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Fourier-Mukai partners of abelian varieties and K3 surfaces in positive and mixed characteristics

Riku Kurama

Abstract

We study Fourier-Mukai equivalences of (families of) abelian varieties and K3 surfaces in positive and mixed characteristics. We first prove in any characteristics that Fourier-Mukai partners of abelian varieties are again abelian varieties. We subsequently focus on the canonical lifts of ordinary abelian varieties and ordinary K3 surfaces. For such schemes, we show that Fourier-Mukai equivalences on the special fibers can be lifted to the canonical lifts. We also prove that the relative Fourier-Mukai partners of the canonical lifts are in bijection with the Fourier-Mukai partners of the special fibers. We conclude by demonstrating that the last result can be used to recover the ordinary case of a result originally proved by Honigs, Lombardi and Tirabassi.

Fourier-Mukai partners of abelian varieties and K3 surfaces in positive and mixed characteristics

Abstract

We study Fourier-Mukai equivalences of (families of) abelian varieties and K3 surfaces in positive and mixed characteristics. We first prove in any characteristics that Fourier-Mukai partners of abelian varieties are again abelian varieties. We subsequently focus on the canonical lifts of ordinary abelian varieties and ordinary K3 surfaces. For such schemes, we show that Fourier-Mukai equivalences on the special fibers can be lifted to the canonical lifts. We also prove that the relative Fourier-Mukai partners of the canonical lifts are in bijection with the Fourier-Mukai partners of the special fibers. We conclude by demonstrating that the last result can be used to recover the ordinary case of a result originally proved by Honigs, Lombardi and Tirabassi.

Paper Structure

This paper contains 13 sections, 43 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Relative Fourier-Mukai transform
  3. Fourier-Mukai partners of abelian varieties
  4. Abelian variety case
  5. Relative de Rham cohomology of abelian schemes
  6. Grothendieck-Messing theory via Mukai structures
  7. Proofs of the main results for abelian varieties
  8. K3 surface case
  9. Crystalline cohomology
  10. De Rham cohomology
  11. Proofs of the main results for K3 surfaces
  12. Sample application to a result in Honigs
  13. Crystalline and de Rham Chern classes

Key Result

Theorem 1.1

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ and $Y$ be ordinary abelian varieties (resp. ordinary K3 surfaces in which case we assume $p>2$), and assume that we have a kernel $E\in\mathop{\mathrm{D}}\nolimits^{\mathop{\mathrm{b}}\nolimits}_{\mathop{\mathrm{coh}}\nolimit

Theorems & Definitions (86)

  • Theorem 1.1: Lifting Fourier-Mukai equivalences; Propositions \ref{['CanLiftAVKer']}, \ref{['K3LiftKer']}
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.5: Relative Fourier-Mukai partners of canonical lifts; Propositions \ref{['FMPofCanLiftAV']}, \ref{['FMPofCanLiftK3']}
  • Corollary 1.6: Special case of Honigs; Proposition \ref{['HonigsApp']}
  • Theorem 1.7: Fourier-Mukai partners of abelian varieties; Proposition \ref{['FMPAV']}
  • Theorem 1.8: Brantner Taelman, BranTael
  • Remark 1.11
  • Definition 2.1: Relative Fourier-Mukai transform
  • ...and 76 more