[Paper Review] Remarks on the nonvanishing of cohomology groups for perverse sheaves on abelian varieties

Rainer Weissauer|arXiv (Cornell University)|Dec 5, 2016

Algebraic Geometry and Number Theory2 references3 citations

TL;DR

This paper establishes sharp lower bounds on the dimension of cohomology groups of irreducible perverse sheaves on abelian varieties, proving that for such sheaves with nonvanishing top cohomology, the dimension of the cohomology in degree $d-1$ exceeds a multiple of the dimension in degree $d$, with the bound depending on the sheaf's support dimension and the abelian variety's dimension. The result extends the Hard Lefschetz theorem and implies the cohomology is supported in a symmetric interval around zero.

ABSTRACT

It is shown that for a perverse sheaf $K$ on an abelian variety $X$ the integers $i$ for which the cohomology $H^i(X,K)$ does not vanish define an interval in the number line (under certain conditions on the field of definition of $K$)

Motivation & Objective

• To establish quantitative lower bounds on the dimensions of cohomology groups of irreducible perverse sheaves on abelian varieties.
• To extend the Hard Lefschetz theorem by showing that nonvanishing cohomology is symmetrically distributed around zero for such sheaves.
• To analyze the cohomology of perverse sheaves under the Albanese morphism of smooth projective varieties, particularly when the image is an abelian variety.
• To derive cohomological estimates for Galois covers of complex varieties via the decomposition of cohomology into irreducible representations.
• To provide new bounds on the multiplicities of irreducible representations in the cohomology of Galois covers, improving upon known Chevalley-Weil type estimates.

Proposed method

• Uses the convolution power of a perverse sheaf $K$ on an abelian variety $X$, decomposing it into a sum of a perverse sheaf $K_r$ and negligible summands $L_ u[n_ u]$.
• Applies the Künneth formula and coefficient comparison in the Laurent polynomial $h_t(X,K)$ to relate cohomology dimensions at degrees $rd$ and $rd-1$.
• Employs the relative Hard Lefschetz theorem and cohomological bounds from [BBD] to constrain the support and degrees of cohomology groups.
• Uses the structure of quotient maps $\pi: X \to X/A$ for abelian subvarieties $A$ to analyze the contribution of negligible sheaves to the convolution.
• Applies perverse analytic continuation and duality to ensure that irreducible constituents of direct images remain irreducible under the Albanese morphism.
• Relies on the trace formula and representation theory to express cohomology multiplicities in terms of characters and Euler characteristics.

Experimental results

Research questions

• RQ1 What is the minimal possible ratio between the dimensions of cohomology groups $H^{d-1}(X,K)$ and $H^d(X,K)$ for an irreducible perverse sheaf $K$ on an abelian variety with $d = d(K) > 0$?
• RQ2 Under what conditions does the cohomology of a perverse sheaf on a smooth projective variety $Y$ supported in degrees $[-d,d]$ when the Albanese morphism is nontrivial?
• RQ3 How do the cohomology multiplicities of irreducible representations in the cohomology of a Galois cover $\tilde{Y} \to Y$ relate to the geometry of $Y$ and its Albanese variety?
• RQ4 Can the symmetric support of cohomology under the Hard Lefschetz theorem be strengthened by quantitative bounds in terms of the abelian variety's dimension and the sheaf's support?
• RQ5 What is the precise contribution of negligible summands in the convolution of perverse sheaves to the cohomology of the resulting complex?

Key findings

• For any irreducible perverse sheaf $K$ on an abelian variety $X$ of dimension $g$ with $d = d(K) > 0$, the inequality $\dim H^{d-1}(X,K) > \frac{2d}{d+g} \cdot \dim H^d(X,K)$ holds.
• If $X$ is simple, then $\dim H^{d-1}(X,K) > d \cdot \dim H^d(X,K)$, which is a stronger bound than in the general case.
• The cohomology of $K$ is nonvanishing precisely in degrees $\nu \in [-d(K), d(K)]$, a consequence of the Hard Lefschetz theorem and the main inequality.
• For a smooth projective variety $Y$ with nontrivial Albanese morphism $f: Y \to X$, if $H^d(Y,L) \neq H^0(X, {}^pH^d(Rf_*L))$, then $H^\nu(Y,L) \neq 0$ if and only if $\nu \in [-d,d]$.
• In the case of a Galois cover $\pi: \tilde{Y} \to Y$ with group $\Gamma$, the multiplicity $m_{d-1}(\phi)$ of an irreducible representation $\phi$ in $H^{d+n}(\tilde{Y}, \mathbb{C})$ satisfies $m_{d-1}(\phi) > \frac{2d}{d+g} m_d(\phi)$.
• For surfaces $Y$ and nontrivial irreducible representations $\phi$, the Chevalley-Weil trace formula yields $m_0(\phi) - 2m_1(\phi) \geq 0$, improving previous bounds.