arXiv research

We give hodge structures on quasitoric orbifolds. We define orbifold hodge numbers and show a correspondence of orbifold hodge numbers for crepant resolutions of quasitoric orbifolds. In short we extend hodge structures to a non complex setting .

Proves Hodge structures on modular functors, providing formulas for Hodge numbers.

We refine the Morgan's work on mixed Hodge structures on Sullivan's $1$--minimal models by using non-abelian Hodge theory. As an application, we give explicit representatives of real unipotent variations of mixed Hodge structures over compact K"ahler manifolds.

We give a way of constructing real variations of mixed Hodge structures over compact Kähler manifolds by using mixed Hodge structures on Sullivan's $1$-minimal models of certain differential graded algebras associated with real variations of Hodge structures.

We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kaehler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and tho…

We consider the geometric properties of Hodge Cousin groups, introduced in an unpublished paper \cite{OVV}, emphasizing the case of Hodge Cousin groups corresponding to polarized $\mathbb{Q}$-Hodge structures. Basing on this consideration, we introduce the class of abelian Cousin groups and prove an analogue of Poincar…

Resolves conjectures on non-abelian Hodge loci for quasi-projective varieties.

Sheng and Zuo's characteristic forms are invariants of a variation of Hodge structure. We show that they characterize Gross's canonical variations of Hodge structure of Calabi-Yau type over (Hermitian symmetric) tube domains.

New Sasaki structures identified by Hodge numbers in odd dimensions.

We study a generalization of Hodge structures which first appeared in the work of Cecotti and Vafa. It consists of twistors, that is, holomorphic vector bundles on P^1, with additional structure, a flat connection on C^*, a real subbundle and a pairing. We call these objects TERP-structures. We generalize to TERP-struc…

Study on complex variation of Hodge structures for non-Kähler manifolds.

We define the notion of a loop Hodge structure -- an infinite dimensional generalization of a Hodge structure -- and prove that a suitable variation of this object over a complex manifold is equivalent to the datum of a harmonic bundle. Hence one can study harmonic bundles using classical tools of Hodge theory, especia…

Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford-Tate domains, arise as open $G_{\mathbb{R}}$--orbits in flag varieties $G/P$. We investigate Hodge--theoretic aspects of the geometry and representation theory associated with these flag varieties. In particular, w…

New inequalities generalize Li's theorem on mixed Hodge structures.

The study proves rigidity for mixed Hodge structures and applies to curve families.

Introduces non-abelian Hodge correspondence linking algebraic structures to geometry.

Proves modular functors for SO(3) have integral Hodge structures.

In this paper we prove the following results: $1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Ho…

Constructs a connection for Hodge theoretic projective structures on Riemann surfaces.

We use Hodge theory and a construction of Merkulov to construct $A_{\infty}$ structures on de Rham cohomology and Dolbeault cohomology.

Study Hodge-de Rham numbers for almost complex 4-manifolds, extending properties from complex surfaces.

The first Betti number for a lattice in a classifying space for variations of Hodge structures vanishes.

Lecture notes from the Concentrated Graduate Course preceding the Workshop on Hodge Theory in String Theory at the Fields Institute in Toronto, November 11--15, 2013.

Let $V$ be a complex projective variety with isolated singularities. Let the smooth part be given the metric induced by a projective imbedding. Then we develop the $L_2$ harmonic theory and construct a pure Hodge structure on the $L_2$-cohomology of $V$. If the dimension of $V$ is two, we put a cohomological Hodge stru…

New natural presentation of supergravity c-map using Hodge structures.

Establishes correspondence between Calabi-Yau and Landau-Ginzburg structures.

We shall construct a natural Higgs bundle structure on the complexified Kähler cone of a compact Kähler manifold, which can be seen as an analogy of the classical Higgs bundle structure associated to a variation of Hodge structure. In the proof of the flat-ness of our Higgs bundle, we find a commutator identity that ca…

A systematic study of the contributions at infinity for the cohomology of variations of polarized Hodge structures over quasicompact Kähler manifolds. Several isomorphisms between different cohomologies given.

Two projective structures on Riemann surfaces are described and shown not to be identical.

In this article we introduce algorithms which compute iterations of Gauss-Manin connections, Picard-Fuchs equations of Abelian integrals and mixed Hodge structure of affine varieties of dimension $n$ in terms of differential forms. In the case $n=1$ such computations have many applications in differential equations and…

For any symmetric collection of natural numbers h^{p,q} with p+q=k, we construct a smooth complex projective variety whose weight k Hodge structure has these Hodge numbers; if k=2m is even, then we have to impose that h^{m,m} is bigger than some quadratic bound in m. Combining these results for different weights, we so…

Study on existence of balanced metrics on non-Kähler manifolds.

The paper studies cyclic covers of rational surfaces and their Hodge structures.

Uniformizes Hodge structures, proving Lyapunov exponents and log-Anosov monodromy.

Study examines $L^p$-boundedness of Hodge projection on manifolds with ends.

Extends Hodge theory to nearly Kähler manifolds of arbitrary dimensions.

We provide a formula describing the G-module structure of the Hurwitz-Hodge bundle for admissible G-covers in terms of the Hodge bundle of the base curve, and more generally, for describing the G-module structure of the push-forward to the base of any sheaf on a family of admissible G-covers. This formula can be interp…

The non-abelian Hodge correspondence identifies complex variations of Hodge structures with certain Higgs bundles. In this work we analyze this relationship, and some of its ramifications, when the variations of Hodge structures are determined by a (complete) one-dimensional family of compact Calabi-Yau manifolds. This…

These are the notes for the talk "Hodge numbers of a hypothetical complex structure on $S^6$" given by the author at the MAM1 "(Non)-existence of complex structures on $S^6$" held in Marburg in March 2017. They are based on [A. Gray, A property of a hypothetical complex structure on the six sphere, Boll. Un. Mat. Ital.…

This paper generalizes L2 cohomology theory for complex manifolds.

Introduces a new Hodge theory using vector fields on manifolds.

We treat two quite different problems related to changes of complex structures on Kähler manifolds by using global geometric method. First, by using operators from Hodge theory on compact Kähler manifold, we present a closed explicit extension formula for holomorphic canonical forms in different complex structures. As …

A generalized complex manifold which satisfies the $\partial \overline{\partial}$-lemma admits a Hodge decomposition in twisted cohomology. Using a Courant algebroid theoretic approach we study the behavior of the Hodge decomposition in smooth and holomorphic families of generalized complex manifolds. In particular we …

Sharp lower bound for Hodge Laplacian on Kähler hyperbolic manifolds.

The transcendental Hodge lattice of a projective manifold $M$ is the smallest Hodge substructure in $p$-th cohomology which contains all holomorphic $p$-forms. We prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this algebra explicitly for a hyperkahler manif…

The Hodge Conjecture is equivalent to a statement about conditions under which a complex vector bundle on a smooth complex projective variety admits a holomorphic structure. I advertise a class of abelian four-folds due to Mumford where this approach could be tested. I construct explicit smooth vector bundles - which c…

New theorems on Hodge numbers and Kähler structures derived from complex differential forms.

This paper (the seventh paper in a series of eight) continues the development of our theory of multivector and extensor calculus on smooth manifolds. Here we deal first with the concepts of ordinary Hodge coderivatives, duality identities, and Hodge coderivative identities. Then, we recall the concept of a Levi-Civita …