Characterizes stable sheaves for equality in orbifold BG inequality.
problem Stability of sheaves on compact Kähler varieties with klt singularities.
method Characterization of stable reflexive sheaves for BG equality.
result Characterizes stable reflexive sheaves for equality in BG inequality.
Analyzes structure of log smooth pairs when equality holds in Bogomolov-Gieseker inequality.
problem Analyzing log smooth pairs under equality in Bogomolov-Gieseker inequality.
method Examines structure when equality holds in the Bogomolov-Gieseker inequality for semistable logarithmic tangent bundle and canonical extension sheaf.
result Provides insights into the structure of log smooth pairs under specific conditions.
The study proves a key inequality for specific types of three-dimensional spaces.
problem Establishing a mathematical inequality for a specific class of three-dimensional spaces.
method Developed the orbifold version of the Bogomolov-Gieseker inequality for stable Q-sheaves on log terminal Kähler threefolds.
result Proved the Bogomolov-Gieseker inequality for log terminal Kähler threefolds.
The paper proves a key inequality for Higgs sheaves on complex manifolds.
problem Understanding the stability of Higgs sheaves on complex manifolds.
method Proves the existence of an approximate Hermitian-Einstein structure for semistable Higgs sheaves.
result Establishes a Bogomolov type inequality for semi-stable Higgs sheaves.
Using gauge theory for Spin(7)-manifolds of dimension 8, we develop a procedure, called Spin-rotation, which transforms a (stable) holomorphic structure on a vector bundle over a complex torus of dimension 4 into a new holomorphic structure over a different complex torus. We show non-trivial examples of this procedure …
The paper proves a theorem and characterizes connections over normal varieties.
problem The study addresses the stability and connections over normal varieties.
method The authors prove a complete version of the Donaldson-Uhlenbeck-Yau theorem and use it to show the polystability of reflexive sheaves.
result An admissible Hermitian-Yang-Mills connection defines a polystable reflexive sheaf and gives a lower bound for discriminants.
The aim of this paper is to consider a possible extension of the Bogomolov--Miyaoka--Yau inequality to differentiable orbifolds. The conjectured extension is related to the Montgomery--Yang problem about circle actions on the 5--sphere and also to the H--cobordism of Seifert fibered 3--manifolds. Related conjectures on…
The article proves a complex analytic inequality for stable Q-sheaves on Kähler varieties.
problem Proving a Bogomolov-Gieseker inequality for stable Q-sheaves on Kähler varieties.
method Complex analytic approach, including a new purely analytical proof and novel interpretation of orbifold Chern classes.
result Characterization of the equality case in the Bogomolov-Gieseker inequality and novel interpretation of the second orbifold Chern class.
The study constructs geometrically decomposable aspherical 4-manifolds with non-zero signature and explores their properties.
problem Characterizing geometrically decomposable aspherical 4-manifolds with non-zero signature.
method Constructing examples and proving inequalities for geometrically decomposable aspherical 4-manifolds.
result All geometrically decomposable aspherical 4-manifolds with non-zero signature satisfy the inequality \( \chi \geq 3|σ| \).
The article characterizes complex torus quotients with numerical conditions.
problem Characterizing quotients of complex tori by finite groups.
method Numerical vanishing condition on Chern classes, Bogomolov-Gieseker inequality for singular spaces.
result Generalization of previous results in projective and three-dimensional settings.
Solved a conjecture about rational homology projective planes with quotient singularities.
problem A conjecture about rational homology projective planes with quotient singularities.
method Combining Donaldson's diagonalization theorem with a distinguished spin^c structure on the smooth locus.
result Proved that rational homology projective planes with quotient singularities have at most three singular points.
The paper establishes a Miyaoka-Yau type inequality for hyperplane arrangements in complex projective space.
problem Finding a lower bound for the total sum of multiplicities of codimension 2 intersection subspaces of a hyperplane arrangement.
method Using a quadratic form defined by the intersection poset of the hyperplane arrangement, and applying the Bogomolov-Gieseker inequality for parabolic bundles.
result The inequality Q(a,…,a)≤0 gives a lower bound for the total sum of multiplicities of codimension 2 intersection subspaces of the hyperplane arrangement, with equality conditions provided. We prove that if a Calabi--Yau manifold M admits a holomorphic Cartan geometry, then M is covered by a complex torus. This is done by establishing the Bogomolov inequality for semistable sheaves on compact Kähler manifolds. We also classify all holomorphic Cartan geometries on rationally connected complex projectiv…
The paper proves inequalities for orbifold second Chern classes in Fujiki's class.
problem Inequalities for orbifold second Chern classes of compact normal analytic varieties.
method Generic nefness theorems for tangent and cotangent sheaves, and an orbifold Bogomolov--Gieseker inequality for mixed polarizations.
result Semipositivity of the orbifold second Chern class for varieties with nef anti-canonical divisor.
Let M be a compact irreducible hyperkahler manifold, from Bogomolov inequality [V1] we obtain forbidden values of the second Betti number b2 in arbitrary dimension. UPD: Unfortunately, decomposition of dual to BBF-form is not right in the main theorem. Instead of this work, take a look on recent preprints of Sawon…
New unoriented versions of Schur and Bogomolov multipliers for finite groups.
problem Defining and analyzing unoriented versions of Schur and Bogomolov multipliers.
method Using cohomology groups and quotient groups to define unoriented multipliers.
result Triviality of unoriented Bogomolov multiplier for certain groups, nontriviality for others.
The paper proves a key inequality for a specific type of complex spaces.
problem Establishing a mathematical inequality for a class of complex spaces.
method Analytical approach involving Higgs sheaves and orbifolds.
result Proves the Miyaoka-Yau inequality for minimal Kähler klt spaces.
Establishes Kobayashi-Hitchin correspondence for nef and big classes.
problem Analyzing stability and positivity in algebraic geometry.
method Introducing adapted currents and metrics to establish correspondence.
result Equality cases of Bogomolov-Gieseker and Miyaoka-Yau inequalities.
The paper proves a Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles.
problem Proving a Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles.
method Analyzing degenerating families of projective normal varieties and studying the limiting behavior of semistable bundles.
result Improves several previously known algebro-geometric results on normalized tautological classes and proves a new version of the singular Donaldson-Uhlenbeck-Yau theorem.
Paper introduces a new type of canonical metric for varieties with intermediate Kodaira dimension.
problem Finding canonical metrics for varieties with intermediate Kodaira dimension.
method Introducing a new notion of canonical metric and proving conditions for existence of relative Kähler-Einstein metric.
result Established conditions for the existence of relative Kähler-Einstein metric.
Classify projective subvarieties in Bogomolov-Guan manifolds using quasi-diagonals.
problem Classify projective subvarieties in non-Kahler holomorphically symplectic manifolds.
method Use quasi-diagonals to classify projective subvarieties.
result Prove that any projective subvariety belongs to a fiber of the Lagrangian fibration.
The Miyaoka-Yau inequality is proven for certain singular varieties with big canonical or anticanonical divisors.
problem Establishing the Miyaoka-Yau inequality for singular varieties with specific divisors.
method Defining the non-pluripolar product and establishing the Bogomolov-Gieseker type inequality for Higgs sheaves; investigating second Chern class inequalities.
result Proven the Miyaoka-Yau inequality for projective klt varieties with big canonical or anticanonical divisors.
In this paper we construct a family of symplectic 4--manifolds with positive signature for any given fundamental group G that approaches the BMY line. The family is used to show that one cannot hope to do better than than the BMY inequality in finding a lower bound for the function f=χ+bσ on the class of all minima…
Proof of complex geometry theorem for specific singular spaces.
problem Proving a complex geometry theorem for a specific type of singular spaces.
method Self-contained proof of singular Beauville-Bogomolov decomposition theorem.
result Proof of singular Beauville-Bogomolov decomposition theorem for compact Kähler varieties with log terminal singularities and zero first Chern class.
This research proves a topological inequality for symplectic four-manifolds using non-Abelian monopoles.
problem Proving the Bogomolov-Miyaoka-Yau inequality for symplectic four-manifolds.
method Using Morse theory on the moduli space of non-Abelian monopoles, focusing on the square of the L2 norm of coupled spinors. result Existence of a projectively anti-self-dual connection on a rank-two Hermitian vector bundle over a blow-up of the four-manifold.
Neural networks approximate Calabi-Yau metrics and curvature.
problem Finding Ricci-flat metrics for Calabi-Yau manifolds.
method Use neural networks to approximate metrics within a Kähler class.
result Neural networks can approximate topological characteristics of Calabi-Yau manifolds.
Develops virtual Morse-Bott indices for four-manifolds, proving inequalities.
problem Proving inequalities for four-manifolds of Seiberg-Witten simple type.
method Uses virtual Morse-Bott indices and Hirzebruch-Riemann-Roch Theorem.
result Proves positivity of virtual Morse-Bott indices, leading to inequalities.
Counterexample disproves log canonical Beauville--Bogomolov decomposition.
problem Disproving the log canonical Beauville--Bogomolov decomposition.
method Constructing a specific log canonical, K-trivial variety with non-birational fibers.
result Provides a counterexample to the Beauville--Bogomolov decomposition in the log canonical setting.
We establish the correspondence between tame harmonic bundles and μL-stable parabolic Higgs bundles with trivial characteristic numbers. We also show the Bogomolov-Gieseker type inequality for μL-stable parabolic Higgs bundles. Then we show that any local system on a smooth quasi projective variety can be deforme…
Study on stable vector bundles over Gauduchon manifolds.
problem Existence and stability of vector bundles over Gauduchon manifolds.
method Uhlenbeck--Yau's continuity method for approximate Hermitian--Einstein structures.
result Equivalence of semi-stability and existence of Hermitian--Einstein structures.
Introduces a new equation for complex surfaces, proving stability and inequalities.
problem Stability conditions involving higher Chern forms on complex surfaces.
method Vector bundle version of the complex Monge-Ampere equation, positivity condition (MA positivity), stability and inequalities.
result Proves stability and a Kobayashi-Lubke-Bogomolov-Miyaoka-Yau type inequality for positively curved solutions.
Constructs 4-manifolds with positive Euler characteristic proving a conjecture.
problem Finding aspherical 4-manifolds with positive Euler characteristic.
method Explicit construction of 4-manifolds with specified properties.
result Proves a conjecture by constructing a 4-manifold with Euler characteristic 1.
Proves finiteness of Lagrangian fibrations with specific invariants.
problem Finiteness of hyperkaehler Lagrangian fibrations with fixed invariants.
method Proves finiteness of fibrations with fixed Fujiki constant, discriminant, and line bundle degree.
result Finiteness of hyperkaehler Lagrangian fibrations in any fixed dimension.
Paper defines new stability and metrics for complex spaces.
problem Stability and metrics for complex spaces with big cohomology classes.
method Introduces slope stability and Hermitian-Einstein metrics for big cohomology classes.
result Kobayashi Hitchin correspondence and Bogomolov Gieseker inequality proved.
Condition for symmetric 3-differentials on complex surfaces.
problem Conditions for representing symmetric 3-differentials as products of holomorphic forms.
method Two coordinate versions of a necessary and sufficient condition.
result Locally representable symmetric 3-differentials as products of holomorphic forms.
Study non-Kahler symplectic manifolds, proving deformation and Torelli theorems.
problem Topology and deformation theory of non-Kahler holomorphically symplectic manifolds.
method Investigation of topology and deformation theory, proving local Torelli theorem and Fujiki formula.
result Holomorphically symplectic deformations of BG-manifolds are unobstructed, and the period map is locally a diffeomorphism.
On compact Kähler manifolds, we classify regular holomorphic foliations of codimension 1 whose canonical bundle is numerically trivial.
The paper examines symplectic structures and their relation to the partial derivative lemma.
problem Understanding the relationship between complex symplectic structures and the partial derivative lemma.
method Analyzes complex symplectic manifolds and their associated Beauville-Bogomolov-Fujiki quadric.
result The quadric is smooth if and only if h2,0(X)=1 and irreducible if and only if h1,1(X)>0 when the partial derivative lemma holds. We generalize Fujiki relation of Beauville-Bogomolov quadratic form on a projective symplectic variety. As an application, we study a fibre space structure of a projective symplectic variety.
Following a suggestion made by J.-P. Demailly, for each k≥1, we endow, by an induction process, the k-th (anti)tautological line bundle OXk(1) of an arbitrary complex directed manifold (X,V) with a natural smooth hermitian metric. Then, we compute recursively the Chern curvature form for this me…
The paper classifies certain singular projective varieties with specific properties.
problem Classifying projective klt pairs with nef anti-log canonical divisors.
method Establishes a structure theorem using locally trivial rationally connected fibrations.
result Projective klt pairs can be decomposed into rationally connected and Calabi-Yau varieties.
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.
Orbifold uniformization of complex algebraic variety via polystable parabolic Higgs bundle
problem Uniformizing complex algebraic varieties using parabolic Higgs bundles
method Constructing a faithful monodromy representation and a period map
result Identifying orbifold toroidal compactification with canonical orbifold toroidal compactification
We finish the proof of the conjecture of F. Bogomolov and F. Pop: Let F1 and F2 be fields finitely-generated and of transcendence degree ≥2 over k1 and k2, respectively, where k1 is either Qˉ or Fˉp, and k2 is algebraically closed. We denote by $G_{…
The paper shows that certain geometric structures remain unchanged under specific twists.
problem The rational Beauville-Bogomolov-Fujiki lattices of related fibrations are similar.
method Analytic and étale topologies, Hodge structures, and degenerate twistor deformations.
result Isomorphisms of graded vector spaces and Hodge-similar lattices.
Let S be an infinite-dimensional manifold of all symplectic, or hyperkahler, structures on a compact manifold M, and Diff0 the connected component of its diffeomorphism group. The quotient $S/\Diff_0$ is called the Teichmuller space of symplectic (or hyperkahler) structures on M. MBM classes on a hyperkahler manifol…
In this short note, we present a construction of new symplectic 4-manifolds with non-negative signature using the complex surfaces on Bogomolov-Miyaoka-Yau line c12=9χh, the fake projective planes and Cartwright-Steger surfaces. Our construction yields an infinite family of fake rational homology $(2n-1)\CP#(2n-…
Paper constructs new symplectic 4-manifolds, improving previous results.
problem Symplectic 4-manifolds with nonnegative signature and nonspin structure.
method Combining complex surfaces and exotic symplectic manifolds.
result Infinitely many new homeomorphic but non-diffeomorphic symplectic manifolds.