arXiv research

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

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

Explains Hodge theory and Kodaira embedding theorem for complex manifolds.

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

The Hodge theorem connects cohomology groups on compact Kähler manifolds.

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

This paper extends Dabrowski-Sitarz-Zalecki theorems to manifolds with boundary.

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

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

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

Promotes spectral functionals to noncommutative fields and proves a theorem.

Study Morse-Novikov cohomology on foliated manifolds and prove Hodge theorem.

This note contains a reformulation of the Hodge index theorem within the framework of Atiyah's $L^2$-index theory. More precisely, given a compact Kähler manifold $(M,h)$ of even complex dimension $2m$, we prove that $σ(M)=\sum_{p,q=0}^{2m}(-1)^ph_{(2),Γ}^{p,q}(M)$ where $σ(M)$ is the signature of $M$ and $h_{(2),Γ}^…

Proves the Hodge conjecture for complex projective manifolds.

We consider the geometric properties of Hodge Cousin groups, introduced in an unpublished paper \cite{OVV}, emphasizing the case of Hodge Cousin groups corresponding to polarized $\mathbb{Q}$-Hodge structures.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

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…

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

Proves cohomology theorems for tropical varieties.

The study shows conditions for Kähler manifolds to have rational cohomology.

The Hodge-de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a nove…

Anabelian geometry reformulated using Hodge theory for hyperbolic curves.

A manifold with fibered cusp metrics $X$ can be considered as a geometrical generalization of locally symmetric spaces of $\mathbb{Q}$-rank one at infinity. We prove a Hodge-type theorem for this class of Riemannian manifolds, i.e. we find harmonic representatives of the de Rham cohomology $H^p(X)$. Similar to the situ…

Discretizes Hodge-Dirac operators on a torus.

Let $\bar{M}$ be a manifold with boundary $Y$ which is the total space of a fibre bundle, and is defined by the vanishing of a boundary defining function, $x$. We prove $L^2$ Hodge and signature theorems for $M$ endowed with a metric of the form $dx^2 + x^{2c} h + k$, where $k$ is the lift to $Y$ of the metric on the b…

Analyzes semi-characteristics on specific manifolds, proving a vanishing theorem.

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

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…

A generalized complex manifold which satisfies the $\partial \overline{\partial}$-lemma admits a Hodge decomposition in twisted cohomology. Using a Courant algebroid theoretic approach we study the behavior of the Hodge decomposition in smooth and holomorphic families of generalized complex manifolds. In particular we …

Uniform eigenvalue bounds for Hodge Laplacian on manifolds with Ricci curvature and injectivity radius bounds.

The paper proves a decomposition theorem for forms on sub-Riemannian contact manifolds.

The transcendental Hodge lattice of a projective manifold $M$ is the smallest Hodge substructure in $p$-th cohomology which contains all holomorphic $p$-forms. We prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this algebra explicitly for a hyperkahler manif…

Uniformizes Hodge structures, proving Lyapunov exponents and log-Anosov monodromy.

We generalise Simpson's nonabelian Hodge correspondence to the context of projective varieties with klt singularities. The proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety X and any resolution of singu…

The paper extends Hodge-de Rham and Lichnérowicz Laplacians to double forms and proves vanishing theorems.

Paper extends Hodge correspondence to singular Kähler spaces.

New theorems on Hodge numbers and Kähler structures derived from complex differential forms.

Paper refines Alesker-Bernig-Schuster theorem, proving Hodge-Riemann relations for Euclidean balls.

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

In this paper, we proved a rigidity theorem of the Hodge metric for concave horizontal slices and a local rigidity theorem for the monodromy representation.

In this paper, we get a Kastler-Kalau-Walze type theorem associated to nonminimal de Rham-Hodge operators on compact manifolds with boundary. We give two kinds of operator-theoretic explanations of the gravitational action in the case of four dimensional compact manifolds with flat boundary.

We study rank $1$ flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank $1$ flat bundles, we represent the structure theorem of Kähler solvmanifolds as extensions of Hasegawa's result and Benson-Gordon's result for nilmanifolds.

On a compact, oriented, Riemannian manifold, the Hodge decomposition theorem associates a smooth primitive to any exact smooth form omega. In this paper, we show that given a smooth family of exact smooth forms omega(t), the family of associated primitives is also a smooth family with respect to t.

We provide a formula describing the G-module structure of the Hurwitz-Hodge bundle for admissible G-covers in terms of the Hodge bundle of the base curve, and more generally, for describing the G-module structure of the push-forward to the base of any sheaf on a family of admissible G-covers. This formula can be interp…

The paper develops $L^2$-Hodge theory on almost Kähler manifolds and proves the Hopf conjecture.

Introduces non-abelian Hodge correspondence linking algebraic structures to geometry.

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

We investigate when the Chevalley-Eilenberg differential of a complex Lie algebroid on a manifold with boundary admits a Hodge decomposition. We introduce the concepts of Cauchy-Riemann structures, elliptic and non-elliptic boundary points and Levi-forms, which we use to define the notion of q-convexity. We show that t…

New formulas with quadratic curvature terms on Kähler manifolds for Hodge number estimates.