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APPEARED IN BULLETIN OF THE

arXiv:math/9204225v1 [math.AG] 1 Apr 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 2, April 1992, Pages 310-314HIGGS LINE BUNDLES, GREEN-LAZARSFELD SETS,AND MAPS OF K¨AHLER MANIFOLDS TO CURVESDonu ArapuraAbstract. Let X be a compact K¨ahler manifold.

The set char(X) of one-dimensionalcomplex valued characters of the fundamental group of X forms an algebraic group.Consider the subset of char(X) consisting of those characters for which the corre-sponding local system has nontrivial cohomology in a given degree d. This set isshown to be a union of finitely many components that are translates of algebraicsubgroups of char(X). When the degree d equals 1, it is shown that some of thesecomponents are pullbacks of the character varieties of curves under holomorphicmaps.

As a corollary, it is shown that the number of equivalence classes (under anatural equivalence relation) of holomorphic maps, with connected fibers, of X ontosmooth curves of a fixed genus > 1 is a topological invariant of X. In fact it dependsonly on the fundamental group of X.Let X denote a compact K¨ahler manifold.

Call two holomorphic maps f : X →Cand f ′ : X →C′, where C and C′ are curves, equivalent if there is an isomorphismσ: C →C′ such that f ′ = σ ◦f. Fix an integer g > 1, and consider the set ofequivalence classes of surjective holomorphic maps, with connected fibers, of X ontosmooth curves of genus g. We will see that this set is finite and that its cardinalityNg(X) depends only on the fundamental group of X.This result is deduced from a structure theorem for certain homologically definedsets of characters.

A character of X is a homomorphism of π1(X) into C∗; it isunitary if the image of π1(X) lies in the unit circle U(1).The set char(X) ofcharacters forms an affine algebraic group. For every character ̺ ∈char(X), welet C̺ denote the local system or locally constant sheaf on X whose monodromyrepresentation is given by ̺.

For each pair of integers i and m, we define the subsetΣim(X) of char(X) to consist of those characters ̺ for which dim Hi(X, C̺) ≥m.We will denote Σi1(X) by Σi(X), and we will suppress the dependence on X whenthere is no danger of confusion.We will call a subset S of char(X) a unitarytranslate of an affine subtorus if there exists a unitary character ̺ ∈char(X) suchthat ̺S is a connected algebraic subgroup.Theorem 1. For X, i, and m as above, the set Σim is a union of finitely manyunitary translates of affine subtori.By a component of Σim, we will mean a unitary translate of an affine subtorusT ⊆Σim that is maximal with respect to inclusion.

Using results of Beauville [B1],[B2], we can explicitly describe the positive dimensional components of Σ1.1991 Mathematics Subject Classification. Primary 14C30.Received by the editors May 22, 1991Partially supported by NSFc⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2DONU ARAPURATheorem 2. Any positive-dimensional component of Σ1 is a translate of an affinesubtorus by a torsion element in char(X).

If T ⊆Σ1 is a positive-dimensional com-ponent containing the trivial character, then there exists a surjective holomorphicmap with connected fibers f : X →C onto a smooth curve of genus at least twosuch that T = f ∗char(C)Corollary. If g ≥2 then Ng(X) is finite and it depends only π1(X).

In otherwords, if X′ is another compact K¨ahler manifold with π1(X′) ∼= π1(X) then Ng(X′) =Ng(X).Sketch of proof. Using the theorem, we see that Ng(X) counts the number of2g-dimensional components of Σ1(X) containing the trivial character.

Σ1 has apurely group theoretic description: ̺ ∈Σ1(X) if and only if H1(π1(X), C̺) ̸= 0.Therefore, an isomorphism ϕ: π1(X) ∼= π1(X′) induces a bijection ϕ∗: char(X′) →char(X) such that ϕ∗(Σ1(X′)) = Σ1(X).□Using Hodge theory, we can give a different, more analytic description of char(X).By a Higgs line bundle, we mean a pair (L, θ) consisting of a holomorphic line bundleL whose first Chern class c1(L) lies in the torsion subgroup H2(X, Z)tors, togetherwith a holomorphic 1-form θ.The set of Higgs lines bundles Higgs(X) can beendowed with the structure of a complex Lie group by identifying it with the productof the Picard torus Pic0(X), H2(X, Z)tors and the vector space of holomorphic 1-forms.We define a map ψ: char(X) →Higgs(X) as follows: ψ(̺) = (L̺, θ̺),where L̺ is the holomorphic bundle whose sheaf of sections is C̺ ⊗C OX and θ̺is the (1, 0) part of log ∥̺∥viewed as a cohomology class under the isomorphismH1(X, R) ∼= Hom(π1(X), R). Then ψ is an isomorphism of topological groups (butnot of complex Lie groups).

Simpson [S] introduced the concept of a Higgs bundle ofarbitrary rank on a K¨ahler manifold; however, the notion of Higgs line bundle alsooccurs implicitly in the work of Green and Lazarsfeld [GL1], [GL2] and Beauville.Before describing the image of Σim under ψ, we need to define the cohomologygroup of a Higgs line bundle (L, θ)Hpq(L, θ) = ker(Hq(X, ΩpX ⊗L)∧θ→Hq(X, Ωp+1X⊗L))im(Hq(X, Ωp−1X⊗L)∧θ→Hq(X, ΩpX ⊗L)).The next theorem follows by combining the results of Green and Lazarsfeld [GL1,3.7] with those of Simpson [S, 3.2].Theorem 3. For each i there is an isomorphismHi(X, C̺) ∼=Mp+q=iHpq(ψ(̺)).We define the setsσpqm = {(L, θ) ∈Higgs(X)| dim Hpq(L, θ) ≥m},Spqm = {L ∈Pic0(X)| dim Hq(X, ΩpX ⊗L) ≥m}.The set Spqm was defined by Green and Lazarsfeld; it equals the intersection of σpqmwith Pic0(X) × {0}.

HIGGS LINE BUNDLES AND GREEN-LAZARSFELD SETS3Corollary. ψ(Σim) = SµT0≤k≤i σk,i−kµ(k) , where µ runs over all partitions of m,i.e., functions µ: {0 · · · i} →{0, 1, 2, .

. .} such that Σµ(k) = m.Let R+ denote the set of positive real numbers viewed as a group under multipli-cation.

A number t ∈R+ acts on a Higgs line bundle by the rule t ∗(L, θ) = (L, tθ).We can transfer this action to char(X) via ψ, namely, t ∗̺ = ψ−1(t ∗ψ(̺)). Af-ter choosing generators for π1(X), we can identify the connected components ofchar(X) with a product of C∗’s.

Under this identification the R+ action is de-scribed byt ∗(r1eiλ1, r2eiλ2, . .

. ) = (r1eitλ1, r2eitλ2, .

. .

)where r1, r2, . .

. λ1, · · · ∈R.We can now indicate the idea of the proof of the first theorem.

Using a Cechcomplex, it is possible to write down equations for Σim, so we conclude that this isan algebraic subset of char(X). The corollary to Theorem 3 shows that this set isstable under the R+ action.

The theorem now follows fromProposition. If V ⊆(C∗)n is a closed irreducible subvariety stable under the aboveR+ action, then V is a unitary translate of an affine subtorus.Sketch of proof.

The Zariski closure of any orbit R+ ∗v, with v ∈(C∗)n, can beshown to be a unitary translate of an affine subtorus. One then checks that for asufficiently general point v ∈V , the orbit R+ ∗v is Zariski dense in V.□As a corollary to Theorem 1, we obtain a new proof of a theorem of Green andLazarsfeld [GL2] about the structure of Spqm .

We say that a subset T of the Picardgroup Pic(X) is a translate of a complex subtorus if there is an element τ ∈Pic(X)such that τ + T is a connected complex Lie subgroup.Corollary. There exist a finite number of translates of complex subtori Ti ofPic(X) and subspaces Vi of the space of holomorphic 1-forms on X with dim Ti =dim Vi, such that σpqm is a union of Ti × Vi.

In particular Spqm is the union of thoseTi contained in Pic0(X).Sketch of proof. σpqm is an analytic subvariety of Higgs(X).

Choose an irreduciblecomponent U of this set. Let i = p + q and for k ∈{0, .

. .

i} defineµ(k) = max{n|U ⊆σk,i−kn}.Then U is an irreducible component of Ti σk,i−kµ(k)that is not contained in Ti σk,i−kµ′(k)for any other partition µ′ of M = Pj µ(j). Thus U is an irreducible componentof ψ(ΣiM).

By the theorem, it can be shown that any irreducible component ofψ(ΣiM) is the image under ψ of a unitary translate of an affine subtorus; such a setis of the form T × V , where T is a translate of a complex subtorus of Pic(X) andV is a subspace of 1-forms of the same dimension.□We will call an unramified cover of X with abelian Galois group an abelian cover.The maximal abelian cover Xab is obtained as the quotient of the universal coverby the commutator subgroup π1(X)′. The Galois group of Xab over X is preciselyH1(X, Z).

The homology groups Hi(Xab, Z) are finitely generated as Z[H1(X, Z)]-modules although not necessarily as abelian groups. Our next theorem give partialsupport to some conjectures of Beauville [B2] and Catanese [C] on the structure ofGreen-Lazarsfeld sets.

4DONU ARAPURATheorem 4. Fix an integer N. Suppose that Hi(Xab, Z) is a finitely generatedabelian group for all i < N. Then(a) Σi(X) consists of a finite set of torsion points of char(X) whenever i < N.(a′) Spq1 (X) consists of a finite set of torsion points in Pic0(X) whenever p+q

(b) There is a finite sheeted abelian cover X′ →X such that Σi(X′) = {1}where 1 is the trivial character whenever i < N.(b′) Spq1 (X′) = {OX} whenever p + q < N.(c) ΣN(X) has a positive-dimensional component if and only if HN(Xab, Q) isinfinite-dimensional. (c′) Spq1 (X) has a positive-dimensional component for some p and q, with p+q =N, if and only if HN(Xab, Q) is infinite-dimensional.Sketch of proof of (a).

Let V be a finite-dimensional C-vector space upon whichA = H1(X, Z) acts. A character ̺ will be called a weight of V if there is a nonzerov ∈V such that for all a ∈A, av = ̺(a)v. We prove a vanishing/nonvanishingtheorem: H0(A, V ⊗CC̺) = 0 if ̺−1 is a weight of V , otherwise Hp(A, V ⊗C C̺) = 0for all p. Let W be the union of the set of weights of Hi(Xab, C) = Hi(Xab, Z)⊗Cwith i < N, and let W −1 be the set of inverses of these weights.

Associated to thecover Xab there is a spectral sequenceEpq2 = Hp(A, Hq(Xab, C) ⊗C C̺) ⇒Hp+q(X, C̺).This together with the vanishing/nonvanishing theorem implies that Si

Since we have shown that the characters in W are alsounitary, it follows by a theorem of Kronecker that they must have finite order.□Corollary. The following are equivalent.

(a) H1(π1(X)′, Q) is infinite-dimensional. (b) There is a finite sheeted abelian cover of X that maps onto a curve of genusat least two.Sketch of proof of (a) ⇒(b).

If H1(π1(X)′, Q) ∼= H1(Xab, Q) ∼= H1(Xab, Q) isinfinite-dimensional then Σ1(X) has a positive-dimensional component. By theorem2, this component is a translate of an affine subtorus by a torsion element.

Thereforethere is a finite abelian cover X′ of X such that the pull back of this component,which lies in Σ1(X′), contains the trivial character. Then Theorem 2 shows thatX′ maps onto a curve of genus at least 2.□AcknowledgmentsI would like to thank A. Beauville, P. Bressler, M. Green, R. Hain, R. Lazars-feld, M. Nori, M. Ramachandran, and C. Simpson for helpful conversations andcorrespondence.

HIGGS LINE BUNDLES AND GREEN-LAZARSFELD SETS5References[A]D. Arapura, Hodge theory with local coefficients on compact varieties, Duke Math. J.

61(1990), 531–543.[B1]A. Beauville, Annulation du H1 et systemes paracanonique sur les surfaces, J. ReineAngew.

Math. 388 (1988), 149–157.

[B2], Annulation du H1 pour fibr´es en droit plats, Proc. Bayreuth Conf.

on Alg. Geom.,Springer-Verlag, New York (to appear).[C]F.

Catanese, Moduli and classification of irregular Kaehler manifolds. .

. , Invent.

Math.104 (1991), 263–289. [GL1] M. Green and R. Lazarsfeld, Deformation theory, generic vanishing theorems..., Invent.Math.

90 (1987), 389–407. [GL2], Higher obstructions of deforming cohomology groups of line bundles, JournalAmer.

Math. Soc.

4 (1991), 87–103.[S]C. Simpson, Higgs bundles and local systems, preprint.Department of Mathematics, Purdue University, West Lafayette, Indiana 47907E-mail address: dvb@math.purdue.edu

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