Let V = R be the pseudo-Euclidean vector space of signature (p, q), p ≥ 3 and W a module over the even Clifford algebra Cl0(V ). A homogeneous quaternionic manifold (M,Q) is constructed for any spin(V )-equivariant linear map Π : ∧2W → V . If the skew symmetric vector valued bilinear form Π is nondegenerate then (M,Q) is endowed with a canonical pseudo-Riemannian metric g such that (M,Q, g) is ...

We study a class of asymptotically cylindrical Ricci-flat Kähler metrics arising on quasiprojective manifolds. Using the Calabi–Yau geometry and analysis and the Kodaira–Kuranishi–Spencer theory and building up on results of N.Koiso, we show that under rather general hypotheses any local asymptotically cylindrical Ricci-flat deformations of such metrics are again Kähler, possibly with respect t...

Prescribing geometric structures of a complex manifold often introduces interesting and important partial differential equations. A typical example of this kind is the problem of finding the Kähler metrics with constant scalar curvature on a Kähler manifold. Such a problem defines a fourth order elliptic partial differential equation. The study of these partial differential equations, including...

Our main results are: (1) The complex and Lagrangian points of a non-complex and nonLagrangian 2n-dimensional submanifold F :M →N , immersed with parallel mean curvature and with equal Kähler angles into a Kähler-Einstein manifold (N, J, g) of complex dimension 2n, are zeros of finite order of sin θ and cos θ respectively, where θ is the common J-Kähler angle. (2) If M is a Cayley submanifold o...

A locally conformally Kähler (LCK) manifold M is one which is covered by a Kähler manifold M̃ with the deck transform group acting conformally on M̃ . If M admits a holomorphic flow, acting on M̃ conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vaisman manifolds is stable under small deformations. We define a new class of LCK-manifolds, called LCK manifo...

It is shown that any compact Kähler manifold M gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second one with the Hodge theory of the Dolbeault complex. In these algebras the product of two harmonic differential forms is again harmonic. If M happens to be a Calabi-Yau manifold, there exists a third st...

Any (global) kähler deformation of a flag manifold F with b2 = 1 is biholomorphic to F .

Yau’s uniformization conjecture states: a complete noncompact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to C. The Kähler-Ricci flow has provided a powerful tool in understanding the conjecture, and has been used to verify the conjecture in several important cases. In this article we present a survey of the Kähler-Ricci flow with focus on its application to...