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.
K3 surfaces get a rational curve when a divisor is big and positive enough.
problem Finding rational curves on K3 surfaces with specific conditions.
method Degeneration technique to prove existence of integral nodal rational curves.
result Generic Λ-polarised K3 surface has an integral nodal rational curve in the linear system ∣L∣. Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.
problem Existence of non-trivial holomorphic vector fields on compact complex manifolds.
method Vanishing result for measure preserving holomorphic vector fields, Gibbs stability, and log terminal singularities.
result No non-trivial holomorphic vector fields on compact complex manifolds with big anti-canonical line bundle.
Study confirms no Kähler-Einstein edge metrics for certain asymptotically log Fano varieties.
problem Existence of Kähler-Einstein edge metrics on asymptotically log Fano varieties.
method Analyzes properties of asymptotically log Fano varieties and their metrics.
result No Kähler-Einstein edge metrics exist for specified varieties.
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.
We generalize the maximal time existence of Kähler-Ricci flow in Tian-Zhang and Song-Tian to conical case. Furthermore, if the twisted canonical bundle KM+(1−β)[D] is big or big and nef, we can expect more on the limit behaviors of such conical Kähler-Ricci flow. Moreover, the results still hold for simple normal …
Study finds existence and non-uniqueness of cone spherical metrics on compact Riemann surfaces.
problem Existence and non-uniqueness of cone spherical metrics with prescribed singularities.
method Utilizing polystable extensions of line bundles, the study establishes three primary results concerning these metrics.
result Existence of multiple irreducible and reducible cone spherical metrics for certain effective divisors.
The paper studies positivity properties of cotangent bundles in complex hyperbolic manifolds with cusps.
problem Positivity properties of cotangent bundles in complex hyperbolic manifolds with cusps.
method Analyzes intrinsic positivity properties of cotangent bundles using toroidal compactifications and ample line bundles.
result The cotangent bundle is ample modulo the boundary divisor for sufficiently small rational numbers.
Study on Kähler-Einstein metrics on quasi-projective manifolds.
problem Constructing and analyzing Kähler-Einstein metrics on quasi-projective manifolds.
method Utilizes singular Kähler-Einstein metrics and conic Kähler-Einstein metrics of negative curvature.
result Established the weak convergence of conic Kähler-Einstein metrics to singular Kähler-Einstein metrics.
We generalize Demailly's construction of projective jet bundles and strictly negatively curved pseudometrics on them to the logarithmic case. We establish this logarithmic generalization explicitly via coordinates, just as Noguchi's generalization of the jets used by Green-Griffiths. As a first application, we give a m…
Study of random sections on complex spaces converging to equilibrium metrics.
problem Understanding the behavior of random holomorphic sections on complex spaces.
method Analyzing the convergence of normalized Fubini-Study currents and integration currents to the equilibrium metric's curvature.
result The normalized currents of integration along zero divisors converge almost surely to the curvature current of the equilibrium metric.
Let X be a smooth complex projective variety of dimension d. It is classical that ample line bundles on X satisfy many beautiful geometric, cohomological, and numerical properties that render their behavior particularly tractable. By contrast, examples due to Cutkosky and others have led to the common impression that t…
Study b-divisors on Kähler manifolds linking them to currents.
problem Intersection theory of b-divisors on Kähler manifolds.
method Established correspondence between closed positive currents and nef b-divisors.
result Intersection theory of nef b-divisors answered.
Study contact geometry of symplectic divisors, invariant under specific transformations.
problem Understanding contact structures on symplectic divisors and their boundaries.
method Invariant analysis of contact structures under toric and interior blow-ups/blow-downs, open book decomposition construction.
result Contact structure on divisor boundaries is invariant under specified transformations.
Abstract: Study motion of divisors on curves with topological constraints.
problem Topological restrictions on divisors' motion.
method Analysis of closed motion of simple real divisors on non-singular real algebraic projective curves.
result Found topological restrictions on divisors' motion.
In this paper, we extend the existence and regularity theorems for Kähler-Einstein metrics having conic singularities along a simple normal crossing divisor to the case of normal crossing divisor, i.e. when components of the divisor are allowed to intersect themselves transversely.
The paper extends intersection theory for b-divisors, proving monotonicity and volume inequalities.
problem Intersection theory for b-divisors and monotonicity of intersection products.
method Developed general intersection theory of nef b-divisors, defined restricted volume, proved monotonicity.
result Proved quantitative monotonicity of intersection product and new volume inequalities.
Study shows how the energy of a metric on a toric variety relates to the volume of holomorphic sections.
problem Understanding the volume of holomorphic sections on projective toric varieties.
method Defined energy at equilibrium and showed its asymptotic behavior as a function of the volume of L2-norm unit balls. result The energy of a metric on a toric variety describes the asymptotic behavior of the volume of holomorphic sections.
The paper proves extension theorems for holomorphic sections from divisors.
problem Extension of holomorphic sections from reduced unions of strata of divisors.
method Proves an Ohsawa--Takegoshi type extension theorem.
result Qualitative results on extension from snc divisors and generic global generation of vector bundles.
Holomorphic families yield metrics with explicit curvature formulas.
problem Positivity of direct images with Poincaré type singularities.
method Analyzes holomorphic families and line bundles with Poincaré type singular metrics.
result Explicit formula for curvature of direct image metrics.
Study Lagrangian Floer theory in smooth divisor complements.
problem Intersection Floer theory of Lagrangians in smooth divisor complements.
method Complete construction of Floer homology.
result Floer homology for Lagrangians in smooth divisor complements constructed.
This paper studies Poisson structures defined by divisor ideals.
problem Understanding Poisson structures with degeneracy captured by divisor ideals.
method Developed a framework using divisor ideals and Lie algebroids.
result Effective methods for studying Poisson structures of divisor-type.
Study locates divisors in Hodge bundle with specific properties.
problem Locating effective divisors in the projectivized Hodge bundle.
method Computing the class of closures of loci of canonical divisors with specific conditions.
result Strata of canonical and bicanonical divisors with double zeros span extremal rays of pseudoeffective cones.
The paper provides a combinatorial criterion for realizing tropical pluri-canonical divisors.
problem Determining when a tropical pair corresponds to a smooth algebraic curve with a pluri-canonical divisor.
method Introducing tropical normalized covers and reducing the problem to their realizability.
result Generalizes previous work on tropical canonical divisors and incorporates recent progress on k-differentials. Solves the realizability problem for tropical canonical divisors.
problem Deciding if effective tropical canonical divisors can be realized by smooth curves.
method Using compactifications of strata of abelian differentials and combinatorial conditions.
result Provides a purely combinatorial condition to decide realizability.
Paper proves positivity for differential classes, similar to earlier work on stable curves.
problem Positivity of divisor classes for families of stable differentials.
method Establishes analogous positivity result to Cornalba-Harris for stable differentials.
result Analogous positivity result for divisor classes of stable differentials.
Solves logarithmic ∂-equation on Kähler manifolds with smooth divisors.
problem Closedness of logarithmic forms and injectivity theorems.
method Cyclic covering trick to solve ∂-equation.
result Unobstructed deformations for smooth divisors.
New effective divisors found in moduli spaces from abelian differentials.
problem Computing effective divisors in moduli spaces Mg,n. method Utilizing maps between moduli spaces and the degeneration of abelian differentials.
result Many new classes of effective divisors computed and reproduced known results.
Study shows normal distribution in divisor counts of random sections on complex manifolds.
problem Distribution of divisors on complex manifolds.
method Central limit theorem for smooth linear statistics of Gaussian sections.
result Asymptotic normality of divisor counts.
Study of zero-divisors in sedenions via determinant factorization.
problem Characterizing zero-divisors in the sedenion algebra.
method Factorization of determinant of left multiplication, reduction to quaternionic normal form, block computation.
result Quartic polynomial factorization of determinant, geometric model of zero-divisor locus.
New complete Calabi-Yau metrics found in complex space.
problem Finding metrics on complex spaces with specific conditions.
method Generalized Calabi ansatz, non-archimedean Monge-Ampère equation.
result Complete Calabi-Yau metrics constructed in Fano manifolds.
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.
Floer theory for Lagrangians in open symplectic manifolds with smooth divisors.
problem Floer homology for Lagrangians in open symplectic manifolds with smooth divisors.
method Compactification of moduli space of pseudo-holomorphic discs, using a stronger boundary condition than stable maps.
result Established fundamental properties of the compactification as a topological space.
For homogeneous simply connected Hodge manifolds it is proved that the set of coherent vectors orthogonal to a given one is the divisor responsible for the homogeneous holomorphic line bundle of the coherent vectors. In particular, for naturally reductive spaces, the divisor is the cut locus.
Floer homology constructed for monotone Lagrangians in smooth divisor complements.
problem Computing Floer homology for Lagrangians in smooth divisor complements.
method Compactification of moduli spaces of holomorphic discs and strips, introduction of Kuranishi structures.
result RGW compactifications admit Kuranishi structures, enabling construction of Floer homology.
The study provides criteria for K-stability of Fano varieties using anticanonical Q-divisors.
problem Determining K-stability of Fano varieties.
method Applying Li and the first author's theorem, proposing and proving conditions related to anticanonical Q-divisors.
result Proposed condition sufficient for K-stability of Fano varieties, related to Berman-Gibbs stability.
The paper proves finiteness for stable Lagrangian fibrations with a given divisor.
problem Finiteness of deformation classes of hyperkähler Lagrangian fibrations.
method Survey and proof of finiteness for stable Lagrangian fibrations with a given discriminant divisor.
result Finiteness for stable Lagrangian fibrations with a given discriminant divisor.
The paper classifies and studies symplectic and contact properties of circular spherical divisors.
problem Investigating symplectic and contact topology of circular spherical divisors.
method Classification and analysis of concave circular spherical divisors, including embedding, Stein fillability, and rational homology type determination.
result All concave circular spherical divisors up to toric equivalence are realized as symplectic log Calabi-Yau pairs with minimal complements.
Introduces algorithm for Weinstein handlebodies of certain divisors.
problem Constructing Weinstein handlebodies for complements of smoothed toric divisors.
method Explicit coordinates and simple example for algorithm.
result Produces Weinstein Kirby diagrams for complements of smoothed toric divisors.
Study Weinstein structures on toric divisors' complements.
problem Understanding Weinstein structures on toric divisors' complements.
method Define a partially-centered condition on Delzant polytopes, develop an algorithm for Weinstein handlebody diagrams.
result Explicit Weinstein structures for complements of smoothed toric divisors.
Recently it was shown by H. Guenancia and M. Paun that a singular metric satisfying the conical Kahler-Einstein equation with a simple normal crossing divisor is equivalent to a conical metric along that divisor. In this note, we present an alternative proof of their theorem.
The study finds infinitely many divisors in a specific space of geometric objects.
problem Understanding the effective cone of moduli spaces of abelian differentials.
method Exhibiting extremal effective divisors from abelian differential strata.
result Infinitely many extremal effective divisors discovered in Mg,n. Normal forms and moduli stacks for flat connections on complex manifolds.
problem Understanding singular flat connections on complex manifolds.
method Introducing homogeneous Lie groupoids and studying their representation theory to prove normal form theorems and moduli space structures.
result Moduli spaces of singular flat connections admit the structure of algebraic quotient stacks.
We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a sufficient and necessary criterion, which is simple and also works in higher dimens…
Study projective klt pairs with nef anti-canonical divisor and their properties.
problem Characterize properties of projective klt pairs with nef anti-log canonical divisors.
method Analyze maximally rationally connected fibration and use numerical dimension analogy.
result Numerical dimension of anti-log canonical divisor on X matches that on a general fiber. The flow contracts cone divisors on Kähler surfaces to points.
problem Analyzing the conical Kähler-Ricci flow on Hirzebruch surfaces.
method Using the momentum construction of Calabi, the conical Kähler-Ricci flow is studied on Hirzebruch surfaces.
result The flow either converges to a sphere or a single point, or contracts the cone divisor to a single point.
We give a simple criterion for slope stability of Fano manifolds X along divisors or smooth subvarieties. As an application, we show that X is slope stable along an ample effective divisor D⊂X unless X is isomorphic to a projective space and D is a hyperplane section. We also give counterexamples to Au…
Study Moishezon twistor spaces using quartic hypersurfaces and Del Pezzo fibrations.
problem Classify Moishezon twistor spaces with specific half-anti-canonical systems.
method Utilize pluri-half-anti-canonical maps and quartic hypersurfaces to investigate twistor spaces.
result Complete classification of Moishezon twistor spaces with half-anti-canonical systems as pencils.