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.
The paper shows how certain complex projective varieties can be broken down into simpler types.
problem Understanding the structure of complex projective varieties with pseudo-effective tangent sheaves.
method Developed a theory of pseudo-effective sheaves and applied the minimal model program.
result Projective klt varieties with pseudo-effective tangent sheaves can be decomposed into Fano varieties and Q-abelian varieties.
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.
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 paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.
problem Understanding the structure of projective klt varieties with nef tangent sheaves.
method Developing theory of positivity of coherent sheaves and proving structure theorem.
result Projective klt varieties with specific tangent sheaf properties admit rationally connected fibrations onto abelian varieties.
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.
Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in 14-dimensional projec…
Each of the four critical Severi varieties arises from a minimal holomorphic nilpotent orbit in a simple regular rank 3 hermitian Lie algebra and each such variety lies as singular locus in a cubic--the chordal variety--in the corresponding complex projective space; the cubic and projective space are identified in term…
Integral points are potentially dense in character varieties of quasi-projective varieties.
problem Density of integral points in character varieties of quasi-projective varieties.
method Reduction to Riemann surfaces and use of Corlette-Simpson work.
result Integral points have Zariski-dense orbit under the mapping class group.
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.
In this paper, we study asymptotic behavior of projective embeddings of Kummer varieties given by theta functions, and their amoebas. We prove that a Lagrangian fibration of the Kummer variety can be approximated by moment maps of the projective spaces.
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.
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.
Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.
problem Understanding geometric structures of singular Kahler spaces.
method Proving RCD spaces homeomorphic to 3D projective varieties with bounded Nash entropy and Ricci curvature.
result Compact RCD spaces are equivalent to underlying projective varieties.
Projective varieties remain stable under close polarizations, extending to Kähler cones.
problem Maintaining stability of projective varieties under close polarizations.
method Uniformly valuative stability definition and extension to Kähler cones.
result Openness of uniformly valuative stability on the Kähler cone of projective manifolds.
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.
Study of groups acting on complex projective varieties.
problem Classifying groups of birational transformations on complex projective varieties.
method Free, properly discontinuous, cocompact action on open sets of complex projective varieties.
result Classification in dimension two.
We prove rigidity of various types of holomorphic parabolic geometry on smooth complex projective varieties.
The study of projective varieties with nef anticanonical divisors and log terminal singularities.
problem Understanding the structure and properties of projective varieties with specific divisor conditions.
method Analyzing the Albanese map and MRC fibration for klt projective varieties, showing locally constant fibrations and product decompositions.
result Generalization of results for smooth projective varieties to the klt case, including decomposition into rationally connected and projective varieties with trivial canonical divisor.
Resolves conjectures on non-abelian Hodge loci for quasi-projective varieties.
problem Understanding non-abelian Hodge loci for quasi-projective varieties.
method Analyzes Z-local systems and polarized variations of Hodge structures. result Proves algebraicity of non-abelian Hodge loci for Q-anisotropic monodromy. Demailly's conjecture, which is a consequence of the Green-Griffiths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly's conjecture by verifying several predictions it makes. We first defi…
By the quantization condition compact quantizable Kaehler manifolds can be embedded into projective space. In this way they become projective varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This all…
Proves certain Calabi-Yau varieties are projective.
problem Compact Calabi-Yau varieties with isolated singularities are not always projective.
method Analysis and Ohsawa's degenerate spectral sequence in higher dimensions.
result Proves compact Calabi-Yau varieties with certain isolated singularities are projective.
Study on unique generalized Gauss maps of minimal surfaces sharing hypersurfaces in projective varieties.
problem Uniqueness of generalized Gauss maps for minimal surfaces with shared hypersurfaces in projective varieties.
method Analysis of minimal surfaces in Rn+1 with inverse images of hypersurfaces in a projective subvariety. result Generalization and improvement of previous results on the uniqueness of generalized Gauss maps.
In this paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying the theory of Variation of Geometric Invariant Theory Quotients ([3]), we show tha…
Complex projective manifolds without rational curves are quotients of Abelian varieties.
problem Characterizing complex projective manifolds without rational curves.
method Using conjectures about rational and entire curves on Calabi-Yau varieties.
result Non-hyperbolic complex projective manifolds contain the image of an Abelian variety.
The study classifies holomorphic projective connections on complex threefolds.
problem Characterizing holomorphic projective connections on complex threefolds.
method Analyzing properties of holomorphic projective connections on complex projective threefolds.
result Holomorphic projective connections on complex threefolds are either flat or translation invariant on abelian threefolds.
We present a new proof of the classification of complex simple Lie algebras via the projective geometry of homogeneous varieties. Our proof proceeds by constructing homogeneous varieties using the ideals of the secant and tangential varieties of homogeneous varieties already constructed. Our algorithms make no referenc…
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.
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.
Uniform bounds prove connection between Kähler metrics and RCD spaces.
problem Bounding Nash entropy and Calabi energy for Kähler metrics.
method Proving uniform Sobolev bounds for Kähler manifolds.
result Establishes connection to RCD spaces and provides examples.
Study complex slices on real algebraic varieties and their properties.
problem Understanding the properties and bounds of complex slices on real algebraic varieties.
method Analyzing intersections and cohomology classes of complex subvarieties with real subvarieties.
result Proved an upper bound for the linking number of certain slices.
A point p∈PN of a projective space is h-identifiable, with respect to a variety X⊂PN, if it can be written as linear combination of h elements of X in a unique way. Identifiability is implied by conditions on the contact locus in X of general linear spaces called non weak defecti…
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.
Non-exact Poisson structures found on toric varieties.
problem Existence of exact Poisson structures on toric varieties.
method Geometric criterion for non-exactness of Poisson structures with finite symplectic leaves.
result Non-exactness of Poisson structures on projective toric varieties.
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.
The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.
problem Extending the Poincaré-Hopf theorem to varieties with isolated singularities.
method Using generalizations of the Poincaré-Hopf index.
result A Poincaré-Hopf type theorem for projective varieties with isolated singularities.
Stability results for complex Monge-Ampère equations in various classes.
problem Stability of solutions to complex Monge-Ampère equations.
method Weak stability results followed by Ck,α stability proofs. result Proves stability of solutions in relative full mass classes and on quasi-projective varieties.
We formulate the equivalence problem, in the sense of E. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form …
The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.
problem Extending the Poincaré-Hopf theorem to projective varieties with isolated singularities.
method Using generalized Poincaré-Hopf indices for a projective variety with isolated determinantal singularities.
result A Poincaré-Hopf type theorem is proven for projective varieties with isolated singularities.
We prove that for every finitely-presented group G there exists a 2-dimensional irreducible complex-projective variety W with the fundamental group G, so that all singularities of W are normal crossings and Whitney umbrellas.
Existence of Kähler-Einstein metrics on toric varieties proven.
problem Existence of Kähler-Einstein metrics on toric varieties.
method Characterization of K-stability using log Cox ring and universal orbifold cover.
result Every Q-factorial normal projective toric variety allows an orbifold Kähler-Einstein metric.
Proves cohomology theorems for tropical varieties.
problem Cohomology of smooth projective tropical varieties.
method Introduces and proves new results in tropical geometry.
result Establishes tropical analogs of three fundamental theorems.
The paper classifies a specific type of quadratic variety with a small codimension.
problem Classifying nondegenerate smooth projective varieties of dimension n=2c−1 defined by quadratic equations. method Using the Hartshorne conjecture on complete intersections and classification techniques.
result The paper classifies varieties with n=2c−1 and proves they are complete intersections. 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…
Assume that a projective variety together with a polarization is uniformly K-stable. If the polarization is canonical or anti-canonical, then the projective variety is uniformly K-stable with respects to any polarization sufficiently close to the original polarization.
In this paper, we study smooth complex projective varieties X such that some exterior power ⋀rTX of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two cases. If TX is strictly nef, then X isomorphic to the projective space $\…
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.