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On lattice-polarized K3 surfaces

Valery Alexeev, Philip Engel

TL;DR

The paper corrects and refines the framework for lattice-polarized and lattice-quasipolarized K3 surfaces by replacing Weyl-chamber data with generalized 'small cones' to stabilize period-map fibers. It extends the theory to ADE K3 surfaces, proving separated moduli stacks and identifying coarse moduli spaces with period-domain quotients D_Λ/Γ, while showing independence from the chosen very irrational vector within a small cone. The approach unifies smooth and ADE cases, clarifies non-separation phenomena, and provides a robust moduli-theoretic foundation likely adaptable to K-trivial and hyperkähler varieties. Key contributions include corrected definitions (mqpol), a detailed small-cone analysis, and a separated moduli-theoretic description for ADE K3 surfaces. The results have broad implications for applications and future generalizations in the realm of K3-type and hyperkähler moduli.

Abstract

We propose modifications to the commonly used definitions of lattice-polarized and lattice-quasipolarized smooth K3 surfaces, collecting various versions of the definition, and determining the effects of these choices on the resulting moduli space. We fill a gap in the theory, by replacing Weyl chambers with the new notion of a ``small cone'': the true datum in the definition of lattice quasipolarized K3 surfaces. In addition, we describe the separated moduli stack and moduli space for lattice-polarized K3 surfaces with $ADE$ singularities, an important notion for applications.

On lattice-polarized K3 surfaces

TL;DR

The paper corrects and refines the framework for lattice-polarized and lattice-quasipolarized K3 surfaces by replacing Weyl-chamber data with generalized 'small cones' to stabilize period-map fibers. It extends the theory to ADE K3 surfaces, proving separated moduli stacks and identifying coarse moduli spaces with period-domain quotients D_Λ/Γ, while showing independence from the chosen very irrational vector within a small cone. The approach unifies smooth and ADE cases, clarifies non-separation phenomena, and provides a robust moduli-theoretic foundation likely adaptable to K-trivial and hyperkähler varieties. Key contributions include corrected definitions (mqpol), a detailed small-cone analysis, and a separated moduli-theoretic description for ADE K3 surfaces. The results have broad implications for applications and future generalizations in the realm of K3-type and hyperkähler moduli.

Abstract

We propose modifications to the commonly used definitions of lattice-polarized and lattice-quasipolarized smooth K3 surfaces, collecting various versions of the definition, and determining the effects of these choices on the resulting moduli space. We fill a gap in the theory, by replacing Weyl chambers with the new notion of a ``small cone'': the true datum in the definition of lattice quasipolarized K3 surfaces. In addition, we describe the separated moduli stack and moduli space for lattice-polarized K3 surfaces with singularities, an important notion for applications.

Paper Structure

This paper contains 5 sections, 18 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Smooth vs ADE K3 surfaces
  3. Analytic moduli
  4. Quasipolarized moduli
  5. ADE moduli

Key Result

Theorem 3.4

Two analytic K3 surfaces $X$ and $X'$ are isomorphic if and only if they are Hodge-isometric, i.e. there exists an isometry $i : H^2(X',\mathbb Z)\to H^2(X,\mathbb Z)$ for which $i(H^{2,0}(X'))=H^{2,0}(X)$. Furthermore, $i=f^*$ for an isomorphism $f:X\to X'$ if and only if $i(\mathcal{K}_{X'})=\math

Theorems & Definitions (61)

  • Definition 2.1
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Theorem 3.4: piateski-shapiro1971torelliburns1975on-the-torellilooijenga1980torelli
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • Definition 3.8
  • Theorem 3.9
  • ...and 51 more