Let $Z'\subset \mathbb{P}^{n}$ be a smooth projective hypersurface of degree $d>1$ and let $Z\to \mathbb{P}^n$ be the $μ_d$-cover totally ramified along $Z'$. We relate full level $d$ structures on the primitive cohomology $Z'$ with full level $d$ structures on the primitive cohomology of $Z$. In the special case, $d=n=3$ this makes a marking of a smooth cubic surface determine a level $3$-structure on the associated cubic threefold, thereby answering a question by Beauville. We expect many more such applications.
Let Z ′ ⊂ P n be a smooth projective hypersurface of degree d > 1 and let Z → P n be the µ d -cover totally ramified along Z ′ . Our main theorem 3.3 states that if we take the µ d -co-invariants of the primitive homology of Z and reduce modulo d, then this naturally surjects onto the image of the primitive homology of Z ′ on the primitive cohomology of Z ′ , followed by reduction modulo d. When d is a prime, we can also make explicit what the kernel of this surjection is (Corollary 3.4). The case when Z ′ is a cubic surface of particular interest because of the way this relates to the ball quotient description of its moduli space by Allcock-Carlson-Toledo [1].
The proof is based on two simple maps we associate with a module M for the group ring of any finite group G (as defined in Section 2). We apply this to a particular Zµ d -module M whose invariants M µ d yield the primitive cohomology of Z ′ and for which M/M µ d gives the primitive homology of Z. We also apply this to the dual of M for which the dual assertion holds. We expect this to have similar applications to other instances of a smooth Gcoverings relating the homology of the cover with that of the branch locus (which coefficients dividing order of the abelianization of G).
We expect our main result to generalize with other applications of interest. We justify this in Remark 3.6, where we briefly discuss the case of a del Pezzo surface of degree 2 as double a cover of P 2 , a K3 surface of degree 4 as a double cover of del Pezzo surface of degree 2 and a K3 surface of degree 6 as a µ 3 -cover of P 1 × P 1 . These last two covers were used by Kondō for his ball quotient description of the moduli space of genus 3 resp. genus 4 curves. We thus find that these ball quotient descriptions subsist we impose on a level 2 structure on a genus 3 curve or a level 3 structure on a genus 4 curve.
Our main result also leads to a precise description of the (co)primitive cohomology of the smooth Fermat hypersurface as a module over its symmetry group (and thus correct an error in [7]). Damián Gvirtz alerted me (back in March 2021) to this (which he and Skorobogatov corrected in the surface case in [4]) and this was later also observed by Nicholas Addington.
The situation discussed here belongs to a classical theme in algebraic topology. Here the conventional approach is for a space X on which a finite group G acts, to compare the equivariant cohomology H • G (X) and the cohomology of its fixed point set H • (X G ). It would nice to see our main result expressed in these terms and obtained along these lines.
I am grateful to Damián Gvirtz and Nicholas Addington for pointing out the issue mentioned above. I am further indebted to Nicholas and Benson Farb for their comments on an earlier draft.
Let Z ⊂ P n+1 be a smooth hypersurface of degree d > 1. It is well-known that its integral homology and cohomology is torsion free (the discussion below will reprove this). Let η ∈ H 2 (P n ) denote the hyperplane class and η Z ∈ H 2 (Z) its restriction to Z. Then η n Z takes on the fundamental class [Z] ∈ H 2n (Z) the value d. The primitive cohomology of Z, denoted here by H P (Z), is the cokernel of the natural map H n (P n+1 ) → H n (Z) and dually, its primitive homology H P (Z) as the kernel of H n (Z) → H n (P n+1 ). Both are torsion free and each other’s dual. The composite
(where the middle map is given by Poincaré duality) defines the intersection pairing restricted to H P (Z). All these maps are isomorphisms when n is odd, whereas for n is even the composite is injective with cokernel canonically isomorphic with Z/d. Note that when n = 0 (in which case Z is a d-element subset of P 1 ), we have identifications H P (Z) H0 (Z) (the linear combinations of the points on Z with zero coefficient sum) and H P (Z) = H0 (Z) (the linear combinations of the points on Z modulo their sum).
We have a perfect pairing between H P (Z) and H P (Z), denoted (α, b) ∈ H P (Z) × H P (Z) → (α|b). This enables us identify one with the other’s dual. The intersection pairing on H P (Z) satisfies a • b = (ι Z (a)|b).
The following lemma must be well-known.
Lemma 1.1. Let Z ′ ⊂ Z be smooth hyperplane section and put Z := Z ∖ Z ′ . Then Z is a smooth affine hypersurface which has the homotopy type of a wedge of (d -1) n+1 n-spheres and we have a short exact sequences fitting in the (self-dual) commutative diagram
of which the vertical maps are the natural ones (defining the intersection pairing on Hn ( Z)). The precomposition of Hn ( Z) → Hn ( Z) with H P (Z ′ ) → Hn ( Z) and its postcomposition with Hn ( Z) → H P (Z ′ ) are both zero.
Proof. Since Z is a Milnor fiber of the affine cone over Z ′ , its homotopy type is that of a wedge of (d -1) n+1 n-spheres whose number is equal to the Milnor number, which is (d -1) n+1 ([9], Thm. 6.5).
We next establish the lower exact sequence. When n = 0, this boils down the observation we just made, namely that H P (Z) H0 ( Z). We therefore assu
This content is AI-processed based on open access ArXiv data.