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 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 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.
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 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. 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.
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…
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 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.
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.
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
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.