Twisted and conical K\"ahler-Ricci soliton on Fano manifolds

TL;DR

This paper explores the existence of twisted and conical K"ahler-Ricci solitons on Fano manifolds, linking their existence to energy functional properness and providing new existence results under specific conditions.

Contribution

It establishes the relationship between twisted K"ahler-Ricci solitons with semi-positive twisting forms and energy functional properness, and proves existence results for conical K"ahler-Ricci solitons under certain assumptions.

Findings

Existence of twisted K"ahler-Ricci solitons is related to energy functional properness.

Properness of the modified log K-energy implies existence of conical K"ahler-Ricci solitons.

Existence results are obtained under assumptions on divisors and $oldsymbol{ ext{α}}$-invariant.

Abstract

In this paper, we consider the twisted K\"ahler-Ricci soliton, and show that the existence of twisted K\"ahler-Ricci soliton with semi-positive twisting form is closely related to the properness of some energy functionals. We also consider the conical K\"ahler-Ricci soliton, and obtain some existence results. In particular, under some assumptions on the divisor and $\alpha$-invariant, we get the properness of the modified log K-energy and the existence of conical K\"ahler-Ricci soliton with suitable cone angle.