Twisted and conical Kähler–Ricci solitons on Fano manifolds

Abstract

In this paper, we consider the twisted Kähler-Ricci soliton, and show that the existence of twisted Kähler-Ricci soliton with semi-positive twisting form is closely related to the properness of some energy functionals. We also consider the conical Kähler-Ricci soliton, and obtain some existence results. In particular, under some assumptions on the divisor and α-invariant, we get the properness of the modified log K-energy and the existence of conical Kähler-Ricci soliton with suitable cone angle.1 XISHEN JIN, JIAWEI LIU, AND XI ZHANG is a smooth Kähler metric invariant under the action of Φ Im X , where Φ Im X is the one-parameter transformations subgroup of Aut(M ) generating by Im X, i.e.> 0 in the sense of current}. As in [5], we define the following function subspace of H(M, ω 0 ):We define K 0 X (ω 0 ) to be the space of smooth semipositive (1, 1)-forms cohomology to ω 0 , i.e.is a subspace of Kähler metrics defined as follow:Definition 1.1. We define the following invariant with respect to X,Note that ω is a closed form and X is holomorphic, we have that ∂(i X ω) = 0. According to the Hodge decomposition theorem and the property of Fano manifold, we can find a smooth real-valued function θ X (ω) such that for ω ∈ K X (ω 0 ),and θ X (ω) satisfies the normalization M e θX (ω) ω n = M ω n 0 .We will take notation that θ X = θ X (ω 0 ) in the whole paper without special instruction. By direct computation, we get that θ X (ω ϕ ) = θ X + X(ϕ).Definition 1.2. For any (1, 1)-form η ∈ (1 − β)K 0 X (ω 0 ), we say a Kähler metric ω ∈ K X (ω 0 ) is a twisted Kähler-Ricci soliton with respect to η if it satisfiesRemark 1.2. It is easy to see that finding the twisted Kähler-Ricci soliton as (1.4) is equivalent to solving the following Monge-Ampère equation:where h ω is the Ricci potential defined by (1.6) Ric(ω) − βω − η = √ −1∂∂h ω normalized such that M e hω ω n = M ω n . And we just consider the case when β is nonnegative, since the equation (1.5) is solvable according to the celebrated work of Aubin [1] and Yau [31] on the other case.

Schauder estimates for equations with cone metrics, II

“…There is also an alternative approach for the conical Schauder estimates using microlocal analysis [23]. There are also various global and local estimates and regularity derived in the conical setting [1,15,9,11,13,12,18,15,24,29,31,44,45].…”

“…Corollary 1.3 treats the general case of conical singularities with simple normal crossings. There have been many results in the analytic aspects of the conical Ricci flow [8,9,15,16,24,32,43]. In [30], the conical Ricci flow on Riemann surfaces is completely classified with jumping conical structure in the limit.…”

Schauder estimates for equations with cone metrics, I

“…[27,4,5,6,29]) has also inspired many works on the study of canonical Kähler metrics with cone singularities and their relation to algebraic geometry (cf. [1,13,25,7,14,19,21,33,34,10]). One of the main difficulties in solving (1.2) is how to derive a suitable Schauder estimate for the linearized equation of (1.2).…”