A note on conical K\"ahler-Ricci flow on minimal elliptic K\"ahler surfaces

TL;DR

This paper proves that under certain assumptions, the conical K"ahler-Ricci flow on minimal elliptic K"ahler surfaces converges to a generalized conical K"ahler-Einstein metric, with smooth convergence outside singular fibers and divisors.

Contribution

It establishes convergence of the conical K"ahler-Ricci flow on minimal elliptic surfaces to a generalized K"ahler-Einstein metric under semi-ampleness assumptions.

Findings

Flow converges to a generalized conical K"ahler-Einstein metric.

Convergence is smooth outside singular fibers and divisors.

Results depend on semi-ampleness assumption on the twisted canonical bundle.

Abstract

We prove that, under a semi-ampleness type assumption on the twisted canonical line bundle, the conical K\"ahler-Ricci flow on a minimal elliptic K\"ahler surface converges in the sense of currents to a generalized conical K\"ahler-Einstein on its canonical model. Moreover, the convergence takes place smoothly outside the singular fibers and the chosen divisor.