Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Kappa classes on KSBA spaces

Valery Alexeev

TL;DR

The paper extends Miller–Morita–Mumford kappa classes to KSBA moduli spaces of stable varieties and pairs via a virtual fundamental class framework, and develops several approaches to define these classes in cohomology. It then computes the resulting kappa classes in explicit settings where the moduli spaces are understood: Burniat surfaces (degrees 6 and 4) and Campedelli surfaces. For Burniat, the author carries out toric and polytope-based intersection calculations to obtain concrete κ0, κ1, and κ2 values on the base del Pezzo surface Z or Σ, illustrating nef/ample behavior through these explicit numbers. For Campedelli, the Chow ring is computed by passing to GL(3,2)-invariants of the SL(3) quotient of a 7-fold product of P^2, and the kappa classes are then expressed in terms of explicit generators in the invariant Chow ring, with detailed formulas linking κ_l to the ring generators. Overall, the work demonstrates concrete KSBA kappa-class computations in higher-dimensional moduli, linking GIT and toric methods to KSBA geometry and providing a blueprint for further calculations in similar KSBA scenarios.

Abstract

We define kappa classes on moduli spaces of KSBA stable varieties and pairs, generalizing the Miller-Morita-Mumford classes on moduli of curves, and compute them in some cases where the virtual fundamental class is known to exist, including Burniat and Campedelli surfaces. For Campedelli surfaces, an intermediate step is finding the Chow (same as cohomology) ring of the GIT quotient $(\mathbb P^2)^7//SL(3)$.

Kappa classes on KSBA spaces

TL;DR

The paper extends Miller–Morita–Mumford kappa classes to KSBA moduli spaces of stable varieties and pairs via a virtual fundamental class framework, and develops several approaches to define these classes in cohomology. It then computes the resulting kappa classes in explicit settings where the moduli spaces are understood: Burniat surfaces (degrees 6 and 4) and Campedelli surfaces. For Burniat, the author carries out toric and polytope-based intersection calculations to obtain concrete κ0, κ1, and κ2 values on the base del Pezzo surface Z or Σ, illustrating nef/ample behavior through these explicit numbers. For Campedelli, the Chow ring is computed by passing to GL(3,2)-invariants of the SL(3) quotient of a 7-fold product of P^2, and the kappa classes are then expressed in terms of explicit generators in the invariant Chow ring, with detailed formulas linking κ_l to the ring generators. Overall, the work demonstrates concrete KSBA kappa-class computations in higher-dimensional moduli, linking GIT and toric methods to KSBA geometry and providing a blueprint for further calculations in similar KSBA scenarios.

Abstract

We define kappa classes on moduli spaces of KSBA stable varieties and pairs, generalizing the Miller-Morita-Mumford classes on moduli of curves, and compute them in some cases where the virtual fundamental class is known to exist, including Burniat and Campedelli surfaces. For Campedelli surfaces, an intermediate step is finding the Chow (same as cohomology) ring of the GIT quotient .
Paper Structure (12 sections, 9 theorems, 18 equations, 2 figures)

This paper contains 12 sections, 9 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definition of kappa classes
  3. Products of curves
  4. Kappa classes on moduli of ${\mathbb Z}_2^k$-covers
  5. Burniat surfaces
  6. Degree $6$
  7. Degree $4$
  8. Kappa classes
  9. Campedelli surfaces
  10. Chow ring
  11. $\mathop{\mathrm{GL}}\nolimits(3,2)$-invariants
  12. Kappa classes

Key Result

Lemma 2.1

For any family $f\colon (X,D)\to S$ of KSBA-stable pairs, there exists a well defined ${\mathbb Q}$-line bundle $K_{X/S}+D$ on $X$ which is functorial, i.e. compatible with base change $S'\to S$.

Figures (2)

  • Figure 1: Burniat configurations of degree 6 and 4 non-nodal cases
  • Figure 2: Surfaces over $Z\subset{\overline M}_6$ of degrees $6$ and $4$

Theorems & Definitions (26)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Definition 2.3: $\kappa_0$ and $\kappa_1$
  • Definition 2.4: Cycle version
  • Definition 2.5: Smooth moduli stack version
  • Definition 2.6: Lci morphisms
  • Definition 2.7: Cohomological version
  • Definition 2.8: Almost lci morphisms
  • Definition 2.9
  • ...and 16 more