arXiv research

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.

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

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.

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

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

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

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.

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 …

Proves cohomology theorems for tropical varieties.

Abstract: Almost Kähler Hodge numbers vary with metric choices.

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…

Extends Hodge theory to nearly Kähler manifolds of arbitrary dimensions.

The paper extends game theory using Hodge theory on graphs.

We examine the relationship between nonabelian Hodge theory for Riemann surfaces and the theory of vector valued modular forms. In particular, we explain how one might use this relationship to prove a conjectural three-term inequality on the weights of free bases of vector valued modular forms associated to complex, fi…

The paper generalizes Hodge theory to semisimple local systems and proves a geometric Decomposition theorem.

Based on some analogies with the Hodge theory of isolated hypersurface singularities, we define Hodge-type numerical invariants (called H-numbers) of any, not necessarily algebraic, link in $S^3$. They contain the same information as the (normalized) real Seifert matrix. We study their basic properties, we express the …

Develops Hodge theory for boundary-value problems on general geometric structures.

Graph theory criterion for Hodge theory to match linearly.

Constructs moduli stacks for quiver connections and extends non-Abelian Hodge theory.

Derives integral formula for Hodge and Teichmüller norms.

This paper constructs a Hodge theory of noncompact topologically tame manifolds $M$. The main result is an isomorphism between the de Rham cohomology with compact supports of $M$ and the kernel of the Hodge--Witten--Bismut Laplacian $\lap_μ$ associated to a measure $dμ$ which has sufficiently rapid growth at infinity o…

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…

We develop Hodge theory for a Riemannian manifold $(M,g)$ with a background closed 3-form, H. Precisely, we prove that if the metric connections with torsion $\pm H$ have holonomy groups $G_\pm$, then the $d^H$-Laplacian preserves the irreducible representations of the Lie algebras of the holonomy groups on the space o…

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 …