Equality in Miyaoka-Yau inequality implies uniformization of Klt pairs.
problem Understanding uniformization of Klt pairs under equality in Miyaoka-Yau inequality.
method Analyzing Kähler klt pairs with specific conditions and using orbifold Miyaoka-Yau inequality.
result Orbifold universal cover is either the unit ball or affine space.
The paper classifies minimal projective varieties satisfying a specific equality.
problem Classifying minimal projective varieties with a specific equality.
method Established a structure theorem for minimal projective klt varieties satisfying Miyaoka's equality.
result Minimal projective klt varieties with Miyaoka's equality have semi-ample canonical divisors and specific Kodaira dimensions.
Uniformizes klt pairs using bounded symmetric domains.
problem Characterizing klt pairs uniformizable by bounded symmetric domains.
method Determines conditions for uniformization using Miyaoka-Yau-type inequalities.
result Characterizations of orbifold quotients of polydisc and classical bounded symmetric domains.
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 \( \chi \geq 3|σ| \).
Study Miyaoka-Yau inequality for certain projective manifolds.
problem Proving Miyaoka-Yau inequality for specific types of manifolds.
method Using recent work by K.~Zhang and delta-invariant introduced by Fujita and Odaka.
result Established Miyaoka-Yau type inequality for projective manifolds with nef anti-canonical line bundle.
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. Uniformizes varieties with log-canonical singularities using ball quotients.
problem Uniformizing complex projective varieties with log-canonical singularities.
method Criteria based on Miyaoka-Yau inequality and log-resolutions of singularities.
result Criteria for isomorphism to Baily-Borel-Mok compactifications.
Miyaoka-Yau inequality proven for smooth minimal models.
problem Proving the Miyaoka-Yau inequality for smooth minimal models.
method Existence of cscK metrics in a neighborhood of the canonical class.
result Miyaoka-Yau inequality holds for compact Kähler manifolds with nef canonical bundle.
In this short note, we prove the Miyaoka-Yau inequality for minimal projective n-manifolds of general type by using Kähler-Ricci flow.
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 prove the Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle. The key point of the proof is the estimate for the L2-norm of the scalar curvature along the Kähler-Ricci flow.
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.
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.
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.
After establishing suitable notions of stability and Chern classes for singular pairs, we use Kähler-Einstein metrics with conical and cuspidal singularities to prove the slope semistability of orbifold tangent sheaves of minimal log-canonical pairs of log general type. We then proceed to prove the Miyaoka-Yau inequali…
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…
On a compact Kähler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a Kähler metric with semi-negative holomorphic sectional curvature. We prove that a compact Kähler manifold of almost nonpositive holomorphic sectional…
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.
Paper extends Hodge correspondence to singular Kähler spaces.
problem Establishing Hodge correspondence over Kähler spaces with singularities.
method Using equivalence of polystable Higgs bundles and semi-simple flat bundles over regular loci, and descent theorem for semistable Higgs bundles.
result Non-abelian Hodge correspondence established over compact Kähler klt spaces and their regular loci.
Using the new diffeomorphism invariants of Seiberg and Witten, a uniqueness theorem is proved for Einstein metrics on compact quotients of irreducible 4-dimensional symmetric spaces of non-compact type. The proof also yields a Riemannian version of the Miyaoka-Yau inequality.
In this paper, we show that the existence of Sasakian-Einstein metrics is closely related to the properness of corresponding energy functionals. Under the condition that admitting no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of Sasakian-Einstein metric implies a Moser-Trudinger type i…
The paper establishes inequalities for Chern classes and numbers on polarized manifolds and nef vector bundles.
problem Chern class and number inequalities on polarized manifolds and nef vector bundles.
method Sharp inequalities derived from polarized pairs and nef vector bundles.
result Bounding Chern numbers of nef vector bundles and classifying compact Kähler manifolds.
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-…
Compact Dupin hypersurfaces without constant Lie curvatures found.
problem Finding compact Dupin hypersurfaces with non-constant Lie curvatures.
method Two constructions of compact proper Dupin hypersurfaces in Sn. result Examples of compact proper Dupin hypersurfaces without constant Lie curvatures.
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…
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.
On a compact complex manifold we study the behaviour of strong Kähler with torsion (strong KT) structures under small deformations of the complex structure and the problem of extension of a strong KT metric. In this context we obtain the analogous result of Miyaoka extension theorem. Studying the blow-up of a strong KT…
The Wu-Yau theorem is proven for Sasakian manifolds with specific curvature conditions.
problem Proving properties of Sasakian manifolds with negative transverse holomorphic sectional curvature.
method Analyzing the curvature properties and applying the Wu-Yau theorem.
result Compact Sasakian manifolds with negative transverse holomorphic sectional curvature have negative transverse Ricci curvature.
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.
We introduce a vector bundle version of the complex Monge-Ampere equation motivated by a desire to study stability conditions involving higher Chern forms. We then restrict ourselves to complex surfaces, provide a moment map interpretation of it, and define a positivity condition (MA positivity) which is necessary for …
The study constructs new minimal surfaces with more ramified values than previously known.
problem Understanding minimal surfaces with finite total curvature and specific ramification properties.
method Systematic construction of meromorphic functions on punctured spheres.
result New minimal surfaces with νg=2.5 and Dg=1 on the four-punctured sphere. We partially confirm a conjecture of Donaldson relating the greatest Ricci lower bound R(X) to the existence of conical Kahler-Einstein metrics on a Fano manifold X. In particular, if D∈∣−KX∣ is a smooth simple divisor and the Mabuchi K-energy is bounded below, then there exists a unique conical Kahler-Eins…
Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character χof P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle …
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.
Let M be a closed symplectic manifold of volume V. We say that the symplectic packings of M by ellipsoids are unobstructed if any collection of disjoint symplectic ellipsoids (possibly of different sizes) of total volume less than V admits a symplectic embedding to M. We show that the symplectic packings by ellipsoids …
In a previous paper, we proved that a projective Kähler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollár-Miyaoka-Mori that a projective Kähler manifold of positive Ricci curvature is rationally connected. In the present work, we …
For projective varieties with definite first Chern class we have one type of canonical metric which is called Kähler-Einstein metric. But for varieties with an intermidiate Kodaira dimension we can have several different types of canonical metrics. In this paper we introduce a new notion of canonical metric for varieti…
First we confirm a conjecture asserting that any compact Kähler manifold N with $\Ric^\perp>0$ must be simply-connected by applying a new viscosity consideration to Whitney's comass of (p,0)-forms. Secondly we prove the projectivity and the rational connectedness of a Kähler manifold of complex dimension n under…
In \cite{AP3, AHP}, the first author and his collaborators constructed the irreducible symplectic 4-manifolds that are homeomorphic but not diffeomorphic to (2n−1)CP2#(2n−1)CP2 for each integer n≥25, and the families of simply connected irreducible nonspin symplectic 4…
Solves equality case in isoperimetric inequality for non-convex domains.
problem Equality case in relative isoperimetric inequality outside convex sets.
method Analyzes non-convex domains to settle the equality case.
result Solves the equality case for relative isoperimetric inequality outside arbitrary convex sets.
Develops spherical density-equalizing maps for closed surfaces.
problem Lack of methods for genus-0 closed surfaces.
method Conformal parameterization onto unit sphere, density equalization, quasi-conformal theory, harmonic energy, landmark constraints.
result Landmark-aligned spherical density-equalizing maps balancing different distortion measures.
Let S be a smooth projective variety and Δ a simple normal crossing Q-divisor with coefficients in (0,1]. For any ample Q-line bundle L over S, we denote by E(L) the extension sheaf of the orbifold tangent sheaf TS(−log(Δ)) by the structure sheaf OS with the …
New partition designs reduce star discrepancy in high-dimensional sampling.
problem Improving the expected star discrepancy in high-dimensional sampling.
method Developed non-equal volume partitions to achieve lower expected star discrepancy.
result Explicit upper bounds for expected star discrepancy under non-equal volume partitions.
FPGA-based multi-layer equalizer adapts to changing channels.
problem Real-time adaptation to time-varying channel impairments.
method Multi-layer machine learning on FPGA with on-chip gradient backpropagation training.
result Real-time adaptation to changing channel conditions achieved.
Sharp inequalities proved for RCD spaces, showing equality conditions.
problem Proving sharp inequalities for RCD spaces and identifying equality conditions.
method Analyzing RCD(1,∞) and RCD(K,∞) spaces to prove inequalities and identify equality conditions. result Equality conditions for Buser's and Cheeger's inequalities in RCD spaces.
The paper proposes a method to measure fairness through equality of effort using algorithmic recourse.
problem Measuring fairness through equality of effort in automated systems.
method Applying algorithmic recourse to quantify equality of effort, overcoming previous limitations.
result An algorithm for assessing equality of effort has been developed and validated.
Schwarz lemma extended to equality cases and curvature on manifolds.
problem Extending Schwarz lemma to equality cases and studying curvature.
method Analyzing Schwarz lemma inequalities and equalities, studying holomorphic sectional curvature.
result Holomorphic maps are totally geodesic and have constant rank when Schwarz lemma equality holds.
Classifies weakly Einstein submanifolds in space forms satisfying specific equalities.
problem Characterizing submanifolds in space forms with certain geometric properties.
method Classification based on Chen's equality and semisymmetric conditions.
result Classification of weakly Einstein submanifolds in space forms.