Projectively flat log smooth pairs

TL;DR

This paper classifies projectively flat log smooth pairs with numerically flat tangent bundles, showing they are either projective spaces or certain blow-ups, and describes their structure as toric fiber bundles over abelian quotients.

Contribution

It generalizes previous classifications by characterizing projectively flat log smooth pairs with numerically flat tangent bundles, including their finite covers and structural properties.

Findings

They are either projective spaces or blow-ups of such pairs.

They are toric fiber bundles over finite étale quotients of abelian varieties.

The structure is well understood and classified.

Abstract

In this article, we study projective log smooth pairs with numerically flat normalized logarithmic tangent bundle. Generalizing works of Jahnke-Radloff and Greb-Kebekus-Peternell, we show that, passing to an appropriate finite cover and up to isomorphism, these are the projective spaces or the log smooth pairs with numerically flat logarithmic tangent bundles blown-up at finitely many points away from the boundary. On the other hand, the structure of log smooth pairs with numerically flat logarithmic tangent bundle is well understood: they are toric fiber bundles over finite \'etale quotients of abelian varieties.