Generalized geometry and the Hodge decomposition

In this paper1 we investigate generalized Kahlerian spaces and find some relations for curvature tensors in these spaces. Also, we define holo- morphically projective mappings of generalized Kahlerian spaces and ob- tain an invariant geometric object for these mappings.

2016

Early Proceedings of the Alterman Conference on Geometric Algebra and Summer School on Kahler Calculus

Algebraic Geometry, 2009

Given a projective morphism of compact, complex, algebraic varieties and a relatively ample line bundle on the domain we prove that a suitable choice, dictated by the line bundle, of the decomposition isomorphism of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber, yields isomorphisms of pure Hodge structures. The proof is based on a new cohomological characterization of the decomposition isomorphism associated with the line bundle. We prove some corollaries concerning the intersection form in intersection cohomology, the natural map from cohomology to intersection cohomology, projectors and Hodge cycles, and induced morphisms in intersection cohomology.

The Michigan Mathematical Journal, 2002

2011

In this paper, we shall generalize the theory of mixed Hodge structures due to Deligne and obtain a subcategory GMHS in the category of mixed Hodge structures such that we have Ext_{GMHS}^2(Q,-)\not=0 in general.

Publications of the Research Institute for Mathematical Sciences, 1988

The technique of dimensional reduction of an integrable system usually requires symmetry arising from a group action. In this paper we study a situation in which a dimensional reduction can be achieved despite the absence of any such global symmetry. We consider certain holmorphic vector bundles over a Kahler manifold which is itself the total space of a fiber bundle over a Kahler manifold. We establish an equivalence between invariant solutions to the Hermitian– Einstein equations on such bundles, and general solutions to a coupled system of equations defined on holomorphic bundles over the base Kahler manifold. The latter equations are the Coupled Vortex Equations. Our results thus generalize the dimensional reduction results of García–Prada, which apply when the fiber bundle is a product and the fiber is the complex projective line. 1991 Mathematics Subject Classification. Primary 58C25; Secondary 58A30, 53C12, 53C21, 53C55, 83C05. Key words and phrases. Coupled vortex equation, ...

Publications of the Research Institute for Mathematical Sciences, 1984

Communications in Mathematical Physics, 2007

We study generalized Kähler manifolds for which the corresponding complex structures commute and classify completely the compact generalized Kähler four-manifolds for which the induced complex structures yield opposite orientations.

2010

These three lectures summarize classical results of Hodge theory concerning algebraic maps, and presumably contain much more material than I'll be able to cover. Lectures 4 and 5, to be delivered by M. A. de Cataldo, will discuss more recent results. I will not try to trace the history of the subject nor attribute the results discussed. Coherently with this policy, the bibliography only contains textbooks and a survey, and no original paper. Furthermore, quite often the results will not be presented in their maximal generality; in particular I'll alway stick to projective maps, even though some results discussed hold more generally. Contents 1 Introduction. 2 2 The smooth case: E 2 -degeneration 4 3 Mixed Hodge structures. 8 3.1 Mixed Hodge structures on the cohomology of algebraic varieties 8 3.2 The global invariant cycle theorem .