Variation of Kahler-Einstein metrics with mixed singularities

TL;DR

This paper studies how Kahler-Einstein metrics with mixed singularities vary in families of complex manifolds, proving positivity of the associated currents and applying this to the surjectivity of the Albanese map.

Contribution

It generalizes previous results on Kahler-Einstein metrics to include mixed cone and Poincare singularities in a fibered setting.

Findings

Positivity of currents induced by Kahler-Einstein metrics with mixed singularities.

Extension of Schumacher's and Guenancia's results to more general singularities.

Surjectivity of the Albanese map for certain log canonical pairs.

Abstract

In this short note, we consider a fiberation f: (X, Delta) to Y between two compact Kahler manifolds with generic fiber of f being a smooth log canonical pair with ample canonical divisor, we prove that the current induced by variation of Kahler Einsteins with mixed cone and Poincare singularities is positive, hence generalize the result of Schumacher in the smooth case [22] and the result of Guenancia in the conic case [14]. As application, we prove the surjectivity of Albanese map for a smooth log canonical pair with -(KX + Delta) being nef.