Structure theorem for projective klt pairs with nef anti-canonical divisor

TL;DR

This paper proves a structure theorem for projective klt pairs with nef anti-canonical divisors, showing they decompose into rationally connected and Calabi-Yau components after a finite cover, extending previous smooth case results.

Contribution

It generalizes structure results to singular varieties with nef anti-log-canonical divisors, linking to the Beauville-Bogomolov decomposition for Calabi-Yau varieties.

Findings

Decomposition of klt pairs into rationally connected and Calabi-Yau parts.

Extension of structure theorems from smooth to singular varieties.

Reduction of structure problems to the singular Beauville-Bogomolov decomposition.

Abstract

In this paper, we establish a structure theorem for projective klt pairs $(X,\Delta)$ with nef anti-log canonical divisor; specifically, we prove that, up to replacing $X$ with a finite quasi-\'etale cover, $X$ admits a locally trivial rationally connected fibration onto a projective klt variety with numerically trivial canonical divisor. This structure theorem generalizes previous works for smooth projective varieties and reduces several structure problems to the singular Beauville-Bogomolov decomposition for Calabi-Yau varieties. As an application, projective varieties of klt Calabi-Yau type, which naturally appear as an outcome of the Log Minimal Model Program, are decomposed into building block varieties: rationally connected varieties and Calabi-Yau varieties.