Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomog…
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Paper describes holomorphic polyvector fields on toric varieties.
problem No specific problem stated; general description of fields.
method Explicit description of holomorphic polyvector fields on smooth compact toric varieties.
result Generalizes Demazure's result of holomorphic vector fields on toric varieties.
The study classifies complex smooth Fano varieties with large pseudoindex.
problem Classifying Fano varieties with specific properties.
method Analyzing Fano varieties with large pseudoindex and Picard number greater than one.
result Classification of Fano varieties with pseudoindex at least n-2 and Picard number greater than one.
Classifies holomorphic parabolic geometries on complex manifolds.
problem Classifying holomorphic parabolic geometries on complex manifolds.
method Bounding numerical dimension and using geometric invariants.
result Uncovering foliations and fibrations on smooth projective varieties.
The paper studies gradings on nilpotent Lie algebras linked to smooth algebraic varieties.
problem Understanding gradings on nilpotent Lie algebras associated with algebraic varieties.
method Analyzing lattice structures in nilpotent Lie groups and their fundamental groups.
result Conditions for a lattice to be the fundamental group of a smooth complex algebraic variety.
Extends Kollár's result to fibered Calabi-Yau varieties with cohomological assumption.
problem Deformations of fibered Calabi-Yau varieties under cohomological constraints.
method Hodge theoretic techniques and -lifting criterion of Kawamata-Ran.
result Small deformations of fibered Calabi-Yau varieties remain fibered.
We construct normal rationally connected varieties (of arbitrarily large dimension) not containing any smooth rational curves.
Segre varieties' hyperplane sections are unstable under certain conditions.
problem Stability of hyperplane sections of Segre varieties under different conditions.
method Proving instability with respect to any polarization for non-smooth or cases.
result Normal hyperplane sections of Segre varieties are K-unstable under specified conditions.
The paper defines and proves conditions for numerical semistability of smooth toric varieties.
problem Understanding the numerical semistability of smooth toric varieties.
method Analyzing the Chow/Hurwitz forms and applying toric degenerations.
result A necessary and sufficient condition for a smooth toric variety to be numerically semistable.
The study classifies Fano varieties with specific pseudoindex.
problem Classifying Fano varieties with a particular pseudoindex.
method Classification based on birational contractions and pseudoindex condition.
result Classification of Fano varieties with pseudoindex equal to half of their dimension.
Kähler-Einstein metrics found on special types of symmetric varieties.
problem Finding Kähler-Einstein metrics on smooth Fano symmetric varieties with specific properties.
method Used a combinatorial criterion for K-stability of Fano spherical varieties and computed algebraic moment polytopes and barycenters.
result Proved all smooth Fano symmetric varieties with Picard number one admit Kähler-Einstein metrics.
We prove the Yau-Tian-Donaldson's conjecture for any -Fano variety that has a log smooth resolution of singularities such that the discrepancies of all exceptional divisors are non-positive. In other words, if such a Fano variety is K-polystable, then it admits a Kähler-Einstein metric. This extends the pre…
Solves Yau-Tian-Donaldson conjecture for smooth projective varieties.
problem Uniform K-stability and existence of cscK metrics.
method Special Fujita approximations and regularization of entropy functional.
result Uniformly K-stable polarized smooth projective varieties admit cscK metrics.
Study Higgs bundles on smooth projective varieties and their restrictions to curves.
problem Interplay between Higgs bundles on smooth projective varieties and their restrictions to curves.
method Investigate the restriction map of Higgs bundles and study branes in moduli spaces.
result Interconnectedness of Higgs bundles and branes on smooth projective varieties and their restrictions.
We prove that for any affine variety S defined over Q there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety Hom(G, PO(3))//PO(3). The subset U contains all real points of S . As an application we construct new examples of…
Generalizes Toledo invariant to singular klt varieties, proving Milnor-Wood inequality.
problem Proving Milnor-Wood inequality for singular klt varieties.
method Generalized Toledo invariant to singular klt varieties and proved Milnor-Wood inequality.
result Milnor-Wood inequality for singular klt varieties.
BV operators of the Gerstenhaber algebras of holomorphic polyvector fields on toric varietiesmath.AG
Study BV operators on holomorphic polyvector fields on toric varieties.
problem Existence of BV operators in Gerstenhaber algebras.
method Analyzing BV operators on holomorphic polyvector fields on smooth compact toric varieties.
result Necessary and sufficient condition for BV operators existence.
Geometric structures modeled on smooth projective horospherical varieties of Picard number onemath.AG
Geometric structures modeled on rational homogeneous manifolds are studied to characterize rational homogeneous manifolds and to prove their deformation rigidity. To generalize these characterizations and deformation rigidity results to quasihomogeneous varieties, we first study horospherical varieties and geometric st…
In this paper we construct all smooth torus fibres of the generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit.
Study identifies subvarieties of projective varieties mapping to models.
problem Understanding mappings of subvarieties to models on projective varieties.
method Analyzes smooth projective varieties with holomorphic locally homogeneous structures.
result Determines all subvarieties mapping to the model.
Survey various symmetry notions for toric varieties.
problem Understanding different types of symmetries in toric varieties.
method Exploring algebraic, complex, representation, combinatorial, convex, and geometric stability perspectives.
result Establishes relationships between different symmetry notions.
Given a singular Schubert variety Z in a compact Hermitian symmetric space it is a longstanding question to determine when Z is homologous to a smooth variety Y. We identify those Schubert varieties for which there exist first-order obstructions to the existence of Y. This extends (independent) work of M. Walters, R. B…
We prove a criterion for the existence of harmonic metrics on Higgs bundles that are defined on smooth loci of klt varieties. As one application, we resolve the quasi-etale uniformisation problem for minimal varieties of general type to obtain a complete numerical characterisation of singular quotients of the unit ball…
We prove rigidity of various types of holomorphic parabolic geometry on smooth complex projective varieties.
Schubert varieties are irreducible subvarieties of homogeneous manifold, which are important to understand the geometry of homogeneous manifold G/P and the action of the semisimple Lie group G. Consider the space of effective cycles in G/P with homology class equal to an integral multiple of the homology class of a Sch…
We study the rate of growth of normalized Hodge numbers along a tower of abelian covers of a smooth projective variety with semismall Albanese map. These bounds are in some cases optimal. Moreover, we compute the -Betti numbers of irregular varieties that satisfy the weak generic Nakano vanishing theorem e.g., var…
The dual variety X* for a smooth n-dimensional variety X of the projective space P^N is the set of tangent hyperplanes to X. In the general case, the variety X* is a hypersurface in the dual space (P^N)*. If dim X* < N - 1, then the variety X is called dually degenerate. The authors refine these definitions for a varie…
Constructs non-Kähler Calabi-Yau manifolds with large Betti numbers.
problem Finding non-Kähler Calabi-Yau manifolds with high Betti numbers.
method Smoothing normal crossing varieties to create K3 fibrations over smooth projective varieties.
result Examples of non-Kähler Calabi-Yau manifolds with arbitrarily large 2nd Betti numbers.
Researchers characterize a specific type of projective variety based on its tangents.
problem Characterizing smooth projective horospherical varieties of Picard number one.
method Using methods of W-normal complete step prolongations and Lie algebra cohomology.
result A uniruled projective manifold of Picard number one is biholomorphic to the variety if its tangents match.
In this note we discuss the problem of resolving conically singular cscK varieties to construct smooth cscK manifolds, showing a glueing result for (some) crepant resolutions of cscK varieties with discrete automorphism groups.
Nilmanifolds are shown to be diffeomorphic to trivial bundles over tori.
problem Characterizing diffeomorphisms between nilmanifolds and smooth quasi-projective varieties.
method Analyzing cohomology and diffeomorphism properties of quasi-projective varieties and nilmanifolds.
result Nilmanifolds are diffeomorphic to trivial bundles over tori under certain conditions.
Study of tangent bundle positivity on complex projective varieties.
problem Positivity of the second exterior power of tangent bundles on smooth complex projective varieties.
method Analyzes properties of tangent bundles and uses algebraic geometry techniques.
result Proves that up to a finite cover, the Albanese map is a locally trivial fibration with nef fibers.
Solves open problems on curved projective varieties.
problem Structure theorems for curved projective varieties.
method Supplements and proposes open problems.
result Provides new insights into structure of curved projective varieties.
The study finds Kähler-Einstein metrics on certain Fano varieties of type AIII.
problem Finding Kähler-Einstein metrics on specific Fano varieties.
method Using combinatorial criteria for K-polystability and properties of Fano varieties.
result Proves existence of Kähler-Einstein metrics on for and on for .
Proves tropical Hodge theory for smooth projective varieties, conditional on Laplacian regularity.
problem Proving log-concavity of characteristic polynomials of matroids.
method Combinatorial approach, conditional proof of Kähler package.
result Conditional proof of Kähler package for tropical cohomology.
The paper studies foliations on smooth projective varieties and their properties.
problem Characterizing and understanding foliations on smooth projective varieties.
method Develops a structure theorem for smooth projective varieties with almost nef regular foliations, using a smooth morphism and MRC fibration.
result An almost nef regular foliation on a smooth projective variety can be decomposed into a numerically flat regular foliation and a smooth morphism.
Classifies Fano varieties with large pseudoindex and non-free rational curves.
problem Classifying Fano varieties with specific properties.
method Extremal contractions and classification of varieties.
result Complete classification of Fano -folds with pseudoindex at least and Picard number greater than one.
Study on transnormal functions and their level sets on Finsler manifolds.
problem Understanding transnormal functions and their geometric properties on Finsler manifolds.
method Proving smoothness of focal varieties and regular level sets of transnormal functions.
result Focal varieties of a C2 transnormal function are smooth submanifolds and regular level sets are tubes over these varieties.
We investigate a method of construction of Calabi--Yau manifolds, that is, by smoothing normal crossing varieties. We develop some theories for calculating the Picard groups of the Calabi--Yau manifolds obtained in this method. Some applications are included, such as construction of new examples of Calabi--Yau 3-folds …
New examples show not all homology fiber bundles are topological.
problem Disprove conjecture about homology fiber bundles.
method Construct flat, projective morphisms that are -homology fiber bundles.
result Disprove conjecture about homology fiber bundles.
By using the equivariant localization formula of toric varieties. We prove the vanishing of the Witten genus of some string complete intersections in smooth toric varieties.
Study torsion-free nilpotent fundamental groups of smooth varieties up to rank 7.
problem Characterize fundamental groups of smooth quasi-projective varieties.
method Analyzes fundamental groups of smooth quasi-projective varieties using topological and Lie group theory.
result Determine fundamental groups for smooth quasi-projective varieties up to rank 7.
Study projective KLT varieties with projectively flat cotangent sheaves.
problem Uniformisation problems on projective varieties with klt singularities.
method Generalising Jahnke-Radloff's work, study torus quotients and varieties with semistable cotangent sheaves and extremal Chern classes.
result Torus quotients are the only klt varieties with semistable cotangent sheaves and extremal Chern classes.
Rigidity of smooth Schubert varieties in a rational homogeneous manifold associated to a short rootmath.AG
We classify smooth Schubert varieties S_0 in a rational homogeneous manifold S associated to a short root, and show that they are rigid in the sense that any subvariety of S having the same homology class as S_0 is induced by the action of Aut_0(S), unless S_0 is linear.
Study Lie algebroid connections on principal bundles over complex projective varieties.
problem Existence and properties of Lie algebroid connections on principal bundles.
method Definition and study of Lie algebroid valued connections on holomorphic principal G-bundles, investigation of existence criteria.
result Investigation of criteria for existence of Lie algebroid connections on principal G-bundles over smooth complex projective curves.
Study Kähler-Einstein metrics on singular varieties, proving metric completion properties.
problem Kähler-Einstein metrics on singular projective varieties.
method Approximation with constant scalar curvature metrics, RCD space analysis.
result Metric completion of smooth part is non-collapsed RCD space and homeomorphic to original variety.
In this paper, I give a new construction of a Kähler-Einstein metrics on a smooth projective variety with ample canonical bundle. This result can be generalized to the construction of a singular Kähler-Einstein metric on a smooth projective variety of general type which gives an AZD of the canonical bundle. Also the va…
Proves a generalized vanishing theorem for quasi-smooth stacks, with applications in K-theory and birational geometry.
problem Vanishing theorems for quasi-coherent sheaves on derived blow-ups of quasi-smooth stacks.
method Derived blow-ups, intrinsic blow-up theory, Kiem-Li-Savvas blow-up theory, virtual localization theorem, desingularization theorem, resolution of diagonal.
result Generalized vanishing theorem for quasi-coherent sheaves on derived blow-ups of quasi-smooth stacks.