We prove that when Hodge theory survives on non-compact symplectic manifolds, a compact symplectic Lie group action having fixed points is necessarily Hamiltonian, provided the associated almost complex structure preserves the space of harmonic one-forms. For example, this is the case for complete Kähler manifolds for which the symplectic form has an appropriate decay at infinity. This extends ...

We find the conditions under which a Riemannian manifold equipped with a closed threeform and a vector field define an on–shell N = (2, 2) supersymmetric gauged sigma model. The conditions are that the manifold admits a twisted generalized Kähler structure, that the vector field preserves this structure, and that a so–called generalized moment map exists for it. By a theorem in generalized comp...

A para-Kähler manifold can be defined as a pseudoRiemannian manifold (M, g) with a parallel skew-symmetric paracomplex structures K, i.e. a parallel field of skew-symmetric endomorphisms with K = Id or, equivalently, as a symplectic manifold (M, ω) with a bi-Lagrangian structure L, i.e. two complementary integrable Lagrangian distributions. A homogeneous manifold M = G/H of a semisimple Lie gro...

We formulate Dirac-Kähler fermion action by introducing a new Clifford product with noncommutative differential form on a lattice. Hermiticity of the Dirac-Kähler action requires to choose the lattice structure having both orientabilities on a link. The Kogut-Susskind fermion and the staggered fermion actions are derived directly from the Dirac-Kähler fermion formulated by the Clifford product....

These notes are based on two talks given at the Arithmetic & Algebraic Geometry Seminar of the Korteweg-de Vriesinstituut for mathematics of the Universiteit van Amsterdam. They are intended to give a short introduction to the theory of Kähler manifolds, with a slight focus of applicability to the subject of K3 surfaces. However, they also include other interesting results not related to K3 sur...

Let L ⊂ V = R be a maximally isotropic subspace. It is shown that any simply connected Lie group with a bi-invariant flat pseudo-Riemannian metric of signature (k, l) is 2-step nilpotent and is defined by an element η ∈ Λ3L ⊂ Λ3V . If η is of type (3, 0)+(0, 3) with respect to a skew-symmetric endomorphism J with J2 = ǫId, then the Lie group L(η) is endowed with a left-invariant nearly Kähler s...

This is a survey of recent contributions to the area of special Kähler geometry. It is based on lectures given at the 21st Winter School on Geometry and Physics held in Srni in January 2001. 1 Remarkable features of special Kähler manifolds A (pseudo-) Kähler manifold (M,J, g) is a differentiable manifold endowed with a complex structure J and a (pseudo-) Riemannian metric g such that (i) J 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...

A contact hypersurface in a Kähler manifold is a real hypersurface for which the induced almost contact metric structure determines a contact structure. We carry out a systematic study of contact hypersurfaces in Kähler manifolds. We then apply these general results to obtain classifications of contact hypersurfaces with constant mean curvature in the complex quadric Q = SOn+2/SOnSO2 and its no...

Suppose M a compact manifold which admits an Einstein metric g which is Kähler with respect to some complex structure J . Is every other Einstein metric h on M also Kähler-Einstein? If the complex dimension of (M,J) is ≥ 3, the answer is generally no; for example, CP3 admits both the FubiniStudy metric, which is Kähler-Einstein, and a non-Kähler Einstein metric [2] obtained by appropriately squ...