arXiv research

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

Develops Hodge theory on ALG$^*$ manifolds, proving existence and vanishing results.

Study of height jumps in Ceresa cycle using asymptotic Hodge theory.

We find G2-manifolds with specific asymptotic properties.

We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the…

Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove…

We consider the Hodge Laplacian on manifolds with incomplete edge singularities, with infinite dimensional von Neumann spaces and intricate elliptic boundary value theory. We single out a class of its algebraic self-adjoint extensions. Our microlocal heat kernel construction for algebraic boundary conditions is guided …

Let X be a non-compact Calabi-Yau manifold and f be a holomorphic function on X with compact critical locus. We introduce the notion of f-twisted Sobolev spaces for the pair (X,f) and prove the corresponding Hodge-to-de Rham degeneration property via L2-Hodge theoretical methods when f satisfies an asymptotic condition…

Study geometric operators on Tian-Yau spaces, finding $L^2$ harmonic forms and asymptotic regularity.

We consider a family of compact, oriented and connected Riemannian manifolds shrinking to a metric graph and describe the asymptotic behaviour of the eigenvalues of the Hodge Laplacian. We apply our results to produce manifolds with spectral gaps of arbitrarily large size in the spectrum of the Hodge Laplacian.

Study on Hodge theory for almost complex manifolds.

Hodge theory applied to tropical curves.

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.

We calculate the second coefficient of the asymptotic expansion of the Bergman kernel of the Hodge-Dolbeault operator associated to high powers of a Hermitian line bundle with non-degenerate curvature, using the method of formal power series developed by Ma and Marinescu.

Lin and Sjamaar have used symplectic Hodge theory to obtain canonical equivariant extensions for Hamiltonian actions on closed symplectic manifolds that have the strong Lefschetz property. Here we obtain canonical equivariant extensions much more generally by means of classical Hodge theory.

The paper uses non-abelian Hodge theory to generalize Kodaira vanishing theorems.

Introduces a new Hodge theory using vector fields on manifolds.

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

We prove that the deformation theory of compactifiable asymptotically cylindrical Calabi-Yau manifolds is unobstructed. This relies on a detailed study of the Dolbeault-Hodge theory and its description in terms of the cohomology of the compactification. We also show that these Calabi-Yau metrics admit a polyhomogeneous…

Discretizes Hodge-Dirac operators on a torus.

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.

Explains Hodge theory and Kodaira embedding theorem for complex manifolds.

Anabelian geometry reformulated using Hodge theory for hyperbolic curves.

We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show th…

We study cohomologies and Hodge theory for complex manifolds with twisted differentials. In particular, we get another cohomological obstruction for manifolds in class $\mathcal{C}$ of Fujiki. We give a Hodge-theoretical proof of the characterization of solvmanifolds in class $\mathcal{C}$ of Fujiki, first proven by D.…

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

We present a method to develop a Hodge theory for tangential cohomology of foliations by mimicing Witten's approach to ordinary Morse theory by perturbations of the Laplacian

Study describes splitting and filtration of Hodge bundle on quadratic differentials.

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

The paper connects Chern-Simons invariants to mixed Tate motives in hyperbolic 3-manifolds.

New class of singular complex manifolds studied with degenerate theory.

This paper generalizes L2 cohomology theory for complex manifolds.

Proves tropical Hodge theory for smooth projective varieties, conditional on Laplacian regularity.

The paper explores Higgs bundles and their moduli spaces on Riemann surfaces.

Here we survey questions and results on the Hodge theory of hyperkaehler quotients, motivated by certain S-duality considerations in string theory. The problems include L^2 harmonic forms, Betti numbers and mixed Hodge structures on the moduli spaces of Yang-Mills instantons on ALE gravitational instantons, magnetic mo…

Smooth bundles with rough data maintain Hodge kernel isomorphism.

Proves hard Lefschetz theorem and Hodge-Riemann relations for convex valuations.

Study representation varieties of twisted Hopf links using combinatorial and Hodge theory.

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 …

Paper introduces magnetic Hodge Laplacian for differential forms.

Proves cohomology theorems for tropical varieties.

Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop …

Abstract: Almost Kähler Hodge numbers vary with metric choices.

Develops calculus for QFB metrics, proving Fredholm properties and decay of harmonic forms.

It is well known that positivity properties of the curvature of a vector bundle have implications on the algebro-geometric properties of the bundle, such as numerical positivity, vanishing of higher cohomology leading to existence of global sections etc. It is also well known that bundles arising in Hodge theory tend t…

The abstract explores analogues of Hodge theory in Lie algebroids.

Semisimplicity proven for conformal blocks representations.

These lecture notes in the De Rham-Hodge theory are designed for a 1-semester undergraduate course (in mathematics, physics, engineering, chemistry or biology). This landmark theory of the 20th Century mathematics gives a rigorous foundation to modern field and gauge theories in physics, engineering and physiology. The…