Related papers: Variation of Hodge structure for generalized compl…
We study a natural Hodge theoretic generalization of rational (or $\mathbb{Q}$-)homology manifolds through an invariant ${\rm HRH(Z)}$ where $Z$ is a complex algebraic variety. The defining property of this notion encodes the difference…
In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections on principal bundles, and complex…
We prove two homotopy decomposition theorems for the loops on co-H-spaces, including a generalization of the Hilton-Milnor Theorem. These are applied to problems arising in algebra, representation theory, toric topology, and the study of…
We introduce a natural map from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the infinitesimal deformations of this complex manifold. By use of this map, we generalize an extension…
The Borel-Weil-Bott theorem can be used to decompose the cohomology of twisted sheaves of holomorphic forms on the complex Grassmannian into irreducible representations of the general linear group. By analyzing this decomposition, we…
We construct a mixed Hodge structure on the topological K-theory of smooth Poisson varieties, depending weakly on a choice of compactification. We establish a package of tools for calculations with these structures, such as functoriality…
Given a compact symplectic toric manifold $(M,\omega, \mathbb{T})$, we identify a class $DGK_{\omega}^{\mathbb{T}}(M)$ of $\mathbb{T}$-invariant generalized K\"ahler structures for which a generalisation the Abreu-Guillemin theory of toric…
We show that closures of families of unitary local systems on quasiprojective varieties for which the dimension of a graded component of Hodge filtration has a constant value can be identified with a finite union of polytopes. We also…
We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi continuity Theorem for the dimensions of kernels of a smooth family of transversely…
We analyze the behavior of polarized complex variations of Hodge structure on the punctured unit disk. For integral variations of Hodge structure, this analysis was first carried out by Wilfried Schmid. We get rid of the assumption that the…
Andreotti-Vesentini, Ohsawa, Gromov, Koll\'ar, among others, have observed that Hodge theory could be extended to (non compact) K\"ahler complete manifolds, within the L^2 framework. Also, many vanishing theorems on projective or K\"ahler…
Generalized geometry finds many applications in the mathematical description of some aspects of string theory. In a nutshell, it explores various structures on a generalized tangent bundle associated to a given manifold. In particular,…
Let $G=\C^{n}\ltimes_{\phi} \C^{m}$ with a semi-simple action $\phi: \C^{n}\to GL_{m}(\C)$ (not necessarily holomorphic). Suppose $G$ has a lattice $\Gamma$. Then we show that in some conditions on $G$ and $\Gamma$, $G/\Gamma$ admits a…
In this paper we study dually flat spaces arising from Delzant polytopes equipped with a symplectic potential together with their corresponding toric K\"ahler manifolds as their torifications.We introduce a dually flat structure and the…
For a certain class of complexes of pre-Hilbert $A$-modules, we prove that their cohomology groups equipped with a canonical quotient structure are again pre-Hilbert $A$-modules and derive the Hodge decomposition for them. We call these…
The cohomology algebra of the canonical bundle of a compact K\"ahler manifold is naturally viewed as a module over an exterior algebra. We use the Bernstein-Gel'fand-Gel'fand correspondence, together with Generic Vanishing theory, in order…
We prove a filtered version of the Homotopy Transfer Theorem which gives an A-infinity algebra structure on any page of the spectral sequence associated to a filtered dg-algebra. We then develop various applications to the study of the…
We describe some L-infinity model for the local period map of a compact Kaehler manifold. Applications include the study of deformations with associated variation of Hodge structure constrained by certain closed strata of the Grassmannian…
A compact K\"ahler manifold is shown to be simply-connected if its `symmetric cotangent algebra' is trivial. Conjecturally, such a manifold should even be rationally connected. The relative version is also shown: a proper surjective…
Cohesive modules give a dg-enhancement of the bounded derived category of coherent sheaves on a complex manifold via superconnections. In this paper we discuss the deformation theory of cohesive modules on compact complex manifolds. This…