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Tropical Poincaré bundle, Fourier-Mukai transform, and a generalized Poincaré formula

Soham Ghosh, Farbod Shokrieh

TL;DR

This work develops a tropical analogue of the classical Fourier–Mukai theory for real tori with integral structures by constructing a tropical Poincaré bundle on $X\times\widehat{X}$ and a cohomological tropical Fourier–Mukai transform $F_{E_X}$. The authors prove a generalized tropical Poincaré formula that expresses $F_{E_X}$-images of powers of the first Chern class of nondegenerate line bundles in terms of pushforwards along $\phi_{\mathscr{L}}$, linking tropical intersection theory with tropical Pontryagin products. They establish a tropical analytic Riemann–Roch formula for ample tropical line bundles via determinant data of the polarization and show that $F_{E_X}$ induces a Poincaré duality-type isomorphism between $H^{p,q}(X)$ and $H^{g-q,g-p}(\widehat{X})$, preserving the bigrading. The paper then derives strong consequences, including a tropical geometric Riemann–Roch theorem for tropical abelian varieties and explicit tropical Poincaré–Prym formulas, resolving conjectures in the tropical Prym setting and providing a bridge to non-archimedean degeneration phenomena.

Abstract

We construct a tropical analogue of the Poincaré bundle and prove a (cohomological) Fourier-Mukai transform for real tori with integral structures. We then prove a tropical analogue of Beauville's generalized Poincaré formula for polarized abelian varieties. Some consequences include a geometric Riemann-Roch theorem for tropical abelian varieties, as well as a tropical Poincaré-Prym formula which was recently conjectured by Röhrle and Zakharov.

Tropical Poincaré bundle, Fourier-Mukai transform, and a generalized Poincaré formula

TL;DR

This work develops a tropical analogue of the classical Fourier–Mukai theory for real tori with integral structures by constructing a tropical Poincaré bundle on and a cohomological tropical Fourier–Mukai transform . The authors prove a generalized tropical Poincaré formula that expresses -images of powers of the first Chern class of nondegenerate line bundles in terms of pushforwards along , linking tropical intersection theory with tropical Pontryagin products. They establish a tropical analytic Riemann–Roch formula for ample tropical line bundles via determinant data of the polarization and show that induces a Poincaré duality-type isomorphism between and , preserving the bigrading. The paper then derives strong consequences, including a tropical geometric Riemann–Roch theorem for tropical abelian varieties and explicit tropical Poincaré–Prym formulas, resolving conjectures in the tropical Prym setting and providing a bridge to non-archimedean degeneration phenomena.

Abstract

We construct a tropical analogue of the Poincaré bundle and prove a (cohomological) Fourier-Mukai transform for real tori with integral structures. We then prove a tropical analogue of Beauville's generalized Poincaré formula for polarized abelian varieties. Some consequences include a geometric Riemann-Roch theorem for tropical abelian varieties, as well as a tropical Poincaré-Prym formula which was recently conjectured by Röhrle and Zakharov.

Paper Structure

This paper contains 35 sections, 28 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Our contribution
  4. Structure of the paper
  5. Preliminaries on tropical spaces
  6. Boundaryless rational polyhedral spaces
  7. Tropical cycles, cycle classes, and (co)homology
  8. Tropical cycles
  9. Line bundles
  10. Tropical (co)homology
  11. The first Chern class map and tropical cycle class map
  12. Real tori with integral structure
  13. Real tori and integral structures
  14. Tropical (co)homology of real tori with integral structures
  15. Line bundles on real tori with integral structures
  16. ...and 20 more sections

Key Result

Theorem 1

Let $L$ be a symmetric ample line bundle on $A$ with $d=h^0(L)$ and let $\phi_L:A\rightarrow \widehat{A}$ be the corresponding polarization map. Let $F: \operatorname{Ch}^\bullet(A)_{\mathbb{Q}}\rightarrow \operatorname{Ch}^\bullet(\widehat{A})_{\mathbb{Q}}$ be the Fourier--Mukai transform on the ra

Theorems & Definitions (63)

  • Theorem : Beauville
  • Theorem : Beauville
  • Theorem A: =Corollary \ref{['cor:FMpq']}
  • Theorem B: =Theorem \ref{['mainthm']}
  • Theorem C: =Proposition \ref{['prop:tropgenpoincare']}
  • Theorem D: =Corollary \ref{['cor:gRR']}
  • Theorem E: =Corollary \ref{['conPP']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 53 more