There at least three families of hyper-K ̈ahler manifolds built from cubic fourfolds, the most recently discovered one being the compactified intermediate Jacobian fibrations I constructed with

Monge-Ampère and Hessian equations on compact Hermitian manifolds

Consider any meromorphic family of endomorphisms of the complex projective plane parameterized by the punctured unit disk. We shall explain how to describe the behaviour of their measures of

I will discuss certain invariants of singularities, the Hodge ideals, that are defined in the context of Saito’s theory of mixed Hodge modules. They can be considered as higher order analogues of

We investigate the holonomy group of singular Kähler-Einstein metrics on klt varieties with numerically trivial canonical divisor. Finiteness of the number of connected components, a Bochner

This is a talk about my works with Damin Wu concerning those manifolds with negative holomorphic sectional curvature. I shall describe our theorem that such manifold must have negative first Chern

Linearly saturated subvarieties on uniruled projective manifolds

I will discuss a method that we recently introduced in collaboration with Chu and Weinkove which gives interior C1,1 estimates for the non-degenerate complex Monge-Ampère equation on compact

Using and extending an approximation process due to Berman, we show that the quasi-psh envelope of a viscosity super-solution is a pluripotential super-solution of a given complex Monge-Ampère

In the lecture we consider the existence of G2 structures on 7-manifolds with boundary, with prescribed data on the boundary. In the first part we will review general background and theory,

Discrepancies and dimensions of secant varieties

A smooth projective variety over the complex numbers is said to be (Brody) hyperbolic if it doesn’t contain any entire curve. Kobayashi conjectured in the 70’s that general hypersurfaces of

Given a Kahler manifold (X, ω), finding smooth solutions to the equation (ø +i∂̄∂u)n=føn goes back to Yau’s solution of the Calabi conjecture in the seventies. In joint work with E.

I will first discuss some earlier work on the curvature of direct images of adjoint line bundles under a smooth proper fibration, or more generally a surjective map and (maybe) some of its

We show that a foliation on a projective complex manifold is algebraic with rationally connected (closure of) leaves exactly when its minimal slope with respect to some movable class is positive.

In this talk I will present a survey of the connections between canonical metrics and random point processes on a complex algebraic variety X. When the variety X has positive Kodaira dimension,