Study anti-invariant submersions from holomorphic statistical manifolds.
problem Understanding submersions in statistical manifolds.
method Introduced and analyzed anti-invariant holomorphic statistical submersions.
result Supported results with examples.
Link invariants, for 3-manifolds, are defined in the context of the Rozansky-Witten theory. To each knot in the link one associates a holomorphic bundle over a holomorphic symplectic manifold X. The invariants are evaluated for b_{1}(M) \geq 1 and X Hyper-Kaehler. To obtain invariants of Hyper-Kaehler X one finds that …
The study classifies holomorphic projective connections on complex threefolds.
problem Characterizing holomorphic projective connections on complex threefolds.
method Analyzing properties of holomorphic projective connections on complex projective threefolds.
result Holomorphic projective connections on complex threefolds are either flat or translation invariant on abelian threefolds.
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.
Analytic linearization and holomorphic extensions for proper groupoids.
problem Analytic linearization and holomorphic extensions of proper groupoids.
method Establish analytic linearization around invariant submanifolds and apply to holomorphic extensions.
result Proper groupoids admit holomorphic extensions.
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. Paper constructs an invariant for a specific type of complex manifolds.
problem Analyzing an invariant for a specific class of complex manifolds.
method Using equivariant analytic torsion, the paper constructs an invariant for irreducible holomorphic symplectic manifolds with antisymplectic involution.
result A formula for the complex Hessian of the logarithm of the invariant is provided.
Defines Milnor number for foliations and shows its topological invariance.
problem Defining and proving invariance of Milnor number for non-isolated singularities of holomorphic foliations.
method Defining Milnor number as intersection number of sections; proving invariance via C1 topological equivalences. result Milnor number is invariant under C1 topological equivalences. 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…
Computes colored HOMFLYPT invariants using holomorphic curves.
problem Counting holomorphic curves in Calabi-Yau 3-folds.
method Computes contributions of multiple covers of holomorphic annuli.
result Agrees with topological string theory predictions and proves Ooguri-Vafa formula.
Classifies holomorphic parabolic geometries on complex manifolds.
problem Classifying holomorphic parabolic geometries on complex manifolds.
method Bounding numerical dimension and using geometric invariants.
result Uncovering foliations and fibrations on smooth projective varieties.
The paper explores constant holomorphic d-scalar curvature on specific manifolds.
problem Existence and prescription of constant holomorphic d-scalar curvature.
method Study of closed, connected almost Hermitian manifolds of dimension n≥6. result Obtained an application and variation formula for a conformal invariant.
The paper computes a tau-invariant for holomorphic curves in Stein domains and links.
problem Computing tau-invariant for holomorphic curves in Stein domains.
method Using pseudo-holomorphic curves and Stein fillable contact structures.
result New proof of Thom conjecture and topological obstructions for link types.
The paper studies invariant complex manifolds in holomorphic slow-fast systems.
problem Existence of invariant complex manifolds in holomorphic systems.
method Geometric singular perturbation theory, Fenichel and Briot-Bouquet theories.
result Conditions are provided to guarantee the existence of one-dimensional invariant complex manifolds.
Holomorphic tensors on algebraic cones are invariant under certain group actions.
problem Holomorphic tensors on products of algebraic cones
method Using algebraic structures and embeddings
result Holomorphic tensors are invariant under group actions
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.
New connections on symmetric spaces with invariant properties.
problem Understanding invariant connections on hermitian symmetric spaces.
method Introduced a class of G-invariant connections on homogeneous bundles over hermitian symmetric spaces. result Parameter space of connections is a normal variety with a canonical anti-holomorphic involution.
The paper explores geometric structures on Hom-Lie groups and algebras.
problem Exploring Kähler-Norden structures on Hom-Lie groups and algebras.
method Analyzing the relationship between holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups.
result Left-invariant holomorphic Hom-Lie groups with abelian complex structures are flat.
In [DM] it was asked whether all flat holomorphic Cartan geometries (G,H) on a complex torus are translation invariant. We answer this affimatively under the assumption that the complex Lie group G is affine. More precisely, we show that every holomorphic Cartan geometry of type (G,H), with G a complex affine Lie group…
New deformations of lattice cohomology help calculate knot invariants.
problem Calculating knot invariants using lattice cohomology.
method Using holomorphic triangles counting and lattice cohomology.
result Combinatorial formulae for the upsilon invariant are derived.
Explains complex analytic invariants of vector fields and foliations.
problem Integrating theories of singular varieties and foliations.
method Expository discussion of invariants.
result Introduces connections between complex analytic singular varieties and foliations.
In this paper, the Bando-Futaki invariants on hypersurfaces are derived in terms of the degree of the defining polynomials, the dimension of the underlying projective space, and the given holomorphic vector field. In addition, the holomorphic invariant introduced by Tian and Chen (Ricci Flow on Kähler-Einstein surfaces…
Extends Gromov invariant to Calabi-Yau 3-folds.
problem Counting embedded curves in Calabi-Yau 3-folds.
method Detailed study of bifurcations of moduli spaces of embedded pseudo-holomorphic curves.
result Integer-valued virtual count of embedded curves defined.
The CR δ-invariant for CR-submanifolds was introduced in a recent article [B. Y. Chen, An optimal inequality for CR-warped products in complex space forms involving CR δ-invariant, Internat. J. Math. 23} (2012), no. 3, 1250045 (17 pages)]. In this paper, we prove two new optimal inequalities for anti-holomorphic su…
Holomorphic torsion invariant for log-Enriques surfaces derived from Borcherds products.
problem Holomorphic torsion invariant for log-Enriques surfaces.
method Introduced a holomorphic torsion invariant using Borcherds products.
result The invariant is given by the Petersson norm of an explicit Borcherds product.
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.
A Kodaira fibration is a compact, complex surface admitting a holomorphic submersion onto a complex curve, such that the fibers have nonconstant moduli. We consider Kodaira fibrations X with nontrivial invariant rational cohomology in degree 1, proving that if the dimension of the holomorphic invariants is 1 or 2, then…
The paper studies chambered invariants of real Cauchy-Riemann operators on Riemann surfaces.
problem Counting pseudo-holomorphic curves in symplectic Calabi-Yau 3-folds.
method Constructs three chambered invariants: nBl, n1,2, n2,1, defined by counting solutions to ADHM vortex equations and pseudo-holomorphic sections of bundles. result Conjectures a relationship between n1,2 and n2,1 and symplectic invariants. Study of holomorphic curves and surfaces using singularity theory.
problem Understanding the geometry of holomorphic curves and complex surfaces.
method Application of singularity theory to holomorphic curves and surfaces.
result Definition of geometric invariants for curves and surfaces.
We study holomorphic foliations of codimension k≥1 on a complex manifold X of dimension n+k from the point of view of the exceptional minimal set conjecture. For n≥2 we show in particular that if the holomorphic normal bundle NF is Griffiths positive, then the foliation does not admit a c…
We studied the axiom of anti-invariant 2-spheres and the axiom of co-holomorphic (2n+1)-spheres. We proved that a nearly Kählerian manifold satisfying the axiom of anti-invariant 2-spheres is a space of constant holomorphic sectional curvature. We also showed that an almost Hermitian manifold M of dimension $2m\geq…
Paper generalizes Schwarz lemma for polydisc mappings with specific metrics.
problem Schwarz lemma for holomorphic mappings with invariant metrics.
method Analyzes mappings between polydiscs with $\mbox{Aut}(P_m)$-invariant Kähler-Berwald metrics.
result Generalizes Schwarz lemma for polydisc mappings with specific metrics.
Minimal discs count as knot invariants in hyperbolic 4-space.
problem Counting minimal discs with a given boundary in hyperbolic 4-space.
method Using J-holomorphic curves in twistor space to count minimal discs, considering singularities at infinity.
result The number of minimal discs with a given boundary is a knot invariant.
Holomorphic tensors on Vaisman manifolds are invariant under the Lee field.
problem Characterizing holomorphic tensors on Vaisman manifolds.
method Using the parallelism of the Lee form and properties of the Lee field.
result The Kodaira dimension of Vaisman manifolds is invariant under certain quotients.
We give a complete characterization of invariant integrable complex structures on principal bundles defined over hermitian symmetric spaces, using the Jordan algebraic approach for the curvature computations. In view of possible generalizations, the general setup of invariant holomorphic principal fibre bundles is desc…
In an earlier work, we investigated some consequences of the existence of a Kähler metric of negative holomorphic sectional curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., non-positive) holomorphic sectional curvature. In doing so, we define a new invar…
We study the relation between J-anti-invariant 2-forms and pseudoholomorphic curves in this paper. We show the zero set of a closed J-anti-invariant 2-form on an almost complex 4-manifold supports a J-holomorphic subvariety in the canonical class. This confirms a conjecture of Draghici-Li-Zhang. A higher di…
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.
Local holomorphic maps preserving (p,p) forms are shown to be isometries.
problem Preserving (p,p) forms under holomorphic maps between Kähler manifolds.
method Analyzing local holomorphic maps between Kähler manifolds, proving isometries up to scalars.
result Holomorphic maps preserving (p,p) forms are isometries under certain conditions.
We define a Floer-homology invariant for links in S3, and study its properties.
The paper introduces new metrics on complex domains with specific geometric properties.
problem Developing metrics on complex domains with strong pseudoconvexity and holomorphic invariance.
method Explicit construction of holomorphic invariant strongly pseudoconvex complex Finsler metrics via deformation of Bergman metrics.
result These metrics have bounded curvature properties similar to Bergman metrics.
The paper generalizes Segre and Verlinde numbers for surfaces with holomorphic 2-forms.
problem Generalizing Segre and Verlinde numbers for surfaces with holomorphic 2-forms.
method Using Mochizuki's formula and Seiberg-Witten invariants, derive universal functions and prove topological invariants.
result Certain canonical virtual Segre and Verlinde numbers of general type surfaces are topological invariants.
Let X be a differentiable manifold endowed with a transitive action α:A×X⟶X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects: {enumerate} equ…
Geometric Invariant Theory applied to Kähler manifolds yields analytic models for vector bundles.
problem Constructing local models for vector bundles on Kähler manifolds.
method Applying Geometric Invariant Theory to Kähler manifolds to construct analytic GIT-quotients.
result Existence of Weil-Petersson forms on parameter spaces for stable vector bundles.
The Bergman metric on symmetrized bidisc has negative curvature properties.
problem Holomorphic curvature properties of the Bergman metric on symmetrized bidisc.
method Analysis of holomorphic sectional and bisectional curvatures.
result Negative pinched holomorphic sectional curvature and non-positive holomorphic bisectional curvature.
We describe the generalized Kuranishi spaces of solvmanifolds with left-invariant complex structures. By using such description, we study the stability of left-invariantness of deformed generalized complex structures and smoothness of generalized Kuranishi spaces on certain classes of solvmanifolds. We also give explic…
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…
Global theory of relative invariants and equivariant line bundles established.
problem Global theory of relative invariants and equivariant line bundles.
method Cohomological description of Pic_{\mathfrak{g}}(M) using Chevalley-Eilenberg complex and Čech complex.
result Characterization of polynomial divisors and multipliers of relative differential invariants.