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An arithmetic Yau-Zaslow formula

Jesse Pajwani, Ambrus Pál

TL;DR

This paper provides an arithmetic refinement of the Yau-Zaslow formula for K3 surfaces by replacing the classical Euler characteristic with a motivic Euler characteristic χ^{mot} valued in the Grothendieck-Witt ring. It develops a robust framework: defining χ^{mot}, establishing Galois descent and transfer compatibility, proving motivic Fubini theorems (including real-closed and group-torsor cases), and applying these to compactified Jacobians, Calabi–Yau varieties, and the Göttsche formula. The main result is an arithmetic Yau-Zaslow formula that encodes both complex-counting (rank) and real/enriched data (signature, discriminant) and specializes to known complex and real YZ formulas while permitting discriminant-based refinements. The approach unifies motivic methods with determinant-of-cohomology techniques, yielding new tools for refined curve counting and connections to Hilbert schemes and moduli of sheaves on K3 surfaces.

Abstract

We prove an arithmetic refinement of the Yau-Zaslow formula by replacing the classical Euler characteristic in Beauville's argument by a "motivic Euler characteristic", related to the work of Levine. Our result implies similar formulas for other related invariants, including a generalisation of a formula of Kharlamov and Rasdeaconu on counting real rational curves on real K3 surfaces, and Saito's determinant of cohomology.

An arithmetic Yau-Zaslow formula

TL;DR

This paper provides an arithmetic refinement of the Yau-Zaslow formula for K3 surfaces by replacing the classical Euler characteristic with a motivic Euler characteristic χ^{mot} valued in the Grothendieck-Witt ring. It develops a robust framework: defining χ^{mot}, establishing Galois descent and transfer compatibility, proving motivic Fubini theorems (including real-closed and group-torsor cases), and applying these to compactified Jacobians, Calabi–Yau varieties, and the Göttsche formula. The main result is an arithmetic Yau-Zaslow formula that encodes both complex-counting (rank) and real/enriched data (signature, discriminant) and specializes to known complex and real YZ formulas while permitting discriminant-based refinements. The approach unifies motivic methods with determinant-of-cohomology techniques, yielding new tools for refined curve counting and connections to Hilbert schemes and moduli of sheaves on K3 surfaces.

Abstract

We prove an arithmetic refinement of the Yau-Zaslow formula by replacing the classical Euler characteristic in Beauville's argument by a "motivic Euler characteristic", related to the work of Levine. Our result implies similar formulas for other related invariants, including a generalisation of a formula of Kharlamov and Rasdeaconu on counting real rational curves on real K3 surfaces, and Saito's determinant of cohomology.
Paper Structure (16 sections, 102 theorems, 198 equations)

This paper contains 16 sections, 102 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Acknowledgements
  4. The motivic Euler characteristic and related invariants
  5. Galois descent for the motivic Euler characteristic
  6. The motivic Euler characteristic and transfer maps
  7. Fubini theorems
  8. The Fubini theorem over real closed fields
  9. The determinant of cohomology
  10. A naïve Fubini theorem for algebraic groups
  11. The motivic Euler characteristic of compactified Jacobians of curves
  12. The motivic Euler characteristics of compactified Jacobians of nodal curves
  13. Motivic Euler characteristics of the compactified Jacobian of the linear system
  14. The motivic Euler characteristic of Calabi-Yau varieties
  15. The motivic Göttsche formula
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $k=\mathbb{C}$. Then $n_g$ depends only on $g$, not on the choice of $K3$ surface $X$ or the choice of linear system of genus $g$ curves $\mathcal{C}$. Moreover there is a generating series for $n_g$ in $\mathbb{Z}[[t]]$

Theorems & Definitions (261)

  • Theorem 1.1: (Complex Yau--Zaslow formula, Be, YZ)
  • Theorem 1.2: (Real Yau--Zaslow formula, KR1)
  • Theorem 1.4: (Arithmetic Yau--Zaslow formula)
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Theorem 2.6: (Theorem 1.3 of LR)
  • ...and 251 more