arXiv research

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.

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

The paper proves a Hard Lefschetz Theorem for push-forwards of polarized twistor modules.

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…

In this paper, we give an expression and some estimates of the curvature tensor of the Hodge metric over the moduli space of a polarized Calabi-Yau threefold. The symmetricity of the Yukawa coupling is also studied. In the last section of this paper, an extra restriction of the limiting Hodge structure for the degenera…

We consider the geometric properties of Hodge Cousin groups, introduced in an unpublished paper \cite{OVV}, emphasizing the case of Hodge Cousin groups corresponding to polarized $\mathbb{Q}$-Hodge structures.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…

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…

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…

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 …

We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized $\mathbb{Z}$-variation of Hodge structure $\mathbb{V}$ on a smooth complex quasi-projective variety $S$, are topologically tame. As an easy corollary of these results and of Peterzil-Starchenko's o-…

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…

The paper describes how Hodge loci are typically equidistributed in complex varieties.

In this paper, we represent the Hodge metric in terms of the Weil-Petersson metric and its Ricci curvature on the moduli spaces of polarized Calabi-Yau threefolds.

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

In this paper, we study the analogue of the Shafarevich conjecture for polarized Calabi-Yau varieties. We use variations of Hodge structures and Higgs bundles to establish a criterion for the {\it rigidity} of families. We then apply the criterion to obtain that some important and typical families of Calabi-Yau varieti…

We establish an unexpected relation among the Weil-Petersson metric, the generalized Hodge metrics and the BCOV torsion. Using this relation, we prove that certain kind of moduli spaces of polarized Calabi-Yau manifolds do not admit complete subvarieties. That is, there is no complete family for certain class of polari…

We study tt*-geometry on the classifying space for regular singular TERP-structures, e.g., Fourier-Laplace transformations of Brieskorn lattices of isolated hypersurface singularities. We show that (a part of) this classifying space can be canonically equipped with a hermitian structure. We derive an estimate for the h…

In this paper, we define and study the Weil-Petersson geometry. Under the framework of the Weil-Petersson geometry, we study the Weil-Petersson metric and the Hodge metric. Among the other results, we represent the Hodge metric in terms of the Weil-Petersson metric and the Ricci curvature of the Weil-Petersson metric f…

Given a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint up to an error w…

Develops moduli theory for Calabi-Yau pairs, constructing a projective space.

We look into a construction of principal abelian varieties attached to certain spin manifolds, due to Witten and Moore-Witten around 2000 and try to place it in a broader framework. This is related to Weil intermediate Jacobians but it also suggests to associate abelian varieties to polarized even weight Hodge structur…

We use superconnections to define and study some natural differential forms on period domains $\mathbb{D}$ that parametrize polarized Hodge structures of given type on a rational quadratic vector space $V$. These forms depend on a choice of vectors $v_1,\ldots,v_r \in V$ and have a Gaussian shape that peaks on the locu…

In this paper, we study the local properties of the moduli space of a polarized Calabi-Yau manifold. Let $U$ be a neighborhood of the moduli space. Then we know the universal covering space $V$ of $U$ is a smooth manifold. Suppose $D$ is the classifying space of a polarized Calabi-Yau manifold with the automorphism gro…

We establish the correspondence between tame harmonic bundles and $μ_L$-stable parabolic Higgs bundles with trivial characteristic numbers. We also show the Bogomolov-Gieseker type inequality for $μ_L$-stable parabolic Higgs bundles. Then we show that any local system on a smooth quasi projective variety can be deforme…

We prove that the first Chern form of the moduli space of polarized Calabi-Yau manifolds, with the Hodge metric or the Weil-Petersson metric, represent the first Chern class of the canonical extensions of the tangent bundle to the compactification of the moduli space with normal crossing divisors.

Constructs projective moduli spaces for Calabi-Yau pairs.

We study the deformations of twisted harmonic maps $f$ with respect to the representation $ρ$. After constructing a continuous "universal" twisted harmonic map, we give a construction of every first order deformation of $f$ in terms of Hodge theory; we apply this result to the moduli space of reductive representations …

We study the asymptotic behaviour of tame harmonic bundles. First of all, we prove a local freeness of the prolongation by an increasing order. Then we obtain the polarized mixed twistor structure. As one of the applications, we obtain the norm estimate of holomorphic or flat sections by weight filtrations of the monod…

On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. The moduli spaces of these objects are obtained by fixing at each singularity the polar part of the connection. We prove that…

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.

Polarized and $G$-polarized CR manifolds are smooth manifolds endowed with a double structure: a real foliation $\Cal F$ (given by the action of a Lie group $G$ in the $G$-polarized case) and a transverse CR distribution $(E,J)$. Polarized means that $(E,J)$ is roughly speaking invariant by $\Cal F$. Both structures ar…

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…

The paper introduces polarizations in symplectic and orthogonal settings.

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.

The paper bounds abnormal and Goh-abnormal sets for metabelian Lie groups with polarizations.

Given any compact Riemann surface $C$, there is a canonical meromorphic 2--form $\widehatη$ on $C\times C$, with pole of order two on the diagonal $Δ\, \subset\, C\times C$, constructed in \cite{cfg}. This meromorphic 2--form $\widehatη$ produces a canonical projective structure on $C$. On the other hand the uniformiza…

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…

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.

Let $E$ be a holomorphic vector bundle. Let $θ$ be a Higgs field, that is a holomorphic section of $End(E)\otimesΩ^{1,0}_X$ satisfying $θ^2=0$. Let $h$ be a pluriharmonic metric of the Higgs bundle $(E,θ)$. The tuple $(E,θ,h)$ is called a harmonic bundle. Let $X$ be a complex manifold, and $D$ be a normal crossing divi…

We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generali…