Polync varieties and multiparameter Kulikov models

TL;DR

This paper investigates polync varieties with nc singularities, introduces d-semistability, and extends Kulikov models to multiparameter bases, leveraging recent advances in logarithmic deformation theory.

Contribution

It generalizes the concept of Kulikov models and smoothability criteria for polync varieties, incorporating multiparameter bases and recent logarithmic deformation results.

Findings

Generalized Kulikov models to multiparameter settings.

Extended smoothability results for d-semistable, K-trivial polync varieties.

Provided new examples illustrating these models.

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.