[Paper Review] Cohomological rank functions on abelian varieties

Zhi Jiang, Giuseppe Pareschi|arXiv (Cornell University)|Jul 18, 2017

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper introduces cohomological rank functions for $\mathbb{Q}$-twisted complexes on abelian varieties, establishing a transformation formula under the Fourier-Mukai-Poincare9 transform. The key contribution is a precise duality relation that enables the analysis of geometric invariants, leading to applications in GV-subschemes and surjectivity of multiplication maps of global sections of ample line bundles.

ABSTRACT

Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper we consider cohomological rank functions of $\mathbb Q$-twisted (complexes of) coherent sheaves on abelian varieties. They satisfy a natural transformation formula with respect to the Fourier-Mukai-Poincaré transform, which has several consequences. In many concrete geometric contexts these functions provide useful invariants. We illustrate this with two different applications, the first one to GV-subschemes and the second one to multiplication maps of global sections of ample line bundles on abelian varieties.

Motivation & Objective

• To generalize the continuous rank function of Barja-Pardini-Stoppino to cohomological rank functions of $\mathbb{Q}$-twisted complexes on abelian varieties.
• To establish a transformation formula under the Fourier-Mukai-Poincare9 transform that relates cohomological ranks at different rational twists.
• To apply the resulting duality to study GV-subschemes and the surjectivity of multiplication maps of global sections of ample line bundles.
• To prove that cohomological rank functions extend continuously to $\mathbb{R}$ and are piecewise polynomial with controlled regularity at rational points.
• To derive quantitative bounds on the critical value $s({\underline{l}})$ for normal generation of line bundles, especially in the case of divisible polarizations.

Proposed method

• Define cohomological rank functions $h^{i}_{\mathcal{F},{\underline{l}}}(x)$ for $x \in \mathbb{Q}$ using the generic cohomology ranks of $\mathcal{F} \otimes L^x$ via isogenies $\mu_b$.
• Establish a transformation formula (Proposition 2.3) relating $h^{i}_{\mathcal{F}}(x{\underline{l}})$ to the cohomological ranks of the Fourier-Mukai transform $\varphi_{\underline{l}}^*\Phi_{\mathcal{P}}(\mathcal{F})$ for $x < 0$ and $x > 0$.
• Use the transformation formula to show that $h^{i}_{\mathcal{F},{\underline{l}}}$ is piecewise polynomial of degree at most $g$ near each rational $x_0$, with explicit polynomial expressions derived from Hilbert polynomials.
• Prove that $h^{i}_{\mathcal{F},{\underline{l}}}$ extends continuously to $\mathbb{R}_{\geq 0}$ and analyze its regularity at rational points via the jump locus $J^{i+}(\mathcal{F}\langle x_0\rangle)$.
• Apply the duality to study the multiplication maps $H^0(L^h) \otimes H^0(L^{hab}_\alpha) \to H^0(\mu_b^*L^h \otimes L^{hab}_\alpha)$, proving surjectivity for $\frac{a}{b} > \frac{1}{h-1}$ when $h{\underline{l}}$ is basepoint free.
• Define $\beta({\underline{l}})$ as the critical value for normal generation and derive the formula $s(h{\underline{l}}) = \frac{\beta({\underline{l}})}{h - \beta({\underline{l}})}$ for the critical value of $h{\underline{l}}$.

Experimental results

Research questions

• RQ1 How do cohomological rank functions behave under the Fourier-Mukai-Poincare9 transform, and what duality relations govern their transformation?
• RQ2 What is the regularity and piecewise polynomial structure of cohomological rank functions near rational points on $\mathbb{Q}$?
• RQ3 How can the transformation formula be used to determine the surjectivity of multiplication maps of global sections of ample line bundles on abelian varieties?
• RQ4 What is the precise value of the critical threshold $s({\underline{l}})$ for normal generation of a line bundle $L$ on an abelian variety?
• RQ5 Under what conditions does the multiplication map $H^0(L^h) \otimes H^0(L^{hab}_\alpha) \to H^0(\mu_b^*L^h \otimes L^{hab}_\alpha)$ remain surjective for general or all $\alpha$?

Key findings

• The cohomological rank function $h^{i}_{\mathcal{F},{\underline{l}}}$ extends continuously to a function $\mathbb{R} \to \mathbb{R}_{\geq 0}$, resolving a question raised in [BPaSt].
• For each $x_0 \in \mathbb{Q}$, $h^{i}_{\mathcal{F},{\underline{l}}}$ is piecewise polynomial with left and right polynomials $P^{-}_{i,\mathcal{F},x_0}$ and $P^{+}_{i,\mathcal{F},x_0}$ of degree at most $g$, and smoothness at $x_0$ holds iff the two polynomials coincide.
• The transformation formula implies that $h^{i}_{\mathcal{F}}(x{\underline{l}})$ for $x < 0$ is proportional to $h^{i}_{\varphi_{\underline{l}}^*\Phi_{\mathcal{P}}(\mathcal{F})}(-\frac{1}{x}{\underline{l}})$, with a factor $\frac{(-x)^g}{\chi({\underline{l}})}$, and similarly for $x > 0$.
• For $h \geq 2$ such that $h{\underline{l}}$ is basepoint free, the critical value for normal generation satisfies $s(h{\underline{l}}) = \frac{\beta({\underline{l}})}{h - \beta({\underline{l}})}$, with equality in $s(h{\underline{l}}) = \frac{1}{h-1}$ if and only if ${\underline{l}}$ has base points.
• The multiplication maps $H^0(L^h) \otimes H^0(L^{hab}_\alpha) \to H^0(\mu_b^*L^h \otimes L^{hab}_\alpha)$ are surjective for all $\alpha \in \widehat{A}$ if $\frac{a}{b} > \frac{1}{h-1}$, and for general $\alpha$ if $\frac{a}{b} \geq \frac{1}{h-1}$, with surjectivity at the critical value $\frac{a}{b} = \frac{1}{h-1}$ holding for all $\alpha$ if ${\underline{l}}$ is basepoint free.
• When ${\underline{l}}$ is a principal polarization, the source and target of the critical map $H^0(L^h) \otimes H^0(L^{h(h-1)}_\alpha) \to H^0(\mu_{h-1}^*L^h \otimes L^{h(h-1)}_\alpha)$ have the same dimension $(h^2(h-1))^g$.