Holomorphic Euler number vanishes for certain Kähler manifolds.
problem Finding obstructions for Kähler manifolds with specific curvature properties.
method Vanishing theorem of Dolbeault-Morse-Novikov cohomology.
result Holomorphic Euler number of Kähler manifolds with almost nonnegative Ricci curvature vanishes.
Complex manifolds can only map to curves, restricting Clemens threefolds and S6.
problem Restricting the mapping properties of complex manifolds and threefolds.
method Analyzing the properties of complex manifolds and their mappings.
result No holomorphic mapping from a specific class of threefolds onto a complex space.
Study Chern number inequalities for negative curvature Kähler manifolds.
problem Chern number inequalities for compact Kähler manifolds with negative sectional curvature.
method Study L2 ∂ˉildeE-harmonic forms on lifting bundle over universal covering space, observe relationship between Laplace-Beltrami eigenvalues and Euler characteristic. result Euler characteristic inequality involving sectional curvature and Laplace-Beltrami eigenvalues.
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.
This article is based on the methods developed in [AGG]. We construct a complex hyperbolic structure on a trivial disc bundle over a closed orientable surface Σ (of genus 2) thus solving a long standing problem in complex hyperbolic geometry (see [Gol1, p. 583] and [Sch, p. 14]). This example answers also [Eli, Open …
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.
Holomorphic connections found on Riemann surfaces with Fuchsian monodromy.
problem Existence of holomorphic connections with Fuchsian monodromy on Riemann surfaces.
method Construction of compact Riemann surfaces with specific holomorphic vector bundles and connections.
result Existence of holomorphic connections with maximal Euler class and Fuchsian monodromy.
In this article we apply the technique of Luttinger surgery to study the complexity of the fundamental group of symplectic 4-manifolds with holomorphic Euler number χh=1. We discuss the topology of symplectic 4-manifolds with b+=1 and provide various constructions of symplectic 4-manifolds with b+=1 and …
The paper extends a conjecture about Euler characteristics of perverse sheaves on certain Kähler manifolds.
problem The conjecture about Euler characteristics of perverse sheaves on compact aspherical Kähler manifolds.
method The method involves expressing the Euler characteristic as an intersection number involving the characteristic cycle, and using curvature conditions to deduce non-negativity. For the second result, the local system is shown to underlie a complex variation of Hodge structures, leading to the desired inequality from curvature properties of the period map.
result The conjecture holds for compact aspherical Kähler manifolds with non-positive holomorphic bisectional curvature and for projective manifolds with a faithful semi-simple rigid local system.
Study quantizes topological numbers on degenerating Einstein manifolds.
problem Quantizing topological numbers on non-collapsed degenerating Einstein manifolds.
method Compactness theory of bubbles, classical vanishing theorems, and Hirzebruch-Riemann-Roch theorems.
result Established quantization results for various topological numbers.
The paper studies geodesic completeness for Lie groups and their metrics.
problem Geodesic completeness of pseudo and holomorphic Riemannian metrics on Lie groups.
method Euler-Arnold formalism, detailed study of geodesics, classification of metrics.
result Full classification of geodesic completeness for the Lie group SL(2, C).
Study complex hyperbolic structures on a disc orbibundle with 5 cone points.
problem Constructing complex hyperbolic structures on a disc orbibundle with vanishing Euler number.
method Analyzing involutions in PU(2,1) and using bending-connectedness. result A 4-dimensional bending-connected family of complex hyperbolic structures on a disc orbibundle.
This paper gives a construction for all minimal immersions f of the Poincaré disc into the complex hyperbolic plane CH2 which are equivariant with respect to an irreducible representation ρ of a hyperbolic surface group into PU(2,1). We exploit the fact that each such immersion is a twisted conformal …
We prove rigidity and vanishing theorems for several holomorphic Euler characteristics on complex contact manifolds admitting holomorphic circle actions preserving the contact structure. Such vanishings are reminiscent of those of LeBrun and Salamon on Fano contact manifolds but under a symmetry assumption instead of a…
Normal forms found for meromorphic connections over a specific F-manifold.
problem Characterizing meromorphic connections over a specific F-manifold.
method Finding normal forms for Euler fields and meromorphic connections.
result Characterized Euler fields induced by (TE)-structures. The paper introduces new functionals and equations for complex vector bundles.
problem Complex vector bundle equations and their solutions.
method Introducing generalized Donaldson's functionals and discussing their Euler-Lagrange equations.
result Uniqueness of solutions to complex vector bundle equations.
Extends Euler class result to symplectic group.
problem Relationship between bounded Euler class and symplectic rotation number.
method Extends Ghys's result to symplectic group.
result Establishes relationship between bounded Euler class and symplectic rotation number.
The paper derives a local formula for the Euler number of circle bundles.
problem Calculating the Euler number of circle bundles over surfaces.
method Derives a local formula for the Euler number using quasisections and singularities.
result The Euler number of the bundle equals the sum of weights of singularities of a quasisection.
The paper studies minimal surfaces in complex hyperbolic spaces, showing moduli space connected components.
problem Understanding the structure of moduli spaces of minimal surfaces in complex hyperbolic spaces.
method Relating the moduli space to nilpotent cones in Higgs bundles, analyzing limit points of actions.
result Connected components of the moduli space of minimal immersions in CH2 are indexed by the Toledo invariant and the Euler number of the normal bundle. A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three only multiples of the top Chern number, which is the Euler characteristic, are invarian…
Study the topological information of map germs using Euler obstruction.
problem Capturing topological information from map germs with complex singular spaces.
method Investigates the Euler obstruction and its relation to local Euler obstruction and Brasselet number.
result Relates Chern number to the number of cusps in a perturbed map-germ.
We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyp…
This paper studies symplectic structures on elliptic surfaces with positive Euler number.
problem Determining symplectic representatives for cohomology classes on elliptic surfaces.
method Analyzes the symplectic cone for elliptic surfaces with positive Euler number.
result Characterizes the symplectic cone for elliptic surfaces with positive Euler number.
In this article we calculate the dimension of the Hilbert space of Kahler quantization of the moduli space of vortices on a Riemann surface. This dimension is given by the holomorphic Euler characteristic of the quantum line bundle.
Develops an L^p theory for Dolbeault-Dirac operators on compact Kähler manifolds.
problem Analyzing Dolbeault-Dirac operators on compact Kähler manifolds with Banach space coefficients.
method Establishes an L^p theory for Dolbeault-Dirac operators, proving bisectoriality, H^\infty functional calculus, and Gaffney-type estimates.
result Identifies the index of the associated Fredholm operator with the holomorphic Euler characteristic, independent of p.
The Euler number of special symplectic hyperbolic manifolds is positive.
problem Understanding the Euler number of symplectic hyperbolic manifolds.
method Study L2-harmonic forms on the universal covering space and prove the Singer conjecture. result The Euler number of a special symplectic manifold satisfies (−1)nχ(X)>0. We define a Floer-homology invariant for knots in an oriented three-manifold, closely related to the holomorphic disk Floer homologies for three-manifolds defined in an earlier paper. We set up basic properties of these invariants, including an Euler characteristic calculation, behaviour under connected sums. Then, we …
The paper investigates the relationship between curvature operator and Euler number on manifolds.
problem Relationship between curvature operator and Euler number on manifolds.
method Analysis based on vanishing theorems for a Dirac operator associated with a smooth 1-form.
result The Euler number of a compact 2m-dimensional manifold with ANCO and nontrivial first de Rham cohomology group vanishes.
Defines new two-variable elliptic genera for manifolds and derives modular forms.
problem Develops new elliptic genera for manifolds.
method Introduces and defines new two-variable elliptic genera for manifolds and derives their properties.
result Derives modular forms from the elliptic genera.
The paper defines new modular forms from almost complex manifolds and derives anomaly cancellation formulas.
problem Anomaly cancellation in almost complex manifolds.
method Defined a generalized elliptic genus and derived SL(2,Z) modular forms.
result Derived anomaly cancellation formulas and divisibility results for holomorphic Euler characteristic.
Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices. It is shown that there are other linear combinations of the numbers of simplices which are combinatorial invariants, but that all such invariants are multiples of the Euler characteristi…
The paper shows how to unknot certain nonorientable surfaces in 4-dimensional spaces.
problem Tackles the unknotting of nonorientable surfaces in 4-spheres and 4-balls.
method Uses topological isotopy and properties of knot groups and normal Euler numbers.
result Proves that certain nonorientable surfaces are topologically unknotted under specific conditions.
The paper establishes a Poisson Poincaré-Dulac theorem for Poisson-flat connections.
problem Analyzing Poisson-flat connections with logarithmic poles.
method Defining an Euler-Poisson principal part and residue theory, establishing a Poisson Poincaré-Dulac theorem.
result Any logarithmic Poisson-flat connection is holomorphically gauge equivalent to a pure Euler-Poisson normal form.
The paper develops a local index formula for complex manifolds with C∗-action.
problem Analyzing the m-index on complex manifolds with C∗-action. method Applying the method of transversal heat kernel asymptotics.
result Obtained a local index formula for the m-index. Researchers found two types of graphs for 6D torus manifolds with Euler number 6.
problem Identifying and constructing 6D almost complex torus manifolds with specific Euler numbers.
method Examined labeled directed graphs associated with fixed points and isotropy spheres, used to construct manifolds and determine Chern numbers.
result Proved the existence of two types of 6D almost complex torus manifolds with Euler number 6.
We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmu…
Study affinely transverse foliations in sphere bundles, finding bounds and vanishing conditions.
problem Understanding affine transverse foliations in sphere bundles and their properties.
method Provided upper bounds for the Euler number and a new proof for vanishing conditions under amenable fundamental group.
result Upper bounds and vanishing conditions for the Euler number of sphere bundles.
First we recall homology groups of prer Lie superalgebras. Then introducing double weighted chain spaces, we deal with pre Lie superalgebra of multi-vector fields with polynomial coefficients on n-dimensional number space. The bracket is Schouten bracket. We have several results about Euler number and Betti numbers of …
The paper proves a criterion for virtual Euler class one in hyperbolic 3-manifolds.
problem Determining the virtual Euler class one in hyperbolic 3-manifolds.
method Analyzing Alexander polynomials and constructing taut foliations.
result Constructing examples of hyperbolic 3-manifolds with virtual Euler class one.
The paper explores how vector fields relate to volume in geometric contexts.
problem Existence of nondiffeomorphic contact forms with identical Reeb vector fields.
method Analyzes geodesible vector fields and their associated Euler classes, applying topological and geometric theorems.
result Proves the Gauss-Bonnet and Poincaré-Hopf theorems for 2D orbifolds using geodesible vector fields.
We use some basic properties of binomial and Stirling numbers to prove that the Euler characteristic is, essentially, the unique numerical topological invariant for compact polyhedra which can be expressed as a linear combination of the numbers of faces of triangulations. We obtain this result converting it into an eig…
In the note, we give a proof, based on the Generalized Thom Conjecture, of Bennequin's Theorem on upper bound for the Euler number of a link which is considered as a closed braid. A lower bound for the Euler number of a link is also given.
In this paper, we construct the moduli space of marked oper structures on a closed, oriented smooth surface of negative Euler characteristic as a holomorphic fiber bundle over Teichmüller space. We prove that the holonomy map from the space of marked oper structures to the moduli space of reductive flat bundles is a ho…
Formula for manifold Euler characteristic using even faces.
problem Calculating Euler characteristic of triangulated manifolds.
method Formula based on even-dimensional faces.
result Universal coefficients for Euler characteristic.
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. Paper defines compact quantum spaces with Kähler structures.
problem Noncommutative Kähler structures on quantum spaces.
method Introduces compact quantum homogeneous Kähler spaces and studies their properties.
result Analytic properties of Dolbeault-Dirac operators and their indices.
Decouples moduli groups in heterotic string theory cohomology.
problem Decomposing moduli groups in heterotic string theory cohomology.
method Computing cohomology groups at the standard embedding and showing their direct sum decomposition.
result Decomposes moduli groups into a direct sum of cohomologies at the standard embedding.
Euler's theorem extended to complex structures.
problem Generalizing Euler's theorem to complex structures.
method Analyzing strongly connected, pure n-dimensional regular CW-complexes. result Evenness of cells is equivalent to generalized cycle decomposition and traversability.