[Paper Review] Asymptotic cohomological functions of toric divisors

Milena Hering, Alex Küronya+1 more|ArXiv.org|Jan 7, 2005

Algebraic Geometry and Number Theory4 references5 citations

TL;DR

This paper establishes that asymptotic cohomological functions of toric divisors—measuring the growth rates of cohomology groups of multiples of divisors—are continuous and piecewise polynomial with respect to a finite polyhedral chamber decomposition of the real Néron-Severi group. The key contribution is a formula for the self-intersection number of a $T$-Cartier divisor in terms of volumes of bounded regions in a hyperplane arrangement, and an asymptotic converse to Serre vanishing is proven in the toric setting.

ABSTRACT

We study functions on the class group of a toric variety measuring the rates of growth of the cohomology groups of multiples of divisors. We show that these functions are piecewise polynomial with respect to finite polyhedral chamber decompositions. As applications, we express the self-intersection number of a T-Cartier divisor as a linear combination of the volumes of the bounded regions in the corresponding hyperplane arrangement and prove an asymptotic converse to Serre vanishing.

Motivation & Objective

• To study the asymptotic behavior of cohomology groups $H^i(X, \mathcal{O}(mD))$ as $m \to \infty$ for toric divisors $D$.
• To establish that the higher asymptotic cohomological functions $\widehat{h}^i(D)$ are continuous and piecewise polynomial on the real Néron-Severi space $A_{n-1}(X)_{\mathbb{R}}$.
• To provide a combinatorial formula for the self-intersection number $(D^n)$ of a $T$-Cartier divisor using volumes of bounded regions in the hyperplane arrangement defined by $D$.
• To prove an asymptotic converse to Serre vanishing in the toric setting, showing that vanishing of higher cohomology for large $m$ implies numerical positivity.

Proposed method

• Utilizes local cohomology computations by Eisenbud, Mustaţǎ, and Stillman to show that the dimensions of graded pieces $H^i(X, \mathcal{O}(D))_u$ are constant on polyhedral regions in $M_{\mathbb{R}}$ indexed by subsets $I \subset \Delta(1)$.
• Defines regions $P_{D,I} = \{ u \in M_{\mathbb{R}} : \langle u, v_\rho \rangle \geq -d_\rho \text{ iff } \rho \in I \}$, which are dilated under $mD$.
• Applies the piecewise constancy of $h^i(mD)$ on these regions to prove the existence of the limit $\widehat{h}^i(D) = \lim_{m \to \infty} \frac{h^i(mD)}{m^n / n!}$.
• Uses the Gelfand-Kapranov-Zelevinsky (GKZ) decomposition of the effective cone into finitely many rational polyhedral chambers where the combinatorial type of the polytope $P_D$ is constant.
• Derives the formula $\chi(\mathcal{O}(D)) = (-1)^n \sum_{P_{D,I} \text{ bounded}} \chi(\Delta_I) \cdot (\# P_{D,I} \cap M)$, linking Euler characteristic to lattice point counts in bounded regions.
• Establishes that the self-intersection number $(D^n)$ equals the sum of $\chi(\Delta_I) \cdot \mathrm{vol}(P_{D,I})$ over bounded $P_{D,I}$, with $\chi(\Delta_I)$ being the Euler characteristic of the fan $\Delta_I$.

Experimental results

Research questions

• RQ1 Are the higher asymptotic cohomological functions $\widehat{h}^i(D)$ continuous and piecewise polynomial on the Néron-Severi space of a toric variety?
• RQ2 Can the self-intersection number $(D^n)$ of a $T$-Cartier divisor be expressed as a sum over combinatorial invariants of bounded regions in a hyperplane arrangement?
• RQ3 Does the vanishing of higher cohomology groups $H^i(X, \mathcal{O}(mD))$ for all $i > 0$ and large $m$ imply that $D$ is numerically positive?
• RQ4 How does the GKZ decomposition of the effective cone relate to the chamber structure of the volume and cohomological functions?

Key findings

• The asymptotic cohomological functions $\widehat{h}^i(D)$ exist as limits and are continuous and piecewise polynomial with respect to a finite polyhedral chamber decomposition of $A_{n-1}(X)_{\mathbb{R}}$.
• The volume function $\mathrm{vol}(D) = \widehat{h}^0(D)$ is given by distinct polynomial expressions on distinct GKZ chambers, with different $n$-th order derivatives on different chambers.
• The self-intersection number $(D^n)$ of a $T$-Cartier divisor is equal to $\sum_{P_{D,I} \text{ bounded}} \chi(\Delta_I) \cdot \mathrm{vol}(P_{D,I})$, where $\chi(\Delta_I)$ is the Euler characteristic of the fan $\Delta_I$.
• The Euler characteristic $\chi(\mathcal{O}(D))$ is computed as $(-1)^n \sum_{P_{D,I} \text{ bounded}} \chi(\Delta_I) \cdot (\# P_{D,I} \cap M)$, linking topological and combinatorial invariants.
• The volume function is polynomial on each GKZ chamber, and the decomposition is finite and rational polyhedral, with chambers corresponding to Mori and GIT chambers.
• An asymptotic converse to Serre vanishing holds: if $H^i(X, \mathcal{O}(mD)) = 0$ for all $i > 0$ and all $m \gg 0$, then $D$ is numerically positive, and $\widehat{h}^0(D) = (D^n)$.