Extends residue theory to flags of holomorphic distributions.
problem Calculating the residue class of flags of holomorphic distributions.
method Developed an effective method to calculate the class in certain cases.
result Established a relation between degrees, tangency order, Euler characteristic, and curve degree.
Formula calculates residues for maps near holomorphic distributions.
problem Calculating residues for maps near holomorphic distributions.
method Residues formula for maps generically transversal to regular holomorphic distributions.
result Established a residues formula for maps near holomorphic distributions.
The study generalizes a specific geometric correspondence to higher dimensions.
problem Understanding nondegenerate lines on holomorphic contact manifolds.
method Analyzing nondegenerate lines and corresponding distributions on higher-dimensional manifolds.
result A generalization of the (2,3,5)-distributions to higher dimensions. Study shows mass distribution of random holomorphic sections follows a central limit theorem.
problem Understanding mass distribution of random holomorphic sections.
method Proved a central limit theorem for mass distribution of random holomorphic sections associated with positive line bundles.
result Almost every sequence of random holomorphic sections exhibits quantum ergodicity.
We classify nonsingular holomorphic foliations of dimension and codimension one on certain Hopf manifolds. More general, we prove that all nonsingular codimension one distributions on intermediary or generic Hopf manifolds are integrable and has holomorphic integral first. Also, we prove some results about singular hol…
Walczak formula is a very nice tool for understanding the geometry of a Riemannian manifold equipped with two orthogonal complementary distributions. Svensson [7] has shown that this formula simplifies to a Bochner type formula when we are dealing with Kähler manifolds and holomorphic (integrable) distributions. Here, …
Study of holomorphic distributions on projective 3-space, focusing on stable tangent sheaves.
problem Characterizing holomorphic distributions and their tangent sheaves on projective 3-space.
method Analysis of singular schemes and tangent sheaves, classification of distributions, use of Grothendieck's Quot-scheme.
result Classification of codimension one distributions with stable tangent sheaves and description of moduli spaces.
The paper studies constant curvature holomorphic two-spheres in complex Grassmann manifold.
problem Investigating constant curvature holomorphic two-spheres in complex Grassmann manifold.
method Exploring the theory of functions of one complex variable to determine curvature distribution and construct examples.
result Explicit characterization and construction of non-homogeneous constantly curved holomorphic two-spheres.
Study shows zeros of random sections are uniformly distributed.
problem Distribution of zeros in random holomorphic sections.
method Equidistribution and moment assumptions for singular Hermitian line bundles.
result Asymptotic distribution of zeros is independent of probability measure.
The paper studies distributions on surfaces and their connection to twistor spaces.
problem Understanding distributions invariant under geodesic flows on surfaces.
method Analyzes transport equations on unit tangent bundles and connects to twistor spaces.
result Holomorphic distributions form a unital algebra and are bijectively related to functions on twistor space.
We propose a new approach to the value distribution theory of entire holomorphic curves. We define a ``packing density'' of an entire holomorphic curve, and show that it has various non-trivial properties. We prove a ``gap theorem'' for holomorphic maps from elliptic curves to the complex projective space, and study th…
The study classifies and analyzes two-dimensional holomorphic distributions on a four-dimensional projective space.
problem Classifying and analyzing two-dimensional holomorphic distributions on P4. method Classification and investigation of distributions with specific properties, including tangent and conormal sheaves.
result The sheaves of distributions are split and the moduli spaces are irreducible quasi-projective varieties.
This paper extends geometric structure theory to infinite type structures.
problem Calculating characteristic class relations in complex Cartan geometries.
method Improves representation theory for infinite type structures.
result Direct calculation of characteristic class relations from structure group representation.
The paper equidistributes zeros of random polynomials and sections on manifolds.
problem Equidistribution of zeros of random polynomials and sections on manifolds.
method Weighted pluripotential theory, asymptotic Bernstein-Markov measures, variance estimation.
result Equidistribution holds for non-i.i.d. random coefficients and non-homogeneous manifolds.
New insights into (2,3,5)-distributions via Legendrian curves.
problem Understanding symmetries of (2,3,5)-distributions. method Exploiting a correspondence between distributions and lines on contact manifolds.
result One-to-one correspondence between equivalence classes of (2,3,5)-distributions and Legendrian curves. Study shows normal distribution in divisor counts of random sections on complex manifolds.
problem Distribution of divisors on complex manifolds.
method Central limit theorem for smooth linear statistics of Gaussian sections.
result Asymptotic normality of divisor counts.
Researchers create non-isomorphic holomorphic Engel structures on C4.
problem Constructing non-isomorphic holomorphic Engel structures on C4.
method Controlled curves and distributions to create Engel structures.
result Existence of uncountably many non-isomorphic holomorphic Engel structures on C4.
Study cone structures on contact manifolds to understand their geometric properties.
problem Characterize cone structures on holomorphic contact manifolds.
method Characterize subadjoint varieties among Legendrian submanifolds in terms of contact prolongations.
result Holomorphic horizontal splitting of the canonical distribution on contact G-structures.
Constructs the moduli space of super J-holomorphic curves.
problem Defines and constructs the moduli space of super J-holomorphic curves.
method Uses component fields of a super differential equation and a transversality argument.
result Constructs the moduli space of super J-holomorphic curves as a smooth subsupermanifold.
Study CR-statistical submanifolds in holomorphic statistical spaces.
problem Characterize CR-statistical submanifolds and their properties.
method Optimization technique to relate Ricci curvature and mean curvature.
result Established relationship between Ricci curvature and mean curvature.
The article proves the existence of horizontal immersions into fat distributions and contact structures.
problem Proving the existence of horizontal immersions in fat distributions and contact structures.
method Gromov's sheaf theoretic and analytic techniques of h-principle. result Existence of horizontal immersions of an arbitrary manifold into degree 2 fat distributions and quaternionic contact structures.
We study anti-holomorphic semi-invariant submersions from Kählerian manifolds onto Riemannian manifolds. We prove that all distributions which are involved in the definition of the submersion are integrable. We also prove that the O'Neill's tensor T vanishes on the invariant vertical distribution. We give n…
Study of flows on 7D manifolds with holomorphic properties.
problem Characterizing transversely holomorphic partially hyperbolic flows.
method Analyzing flows with biholomorphic holonomy pseudo-group, proving properties under integrable subcenter distribution.
result Flow projects to a transversely holomorphic Anosov flow on a 5D manifold.
The paper studies random systems of holomorphic sections on compact Kähler manifolds and proves equidistribution results.
problem Estimating the distribution of zeros of random holomorphic sections on compact Kähler manifolds.
method Asymptotic variance estimate for smooth linear statistics, equidistribution result derivation.
result Smooth positive closed form ω^k can be approximated by currents of integration along analytic subsets of X.
Study Bergman kernels and zero distributions of random sections on Kähler manifolds.
problem Asymptotic distribution of common zeros of random sections on Kähler manifolds.
method Analysis of Bergman kernels and equidistribution for sequences of line bundles.
result Established asymptotic expansion of Bergman kernels and equidistribution of zeros.
Study of flows on complex manifolds with holomorphic properties.
problem Global rigidity of transversely holomorphic Anosov flows on smooth compact manifolds.
method Analyzing the integrability of unstable and stable distributions, proving uniqueness in low dimensions.
result For topologically transitive flows, they are either orbit equivalent to a hyperbolic automorphism or geodesic flow.
This paper studies ruled real hypersurfaces in indefinite complex projective space.
problem Characterizing and classifying ruled real hypersurfaces in indefinite complex projective space.
method Introduced and studied ruled real hypersurfaces with maximal holomorphic distribution integrable and leaves totally geodesic holomorphic hyperplanes. Detailed shape operator computation and method of construction by gluing totally geodesic hyperplanes along a curve.
result Classification of all minimal ruled real hypersurfaces in terms of three main families of curves.
Study on real hypersurfaces with special solitons in complex space forms.
problem Characterizing real hypersurfaces with ∗-Ricci solitons in non-flat complex space forms. method Analyzing real hypersurfaces with ∗-Ricci solitons where the potential vector field is in the principal curvature space and holomorphic distribution. result Characterized real hypersurfaces with ∗-Ricci solitons in non-flat complex space forms. The Kaehler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kaehler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizi…
Using as an underlying manifold an alpha-Sasakian manifold we introduce warped product Kaehler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kaehler manifold is of quasi-constant holomorphic sectional curvatures with special distribution. Conversely, we prov…
We study the classification of singularities of holomorphic foliations and non-integrable one-forms under the hypothesis of transversality with real hypersurfaces.
Holomorphic automorphisms on hyperkähler manifolds with high entropy are Kummer examples.
problem Characterizing holomorphic automorphisms with high entropy on hyperkähler manifolds.
method Using Jensen's inequality and properties of stable and unstable distributions, the authors show uniform contraction and expansion, leading to the conclusion that the manifold is birational to a torus quotient.
result Holomorphic automorphisms with high entropy on hyperkähler manifolds are Kummer examples.
Study on sections of line bundles vanishing along subvarieties in complex spaces.
problem Conditions for the dimension of sections vanishing along subvarieties.
method Analyzes necessary and sufficient conditions for the dimension of sections vanishing along subvarieties.
result Find necessary and sufficient conditions for dimH00(X,Lp)∼pn. 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.
The singular set of a foliation is always connected under certain conditions.
problem Understanding the connectedness of singular sets in foliations.
method Analyzing the normal sheaf and dimension properties of the singular set.
result The union of irreducible components of dimension k−1 in the singular set is connected. This paper bridges Kahler geometry and quantum mechanics in lognormal statistical models.
problem Evolution of spectral curves in Siegel Jacobi space through Schrodinger equation.
method Kahler geometry induced on lognormal statistical manifold, Dombrowski's construction.
result Time-dependent Schrodinger equation with varying energy.
Study on Hermitian manifolds with curvature, finding geometric properties.
problem Understanding the structure of Hermitian manifolds with semipositive Griffiths curvature.
method Combining HCF, torsion-twisted connection properties, and geometric observations.
result Null spaces of the Chern-Ricci form generate a holomorphic, integrable distribution.
Study horizontal discs in fat distributions, proving their existence.
problem Existence of embedded horizontal discs in fat distributions.
method Analyzing nonlinear PDEs and proving local invertibility.
result Existence of germs of embedded horizontal discs.
We study the geometric properties of holomorphic distributions of totally null m-planes on a (2m+ε)-dimensional complex Riemannian manifold (M,g), where ε∈0,1 and m≥2. In particular, given such a distribution N, say, we obtain algebraic conditions on the Weyl tensor and t…
Two new proofs of Gromov's non-squeezing theorem using curve reparametrization and gradient bounds.
problem Gromov's non-squeezing theorem in symplectic geometry.
method Reparametrization of pseudo-holomorphic curves and application of mean value inequality or Gromov-Schwarz lemma.
result Uniform bounds on the gradient of pseudo-holomorphic curves leading to compactness of moduli space.
We prove that every Kaehler metric, whose potential is a function of the time-like distance in the flat Kaehler-Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kaehler manifolds with the above mentioned metrics. New examples o…
Tian's theorem connects Chern classes of bundles to random section zeros and degeneracy sets.
problem Understanding the distribution of zeros and degeneracy sets of random holomorphic sections.
method Analyzing the pullback of Chern classes and computing currents of integration.
result The limit distribution of zeros of random sections is determined by the Chern form.
Holomorphic Jacobi structures enrich the theory of Poisson manifolds.
problem Holomorphic Poisson structures are limited; holomorphic Jacobi structures offer more.
method Developed holomorphic Jacobi structures and their relationship with other structures.
result Holomorphic Jacobi structures provide a broader framework than holomorphic Poisson structures.
Study of Hopf hypersurfaces in a specific nearly Kähler manifold without multiple principal curvatures.
problem Characterizing Hopf hypersurfaces in the homogeneous nearly Kähler S3imesS3. method Analyzing properties of Hopf hypersurfaces with respect to principal curvatures and holomorphic distributions.
result Complete classification of Hopf hypersurfaces with three distinct principal curvatures in the homogeneous NK S3imesS3. Develops theory of d-holomorphic connections on Klein surfaces.
problem No specific problem stated; focuses on theory development.
method Constructs Atiyah exact sequence for d-holomorphic bundles and provides existence criterion.
result Established theory of d-holomorphic connections and existence criterion.
In this paper, the HyperKahler contact distribution of a 3-Sasakian manifold is studied. To analyze the curvature properties of this distribution, the special metric connection ∇ˉ is defined. This metric connection is completely determined by HyperKahler contact distribution. We prove that HyperKahler conta…
Survey on holomorphic structures on complex manifolds.
problem Holomorphic foliations with transverse holomorphic Cartan geometries.
method Analyzes G-structures and Cartan geometries on compact complex manifolds.
result Foliated holomorphic Cartan geometries on compact complex manifolds.
Using the twistor correspondence, this article gives a one-to-one correspondence between germs of toric anti-self-dual conformal classes and certain holomorphic data determined by the induced action on twistor space. Recovering the metric from the holomorphic data leads to the classical problem of prescribing the Cech …