Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hodge numbers are not derived invariants in positive characteristic

Nicolas Addington, Daniel Bragg

TL;DR

The paper shows that in positive characteristic, Hodge numbers need not be preserved under derived equivalences by constructing derived-equivalent Calabi–Yau threefolds X and M in characteristic 3 with different Hodge diamonds; X and M lift to characteristic zero with equal Hodge diamonds, but reduction modulo 3 yields distinct Hodge numbers, including a nonzero h^{0,1} and a nontrivial jump in h^{2,0}. The authors build X as a Calabi–Yau threefold fibered by Abelian surfaces, and M as its dual fibration via a moduli-space construction of sheaves, and prove D^b(X)  D^b(M) using a generalized Bridgeland–Maciocia framework. They further compute Picard torsion, fundamental groups, and perform a detailed analysis using Hodge–Witt, crystalline, and TR invariants to understand the positive-characteristic behavior; they also discuss broader implications for derived-invariance questions and related higher-dimensional phenomena (Petrov) and Abuaf-type invariants. Altogether, the work highlights that Betti numbers remain robust derived-invariants in certain regimes, while Hodge numbers can jump in characteristic p, thus challenging naive generalizations of characteristic-zero expectations to positive characteristic.

Abstract

We study a pair of Calabi-Yau threefolds X and M, fibered in non-principally polarized Abelian surfaces and their duals, and an equivalence D^b(X) = D^b(M), building on work of Gross, Popescu, Bak, and Schnell. Over the complex numbers, X is simply connected while pi_1(M) = (Z/3)^2. In characteristic 3, we find that X and M have different Hodge numbers, which would be impossible in characteristic 0. In an appendix, we give a streamlined proof of Abuaf's result that the ring H^*(O) is a derived invariant of complex threefolds and fourfolds. A second appendix by Alexander Petrov gives a family of higher-dimensional examples to show that h^{0,3} is not a derived invariant in any positive characteristic.

Hodge numbers are not derived invariants in positive characteristic

TL;DR

The paper shows that in positive characteristic, Hodge numbers need not be preserved under derived equivalences by constructing derived-equivalent Calabi–Yau threefolds X and M in characteristic 3 with different Hodge diamonds; X and M lift to characteristic zero with equal Hodge diamonds, but reduction modulo 3 yields distinct Hodge numbers, including a nonzero h^{0,1} and a nontrivial jump in h^{2,0}. The authors build X as a Calabi–Yau threefold fibered by Abelian surfaces, and M as its dual fibration via a moduli-space construction of sheaves, and prove D^b(X)  D^b(M) using a generalized Bridgeland–Maciocia framework. They further compute Picard torsion, fundamental groups, and perform a detailed analysis using Hodge–Witt, crystalline, and TR invariants to understand the positive-characteristic behavior; they also discuss broader implications for derived-invariance questions and related higher-dimensional phenomena (Petrov) and Abuaf-type invariants. Altogether, the work highlights that Betti numbers remain robust derived-invariants in certain regimes, while Hodge numbers can jump in characteristic p, thus challenging naive generalizations of characteristic-zero expectations to positive characteristic.

Abstract

We study a pair of Calabi-Yau threefolds X and M, fibered in non-principally polarized Abelian surfaces and their duals, and an equivalence D^b(X) = D^b(M), building on work of Gross, Popescu, Bak, and Schnell. Over the complex numbers, X is simply connected while pi_1(M) = (Z/3)^2. In characteristic 3, we find that X and M have different Hodge numbers, which would be impossible in characteristic 0. In an appendix, we give a streamlined proof of Abuaf's result that the ring H^*(O) is a derived invariant of complex threefolds and fourfolds. A second appendix by Alexander Petrov gives a family of higher-dimensional examples to show that h^{0,3} is not a derived invariant in any positive characteristic.

Paper Structure

This paper contains 8 sections, 33 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Construction of X and calculation of its Hodge numbers
  3. Construction of M and the derived equivalence
  4. The Picard scheme of M
  5. Hodge numbers of M
  6. Hodge-Witt and crystalline cohomology
  7. A result of Abuaf
  8. A higher-dimensional example in any characteristic (by Alexander Petrov)

Key Result

Theorem 1.1

There are smooth projective threefolds $X$ and $M$ defined over $\bar{\mathbf F}_3$, with Hodge numbers $h^{i,j} = h^j(\Omega^i)$ as shown, and an $\bar{\mathbf F}_3$-linear exact equivalence $D^b(X) \cong D^b(M)$.

Theorems & Definitions (67)

  • Theorem 1.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Corollary 2.4
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 57 more