arXiv research

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

Derives integral formula for Hodge and Teichmüller norms.

The paper explores Hodge decomposition and Hard Lefschetz Condition on almost Kähler manifolds.

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…

This paper is a survey of the subject of variations of Hodge structure (VHS) considered as exterior differential systems (EDS). We review developments over the last twenty-six years, with an emphasis on some key examples. In the penultimate section we present some new results on the characteristic cohomology of a homog…

Derives integral formula for differential forms on compact spaces with applications.

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…

The paper extends game theory using Hodge theory on graphs.

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…

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

The paper classifies Lie algebroids and their connections, modulating principal objects.

The paper proves wave operator existence and completeness for Hodge Laplacians.

A new geometric framework resolves singularities in anomalous transport.

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…

Reformulates elasticity complex with new differential and Hodge star operators.

There are a number of examples of variations of Hodge structure of maximum dimension. However, to our knowledge, those that are global on the level of the period domain are totally geodesic subspaces that arise from an orbit of a subgroup of the group of the period domain. That is, they are defined by Lie theory rather…

In the present notes we explain the relationship between Calabi-Yau integrable systems and Hitchin systems based on work by Diaconescu-Donagi-Pantev and the author. Besides a review of these integrable systems, we highlight related topics, for example variations of Hodge structures, cameral curves and Slodowy slices, a…

An essentially unique homeomorphic solution to the Beltrami equation was found in the 1960s using the theory of Calderón-Zygmund and singular integral operators in $L^p(\mathbb C)$. We will present an alternative method to solve the Beltrami equation using the Hodge star operator and standard elliptic PDE theory. We wi…

In this paper, we show that the generating function for linear Hodge integrals over moduli spaces of stable maps to a nonsingular projective variety $X$ can be connected to the generating function for Gromov-Witten invariants of $X$ by a series of differential operators $\{ L_m \mid m \geq 1 \}$ after a suitable change…

The study finds arithmetic groups often in square-tiled surface monodromies.

Inspired by Katz-Mazur theorem on crystalline cohomology and by Eskin-Kontsevich-Zorich's numerical experiments, we conjecture that the polygon of Lyapunov spectrum lies above (or on) the Harder-Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons a…

In the 80's H. Masur and W. Veech defined two numerical invariants of strata of abelian differentials: the volume and the Siegel-Veech constant. Based on numerical experiments, A. Eskin and A. Zorich proposed a series of conjectures for the large genus asymptotics of these invariants. By a careful analysis of the asymp…

The Hodge star mean curvature flow on a 3-dimension Riemannian or pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold, the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural non-linear dispersive curve flows in geometric analysis. A curve flo…

Formula for Masur-Veech volumes in quadratic differentials with odd zeros.

The $L^2$-$\partial\overline\partial$-Lemma is extended to complete Kähler manifolds with a gap in the spectrum.

The paper proves inequalities for twisted differential forms on manifolds.

A closed formula is obtained for the integral $\int_{\mathcal{\bar{H}}_g^1}κ_{1}ψ^{2g-2}$ of tautological classes over the locus of hyperelliptic Weierstraß points in the moduli space of curves. As a corollary, a relation between Hodge integrals is obtained. The calculation utilizes the homeomorphism between the moduli…

Formula connects $G_2$-structure geometry to Poisson equation.

Study actions of mapping class groups on surface representations, proving finite image for certain representations.

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

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 .

The well-known Kähler identities naturally extend to the non-integrable setting. This paper deduces several geometric and topological consequences of these extended identities for compact almost Kähler manifolds. Among these are identities of various Laplacians, generalized Hodge and Serre dualities, a generalized hard…

We started from computer experiments with simple one-dimensional ergodic dynamical systems called interval exchange transformations. Correlators in these systems decay as a power of time. In the simplest non-trivial case the exponent is equal to 1/3. We found a formula connecting characteristic exponents with explicit …

Study on Hodge theory for almost complex manifolds.

We study pointwise and $L^p$ gradient estimates of the heat kernel, on manifolds that may have some amount of negative Ricci curvature, provided it is not too negative (in an integral sense) at infinity. We also prove uniform boundedness results on $L^p$ spaces for the heat operator of the Hodge Laplacian on differenti…

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…

Study on $GL(1|1)$ Higgs bundles over Riemann surfaces.

This thesis explores algebraic cycles and moduli spaces over real 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.

Starting from the candidate Bloch-Beilinson filtration on Chow groups of 0-cycles constructed by J. Lewis, we develop and describe geometrically a series of Hodge-theoretic invariants defined on the graded pieces. Explicit formulas (in terms of currents and membrane integrals) are given for certain quotients of the inv…

Novel Hodge theory developed for Carrollian bundles, addressing black hole event horizons.

A graph theory approach defines curl and decomposes vector fields.

Geometric model for Hodge filtered complex cobordism constructed.

Introduces a new Hodge theory using vector fields on manifolds.

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

Proves a tropical version of Clemens-Schmid sequence for tropical varieties.

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.

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…