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Tropical intersection homology

Ryota Mikami

TL;DR

This work defines a tropical analogue of intersection homology to bridge algebraic cycle numerical equivalence with tropical data. It constructs geometric tropical intersection chains with graded coefficients, builds IH_Trop^{p,q}(X;Q) and proves a nondegenerate pairing with compact support, yielding an isomorphism IH_Trop^{p,q}(Trop(φ(Y));Q) ≅ CH_Num^{p}(Y)⊗Q(p=q)0(p≠q) under a surjectivity hypothesis. The paper develops a comprehensive sheaf-theoretic framework based on derived categories of locally graded sheaves, truncation functors, and Verdier duality to realize tropical intersection homology as a robust dual theory, even beyond toric cases. This provides a rigorous tropical-algebraic conduit for understanding numerical equivalence via tropicalization and paves the way for applying tropical methods to motive-theoretic questions.

Abstract

Numerical equivalence of algebraic cycles is defined abstractly by intersection numbers. Classically, for smooth complex proper toric varieties, the quotients by numerical equivalence with rational coefficients can be described geometrically as singular cohomology. They are also expressed in terms of tropical geometry, tropical cohomology, introduced by Itenberg-Katzarkov-Mikhalkin-Zharkov. This paper aims to generalize this to suitable pairs of smooth proper varieties and divisors by introducing a tropical analog of Goresky-MacPherson's intersection homology.

Tropical intersection homology

TL;DR

This work defines a tropical analogue of intersection homology to bridge algebraic cycle numerical equivalence with tropical data. It constructs geometric tropical intersection chains with graded coefficients, builds IH_Trop^{p,q}(X;Q) and proves a nondegenerate pairing with compact support, yielding an isomorphism IH_Trop^{p,q}(Trop(φ(Y));Q) ≅ CH_Num^{p}(Y)⊗Q(p=q)0(p≠q) under a surjectivity hypothesis. The paper develops a comprehensive sheaf-theoretic framework based on derived categories of locally graded sheaves, truncation functors, and Verdier duality to realize tropical intersection homology as a robust dual theory, even beyond toric cases. This provides a rigorous tropical-algebraic conduit for understanding numerical equivalence via tropicalization and paves the way for applying tropical methods to motive-theoretic questions.

Abstract

Numerical equivalence of algebraic cycles is defined abstractly by intersection numbers. Classically, for smooth complex proper toric varieties, the quotients by numerical equivalence with rational coefficients can be described geometrically as singular cohomology. They are also expressed in terms of tropical geometry, tropical cohomology, introduced by Itenberg-Katzarkov-Mikhalkin-Zharkov. This paper aims to generalize this to suitable pairs of smooth proper varieties and divisors by introducing a tropical analog of Goresky-MacPherson's intersection homology.
Paper Structure (24 sections, 82 theorems, 452 equations)

This paper contains 24 sections, 82 theorems, 452 equations.

Table of Contents

  1. Introduction
  2. Tropical varieties
  3. Geometric tropical intersection homology
  4. Geometric definition
  5. Local computations
  6. Derived categories of locally graded sheaves
  7. The abelian category of graded modules
  8. Sheaves of graded modules
  9. Derived categories of sheaves of graded modules
  10. Derived categories of locally graded sheaves
  11. A finiteness condition
  12. Truncation functors
  13. Sheaf-theoretic construction and Poincaré-Verdier duality
  14. On smooth part
  15. Sheaf-theoretic construction and Poincaré-Verdier duality
  16. ...and 9 more sections

Key Result

Theorem 1.2

(Corollary cor:poincare duality for sheaf def-1, Corollary cor:IH =00003D Ch/num ) For $p,q\geq0$, there is a non-degenerate pairing Moreover, when $X=\mathop{\mathrm{Trop}}\nolimits(\varphi(Y))$, under assumption (eq:Intro Assump ch num surj), we have a natural isomorphism under which the pairing of $IH_{\mathop{\mathrm{Trop}}\nolimits}^{*,*}(\mathop{\mathrm{Trop}}\nolimits(\varphi(Y));\mathbb{

Theorems & Definitions (273)

  • Remark 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Example 2.8
  • ...and 263 more