Study characterizes totally geodesic submanifolds in quotient spaces.
problem Characterizing totally geodesic submanifolds in quotient spaces.
method Characterization through totally geodesic submanifolds and holomorphic tangent sequence splitting.
result Characterization of totally geodesic submanifolds in quotient spaces.
A holomorphic Poisson structure induces a deformation of the complex structure as Hitchin's generalized geometry. Its associated cohomology naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is the Dolbeault cohomology with coefficients in the exterior a…
Paper proves unique tangent maps for complex maps into algebraic varieties.
problem Proving uniqueness of tangent maps for complex maps into algebraic varieties.
method Developed techniques to prove uniqueness of tangent maps for weakly holomorphic and locally approximable maps.
result Established unique tangent cone property for weakly holomorphic maps into projective algebraic varieties.
The paper proves a splitting theorem for sheaves of holomorphic k-vectors on complex contact manifolds.
problem Understanding the structure of sheaves of holomorphic k-vectors on complex contact manifolds.
method Proving a splitting theorem using sheaves and cohomology.
result The sheaf of holomorphic k-vectors splits into two components.
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.
Embeddings of 3-manifolds into complex 3-space with specific tangents.
problem Embedding closed 3-manifolds into C3 with specified complex tangents. method Constructing embeddings based on link and 2-plane field properties.
result Existence of embeddings with specified complex tangents and holomorphic tangent spaces.
Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider says that there is a finite unramified Galois covering M --> X, a complex torus T, and a holomorphic surjective submersion f: M --> T, such that the fibers o…
Let M be a close complex manifold and TM its holomorphic tangent bundle. We prove that if the global holomorphic sections of tangent bundle generate each fibre, then M is a complex homogeneous manifold. Our proof depends on the complex version of Chow-Rashevskii theorem in Carnot-Caratheodory spaces.
Logarithmic connections on complex manifolds with trivial tangent bundle.
problem Finding logarithmic connections on complex manifolds with specific properties.
method Analyzing holomorphic Cartan geometries and their connections.
result Logarithmic connections preserve holomorphic Cartan geometries.
Study splitting submanifolds in specific homogeneous spaces.
problem Classify splitting submanifolds in rational homogeneous spaces of Picard number one.
method Use global holomorphic vector fields and projection maps to analyze submanifolds.
result Proves submanifolds in certain spaces are rational or Hermitian symmetric.
Constructs equivariant embeddings of Hermitian symmetric spaces into tangent spaces.
problem Embedding Hermitian symmetric spaces into their tangent spaces.
method Using polarity of the K-action to construct equivariant embeddings.
result Characterizes holomorphic/symplectic embeddings and realizes submanifolds.
Classifies 3D F-manifolds with or without Euler fields.
problem Local classification of 3D F-manifolds.
method Integrability condition on multiplication in holomorphic tangent bundle.
result Local classification of 3D F-manifolds.
No non-constant holomorphic maps between certain complex manifolds with specific properties.
problem Existence of non-constant holomorphic maps between complex manifolds.
method Analyzing properties of tangent and cotangent bundles, pseudo-effectiveness, and nefness.
result Holomorphic maps between certain complex manifolds are constant.
Vanishing theorems show holomorphic tensor fields on certain Kähler manifolds are trivial.
problem Understanding properties of holomorphic tensor fields on Kähler manifolds.
method Established vanishing theorems for uniformly rational connected (RC) k-positive Hermitian holomorphic vector bundles. result Holomorphic tangent bundles of Kähler manifolds with positive k-Ricci curvature are uniformly RC k-positive. Study of holomorphic Lie algebroids and their geometric properties.
problem Understanding the geometry of holomorphic Lie algebroids and their connections.
method Introducing and analyzing holomorphic Lie algebroids, studying their tangent and prolongation structures, and examining induced Lagrangian structures.
result Developed a comprehensive understanding of holomorphic Lie algebroids and their geometric properties.
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.
Two rigidity results for Legendrian singularities in complex-analytic category.
problem Understanding singularities of Legendrian subvarieties in contact manifolds.
method Using the relation between infinitesimal contactomorphisms and holomorphic sections of the natural line bundle.
result Normal Legendrian singularities are deformation-rigid.
The authors give a complete classification of projective threefolds admitting a holomorphic conformal structure. A Corollary is the complete list of projective threefolds, whose tangent bundle is a symmetric square.
Study on non-Kähler Calabi-Yau manifolds and their geometric structures.
problem Holomorphic geometric structures on non-Kähler compact complex manifolds.
method Beauville-Bogomolov decomposition and weak Bochner principle.
result Rigidity of Vaisman Calabi-Yau manifolds implies they are Kodaira manifolds.
We classify compact Kähler manifolds with semi-positive holomorphic bisectional and big tangent bundles. We also classify compact complex surfaces with semi-positive tangent bundles and compact complex 3-folds of the form P(T∗X) whose tangent bundles are nef. Moreover, we show that if X is a Fano manifold such t…
Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.
problem Understanding foliations with numerically flat tangent bundles on Kähler manifolds.
method Analyzing the structure of foliations on compact Kähler manifolds, extending earlier results.
result Smooth foliations with numerically flat tangent bundles induce a decomposition of the ambient manifold's tangent bundle.
Flat holomorphic connections on stable bundles over LVMB manifolds are always flat.
problem Characterizing LVMB manifolds and their holomorphic connections.
method Analyzing LVMB manifolds and their tangent bundles, deducing properties of holomorphic connections.
result Holomorphic connections on semi-stable bundles over LVMB manifolds are always flat.
The paper extends tangent space theory for diffeological spaces and bundles.
problem Understanding tangent spaces of diffeological bundles and spaces.
method Introducing weakly filtered and filtered diffeological spaces, extending exact sequences, and defining Hector's tangent bundle.
result Tangent bundles of filtered diffeological spaces are diffeological vector spaces.
Formula derived for holomorphic Poisson blow-ups.
problem Invariance of Koszul-Brylinski homology under Poisson blow-ups.
method Blow-up formula derivation for holomorphic Koszul-Brylinski homologies.
result Invariance of E1-degeneracy of Dolbeault-Koszul-Brylinski spectral sequence. Proves uniqueness of tangent cones for specific types of connections.
problem Uniqueness of tangent cones for Hermitian Yang-Mills connections with isolated singularities.
method Simple direct proof using μ-polystable holomorphic bundles over P^n-1.
result Uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections.
Classifies tangent cones of Kähler metrics with cone singularities.
problem Understanding the behavior of Kähler metrics near singular points.
method Constructs and classifies tangent cones for non-collapsed sequences of Kähler-Einstein metrics.
result Classifies possible tangent cones in two complex dimensions.
Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.
problem Finding upper bounds for dimensions of subspaces where holomorphic sectional curvature vanishes.
method Connection with D'Angelo's work on complex subvarieties of real algebraic varieties and decomposition of polynomials into differences of squares.
result An upper bound for the dimensions of these subspaces is found.
We prove that a compact Hermitian manifold with semi-positive but not identically zero holomorphic sectional curvature has Kodaira dimension −∞. As applications, we show that Kodaira surfaces and hyperelliptic surfaces can not admit Hermitian metrics with semi-positive holomorphic sectional curvature although th…
The study explores extensions of Kähler manifolds and their properties.
problem Analyzing extensions of Kähler manifolds and their properties.
method Formulating conditions for affine bundles and studying their properties.
result Established a relation to adapted complex structures and good complexifications.
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.
The paper proves rational connectedness for certain Kähler manifolds.
problem Rational connectedness of compact Kähler manifolds.
method Uniform weak RC-positivity of the tangent bundle.
result Compact Kähler manifolds with uniformly weakly RC-positive tangent bundles are projective and rationally connected.
In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tange…
Paper develops methods for k-semisprays and nonlinear connections in k-tangent bundles.
problem Developing methods for k-semisprays and nonlinear connections in k-tangent bundles. method Starting from a given nonlinear connection, the paper presents a method to obtain sequences of k-semisprays and nonlinear connections on the k-tangent bundle TkM. result Interesting particular cases of Lagrange and Finsler spaces of order k are explored. Study on Klt varieties with trivial canonical class, focusing on holonomy and stability.
problem Holonomy group of singular Kähler-Einstein metrics on klt varieties with numerically trivial canonical divisor.
method Investigation of holonomy group properties, finiteness of connected components, Bochner principle for holomorphic tensors, and connections between irreducibility of holonomy representations and stability of the tangent sheaf.
result Refinement of known decompositions for tangent sheaves of varieties with trivial canonical divisor, showing that up to finite quasi-étale covers, varieties with strongly stable tangent sheaves are either Calabi-Yau or irreducible holomorphic symplectic.
Study leafwise holomorphic automorphisms of Reeb components.
problem Characterize automorphisms of Reeb components of leafwise complex foliations.
method Analyze Hopf construction and boundary holonomy properties.
result Determine structure of leafwise holomorphic automorphisms.
Investigates conditions for spectral sequence degeneracy in holomorphic Poisson structures.
problem Conditions for spectral sequence degeneracy in holomorphic Poisson structures.
method Uses Lie bi-algebroids, generalized complex structures, and hypercohomology of bi-complexes.
result Investigates conditions for spectral sequence degeneracy on the first page.
Holomorphic Koszul-Brylinski homology studied via Dolbeault cohomology.
problem Investigating holomorphic Koszul-Brylinski homology on Poisson manifolds.
method Using Dolbeault cohomology to derive properties of holomorphic Koszul-Brylinski homology.
result Obtained the Leray-Hirsch theorem, Mayer-Vietoris sequence, and Künneth theorem for holomorphic Koszul-Brylinski homology.
Triangle comparison for Kaehler manifolds with curvature bounds.
problem Understanding curvature bounds in Kaehler manifolds and their limits.
method Analog of triangle comparison for Kaehler manifolds with holomorphic bisectional curvature.
result Curvature bounds pass to noncollapsed Gromov-Hausdorff limits.
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 prove that tangent cones to 2-dimensional calibrated cycles are unique. Using this result we prove a rate of convergence for the mass of the blow-up of a calibrated integral 2-cycle towards the limiting density. With the same techniques, we can also prove such a rate for J-holomorphic maps between almost complex man…
Authors find a Hodge-type decomposition for holomorphic Poisson cohomology on nilmanifolds.
problem Investigating conditions for spectral sequence degeneration in holomorphic Poisson cohomology.
method Analyzing spectral sequences associated with bi-complexes on nilmanifolds.
result A Hodge-type decomposition of holomorphic Poisson cohomology is established for a specific class of structures.
Let G be a connected complex Lie group and Γ⊂G a cocompact lattice. Let H be a complex Lie group. We prove that a holomorphic principal H-bundle EH over G/Γ admits a holomorphic connection if and only if EH is invariant. If G is simply connected, we show that a holomorphic principal H-bundle …
The paper studies complex Finsler metrics on complex Lie groups.
problem Characterizing properties of left invariant complex Finsler metrics on complex Lie groups.
method Using invariant frames, the paper proves properties of the metric and its spray.
result The strongly Kähler, Kähler, and weakly Kähler properties are equivalent for the metric.
The subject for investigation in this note is concerned with holomorphic Poisson structures on nilmanifolds with abelian complex structures. As a basic fact, we establish that on such manifolds, the Dolbeault cohomology with coefficients in holomorphic polyvector fields is isomorphic to the cohomology of invariant form…
We classify the holomorphic structures of the tangent vertical bundle T of the twistor fibration of a quaternionic manifold (M,Q) of dimension bigger than four. In particular, we show that any self-dual quaternionic connection on (M, Q) induces an holomorphic structure on T. We prove that the positive tensor powers of …
Study holomorphic affine connections on non-Kähler manifolds.
problem Investigate geometric structures on non-Kähler compact complex manifolds.
method Prove properties of holomorphic affine connections on Calabi-Yau manifolds and compact complex manifolds of algebraic dimension one.
result Holomorphic affine structures on Calabi-Yau manifolds with polystable tangent bundles are locally homogeneous.
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 study proves a central limit theorem for Gaussian holomorphic sections on Kähler manifolds.
problem Understanding statistical properties of zeros of random holomorphic sections.
method Proves a central limit theorem for smooth linear statistics of zero divisors of Gaussian sections in line bundles over Kähler manifolds.
result Derives first-order asymptotics and upper decay estimates for Bergman kernels.