Using the idea of a generalized Kähler structure, we construct bihermitian metrics on CP2 and CP1×CP1, and show that any such structure on a compact 4-manifold M defines one on the moduli space of anti-self-dual connections on a fixed principal bundle over M . We highlight the role of holomorphic Poisson structures in all these constructions.

We study a sequence of connections which is associated with a Riemannian metric and an almost symplectic structure on a manifold M according to a construction in [5]. We prove that if this sequence is trivial (i.e. constant) or 2-periodic, then M has a canonical Kähler structure.

We prove that a compact quaternionic-Kähler manifold of dimension 4n ≥ 8 admitting a conformal-Killing 2-form which is not Killing, is isomorphic to the quaternionic projective space, with its standard quaternionicKähler structure.

Let (M, I) be a compact Kähler manifold admitting a hypercomplex structure (M, I, J,K). We show that (M, I, J,K) admits a natural HKT-metric. This is used to construct a holomorphic symplectic form on (M, I).

A hypothesis is introduced under which a compact complex analytic space, X, viewed as a structure in the language of analytic sets, is essentially saturated. It is shown that this condition is met exactly when the irreducible components of the restricted Douady spaces of all the cartesian powers of X are compact. Some implications of saturation on Kähler-type spaces, which by a theorem of Fujik...

We prove that every Kähler metric, whose potential is a function of the timelike distance in the flat Kähler-Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kähler manifolds with the above mentioned metrics. New examples of Sasakian space forms are obtained as real hypersurfaces of a Kähler space form ...

where Ric(ωKS) is the Ricci form of gKS and LXωKS denotes the Lie derivative of ωKS along a holomorphic vector field X on M . If X = 0, then gKS is a Kähler-Einstein metric with positive scalar curvature. We will show that the second variation of Perelman’s Wfunctional is non-positive in the space of Kähler metrics with 2πc1(M) as Kähler class. Furthermore, if (M, gKS) is a Kähler-Einstein mani...

We show that the existence of a left-invariant pluriclosed Hermitian metric on unimodular Lie group with abelian complex structure forces to be 2-step nilpotent. Moreover, we prove flow starting from nilpotent preserves Strominger Kähler–like condition.

Classical Hodge theory gives a decomposition of the complex cohomology of a compact Kähler manifold M , which carries the standard Hodge structure {H(M), p + q = k} of weight k. Deformations of M then lead to variations of the Hodge structure. This is best understood when reformulating the Hodge decomposition in an abstract manner. Let HC = HR ⊕ C be a complex vector space with a real structure...

I demonstrate the existence of quasi–realistic heterotic–string models in which all the untwisted Kähler and complex structure moduli are projected out by the generalized GSO projections. I discuss the conditions and characteristics of the models that produce this result. The existence of such models offers a novel perspective on the realization of extra dimensions in string theory. In this vie...