We announce a proof of Calabi's conjectures on the Ricci curvature of a compact Kähler manifold and then apply it to prove some new results in algebraic geometry and differential geometry. For example, we prove that the only Kähler structure on a complex projective space is the standard one.

We consider the moduli space MN of flat unitary connections on an open Kähler manifold U (complement of a divisor with normal crossings) with restrictions on their monodromy transformations. Using intersection cohomology with degenerating coefficients we construct a natural closed 2-form F on MN . When U is quasi-projective we prove that F is actually a Kähler form.

We prove that any simply connected special Kähler manifold admits a canonical immersion as a parabolic affine hypersphere. As an application, we associate a parabolic affine hypersphere to any nondegenerate holomorphic function. Also we show that a classical result of Calabi and Pogorelov on parabolic spheres implies Lu’s theorem on complete special Kähler manifolds with a positive definite met...

We show that the Kähler structure can be naturally incorporated in the Batalin-Vilkovisky formalism. The phase space of the BV formalism becomes a fermionic Kähler manifold. By introducing an isometry we explicitly construct the fermionic irreducible hermitian symmetric space. We then give some solutions of the master equation in the BV formalism.

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is a sharp vanishing theorem for the second L2 cohomology of such manifolds under certain assumptions. The borderline case characterizes a Kähler-Einstein manifold constructed by Calabi.

A Hermitian metric on a complex manifold of complex dimension n is called astheno-Kähler if its fundamental 2-form F satisfies the condition ∂∂F n−2 = 0 and it is strong KT if F is ∂∂-closed. We prove that a conformally balanced astheno-Kähler metric on a compact manifod of complex dimension n ≥ 3, whose Bismut connection has (restricted) holonomy contained in SU(n), is necessarily Kähler. We p...

Let (M, I) be an almost complex 6-manifold. The obstruction to integrability of almost complex structure (socalled Nijenhuis tensor) N : Λ0,1(M)−→ Λ(M) maps a 3-dimensional bundle to a 3-dimensional one. We say that Nijenhuis tensor is non-degenerate if it is an isomorphism. An almost complex manifold (M, I) is called nearly Kähler if it admits a Hermitian form ω such that ∇(ω) is totally antis...

In this article we study compact Kähler manifolds admitting nonsingular holomorphic vector fields with the aim of extending to this setting the classical birational classification of projective varieties with tangent vector fields. We introduce and analyze a particular type of deformations, that we call tangential deformations, and we prove that each compact Kähler manifold X with nowhere vanis...

On a compact connected 2m-dimensional Kähler manifold with Kähler form ω, given a smooth function f : M → R and an integer 1 < k < m, we want to solve uniquely in ω the equation ω̃ ∧ωm−k eω, relying on the notion of k-positivity for ω̃ ∈ ω the extreme cases are solved: k m by Yau in 1978 , and k 1 trivially . We solve by the continuity method the corresponding complex elliptic kthHessian equation...

LetM be a compact, holomorphically symplectic Kähler manifold, and η a (1,1)-current which is nef (a limit of Kähler forms). Assume that the cohomology class of η is parabolic, that is, its top power vanishes. We prove that all Lelong sets of η are coisotropic. When M is generic, this is used to show that all Lelong numbers of η vanish. We prove that any hyperkähler manifold with Pic(M) = Z has...