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Polync varieties and multiparameter Kulikov models

Philip Engel

TL;DR

The paper develops a general framework for smoothability of multiparameter d-semistable, K-trivial polync degenerations using log structures and log Bogomolov–Tian–Todorov theory, extending Friedman and Kawamata–Namikawa to arbitrary-base settings. It identifies d-semistability with the existence of a log structure of semistable type and shows unobstructed log-smooth deformations under the log Calabi–Yau condition, yielding a universal deformation over $\mathrm{Spf}\, \mathbb{C}[[u_s]]_{s\in S}$ and, in the projective case, a projective smoothing by algebraization. For polysnc Type III K3 surfaces, the work introduces a combinatorial framework of slabs and a monodromy cone generated by $\{\lambda_s\}_{s\in S}$, links period data to a gluing of line bundles via $\varphi_{X_0}$, and proves that a 1-parameter base change produces a Kulikov model, with a base-dimension count of $20$ aligning with the universal deformation. The paper then constructs explicit multiparameter Kulikov models—notably using rhombicuboctahedron (4 colors, yielding a 4+16 parameter base) and rainbow colorings (10 colors, yielding a 10-parameter base)—and demonstrates how fibers over coordinate strata exhibit Type II/III behaviors, offering concrete realizations of these refined degenerations. These results broaden the scope of Kulikov-type degenerations beyond 1-parameter families and provide new tools for studying degenerations of K3 and abelian varieties, including transversal cycle considerations and moduli implications.

Abstract

We study "polync varieties", whose singularities are locally products of normal crossing (nc) singularities. We introduce the notion of d-semistability of such varieties, and generalize work of Friedman and Kawamata-Namikawa to address the smoothability of d-semistable, K-trivial, polync varieties. These results are applications of recent breakthroughs on the logarithmic Bogomolov-Tian-Todorov theorem, due to Chan-Leung-Ma and Felten-Filip-Ruddat. We generalize the combinatorial description of Kulikov models for K3 surfaces to the setting of a multiparameter base and describe some interesting examples.

Polync varieties and multiparameter Kulikov models

TL;DR

The paper develops a general framework for smoothability of multiparameter d-semistable, K-trivial polync degenerations using log structures and log Bogomolov–Tian–Todorov theory, extending Friedman and Kawamata–Namikawa to arbitrary-base settings. It identifies d-semistability with the existence of a log structure of semistable type and shows unobstructed log-smooth deformations under the log Calabi–Yau condition, yielding a universal deformation over and, in the projective case, a projective smoothing by algebraization. For polysnc Type III K3 surfaces, the work introduces a combinatorial framework of slabs and a monodromy cone generated by , links period data to a gluing of line bundles via , and proves that a 1-parameter base change produces a Kulikov model, with a base-dimension count of aligning with the universal deformation. The paper then constructs explicit multiparameter Kulikov models—notably using rhombicuboctahedron (4 colors, yielding a 4+16 parameter base) and rainbow colorings (10 colors, yielding a 10-parameter base)—and demonstrates how fibers over coordinate strata exhibit Type II/III behaviors, offering concrete realizations of these refined degenerations. These results broaden the scope of Kulikov-type degenerations beyond 1-parameter families and provide new tools for studying degenerations of K3 and abelian varieties, including transversal cycle considerations and moduli implications.

Abstract

We study "polync varieties", whose singularities are locally products of normal crossing (nc) singularities. We introduce the notion of d-semistability of such varieties, and generalize work of Friedman and Kawamata-Namikawa to address the smoothability of d-semistable, K-trivial, polync varieties. These results are applications of recent breakthroughs on the logarithmic Bogomolov-Tian-Todorov theorem, due to Chan-Leung-Ma and Felten-Filip-Ruddat. We generalize the combinatorial description of Kulikov models for K3 surfaces to the setting of a multiparameter base and describe some interesting examples.
Paper Structure (5 sections, 11 theorems, 33 equations, 7 figures, 1 table)

This paper contains 5 sections, 11 theorems, 33 equations, 7 figures, 1 table.

Table of Contents

  1. Polync varieties
  2. Interpretation via log structures
  3. Polysnc $K$-trivial surfaces
  4. Examples
  5. Acknowledgments

Key Result

Proposition 1.12

Let $X_0$ be an $S$-colored, polync variety. The sheaf ${\mathcal{E}} xt^1(\Omega^1_{X_0},{\mathcal{O}}_{X_0})\simeq \bigoplus_{s\in S} {\mathcal{L}}_s$ is a direct sum of line bundles ${\mathcal{L}}_s$ supported on $(X_0)_{\rm sing}^s$.

Figures (7)

  • Figure 1: Two $2$-dimensional polysimplices and one $3$-dimensional polysimplex
  • Figure 2: Left: Colorable polysimplicial complex of dimension $2$, colored by the set $S = \{{\rm blue},\,{\rm green},\,{\rm red}\}$. Right: $2$-dimensional polysimplicial complex that admits no coloring; the chain of $2$-simplices forces the two $1$-simplex factors of the square to be colored by the same element of $S$.
  • Figure 3: Some snc surfaces built from platonic solids
  • Figure 4: A Kulikov degeneration of degree $2$ K3 surfaces from engel2. Triple points in yellow, double curves in grey, self-intersections $D_{ij}^2$ and $D_{ji}^2$ in purple.
  • Figure 5: Rhombicuboctahedron, colored by four colors
  • ...and 2 more figures

Theorems & Definitions (50)

  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Example 1.4
  • Definition 1.5
  • Definition 1.6
  • Example 1.8
  • Definition 1.9
  • Definition 1.10
  • Remark 1.11
  • ...and 40 more