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Differential forms and cohomology in tropical and complex geometry

Ryota Mikami

Abstract

Ducros, Hrushovski, and Loeser gave maps from families of archimedean diffrential forms to non-archiemedean (or tropical) ones, which are compatible with integrals on algebraic varieties. In this paper, we introduce slight modifications of their maps for complex projective varieties which give natural maps from tropical to the usual Dolbeault cohomology. We also show that our maps are compatible with integrals on generic semi-algebraic subsets and those on their weighted tropicalizations. Weighted tropicalizations induce the dual maps of the above maps of Dolbeault cohomology groups under some assumptions.

Differential forms and cohomology in tropical and complex geometry

Abstract

Ducros, Hrushovski, and Loeser gave maps from families of archimedean diffrential forms to non-archiemedean (or tropical) ones, which are compatible with integrals on algebraic varieties. In this paper, we introduce slight modifications of their maps for complex projective varieties which give natural maps from tropical to the usual Dolbeault cohomology. We also show that our maps are compatible with integrals on generic semi-algebraic subsets and those on their weighted tropicalizations. Weighted tropicalizations induce the dual maps of the above maps of Dolbeault cohomology groups under some assumptions.

Paper Structure

This paper contains 16 sections, 23 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Tropicalizations of algebraic varieties
  4. Fans
  5. Tropicalizations of complex algeraic varieties
  6. Schön compactifications
  7. Tropical cohomology and superforms
  8. Superforms and tropical Dolbeault cohomology
  9. Tropical homology
  10. Tropical $K$-groups and cycle class maps
  11. Construction of $\mathop{\mathrm{Trop}}\nolimits^*$
  12. Semi-algebraic geometry and logarithmic integrals
  13. Weighted tropcializations of semi-algebraic chains
  14. Weighted tropicalizations
  15. Comparison of integrals
  16. ...and 1 more sections

Key Result

Proposition 1.1

Let $p \geq 0$ be an integer, and $L$ a finitely generated extension of $\mathbb{C}$. Then there exists an isomorphism where $\Omega_{\mathop{\mathrm{Kah}}\nolimits}$ is the Kähler differential.

Theorems & Definitions (60)

  • Proposition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark 1.8
  • Definition 3.1
  • Definition 3.2
  • ...and 50 more