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.

for any tangent vectors X,Y to M . If the almost complex structure J is integrable we obtain a Kähler structure. Many efforts have been done in the direction of finding curvature conditions on the metric which insure the integrability of the almost complex structure. A famous conjecture of Goldberg [26] states that a compact almost Kähler, Einstein manifold is in fact Kähler. Important progress...

We prove the existence of a (unique) S-invariant Ricci-flat Kähler metric on a neighbourhood of the zero section in the canonical bundle of a realanalytic Kähler manifold X, extending the metric on X. In the important paper [3], Calabi proved existence of Ricci-flat Kähler metrics on two classes of manifolds: a) cotangent bundles of projective spaces; b) canonical bundles of Kähler-Einstein man...

Let (M, J) be a minimal compact complex surface of Kähler type. It is shown that the smooth 4-manifold M admits a Riemannian metric of positive scalar curvature iff (M,J) admits a Kähler metric of positive scalar curvature. This extends previous results of Witten and Kronheimer. A complex surface is a pair (M,J) consisting of a smooth compact 4-manifold M and a complex structure J on M ; the la...

We review the theory of quaternionic Kähler and hyperkähler structures. Then we consider the tangent bundle of a Riemannian manifold M with a metric connection D (with torsion) and with its well estabilished canonical complex structure. With an extra almost Hermitian structure on M it is possible to find a quaternionic Hermitian structure on TM , which is quaternionic Kähler if, and only if, D ...

In recent years, there are many progress made in Kähler geometry. In particular, the topics related to the problems of the existence and uniqueness of extremal Kähler metrics, as well as obstructions to the existence of such metrics in general Kähler manifold. In this talk, we will report some recent developments in this direction. In particular, we will discuss the progress recently obtained i...

We study relations between quaternionic Riemannian manifolds admitting different types of symmetries. We show that any hyperKähler manifold admitting hyperKähler potential and triholomorphic action of S1 can be constructed from another hyperKähler manifold (of lower dimention) with an action of S1 which fixes one complex structure and rotates the other two and vice versa. On the other hand, the...

On a Kähler spin manifold Kählerian twistor spinors are a natural analogue of twistor spinors on Riemannian spin manifolds. They are defined as sections in the kernel of a first order differential operator adapted to the Kähler structure, called Kählerian twistor (Penrose) operator. We study Kählerian twistor spinors and give a complete description of compact Kähler manifolds of constant scalar...

The Kähler potential is the least understood part of effective N = 1 supersymmetric theories derived from string compactifications. Even at tree-level, the Kähler potential for the physical matter fields, as a function of the moduli fields, is unknown for generic Calabi-Yau compactifications and has only been computed for simple toroidal orientifolds. In this paper we describe how the modular d...

Let (X, ωX) be a compact Kähler manifold such that the anticanonical bundle −KX is nef. We prove that the slopes of the HarderNarasimhan filtration of the tangent bundle with respect to a polarization of the form ω X are semi-positive. As an application, we give a characterization of rationally connected compact Kähler manifolds with nef anticanonical bundles. As another application, we give a ...