arXiv research

We construct Hodge filtered function spaces associated to infinite loop spaces. For Brown-Peterson cohomology, we show that the corresponding Hodge filtered spaces satisfy an analog of Wilson's unstable splitting. As a consequence, we obtain an analog of Quillen's theorem for Hodge filtered Brown-Peterson cohomology fo…

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

We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties. The result is deduced from a generalization of a theorem of Möller. Namely, we prove that the image of a certain twisted Abel-Jacobi map lands in the torsion of a factor of the Jacobians. This statement can be viewed as a split…

The paper extends Einstein condition to 4-manifolds using Hodge splittings.

Study on Kähler manifolds with non-negative mixed curvature, proving splitting and structure theorems.

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

A new method splits surface flow discretizations into streamfunctions and harmonic fields.

In this paper we show that, after completion in the I-adic topology, the Goldman bracket on the space spanned by homotopy classes of loops on a smooth, complex algebraic curve is a morphism of mixed Hodge structure. We prove similar statements for the natural action (defined by Kawazumi and Kuno) of the loops in X on p…

Researchers compute the spectrum of Hodge-Laplacian on 1-forms for SU(2) and SO(3).

Decomposes financial networks to reveal cause-effect hierarchies during crises.

Recent years have witnessed a trend that advanced mathematical tools, such as algebraic topology, differential geometry, graph theory, and partial differential equations, have been developed for describing biological macromolecules. These tools have considerably strengthened our ability to understand the molecular mech…

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…

Study low energy resolvent behavior on fibred boundary metrics.

Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.

On a compact $\partial\bar\partial$-manifold $X$, one has the Hodge decomposition: the de Rham cohomology groups split into subspaces of pure-type classes as $H_{dR}^k (X)=\oplus_{p+q=k}H^{p,\,q}(X)$, where the $H^{p,\,q}(X)$ are canonically isomorphic to the Dolbeault cohomology groups $H_{\bar\partial}^{p,\,q}(X)$. F…

In this paper we introduce the notion of Poincaré DGCAs of Hodge type, which is a subclass of Poincaré DGCAs encompassing the de Rham algebras of closed orientable manifolds. Then we introduce the notion of the small algebra and the small quotient algebra of a Poincaré DGCA of Hodge type. Using these concepts, we inves…

The Dirac operator d+delta on the Hodge complex of a Riemannian manifold is regarded as an annihilation operator A. On a weighted space L_mu^2 Omega, [A,A*] acts as multiplication by a positive constant on excited states if and only if the logarithm of the measure density of mu satisfies a pair of equations. The equati…

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 .

Study on Hodge theory for almost complex manifolds.

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 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.

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

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.

Discretizes Hodge-Dirac operators on a torus.

Paper introduces magnetic Hodge Laplacian for differential forms.

For any positive integer m and any dimension n, we show that any n-dimensional Hodge diamond with values in Z/mZ is attained by the Hodge numbers of an n-dimensional smooth complex projective variety. As a corollary, there are no polynomial relations among the Hodge numbers of n-dimensional smooth complex projective va…

The paper proves formulas and theorems for statistical de Rham Hodge operators on manifolds with boundary.

Hodge theory applied to tropical curves.

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…

The paper studies perturbations of de Rham Hodge operators on manifolds with or without boundaries.

Study shows equality in Hodge Laplacian bound occurs only on spheres.

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

Proves Kato manifolds satisfy Hodge decomposition.

The paper broadens a mathematical correspondence to include more balanced metrics.

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.

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

New proof classifies orbit closures in Hodge bundle.

Lower bounds for Hodge-Laplacian spectrum on orbifolds.

Geometric structures on 5-manifolds from surface group representations of G2'.

Smooth bundles with rough data maintain Hodge kernel isomorphism.

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

This paper generalizes L2 cohomology theory for complex manifolds.