### A note on the holomorphic invariants of Tian-Zhu.

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This is the content of the lectures given by the author at the winter school KAWA3 held at the University of Barcelona in 2012 from January 30 to February 3. The main goal was to give an account of viscosity techniques and to apply them to degenerate Complex Monge-Ampère equations.We will survey the main techniques used in the viscosity approach and show how to adapt them to degenerate complex Monge-Ampère equations. The heart of the matter in this approach is the “Comparison Principle" which allows...

This paper is one in a series generalizing our results in [12, 14, 15, 20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem...

Let (X, P) be a toric variety. In this note, we show that the C0-norm of the Calabi flow φ(t) on X is uniformly bounded in [0, T) if the Sobolev constant of φ(t) is uniformly bounded in [0, T). We also show that if (X, P) is uniform K-stable, then the modified Calabi flow converges exponentially fast to an extremal Kähler metric if the Ricci curvature and the Sobolev constant are uniformly bounded. At last, we discuss an extension of our results to a quasi-proper Kähler manifold.

2000 Mathematics Subject Classification: 11G15, 11G18, 14H52, 14J25, 32L07.We call a complex (quasiprojective) surface of hyperbolic type, iff – after removing finitely many points and/or curves – the universal cover is the complex two-dimensional unit ball. We characterize abelian surfaces which have a birational transform of hyperbolic type by the existence of a reduced divisor with only elliptic curve components and maximal singularity rate (equal to 4). We discover a Picard modular surface of...

We study representations of lattices of $\mathrm{PU}(m,1)$ into $\mathrm{PU}(n,1)$. We show that if a representation is reductive and if $m$ is at least 2, then there exists a finite energy harmonic equivariant map from complex hyperbolic $m$-space to complex hyperbolic $n$-space. This allows us to give a differential geometric proof of rigidity results obtained by M. Burger and A. Iozzi. We also define a new invariant associated to representations into $\mathrm{PU}(n,1)$ of non-uniform lattices in $\mathrm{PU}(1,1)$, and more generally of fundamental groups of orientable...

We define a functional for Hermitian metrics using the curvature of the Chern connection. The Euler–Lagrange equation for this functional is an elliptic equation for Hermitian metrics. Solutions to this equation are related to Kähler–Einstein metrics, and are automatically Kähler–Einstein under certain conditions. Given this, a natural parabolic flow equation arises. We prove short time existence and regularity results for this flow, as well as stability for the flow near Kähler–Einstein metrics...

We determine all anticanonically embedded quasi smooth log del Pezzo surfaces in weighted projective 3-spaces. Many of these admit a Kähler-Einstein metric and most of them do not have tigers.

In this note we discuss some recent and ongoing joint work with Thalia Jeffres concerning the existence of Kähler-Einstein metrics on compact Kähler manifolds which have a prescribed incomplete singularity along a smooth divisor $D$. We shall begin with a general discussion of the problem, and give a rough outline of the “classical” proof of existence in the smooth case, due to Yau and Aubin, where no singularities are prescribed. Following this is a discussion of the geometry of the conical or edge...

Let $X$ be a compact Kähler manifold and $\Delta $ be a $\mathbb{R}$-divisor with simple normal crossing support and coefficients between $1/2$ and $1$. Assuming that ${K}_{X}+\Delta $ is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on $X\setminus \mathrm{Supp}\left(\Delta \right)$ having mixed Poincaré and cone singularities according to the coefficients of $\Delta $. As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair $(X,\Delta )$.