We consider hyper-Kähler manifolds of complex dimension 4 which are fibrations. It is known that the fibers are abelian varieties and the base is P. We assume that the general fiber is isomorphic to a product of two elliptic curves. We prove that such a hyper-Kähler manifold is deformation equivalent to a Hilbert scheme of two points on a K3 surface. 1. Preliminaries First we define our main ob...

Asking that a manifold admit a Kähler structure places swingeing restrictions on its topology. The best-known illustration of this comes from Hodge theory: one knows that when a complex structure on a manifold has a compatible symplectic structure, the decomposition of forms into Hodge types descends to cohomology, with far-reaching consequences. However, this turns out to be just the tip of th...

We investigate the low energy structure of the Kähler potential in SUSY QCD with Nf = Nc + 1 quark flavors. Since this theory’s moduli space is everywhere smooth, a systematic power series expansion of its Kähler potential can be developed in terms of confined meson and baryon fields. Perturbation theory in the supersymmetric sigma model based upon a momentum expansion consistent with naive dim...

Motivated by the classical statements of Mirror Symmetry, we study certain Kähler metrics on the complexified Kähler cone of a Calabi– Yau threefold, conjecturally corresponding to approximations to the Weil– Petersson metric near large complex structure limit for the mirror. In particular, the naturally defined Riemannian metric (defined via cup-product) on a level set of the Kähler cone is se...

inherits from M a Kähler structure; and in the first part of this paper we will describe what the Kähler form and Ricci form look like locally on coordinate patches in Mλ. Then in the second part of this paper we will discuss some global implications of these results. This will include (1) A Kählerian proof of the Duistermaat-Heckman theorem. (2) A formula, due to Biquard and Gauduchon, for the...

Okumura gave a necessary and sufficient condition for an oriented real hypersurface of a Kähler manifold to be a contact metric manifold with respect to the naturally induced almost contact metric structure. In this paper, we discuss an oriented hypersurface of a quasi Kähler manifold and give a necessary and sufficient condition for such a hypersurface to be a quasi contact metric manifold wit...

A self-similar Hessian (special Kähler) manifold is a $$(M,\nabla ,g)$$ endowed with an affine (holomorphic) homothetic vector field $$\xi $$ . Consider action of group G on ,g,\xi )$$ by isometries preserving such that acts the level set $$\{g(\xi ,\xi )=1\}$$ simply transitively. Then, we construct homogeneous conformally Kähler (hyper structure TM $$(T^*M)$$

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 introduce certain homology and cohomology subgroups for any almost complex structure and study their pureness, fullness and duality properties. Motivated by a question of Donaldson, we use these groups to relate J−tamed symplectic cones and J−compatible symplectic cones over a large class of almost complex manifolds, including all Kähler manifolds, almost Kähler 4−manifolds and complex surfa...