − 1 n − 1 X i =2 n − 1 X j = i i − 2 X k =0 τ j n − 1 µ ( k ) n γ ( i ) n = − h e n − 1 n − 2 X i =1 n − 1 X j = i +1 n − 1 X k = j τ j n − 1 µ ( k ) n γ ( i ) n − h e n − 1 n − 1 X i =1 n − 1 X j = i j − 1 X k =0 τ j n − 1 µ ( k ) n γ ( i ) n + h e n − 1 n − 2 X i =1 n − 1 X j = i +1 n − 1 X k = i τ j n − 1 µ ( k ) n γ ( i ) n + h e n − 1 n − 1 X i =2 n − 1 X j = i i − 1 X k =0 τ j n − 1 µ ( k ) n γ ( i ) n = − h e n − 1 n − 1 X j =1 τ j n − 1 µ (0) n γ (1) n = − h e n − 1 N n − 1 µ (0) n γ (1) n This, together with (C.9), (C.10), (C.11), and (C.12) shows th at ( b e ( δ ) + b e ( µ ) + uB e ) b H ( u ) 2 u 2 + b H ( u ) 2 u 2 ( b e ( δ ) + b e ( µ ) + uB e ) = Γ ′ 2 u − U ( δ ) u 2 − V ( δ ) + Γ u Appendix D. Proofs for Se ction 4 D.1. Auxiliary lemma. Let us derive some basic pr op erties of op erators similar to b e ( w ) and b e ( D ( w )). NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 45 Let ( A, d ) b e an (abstract) dg algebra and a ∈ A . Con s ider the follo wing op erator on C e ( A ): b e ( a ) n +1 = n +1 X i =1 a ( i ) n +1 where a ( i ) n +1 ( a 0 [ a 1 | . . . | a n ]) = ( − 1) | sa | P i − 1 j =0 | sa j | a 0 [ a 1 | . . . | a i − 1 | a | a i | . . . | a n ] (D.1) Note that th e parit y of b e ( a ) is opp osite to th at of a . C learly , b e ( a ) preserve s the sub spaces C ( A ) and C + ( A ), and ther efore we may also think of it as r epresen ted by the matrix  b ( a ) 0 0 b ( a )  , where the tw o b ( a )’s are the r estrictions of b e ( a ) on to C ( A ) and C + ( A ), resp ectiv ely . It is also ob vious that b e ( a ) extends to C Π , e ( A ). W e will need y et another op erator ad( a ) on C e ( A ) deﬁned as follo ws ad( a ) n +1 := n +1 X i =0 ad( a ) ( i ) n +1 where ad( a ) ( i ) n +1 ( a 0 [ a 1 | . . . | a n ]) = ( − 1) | a | P i − 1 j =0 | sa j | a 0 [ a 1 | . . . | [ a, a i ] | . . . | a n ] ([ , ] stands for the s up er-commutato r). Lemma D.1. a ) [ b e ( δ ) , b e ( a )] = b e ( da ) b ) [ b e ( µ ) , b e ( a )] = ( − 1) | a | ad( a ) c ) [ B e , b e ( a )] = 0 d ) [ b e ( a ′ ) , b e ( a )] = 0 , ∀ a, a ′ Pro of. As in section B , it will b e conv enient to extend all the op erators to C ( A e ) and establish the relations on the whole of C ( A e ). Observe that the extensions of a ( i ) n +1 and ad( a ) ( i ) n +1 to C ( A e ) satisfy the p r op erties a ( i ) n +1 = τ − i n +2 a (0) n +1 τ i n +1 , ad( a ) ( i ) n +1 = τ − i n +1 ad( a ) (0) n +1 τ i n +1 where a (0) n +1 ( a 0 [ a 1 | . . . | a n ]) = a [ a 0 | a 1 | . . . | a n ] . a) By (3.1 ) δ ( i ) n +2 a (0) n +1 = ( da (0) n +1 , i = 0 ( − 1) | a | +1 a (0) n +1 δ ( i − 1) n +1 , i 6 = 0 46 D. SHKL Y ARO V Therefore, by Lemma B.1 a) δ ( i ) n +2 a ( j ) n +1 = δ ( i ) n +2 τ − j n +2 a (0) n +1 τ j n +1 = τ − j n +2 δ ( i − j ) n +2 a (0) n +1 τ j n +1 = ( da ( j ) n +1 , i = j ( − 1) | a | +1 τ − j n +2 a (0) n +1 δ ( i − j − 1) n +1 τ j n +1 , i 6 = j =      da ( j ) n +1 , i = j ( − 1) | a | +1 a ( j ) n +1 δ ( i − 1) n +1 , i > j ( − 1) | a | +1 a ( j ) n +1 δ ( i ) n +1 , i < j Then b e ( δ ) n +2 b e ( a ) n +1 = n +1 X i =0 n +1 X j =1 δ ( i ) n +2 a ( j ) n +1 = n +1 X i =1 δ ( i ) n +2 a ( i ) n +1 + n X i =0 n +1 X j = i +1 δ ( i ) n +2 a ( j ) n +1 + n +1 X i =2 i − 1 X j =1 δ ( i ) n +2 a ( j ) n +1 = b e ( da ) n +1 + ( − 1) | a | +1 n X i =0 n +1 X j = i +1 a ( j ) n +1 δ ( i ) n +1 + ( − 1) | a | +1 n +1 X i =2 i − 1 X j =1 a ( j ) n +1 δ ( i − 1) n +1 = b e ( da ) n +1 + ( − 1) | a | +1 n X i =0 n +1 X j = i +1 a ( j ) n +1 δ ( i ) n +1 + ( − 1) | a | +1 n X i =1 i X j =1 a ( j ) n +1 δ ( i ) n +1 = b e ( da ) n +1 + ( − 1) | a | +1 b e ( a ) n +1 b e ( δ ) n +1 b) Set l ( a ) (0) n +1 ( a 0 [ a 1 | . . . | a n ]) = aa 0 [ a 1 | . . . | a n ] , l ( a ) ( i ) n +1 := τ − i n +1 l ( a ) (0) n +1 τ i n +1 r ( a ) (0) n +1 ( a 0 [ a 1 | . . . | a n ]) = ( − 1) | a || a 0 | a 0 a [ a 1 | . . . | a n ] , r ( a ) ( i ) n +1 := τ − i n +1 r ( a ) (0) n +1 τ i n +1 Then, clearly , ad( a ) ( i ) n +1 = l ( a ) ( i ) n +1 − r ( a ) ( i ) n +1 . Th e follo wing is easy to chec k usin g (3.2) µ ( i ) n +2 a (0) n +1 =      ( − 1) | a | l ( a ) (0) n +1 , i = 0 ( − 1) | a | +1 a (0) n µ ( i − 1) n +1 , 1 ≤ i ≤ n ( − 1) | a | +1 r ( a ) (0) n +1 τ − 1 n +1 , i = n + 1 In com bination w ith Lemma B.1 b) this gives µ ( i ) n +2 a ( j ) n +1 = µ ( i ) n +2 τ − j n +2 a (0) n +1 τ j n +1 = µ ( i ) n +2 τ n +2 − j n +2 a (0) n +1 τ j n +1 = ( τ 1 − j n +1 µ ( n +2+ i − j ) n +2 a (0) n +1 τ j n +1 i − j < 0 τ − j n +1 µ ( i − j ) n +2 a (0) n +1 τ j n +1 i − j ≥ 0 NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 47 =                ( − 1) | a | +1 τ 1 − j n +1 a (0) n µ ( n +1+ i − j ) n +1 τ j n +1 − n − 1 ≤ i − j ≤ − 2 ( − 1) | a | +1 τ 1 − j n +1 r ( a ) (0) n +1 τ − 1 n +1 τ j n +1 i − j = − 1 ( − 1) | a | τ − j n +1 l ( a ) (0) n +1 τ j n +1 i − j = 0 ( − 1) | a | +1 τ − j n +1 a (0) n µ ( i − j − 1) n +1 τ j n +1 1 ≤ i − j ≤ n ( − 1) | a | +1 τ − j n +1 r ( a ) (0) n +1 τ − 1 n +1 τ j n +1 i − j = n + 1 =                ( − 1) | a | +1 a ( j − 1) n µ ( i ) n +1 − n − 1 ≤ i − j ≤ − 2 ( − 1) | a | +1 r ( a ) ( j − 1) n +1 i − j = − 1 ( − 1) | a | l ( a ) ( j ) n +1 i − j = 0 ( − 1) | a | +1 a ( j ) n µ ( i − 1) n +1 1 ≤ i − j ≤ n ( − 1) | a | +1 r ( a ) ( j ) n +1 τ − 1 n +1 i − j = n + 1 Then b e ( µ ) n +2 b e ( a ) n +1 = n +1 X i =0 n +1 X j =1 µ ( i ) n +2 a ( j ) n +1 = n +1 X j =1 µ ( j ) n +2 a ( j ) n +1 + n +1 X j =1 µ ( j − 1) n +2 a ( j ) n +1 + n +1 X i =0 n +1 X j = i +2 µ ( i ) n +2 a ( j ) n +1 + n +1 X i =0 i +1 X j =1 µ ( i ) n +2 a ( j ) n +1 = ( − 1) | a | n +1 X j =1 l ( a ) ( j ) n +1 + ( − 1) | a | +1 n +1 X j =1 r ( a ) ( j − 1) n +1 +( − 1) | a | +1 n +1 X i =0 n +1 X j = i +2 a ( j − 1) n µ ( i ) n +1 + ( − 1) | a | +1 n +1 X i =0 i − 1 X j =1 a ( j ) n µ ( i − 1) n +1 = ( − 1) | a | ad( a ) n +1 + ( − 1) | a | +1 n +1 X i =0 n +1 X j = i +2 a ( j − 1) n µ ( i ) n +1 + ( − 1) | a | +1 n +1 X i =0 i − 1 X j =1 a ( j ) n µ ( i − 1) n +1 = ( − 1) | a | ad( a ) n +1 + ( − 1) | a | +1 n − 1 X i =0 n X j = i +1 a ( j ) n µ ( i ) n +1 + ( − 1) | a | +1 n X i =1 i X j =1 a ( j ) n µ ( i ) n +1 = ( − 1) | a | ad( a ) n +1 + ( − 1) | a | +1 b e ( a ) n b e ( µ ) n +1 c) First, note that B e is the r estriction of h e N on to C e ( A ). Then observ e that N n +1 b e ( a ) n = N n +1 n X i =1 τ − i n +1 a (0) n τ i n = n X i =1 N n +1 a (0) n τ i n = N n +1 a (0) n n X i =1 τ i n = N n +1 a (0) n N n and similarly ( a (0) n + b e ( a ) n ) N n = n X i =0 τ − i n +1 a (0) n τ i n N n = n X i =0 τ − i n +1 a (0) n N n = N n +1 a (0) n N n F ur thermore, one chec ks easily that h e n +1 a ( i ) n = ( − 1) | a | +1 a ( i +1) n +1 h e n (D.2) 48 D. SHKL Y ARO V Th us, h e n +1 N n +1 b e ( a ) n = h e n +1 ( a (0) n + b e ( a ) n ) N n = h e n +1 n X i =0 a ( i ) n N n = ( − 1) | a | +1 n +1 X i =1 a ( i ) n +1 h e n N n = ( − 1) | a | +1 b e ( a ) n +1 h e n N n d) By (D.1 ) a ′ ( j ) n +2 a ( i ) n +1 ( a 0 [ a 1 | . . . | a n ]) = ( ( − 1) | sa | P i − 1 k =0 | sa k | ( − 1) | sa ′ | P j − 1 l =0 | sa l | a 0 [ a 1 | . . . | a j − 1 | a ′ | . . . | a | a i | . . . | a n ] j ≤ i ( − 1) | sa | P i − 1 k =0 | sa k | ( − 1) | sa ′ | P j − 2 l =0 | sa l | ( − 1) | sa ′ || sa | a 0 [ a 1 | . . . | a i − 1 | a | . . . | a ′ | a j − 1 | . . . | a n ] j > i whic h means a ′ ( j ) n +2 a ( i ) n +1 = ( − 1) | sa ′ || sa | ( a ( i +1) n +2 a ′ ( j ) n +1 , j ≤ i a ( i ) n +2 a ′ ( j − 1) n +1 , j > i (D.3) The latter imp lies the statemen t. D.2. Proof of Prop osition 4.2. Part a) follo ws from Lemma D.1 immediately since in our case [ D ( w ) , w ] = 0, ad( w ) = 0, and b e ( w ) 2 = 0 (the latter is a sp ecial case of P art d) of the Lemma). Part b): On e needs to s ho w that b e exp( b e ( D ( w ))) = exp( b e ( D ( w )))( b e ( µ ) + b e ( w ) ) , B e exp( b e ( D ( w ))) = exp( b e ( D ( w ))) B e The pro of is straigh tforw ard application of th e follo wing f ormulas [ b e ( δ ) , b e ( D ( w ))] = 2 b e ( w ) , [ b e ( µ ) , b e ( D ( w ))] = − b e ( δ ) , [ B e , b e ( D ( w ))] = 0 , [ b e ( w ) , b e ( D ( w ))] = 0 whic h, in tu r n, follo w fr om [ D ( w ) , D ( w )] = 2 w , ad( D ( w )) = b e ( δ ) and Lemma D.1. Part c ): That str is a morph ism of complexes ( C Π , e ( C [ Y ] ⊗ En d C V ) , b e ( µ )) → ( C Π , e ( C [ Y ]) , b e ( µ )) is a sp ecial case of a more general fact (see, for instance, [27]) bu t it is also easy to pro v e in our sp ecial case. It is enough to sho w that str · µ (0) = µ (0) · str and str · τ = τ · str : str · µ (0) (( φ 0 ⊗ E i 0 i 1 )[ φ 1 ⊗ E i 1 i 2 | . . . | φ n ⊗ E i n i 0 ]) = ( − 1) | v i 0 | + | v i 1 | str (( φ 0 φ 1 ⊗ E i 0 i 2 )[ φ 2 ⊗ E i 2 i 3 | . . . | φ n ⊗ E i n i 0 ]) = ( − 1) | v i 0 | + | v i 1 | ( − 1) ( n − 2) | v i 0 | + P n s =2 | v i s | φ 0 φ 1 [ φ 2 | . . . | φ n ] = ( − 1) ( n − 1) | v i 0 | + P n s =1 | v i s | φ 0 φ 1 [ φ 2 | . . . | φ n ] = µ (0) · str (( φ 0 ⊗ E i 0 i 1 )[ φ 1 ⊗ E i 1 i 2 | . . . | φ n ⊗ E i n i 0 ]) , NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 49 str · τ n +1 (( φ 0 ⊗ E i 0 i 1 )[ φ 1 ⊗ E i 1 i 2 | . . . | φ n ⊗ E i n i 0 ]) = ( − 1) ( | sφ 0 | + | v i 0 | + | v i 1 | )( P n − 1 t =1 ( | sφ t | + | v i t | + | v i t +1 | )+ | sφ n | + | v i n | + | v i 0 | ) str (( φ 1 ⊗ E i 1 i 2 )[ . . . | φ 0 ⊗ E i 0 i 1 ]) = ( − 1) (1+ | v i 0 | + | v i 1 | )( n + | v i 0 | + | v i 1 | ) str (( φ 1 ⊗ E i 1 i 2 )[ . . . | φ 0 ⊗ E i 0 i 1 ]) = ( − 1) n (1+ | v i 0 | + | v i 1 | ) str (( φ 1 ⊗ E i 1 i 2 )[ . . . | φ 0 ⊗ E i 0 i 1 ]) = ( − 1) n (1+ | v i 0 | + | v i 1 | ) ( − 1) ( n − 1) | v i 1 | + | v i 0 | + P n s =2 | v i s | φ 1 [ . . . | φ 0 ] = ( − 1) n +( n +1) | v i 0 | + P n s =1 | v i s | φ 1 [ . . . | φ 0 ] = ( − 1) n +( n − 1) | v i 0 | + P n s =1 | v i s | φ 1 [ . . . | φ 0 ] = ( − 1) ( n − 1) | v i 0 | + P n s =1 | v i s | τ n +1 ( φ 0 [ φ 1 | . . . | φ n ]) = τ n +1 · str (( φ 0 ⊗ E i 0 i 1 )[ φ 1 ⊗ E i 1 i 2 | . . . | φ n ⊗ E i n i 0 ]) It remains to v erify that str commutes with b e ( w ) and B e : str · w ( t ) n +1 (( φ 0 ⊗ E i 0 i 1 )[ φ 1 ⊗ E i 1 i 2 | . . . | φ n ⊗ E i n i 0 ]) = ( − 1) t + | v i 0 | + | v i t | str (( φ 0 ⊗ E i 0 i 1 )[ φ 1 ⊗ E i 1 i 2 | . . . | φ t − 1 ⊗ E i t − 1 i t | w ⊗ 1 | . . . | φ n ⊗ E i n i 0 ]) = ( − 1) t + | v i 0 | + | v i t | str (( φ 0 ⊗ E i 0 i 1 )[ φ 1 ⊗ E i 1 i 2 | . . . | φ t − 1 ⊗ E i t − 1 i t | w ⊗ E i t i t | . . . | φ n ⊗ E i n i 0 ]) = ( − 1) t + | v i 0 | + | v i t | ( − 1) n | v i 0 | + | v i t | + P n s =1 | v i s | φ 0 [ φ 1 | . . . | φ t − 1 | w | . . . | φ n ]) = ( − 1) t ( − 1) ( n − 1) | v i 0 | + P n s =1 | v i s | φ 0 [ φ 1 | . . . | φ t − 1 | w | . . . | φ n ]) = w ( t ) n +1 · str (( φ 0 ⊗ E i 0 i 1 )[ φ 1 ⊗ E i 1 i 2 | . . . | φ n ⊗ E i n i 0 ]) (in the ab o v e computation, w e implicitly use the fact that | E ij | = | v i | + | v j | ), str · h e (( φ 0 ⊗ E i 0 i 1 )[ φ 1 ⊗ E i 1 i 2 | . . . | φ n ⊗ E i n i 0 ]) = str ( e [ φ 0 ⊗ E i 0 i 1 [ φ 1 ⊗ E i 1 i 2 | . . . | φ n ⊗ E i n i 0 ]) = ( − 1) ( n +1) | v i 0 | + P n s =1 | v i s | e [ φ 0 | . . . | φ n ] = h e · str (( φ 0 ⊗ E i 0 i 1 )[ φ 1 ⊗ E i 1 i 2 | . . . | φ n ⊗ E i n i 0 ]) Part d): The statemen t is an easy v ariatio n on the classical Ho c hsc hild-Kostan t-Rosen b erg map (cf. [4]). Part e): The quasi-isomorphism of complexes constructed in [26] is the comp osition of ǫ · str · exp( − b e ( D ( w ))) with the embed ding ( C ( A w ) , b ) → ( C e ( A w ) , b e ) Since th e latter emb edding is a quasi-isomorphism (see Prop osition 3.1), we conclude that ǫ · str · exp( − b e ( D ( w ))) : ( C e ( A w ) , b e ) → (Ω( Y ) , − dw ) is a quasi-isomorph ism . It r emains to apply Lemma 4.1.  50 D. SHKL Y ARO V D.3. Proof of Prop osition 4.3. It will b e con v enien t for u s to work with e ∇ instead of ∇ (see Corollary 3.6). T he pro of of Prop osition 4.3 will b e a com bination of t w o Lemmas. Let us extend exp( − b e ( D ( w ))) to a map f rom C Π , e ( A w ) to itself. Lemma D.2. exp( − b e ( D ( w )))( U ( δ ) + U ( µ ))exp( b e ( D ( w ))) = U ( µ ) + U ( w ) , exp( − b e ( D ( w )))( V ( δ ) + V ( µ ))exp( b e ( D ( w ))) = V ( µ ) + V ( w ) , In p articular 11 (D.4) d du + U ( µ ) + U ( w ) u 2 + V ( µ ) + V ( w ) u is a u -c onne ction on ( C Π , e ( A w ) , b e ( µ ) + b e ( w ) , B e ) homotopy gauge e quivalent to e ∇ and ∇ . Pro of. W e will b e using the f ollo w in g form ulas obtained earlier (see the pro of of Lemma D.1): δ ( i ) n +2 D ( w ) ( j ) n +1 =      2 w ( j ) n +1 , i = j D ( w ) ( j ) n +1 δ ( i − 1) n +1 , i > j D ( w ) ( j ) n +1 δ ( i ) n +1 , i < j (D.5) µ ( i ) n +2 D ( w ) ( j ) n +1 =                D ( w ) ( j − 1) n µ ( i ) n +1 − n − 1 ≤ i − j ≤ − 2 r ( D ( w )) ( j − 1) n +1 i − j = − 1 − l ( D ( w )) ( j ) n +1 i − j = 0 D ( w ) ( j ) n µ ( i − 1) n +1 1 ≤ i − j ≤ n r ( D ( w )) ( j ) n +1 τ − 1 n +1 i − j = n + 1 (D.6) µ ( i ) n +2 w ( j ) n +1 =                − w ( j − 1) n µ ( i ) n +1 − n − 1 ≤ i − j ≤ − 2 − r ( w ) ( j − 1) n +1 i − j = − 1 l ( w ) ( j ) n +1 i − j = 0 − w ( j ) n µ ( i − 1) n +1 1 ≤ i − j ≤ n − r ( w ) ( j ) n +1 τ − 1 n +1 i − j = n + 1 (D.7) In wh at f ollo ws, we r epresen t b e ( D ( w )) by the matrix  b ( D ( w )) 0 0 b ( D ( w ))  as w e agreed in section D.1. 11 Note th e similarit y b etw een (3.11) and (D.4); the connections seem to b e sp ecial cases of a more general form ula which should work for arbitrary curved A ∞ algebras. NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 51 Let us simplify exp( − b e ( D ( w ))) U ( δ ) exp ( b e ( D ( w ))). Clearly , 2[ U ( δ ) , b e ( D ( w ))] n =  − µ (0) n +1 δ (1) n +1 b ( D ( w )) n + b ( D ( w )) n − 1 µ (0) n δ (1) n − µ (0) n +1 δ (1) n +1 b ( D ( w )) n + b ( D ( w )) n − 1 µ (0) n δ (1) n 0 0  By (D.5), (D.6) − µ (0) n +1 δ (1) n +1 b ( D ( w )) n = − n X j =1 µ (0) n +1 δ (1) n +1 D ( w ) ( j ) n = − µ (0) n +1 δ (1) n +1 D ( w ) (1) n − n X j =2 µ (0) n +1 δ (1) n +1 D ( w ) ( j ) n = − 2 µ (0) n +1 w (1) n − n X j =2 µ (0) n +1 D ( w ) ( j ) n δ (1) n = − 2 µ (0) n +1 w (1) n − b ( D ( w )) n − 1 µ (0) n δ (1) n and therefore [ U ( δ ) , b e ( D ( w ))] = 2 U ( w ) (D.8) The op erators b e ( D ( w )) and − µ (0) n +1 w (1) n = r ( w ) (0) n are easily seen to comm ute, so w e conclud e that exp( − b e ( D ( w ))) U ( δ ) exp ( b e ( D ( w ))) = U ( δ ) + 2 U ( w ) (D.9) F ur thermore, u s ing − µ (0) n +1 µ (1) n +2 = µ (0) n +1 µ (0) n +2 2[ U ( µ ) , b e ( D ( w ))] n +1 =  µ (0) n +1 µ (0) n +2 b ( D ( w )) n +1 − b ( D ( w )) n − 1 µ (0) n µ (0) n +1 µ (0) n +1 µ (0) n +2 b ( D ( w )) n +1 − b ( D ( w )) n − 1 µ (0) n µ (0) n +1 0 0  By (D.6) µ (0) n +1 µ (0) n +2 b ( D ( w )) n +1 = n +1 X j =1 µ (0) n +1 µ (0) n +2 D ( w ) ( j ) n +1 = µ (0) n +1 µ (0) n +2 D ( w ) (1) n +1 + n +1 X j =2 µ (0) n +1 µ (0) n +2 D ( w ) ( j ) n +1 = µ (0) n +1 r ( D ( w )) (0) n +1 + n +1 X j =2 µ (0) n +1 D ( w ) ( j − 1) n µ (0) n +1 = µ (0) n +1 r ( D ( w )) (0) n +1 + n X j =1 µ (0) n +1 D ( w ) ( j ) n µ (0) n +1 = µ (0) n +1 r ( D ( w )) (0) n +1 + µ (0) n +1 D ( w ) (1) n µ (0) n +1 + n X j =2 µ (0) n +1 D ( w ) ( j ) n µ (0) n +1 = µ (0) n +1 r ( D ( w )) (0) n +1 + r ( D ( w )) (0) n µ (0) n +1 + b ( D ( w )) n − 1 µ (0) n µ (0) n +1 Observe that µ (0) n +1 r ( D ( w )) (0) n +1 ( a 0 [ a 1 | . . . | a n ]) + r ( D ( w )) (0) n µ (0) n +1 ( a 0 [ a 1 | . . . | a n ]) = − a 0 [ D ( w ) , a 1 ][ a 2 | . . . | a n ] = µ (0) n +1 δ (1) n +1 ( a 0 [ a 1 | . . . | a n ]) Hence [ U ( µ ) , b e ( D ( w ))] = − U ( δ ) 52 D. SHKL Y ARO V Therefore, by (D.8) [[ U ( µ ) , b e ( D ( w ))] , b e ( D ( w ))] = − [ U ( δ ) , b e ( D ( w ))] = − 2 U ( w ) and w e conclude that exp( − b e ( D ( w ))) U ( µ )exp ( b e ( D ( w ))) = U ( µ ) − U ( δ ) − U ( w ) By (D.9) and the latter formula exp( − b e ( D ( w )))( U ( δ ) + U ( µ ))exp( b e ( D ( w ))) = U ( µ ) + U ( w ) Let us compute exp( − b e ( D ( w ))) V ( δ ) exp( b e ( D ( w ))) n ow. W e ha v e 2[ V ( δ ) , b e ( D ( w ))] n =  0 0 − h e n +1 P n i =1 P n +1 l = i +1 τ l n +1 δ ( i ) n +1 b ( D ( w )) n + b ( D ( w )) n +1 h e n P n − 1 i =1 P n l = i +1 τ l n δ ( i ) n 0  By (D.5) − h e n +1 n X i =1 n +1 X l = i +1 τ l n +1 δ ( i ) n +1 b ( D ( w )) n = − h e n +1 n X i =1 n +1 X l = i +1 n X j =1 τ l n +1 δ ( i ) n +1 D ( w ) ( j ) n = − h e n +1 n − 1 X i =1 n +1 X l = i +1 n X j = i +1 τ l n +1 D ( w ) ( j ) n δ ( i ) n − h e n +1 n X i =2 n +1 X l = i +1 i − 1 X j =1 τ l n +1 D ( w ) ( j ) n δ ( i − 1) n − 2 h e n +1 n X i =1 n +1 X l = i +1 τ l n +1 w ( i ) n Using the f ormula D ( w ) ( j ) n = τ − j n +1 D ( w ) (0) n τ j n , the latter equals − h e n +1 n − 1 X i =1 n +1 X l = i +1 n X j = i +1 τ l − j n +1 D ( w ) (0) n τ j n δ ( i ) n − h e n +1 n X i =2 n +1 X l = i +1 i − 1 X j =1 τ l − j n +1 D ( w ) (0) n τ j n δ ( i − 1) n − 2 h e n +1 n X i =1 n +1 X l = i +1 τ l n +1 w ( i ) n = − h e n +1 n − 1 X i =1 n +1 X l = i +1 l − 1 X j = i +1 D ( w ) ( n +1+ j − l ) n τ l − 1 n δ ( i ) n − h e n +1 n − 1 X i =1 n +1 X l = i +1 n X j = l D ( w ) ( j − l ) n τ l n δ ( i ) n − h e n +1 n X i =2 n +1 X l = i +1 i − 1 X j =1 D ( w ) ( n +1+ j − l ) n τ l − 1 n δ ( i − 1) n − 2 h e n +1 n X i =1 n +1 X l = i +1 τ l n +1 w ( i ) n After reindexing we get − h e n +1 n − 1 X i =1 n X l = i +1 n X j = n + i +1 − l D ( w ) ( j ) n τ l n δ ( i ) n − h e n +1 n − 1 X i =1 n X l = i +1 n − l X j =0 D ( w ) ( j ) n τ l n δ ( i ) n NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 53 − h e n +1 n − 1 X i =1 n X l = i +1 n + i − l X j = n +1 − l D ( w ) ( j ) n τ l n δ ( i ) n − 2 h e n +1 n X i =1 n +1 X l = i +1 τ l n +1 w ( i ) n = − h e n +1 n − 1 X i =1 n X l = i +1 n X j =0 D ( w ) ( j ) n τ l n δ ( i ) n − 2 h e n +1 n X i =1 n +1 X l = i +1 τ l n +1 w ( i ) n By (D.2) the latter equals − n X j =0 D ( w ) ( j +1) n +1 h e n n − 1 X i =1 n X l = i +1 τ l n δ ( i ) n − 2 h e n +1 n X i =1 n +1 X l = i +1 τ l n +1 w ( i ) n = − b ( D ( w )) n +1 h e n n − 1 X i =1 n X l = i +1 τ l n δ ( i ) n − 2 h e n +1 n X i =1 n +1 X l = i +1 τ l n +1 w ( i ) n Th us, [ V ( δ ) , b e ( D ( w ))] = 2 V ( w ) (D.10) By essentia lly rep eating th e computation we just did word for word, b ut using (D.3) instead of (D.5), one can show that [[ V ( δ ) , b e ( D ( w ))] , b e ( D ( w ))] = 0 Therefore, exp( − b e ( D ( w ))) V ( δ ) exp( b e ( D ( w ))) = V ( δ ) + 2 V ( w ) (D.11) It remains to compute exp( − b e ( D ( w ))) V ( µ )exp( b e ( D ( w ))). The computation is very similar to the pr evious one. 2[ V ( µ ) , b e ( D ( w ))] n =  0 0 − h e n P n − 1 i =1 P n l = i +1 τ l n µ ( i ) n +1 b ( D ( w )) n + b ( D ( w )) n +1 h e n P n − 1 i =1 P n l = i +1 τ l n µ ( i ) n +1 0  By (D.6) − h e n n − 1 X i =1 n X l = i +1 τ l n µ ( i ) n +1 b ( D ( w )) n = − h e n n − 1 X i =1 n X l = i +1 n X j =1 τ l n µ ( i ) n +1 D ( w ) ( j ) n = − h e n n − 2 X i =1 n X l = i +1 n X j = i +2 τ l n D ( w ) ( j − 1) n − 1 µ ( i ) n − h e n n − 1 X i =2 n X l = i +1 i − 1 X j =1 τ l n D ( w ) ( j ) n − 1 µ ( i − 1) n + h e n n − 1 X i =1 n X l = i +1 τ l n l ( D ( w )) ( i ) n − h e n n − 1 X i =1 n X l = i +1 τ l n r ( D ( w )) ( i ) n 54 D. SHKL Y ARO V = − h e n n − 2 X i =1 n X l = i +1 n − 1 X j = i +1 τ l n D ( w ) ( j ) n − 1 µ ( i ) n − h e n n − 2 X i =1 n X l = i +2 i X j =1 τ l n D ( w ) ( j ) n − 1 µ ( i ) n + h e n n − 1 X i =1 n X l = i +1 τ l n δ ( i ) n Using D ( w ) ( j ) n − 1 = τ − j n D ( w ) (0) n − 1 τ j n − 1 , we get − h e n n − 2 X i =1 n X l = i +1 n − 1 X j = i +1 τ l − j n D ( w ) (0) n − 1 τ j n − 1 µ ( i ) n − h e n n − 2 X i =1 n X l = i +2 i X j =1 τ l − j n D ( w ) (0) n − 1 τ j n − 1 µ ( i ) n + h e n n − 1 X i =1 n X l = i +1 τ l n δ ( i ) n = − h e n n − 2 X i =1 n X l = i +1 l − 1 X j = i +1 D ( w ) ( n + j − l ) n − 1 τ l − 1 n − 1 µ ( i ) n − h e n n − 2 X i =1 n X l = i +1 n − 1 X j = l D ( w ) ( j − l ) n − 1 τ l n − 1 µ ( i ) n − h e n n − 2 X i =1 n X l = i +2 i X j =1 D ( w ) ( n + j − l ) n − 1 τ l − 1 n − 1 µ ( i ) n + h e n n − 1 X i =1 n X l = i +1 τ l n δ ( i ) n Just as in the pr evious compu tation, re-ind exing sho ws that the latter expression equals − h e n n − 1 X j =0 D ( w ) ( j ) n − 1 n − 2 X i =1 n − 1 X l = i +1 τ l n − 1 µ ( i ) n + h e n n − 1 X i =1 n X l = i +1 τ l n δ ( i ) n and, by (D.2), w e ﬁn ally get − b ( D ( w )) n h e n − 1 n − 2 X i =1 n − 1 X l = i +1 τ l n − 1 µ ( i ) n + h e n n − 1 X i =1 n X l = i +1 τ l n δ ( i ) n Th us, [ V ( µ ) , b e ( D ( w ))] = − V ( δ ) and b y (D.10 ) [[ V ( µ ) , b e ( D ( w ))] , b e ( D ( w ))] = − [ V ( δ ) , b e ( D ( w ))] = − 2 V ( w ) Therefore, exp( − b e ( D ( w ))) V ( µ )exp( b e ( D ( w ))) = V ( µ ) − V ( δ ) − V ( w ) This, along w ith (D.11), p r o v es that exp( − b e ( D ( w )))( V ( δ ) + V ( µ ) )exp( b e ( D ( w ))) = V ( µ ) + V ( w ) Lemma D.2 is prov ed.  T o ﬁnish the pr o of of Pr op osition 4.3, one needs to add the follo wing u -morphism to the u -connection (D.4): NC HODGE STRUCTURES: M A TCHING CA TEGORICAL A ND GEOM ETRIC EXAMPLES 55 Lemma D.3. The u -morphism 1 u 2 ( − U ( µ ) + U ( w )) + 1 u ( − V ( µ ) + V ( w ) + Γ) fr om ( C Π , e ( A w ) , b e ( µ ) + b e ( w ) , B e ) to itself is u -homotopic to 0. Pro of is s im ilar to that of Corollary 3.6. W e will sho w that (D.12) 2 ( U ( µ ) − U ( w )) + 2 u ( V ( µ ) − V ( w ) − Γ ) is u -homotopic to 0. W e will b e using the notation introdu ced in the pro of of C orollary 3.6. Let u s compute ( b e ( µ ) + b e ( w ) + uB e ) H 0 + H 0 ( b e ( µ ) + b e ( w ) + uB e ) ﬁr st. W e already know that b e ( µ ) H 0 + H 0 b e ( µ ) = 2 U ( µ ) and B e H 0 + H 0 B e = T By (D.7) b e ( w ) H 0 + H 0 b e ( w ) =  b ( w ) µ (0) + µ (0) b ( w ) b ( w ) µ (0) + µ (0) b ( w ) 0 0  = − 2 U ( w ) Th us, ( b e ( µ ) + b e ( w ) + uB e ) H 0 + H 0 ( b e ( µ ) + b e ( w ) + uB e ) = (2 U ( µ ) − 2 U ( w ) ) + uT (D.13) Let us compute ( b e ( µ ) + b e ( w ) + uB e ) H 1 + H 1 ( b e ( µ ) + b e ( w ) + uB e ). W e already kn o w that B e H 1 + H 1 B e = 0 and b e ( µ ) H 1 + H 1 b e ( µ ) = − T − 2Γ + 2 V ( µ ) (D.14) It remains to simp lify b e ( w ) H 1 + H 1 b e ( w ) . The computation is v ery similar to that of b e ( µ ) H 1 + H 1 b e ( µ ), so let u s simply presen t the result: b e ( w ) H 1 + H 1 b e ( w ) = − 2 V ( w ) This, together with (D.13) and (D.14) ﬁnish es the pr o of.  D.4. Proof of Prop osition 4.4. W e need to sh o w that str · U ( w ) = U ( w ) · str , str · V ( w ) = V ( w ) · st r This is a consequ en ce of the pr o of of Pa rt c) in section D.2 . 56 D. SHKL Y ARO V D.5. Proof of Prop osition 4.5. W e need to sh o w that the morphism of complexes  w u 2 + Γ u  ǫ − ǫ  2 U ( w ) u 2 + 2 V ( w ) + Γ u  from ( C Π , e ( C [ Y ])(( u )) , b e ( µ ) + b e ( w ) + uB e ) to (Ω( Y )(( u )) , − dw + ud ) is homotopic to 0. Both wǫ − 2 ǫ U ( w ) and Γ ǫ − ǫ Γ are easily seen to v anish , so w e only need to sh o w that ǫ V ( w ) is homotopic to 0. Observe that ǫ n +2 h e n +1 τ j n +1 ( φ 0 [ φ 1 | . . . | φ n ]) = ( − 1) j n ǫ n +2 h e n +1 ( φ j [ φ j +1 | . . . | φ j − 1 ]) = ( − 1) j n 1 ( n + 1)! dφ j ∧ dφ j +1 ∧ . . . ∧ dφ j − 1 = 1 ( n + 1)! dφ 0 ∧ dφ 1 ∧ . . . ∧ dφ n = 1 n + 1 d ( ǫ n +1 ( φ 0 [ φ 1 | . . . | φ n ])) and d ( ǫ n +1 ( w ( i ) n ( φ 0 [ φ 1 | . . . | φ n − 1 ]))) = ( − 1) i d ( ǫ n +1 ( φ 0 [ φ 1 | . . . | φ i − 1 | w | . . . | φ n − 1 ])) = ( − 1) i 1 n ! d ( φ 0 dφ 1 ∧ . . . ∧ dφ i − 1 ∧ dw ∧ . . . ∧ dφ n − 1 ) = 1 n ! dw ∧ dφ 0 ∧ dφ 1 ∧ . . . ∧ dφ n − 1 = 1 n dw ∧ d ( ǫ n ( φ 0 [ φ 1 | . . . | φ n − 1 ])) Therefore, 2 ǫ n +2 V ( w ) n = − n X i =1 n +1 X j = i +1 ǫ n +2 · h e n +1 · τ j n +1 · w ( i ) n = − n X i =1 n + 1 − i n + 1 d · ǫ n +1 · w ( i ) n = − n X i =1 n + 1 − i n ( n + 1) ( dw · d ) · ǫ n = 1 2 ( dw · d ) · ǫ n It remains to sho w that ( dw · d ) : (Ω( Y )(( u )) , − dw + ud ) → (Ω( Y )(( u )) , − dw + ud ) is homotopic to 0, wh ic h is obvio us: dw · d = ( − dw + ud ) H + H ( − dw + ud ) , H := 1 u w · d  Referen ces [1] S. 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Non-commutative Hodge structures: Towards matching categorical and geometric examples

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