We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi-Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomor…
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48 results for “holomorphic differential forms”
This paper provides an explicit form for symmetric differentials and their corresponding holomorphic functions.
problem Understanding the correspondence between symmetric differentials and holomorphic functions on quotient spaces.
method Explicit description of the correspondence between symmetric differentials and weighted -holomorphic functions.
result Derivation of several applications based on the explicit form of the correspondence.
We prove that the only natural differential operations between holomorphic forms on a complex manifold are those obtained using linear combinations, the exterior product and the exterior differential. In order to accomplish this task we first develop the basics of the theory of natural holomorphic bundles over a fixed …
Defines holomorphic differential forms on complex manifolds over commutative Banach algebras.
problem Defines holomorphic differential forms on complex manifolds over commutative Banach algebras.
method Defines -holomorphic vector bundles and differential forms on -manifolds.
result Defines cohomology groups as -modules, providing a new perspective on cohomology.
Holomorphic QDs dual to Fenchel-Nielsen coords for hyperbolic metrics.
problem Eigenvalue estimates on degenerating surfaces.
method Dual bases of holomorphic quadratic differentials.
result Derived precise estimates for eigenvalues.
Survey of compactifications for differential strata.
problem Compactify strata of holomorphic 1-forms on Riemann surfaces.
method Discuss relations between different compactifications from a geometric perspective.
result Relations between compactifications defined from a flat geometric perspective.
The study generalizes twistor spinors to Kähler manifolds and finds bilinear form equations.
problem Generalizing twistor spinors to Kähler manifolds.
method Finding differential equations and reducing them to conformal Killing-Yano equations.
result Bilinear forms of Kählerian twistor spinors reduce to Kählerian conformal Killing-Yano equations under certain conditions.
We consider the local analytic behavior for a family of holomorphic differentials on a family of degenerating annuli. Three results and discussion are presented. The first is the normal families Lemma 1. The second is an isomorphism of sheaves, formula (3), giving a direct description of families of regular -differe…
Refines Haupt's theorem for surface characters.
problem Characterization of holomorphic 1-forms on surfaces.
method Refines Haupt's theorem with divisor data.
result Necessary and sufficient conditions for surface characters.
Analytic plane curves determine unique conformal coordinates.
problem Determining a conformal coordinate system for analytic plane curves.
method Holomorphic continuation of the Frenet curvature form.
result Holomorphic continuation of the curvature form uniquely determines a conformal coordinate net.
The paper extends holomorphic forms on generalized Hermitian manifolds.
problem Understanding holomorphic forms on generalized Hermitian manifolds.
method Developed a criterion for holomorphic forms and used it to extend -closed forms.
result Invariance of generalized Hodge numbers in deformations of compact generalized Hermitian manifolds.
A new cohomology, induced by a vector field, is defined on pairs of differential forms (--differentiable forms) in a manifold. It is proved a link with the classical de Rham cohomology and an -differentable cohomology of Lichnerowicz type associated to an one form. Also, the case when the manifold is complex and …
Let M be a complex nilmanifold, that is, a compact quotient of a nilpotent Lie group endowed with an invariant complex structure by a discrete lattice. A holomorphic differential on M is a closed, holomorphic 1-form. We show that , where is the algebraic dimension (i.e. the transcendence degre…
We introduce a general setting for multidimensional dispersionless integrable hierarchy in terms of differential -form with the coefficients satisfying the Plücker relations, which is gauge-invariantly closed and its gauge-invariant coordinates (ratios of coefficients) are (locally) holomorphic with respect to…
The paper counts ends of differential forms on surfaces.
problem Counting ends of meromorphic 1-forms on Riemann surfaces.
method Degeneration techniques and moduli space construction.
result Enumeration of ends for meromorphic 1-forms.
Study on opers over complex manifolds of dimension one.
problem Investigating opers over complex manifolds of dimension one.
method Introducing relative opers and differential operators, analyzing their equivalence.
result Bijective correspondence between relative opers and differential operators.
Paper investigates pluripotential theory on Teichmüller space using new methods.
problem Understanding pluripotential theory on Teichmüller space.
method Alternative approach to Krushkal formula, natural stratified structure, Levi form description.
result Natural stratified structure and description of Levi form.
We construct connections and characteristic forms for principal bundles over groupoids and stacks in the differentiable, holomorphic and algebraic category using Atiyah sequences associated to transversal tangential distributions.
Normal forms, moving frames, and differential invariants for nondegenerate hypersurfaces in C^2math.DG
Normal forms and invariants for nondegenerate hypersurfaces in C^2.
problem Equivalence problem for nondegenerate real hypersurfaces in C^2.
method Equivariant moving frames and invariant differentiation.
result A single real differential invariant of order 7 generates the entire algebra of differential invariants for nondegenerate real hypersurfaces at singularly umbilic points.
The paper extends inequalities to twisted differential forms on Kähler manifolds.
problem Generalizing Sobolev-type inequalities to twisted differential forms.
method Establishing heat kernel estimates for differential forms on Kähler manifolds.
result Proves vanishing theorem and -estimates for -operator.
We obtain a unified theory of discrete minimal surfaces based on discrete holomorphic quadratic differentials via a Weierstrass representation. Our discrete holomorphic quadratic differential are invariant under Möbius transformations. They can be obtained from discrete harmonic functions in the sense of the cotangent …
We show that if a generator of a differential Gerstenhaber algebra satisfies certain Cartan-type identities, then the corresponding Lie bracket is formal. Geometric examples include the shifted de Rham complex of a Poisson manifold and the subcomplex of differential forms on a symplectic manifold vanishing on a Lagrang…
A new method integrates forms on Riemann surfaces, leading to modular forms.
problem Integrating differential forms with poles on Riemann surfaces.
method Simple procedure to integrate differential forms with arbitrary holomorphic poles, establishing an analytic theory for integrals over configuration spaces.
result Regularized graph integrals on elliptic curves are almost-holomorphic modular forms.
A new method shows open Riemann surfaces have finite genus.
problem Determining the genus of open Riemann surfaces.
method Introducing a quotient space of forms to determine finite genus.
result Open Riemann surfaces have finite genus and can be embedded in a compact surface.
The study finds bounds on systoles for genus two Riemann surfaces with abelian differentials.
problem Bounding systoles on genus two Riemann surfaces with abelian differentials.
method Analyzing holomorphic 1-forms on Riemann surfaces, providing upper bounds on systoles.
result For genus two, there are at most 10 systolic loops, with a unique realization.
Smooth complex surfaces with triple intersections using differential geometry.
problem Smooth complex surfaces with trivial canonical bundle and triple intersections.
method Explicit construction of local smoothings and solutions to nonlinear elliptic PDEs.
result Existence of smoothings for -semistable SNC complex surfaces with trivial canonical bundle.
We study holomorphic foliations with an affine homogeneous transverse structure. We give a friendly characterization of the case of transversely affine foliations in terms of matrix valued pairs of differential forms. This leads naturally to the study of the case of foliations with singularities. A first extension theo…
Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings and a unitary local system V on it. We consider a differential graded Lie algebra (DGLA) of forms with holomorphic logarithmic singularities and vanishing residues. We construct a spectral sequence corresponding to the ant…
Study surfaces with constant mean curvature in 4D spaces.
problem Classify surfaces with parallel mean curvature in homogeneous 4-manifolds.
method Survey and provide a common framework for classification results.
result Existence of holomorphic quadratic differentials on these surfaces.
New theory connects string theory to swampland distance conjecture.
problem Connecting string theory to swampland distance conjecture.
method Deformations of the heterotic superpotential, treating separately for large fluxes or large distances, integrating out fields to obtain a new field theory.
result New holomorphic theory defined, connects to swampland distance conjecture.
We give a necessary and sufficient condition for a non-degenerate symmetric 3-differential with nonzero Blaschke curvature on a complex surface to be locally representable as a product of three closed holomorphic 1-forms. We give two versions of this condition corresponding to different choices of coordinates, one of w…
Flat connections on manifolds with group actions derived from algebra models.
problem Constructing flat connections on manifolds with group actions.
method Using -algebra models and differential forms.
result Flat connections are uniquely determined by -minimal models.
Numerical experiments support conjecture about opers and nonabelian Hodge.
problem Testing predictions of Gaiotto-Moore-Neitzke and Gaiotto conjectures.
method Numerical experiments on polynomial holomorphic differentials.
result Supports conjectural formulas for Stokes data and Hitchin metric tensor.
Elliptic systems are characterized by Darboux integrability.
problem Characterizing elliptic differential systems with holomorphic solutions.
method Using a complex manifold and associated holomorphic Pfaffian system.
result Elliptic systems are Darboux integrable under generic conditions.
Study properties of holomorphic -contact manifolds, including non-Kähler hyperbolicity and deformations.
problem Characterize the geometric and algebraic properties of holomorphic -contact manifolds.
method Explores non-Kähler hyperbolicity, differential calculus, and -contact deformations, proving unobstructedness theorems.
result Proves a Bogomolov-Tian-Todorov-type unobstructedness theorem for -contact deformations up to order two.
Paper proves flat metrics from holomorphic quadratic differentials can be identified by length spectrum.
problem Identifying flat metrics from holomorphic quadratic differentials.
method Proved using length spectrum on closed oriented surfaces.
result Flat metrics from holomorphic quadratic differentials can be distinguished by their length spectrum.
The paper explores gauge theory invariants and their duals via topological-holomorphic twist.
problem Understanding gauge theory invariants and their duals in 4d and 2d.
method Topological-holomorphic twist of N=4 supersymmetric gauge theory.
result Derived novel topological and holomorphic invariants and their Langlands duals.
We calculate relations on characteristic classes which are obstructions preventing closed Kähler manifolds from carrying holomorphic Cartan geometries. We apply these relations to give global constraints on the phase spaces of complex analytic determined and underdetermined systems of differential equations.
We give a definition of differentiable cohomology of a Lie group G (possibly infinite-dimensional) with coefficients in any abelian Lie group. This differentiable cohomology maps both to the cohomology of the group made discrete and to Lie algebra cohomology. We show that the secondary characteristic classes of Beilins…
Study describes how to realize periods of holomorphic differentials with specific properties.
problem Realizing periods of holomorphic differentials with given zeros and invariants.
method Complete description of realizable relative period representations.
result Answers a question posed by Simion Filip about realizing periods of holomorphic differentials.
We study the global geometry of surfaces in Sasakian space forms whose mean curvature vector is parallel in the normal bundle (these include the Riemannian Heisenberg space of dimension ). We prove a codimension reduction theorem. We introduce two holomorphic quadratic differentials on anti-invariant such surface…
Constructs moduli space for discs mapping to homogeneous spaces.
problem Stable holomorphic discs mapping to homogeneous spaces.
method Orbifold with corners, fibered products, pushforward, pullback of differential forms.
result Moduli space of stable holomorphic discs as an orbifold with corners.
We discuss the local differential geometry of convex affine spheres in $\re^3$ and of minimal Lagrangian surfaces in Hermitian symmetric spaces. In each case, there is a natural metric and cubic differential holomorphic with respect to the induced conformal structure: these data come from the Blaschke metric and Pick f…
The article studies critical points of a new energy functional in higher dimensions.
problem Investigating critical points of a new energy functional in higher dimensions.
method Holomorphic deformations, closed and open properties, differential of the functional.
result Properties of critical points under holomorphic deformations are closed and open.
Generalizes Kawai theorem for orbifold Riemann surfaces.
problem Proving a generalization of Kawai theorem for orbifold Riemann surfaces.
method Using a formula for the differential of a holomorphic map from the cotangent bundle of the Teichmüller space to the -character variety, and evaluating the pullback of Goldman symplectic form.
result Generalization of Kawai theorem for orbifold Riemann surfaces and Goldman's theorem.
New algebra defined for Legendrian submanifolds, preserving key invariants.
problem Defining a new algebra to preserve invariants of Legendrian submanifolds.
method Combining string topology techniques with combinatorial methods to count holomorphic disks.
result The new algebra is a filtered, differential graded algebra that captures invariants of Legendrian submanifolds.
Survey on quadratic differentials in Teichmüller theory.
problem Understanding quadratic differentials in Teichmüller theory.
method Expository survey of quadratic differentials' roles.
result Summarizes results for non-compact surfaces.
Abstract compares two norms in holomorphic quadratic differentials.
problem Comparing two norms in holomorphic quadratic differentials.
method Comparison between Avila-Gouëzel-Yoccoz norm and Teichmüller norm.
result Comparison of two norms in holomorphic quadratic differentials.