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Atoms meet symbols

Leonardo F. Cavenaghi, Ludmil Katzarkov, Maxim Kontsevich

Abstract

This paper introduces a novel framework for constructing invariants in $G$-equivariant birational geometry by unifying two recent approaches: the theory of atoms recently developed by Katzarkov, Kontsevich, Pantev, and Yu, and the theory of modular symbols due to Kontsevich, Tschinkel, and Pestun. We initiate the theory of Chen-Ruan atoms. Assuming the blowup formula for the quantum Chen-Ruan cohomology, we outline how to extend the theory of atoms to global quotient orbifolds and present some striking applications. In addition, we develop a separate class of purely geometric invariants for $\mathbb{Z}/2$- and $\mathbb{Z}/3$-actions on surfaces and threefolds. We provide many examples of non-$G$-linearizable $G$-actions on projective varieties treated with these new techniques.

Atoms meet symbols

Abstract

This paper introduces a novel framework for constructing invariants in -equivariant birational geometry by unifying two recent approaches: the theory of atoms recently developed by Katzarkov, Kontsevich, Pantev, and Yu, and the theory of modular symbols due to Kontsevich, Tschinkel, and Pestun. We initiate the theory of Chen-Ruan atoms. Assuming the blowup formula for the quantum Chen-Ruan cohomology, we outline how to extend the theory of atoms to global quotient orbifolds and present some striking applications. In addition, we develop a separate class of purely geometric invariants for - and -actions on surfaces and threefolds. We provide many examples of non--linearizable -actions on projective varieties treated with these new techniques.

Paper Structure

This paper contains 29 sections, 35 theorems, 124 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The Theory of Atoms
  3. Non-archimedean A-model F-bundles
  4. Action of the Mumford-Tate group
  5. The Spectral Decomposition and Local Hodge Atoms
  6. The Global Set of Hodge Atoms
  7. Motivation from Birational Geometry
  8. Definition of Global Hodge Atoms
  9. $G$-equivariant atoms
  10. Filtration by dimension
  11. Why atoms? Hypothetical atomic semi-orthogonal decomposition
  12. Some examples and first applications
  13. Use of classical monodromy
  14. Atoms meet symbols: A new $\mathbb Z$-module
  15. A recap of modular symbols
  16. ...and 14 more sections

Key Result

Theorem 1

Assume that $X(1,1,1,1)$ is invariant under the $\mathbb Z/2$-action induced by swapping the first and second $\mathbb{P}^1$ factors and the third and fourth $\mathbb{P}^1$ factors. Then, there is no $\mathbb Z/2$-linearizable action on $\mathbb{P}^3$ making it $\mathbb Z/2$-equivariantly birational

Figures (1)

  • Figure 1: Clusters

Theorems & Definitions (77)

  • Theorem 1: =Theorem \ref{['thm:X(1,1,1,1)-withZ_2']}
  • Corollary 2: =Corollary \ref{['thm:X(1,1,1,1)']}
  • Corollary 3: =Corollary \ref{['cor:Kuznetsov']}
  • Theorem 4
  • Theorem 5: =Example \ref{['ex:cubic-surface']}
  • Theorem 6: =Example \ref{['ex:multi-degree-times']}
  • Theorem 7: =Theorem \ref{['thm:invariant']}
  • Theorem 8: =Theorem \ref{['thm:invariant-fine']}
  • Theorem 9: =Example \ref{['ex:fixed-higher-genus-curve']}
  • Theorem 10: =Example \ref{['ex:product=with-line']}
  • ...and 67 more