Proves Hodge structures on modular functors, providing formulas for Hodge numbers.
problem Existence and uniqueness of Hodge structures on modular functors.
method Non-Abelian Hodge correspondence and Ocneanu rigidity.
result Explicit formulas for Hodge numbers in modular functors of level 2 times an odd number.
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 .
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 Hodge theory for almost complex manifolds.
problem Determining Hodge numbers for almost complex manifolds.
method Review and analysis of recent developments in Hodge theory for almost complex manifolds.
result Hodge numbers are almost complex, almost Kähler, or birational invariants in dimension four.
Study Hodge-de Rham numbers for almost complex 4-manifolds, extending properties from complex surfaces.
problem Understanding Hodge-de Rham numbers for almost complex 4-manifolds.
method Introduced and studied Hodge-de Rham numbers, extending properties from complex surfaces.
result All Hodge-de Rham numbers for compact almost complex 4-manifolds are determined by the cohomology, except for one (the irregularity).
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…
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…
The paper finds diffeomorphic complex intersections with distinct Hodge numbers.
problem Identifying complex intersections with different Hodge numbers.
method Provided three pairs of 3-dimensional and one pair of 5-dimensional complex complete intersections, all diffeomorphic but with different Hodge numbers.
result Diffeomorphic complex intersections can have different Hodge numbers.
New Sasaki structures identified by Hodge numbers in odd dimensions.
problem Identifying Sasaki structures with distinct Hodge numbers.
method Producing examples of manifolds with pairs of Sasaki structures having different basic Hodge numbers.
result Examples of manifolds with pairs of Sasaki structures having different basic Hodge numbers in odd dimensions.
Paper shows examples of almost Kähler manifolds satisfying Hard Lefschetz but not Betti-Hodge equality.
problem Understanding the Hard Lefschetz condition in almost Kähler manifolds.
method Examples and counterexamples of compact almost Kähler manifolds.
result The Hard Lefschetz condition does not imply the equality between Betti and Hodge numbers in almost Kähler manifolds.
Abstract: Almost Kähler Hodge numbers vary with metric choices.
problem Variation of almost Kähler Hodge numbers with metrics.
method Analysis of almost complex Hodge numbers under different almost Kähler metrics.
result The almost Kähler Hodge number h0,1 varies with metric choices. The first Betti number for a lattice in a classifying space for variations of Hodge structures vanishes.
Machine learning predicts Hodge numbers of Calabi-Yau four-folds.
problem Predicting Hodge numbers of Calabi-Yau manifolds using machine learning.
method Supervised learning of Hodge numbers h^1,1 and h^3,1 for complete intersection Calabi-Yau four-folds.
result A two-branch network with feature enhancement achieves 98% precision for predicting h^3,1.
The study shows conditions for Kähler manifolds to have rational cohomology.
problem Conditions for Kähler manifolds to have rational cohomology.
method Analyzes the Kähler curvature operator and its eigenvalues.
result Compact Kähler manifolds have rational cohomology under certain conditions.
Extends Hodge theory to nearly Kähler manifolds of arbitrary dimensions.
problem Generalize Hodge-theoretic results to nearly Kähler manifolds of arbitrary dimensions.
method Apply Hodge theory to nearly Kähler manifolds of arbitrary dimensions, relating Hodge numbers to Betti numbers.
result Hodge numbers of compact nearly Kähler manifolds are related to Betti numbers in the same way as on a compact Kähler manifold.
Study Betti and Hodge numbers of solvmanifolds from integer polynomials.
problem Computing Betti and Hodge numbers of solvmanifolds constructed from integer polynomials.
method Analyzing de Rham and Dolbeault cohomology of solvmanifolds under algebraic conditions.
result Explicit generating polynomials for Hodge numbers in quasi full rank case.
Anabelian geometry reformulated using Hodge theory for hyperbolic curves.
problem Determining varieties over number fields using their étale fundamental groups.
method Formulating a Hodge-theoretic version of anabelian conjecture, replacing Galois action with Cimes-action. result Proved a Hodge-theoretic analog of Mochizuki's theorem for smooth projective hyperbolic curves over C. Study Hodge Laplacians for manifold data, improving error bounds.
problem Approximating Laplace-Beltrami operator on differential forms.
method Higher-order graph Laplacians (Hodge Laplacians) as approximations.
result High-probability error bound for Dirichlet forms.
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 S3. They contain the same information as the (normalized) real Seifert matrix. We study their basic properties, we express the …
The basic Dolbeault cohomology groups of a Sasakian manifold M are invariants of its characteristic foliation F (the orbit foliation of the Reeb flow). We show some fundamental properties of this cohomology, which are useful for its computation. In the first part of the article, we show that the basic Hodge numbers, th…
New theorems on Hodge numbers and Kähler structures derived from complex differential forms.
problem Deformation invariance and local stability of Hodge numbers and Kähler structures.
method Using the exponential operator and power series method, the approach focuses on d-closed extensions and foliated cases. result Local stabilities of transversely p-Kähler structures and new theorems on Hodge numbers. These are the notes for the talk "Hodge numbers of a hypothetical complex structure on S6" given by the author at the MAM1 "(Non)-existence of complex structures on S6" held in Marburg in March 2017. They are based on [A. Gray, A property of a hypothetical complex structure on the six sphere, Boll. Un. Mat. Ital.…
This note contains a reformulation of the Hodge index theorem within the framework of Atiyah's L2-index theory. More precisely, given a compact Kähler manifold (M,h) of even complex dimension 2m, we prove that σ(M)=∑p,q=02m(−1)ph(2),Γp,q(M) where σ(M) is the signature of M and $h_{(2),Γ}^…
Research connects geometric structures to knot theory and algebraic combinatorics.
problem Understanding the mixed Hodge structure on cohomology of open positroid varieties.
method Relates mixed Hodge structure to Khovanov-Rozansky homology of associated links.
result Rational q,t-Catalan numbers are derived from mixed Hodge polynomials of open positroid varieties. Constructs metrics with zero eigenvalues for Hodge-Laplacian.
problem Eigenvalues of Hodge-Laplacian under sectional curvature constraints.
method One-parameter family of metrics with bounded sectional curvature.
result k-th positive eigenvalue converges to zero.
New operators help focus on specific areas in complex math problems.
problem Concentration in complex mathematical structures.
method Construct conjugate-linear perturbations of twisted spinc Dirac operators using the conjugate-linear Hodge star operator.
result These perturbations satisfy the concentration principle.
The study proves rigidity for mixed Hodge structures and applies to curve families.
problem Rigidity of period maps for mixed Hodge structures.
method Holomorphic bisectional curvature approach.
result Establishes rigidity in various cases, including curve families.
Machine learning approximates Calabi-Yau Hodge numbers from weight systems.
problem Approximating Hodge numbers of Calabi-Yau manifolds from weight systems.
method Neural networks learned Hodge numbers from weight systems, symbolic regression inspired truncation, and machine learning generated new datasets.
result Approximation provides tight lower bounds and dramatically faster computation.
The paper studies Hodge Laplacians from point clouds, proving spectral convergence and harmonic form consistency.
problem Analyzing Riemannian submanifolds from point cloud data.
method Constructing deformed Hodge Laplacians and empirical operators from point clouds, proving convergence properties.
result Empirical spectral cluster contains the k-th Betti number and converges to harmonic k-forms. 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…
Eigenvalue estimates for weighted manifolds with applications.
problem Eigenvalue estimates for weighted Riemannian manifolds.
method Derivation of various eigenvalue estimates for the Hodge Laplacian acting on differential forms.
result Derivation of an inequality relating eigenvalues of the Jacobi operator and the spectrum of the Hodge Laplacian.
We establish a compact analog of the P = W conjecture. For a holomorphic symplectic variety with a Lagrangian fibration, we show that the perverse numbers associated with the fibration match perfectly with the Hodge numbers of the total space. This builds a new connection between the topology of Lagrangian fibrations a…
Study of section conjecture analogues over complex numbers and Kodaira fibrations.
problem Investigate Grothendieck's section conjecture over complex numbers and Kodaira fibrations.
method Topological and Hodge theoretic analogues of the section conjecture over complex numbers, studied in the context of Kodaira fibrations and families of Jacobians.
result Both topological and Hodge-theoretic analogues of the injectivity part of the section conjecture hold for families of curves, but the topological analogue of the surjectivity part does not hold in general.
Develops Hodge theory on ALG∗ manifolds, proving existence and vanishing results.
problem Existence and vanishing of certain cohomology groups on ALG∗ manifolds. method Fredholm Theory for Hodge Laplacian in weighted spaces on ALG∗ manifolds. result Non-existence of ALG∗ manifolds with non-negative Ricci curvature at infinity. Study cohomology of Bigolin complex on complex manifolds.
problem Characterize cohomology of Bigolin complex on compact complex manifolds.
method Analyze the decomposition of the double complex into squares and zigzags, focusing on the zigzags contributing to cohomology.
result In complex dimension 3, multiplicities of zigzags are characterized by Betti, Hodge, Aeppli numbers plus Bigolin numbers.
We introduce a canonical isomorphism from the space of pure-type complex differential forms on a compact complex manifold to the one on its infinitesimal deformations. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang and the second author. As a direct corollary of the extension…
We study the moduli space M(G,A) of flat G-bundles on an Abelian surface A, where G is a compact, simple, simply connected, connected Lie group. Equivalently, M(G,A) is the (coarse) moduli space of s-equivalence classes of holomorphic semi-stable G_C-bundles with trivial Chern classes where G_C is the complexified grou…
This article deals with a number of topics which are, somewhat surprisingly, related. Firstly, the fundamental theorem of skew invariant theory for the symplectic group giving the generators and relations of symplectic invariants is established. The relations are the so called P_n relations which appear in the study of…
Study on Lp cohomology and Hodge decomposition for ALE manifolds.
problem Understanding Lp cohomology dimensions and harmonic forms in ALE manifolds. method Relating dimensions of Lp cohomology spaces to decaying harmonic forms, proving independence and jumps in dimensions, and providing Hodge decompositions. result Dimension of Lp reduced cohomology spaces in degree k is independent of p for k not equal to 1 or n-1, and jumps by a factor N-1 for k equal to 1 or n-1. New formulas with quadratic curvature terms on Kähler manifolds for Hodge number estimates.
problem Estimating Hodge numbers under weak curvature conditions.
method Established new Bochner-Kodaira formulas with quadratic curvature terms.
result Derivation of Weitzenböck-Bochner-Kodaira formulas with quadratic curvature terms on compact Kähler manifolds.
Study shows no hyperkähler fourfolds in specified conditions.
problem Identifying hyperkähler fourfolds in specific geometric settings.
method Classification and computation of Hodge numbers for fourfolds over rational homogeneous varieties.
result No hyperkähler fourfolds found in the specified conditions.
We prove a blow-up formula for Dolbeault cohomologies of compact complex manifolds by introducing relative Dolbeault cohomology. As corollaries, we present a uniform proof for bimeromorphic invariance of (∙,0)- and (0,∙)-Hodge numbers on a compact complex manifold, and obtain the equality for the number…
Semisimplicity proven for conformal blocks representations.
problem Semisimplicity of conformal blocks representations.
method Theory of extensions in non-Abelian Hodge theory and Ocneanu rigidity.
result Semisimplicity of braid group and mapping class group representations.
Proves L2 Frölicher inequality on complex manifolds.
problem Calculating L2 Betti and Hodge numbers. method Uses spectral projectors of (Dh)2 to build an injection. result New proof of classical Frölicher inequality.
The paper describes how Hodge loci are typically equidistributed in complex varieties.
problem Understanding the distribution of Hodge loci in complex varieties.
method Analyzing polarized variations of Hodge structures over smooth complex quasi-projective varieties.
result Hodge loci are either empty or equidistributed with respect to a pull-push form.
Uniformizes Hodge structures, proving Lyapunov exponents and log-Anosov monodromy.
problem Analyzing weight 3 variations of Hodge structures and their Lyapunov exponents.
method Developed uniformizations and used analytic properties to prove conjectures and properties of monodromy representations.
result Proved log-Anosov property and established strong Torelli theorem for the VHS.
Paper extends Hodge correspondence to singular Kähler spaces.
problem Establishing Hodge correspondence over Kähler spaces with singularities.
method Using equivalence of polystable Higgs bundles and semi-simple flat bundles over regular loci, and descent theorem for semistable Higgs bundles.
result Non-abelian Hodge correspondence established over compact Kähler klt spaces and their regular loci.
Constructs moduli stacks for quiver connections and extends non-Abelian Hodge theory.
problem Extending non-Abelian Hodge theory to moduli stacks of quiver connections.
method Formalizes and constructs moduli stacks of bundles with λ-connections over prestacks.
result Shows moduli stacks are algebraic and locally of finite presentation when base is smooth and projective.