Counterexample disproves log canonical Beauville--Bogomolov decomposition.
problem Disproving the log canonical Beauville--Bogomolov decomposition.
method Constructing a specific log canonical, K-trivial variety with non-birational fibers.
result Provides a counterexample to the Beauville--Bogomolov decomposition in the log canonical setting.
Generalizes nonabelian Hodge theory to klt singularities.
problem Applying nonabelian Hodge theory to spaces with klt singularities.
method Uses descent theorems for numerically flat vector bundles and a new restriction theorem for semistable Higgs sheaves.
result Establishes a new restriction theorem for semistable Higgs sheaves.
In this paper we develop a Morse-like theory in order to decompose birational maps and morphisms of smooth projective varieties defined over a field of characteristic zero into more elementary steps which are locally étale isomorphic to equivariant flips, blow-ups and blow-downs of toric varieties. A crucial role in th…
We study the group of symplectic birational transformations of the plane. It is proved that this group is generated by SL(2,Z), the torus and a special map of order 5, as it was conjectured by A. Usnich. Then we consider a special subgroup H, of finite type, defined over any field which admits a…
The intersection of certain submanifolds yields J-holomorphic curves.
problem Characterizing intersections of almost complex submanifolds.
method Using differential geometry of almost Hermitian manifolds to produce J-holomorphic curves.
result Degree one pseudoholomorphic maps between almost complex 4-manifolds are birational morphisms.
We use a counting argument and surgery theory to show that if D is a sufficiently general algebraic hypersurface in Cn, then any local diffeomorphism F:X→Cn of simply connected manifolds which is a d-sheeted cover away from D has degree d=1 or d=∞ (however all degrees d>1 are poss…
Study Kodaira dimensions on compact almost complex manifolds.
problem Understanding invariants on almost complex manifolds.
method Introduce plurigenera, Kodaira dimension, and Iitaka dimension based on Hodge theory. Prove Hartogs extension theorem using foliation-by-disks technique.
result Show that plurigenera and Kodaira dimension are birational invariants in almost complex category, especially in dimension 4.
Overview of algebraic geometry for almost complex manifolds.
problem Developing algebraic geometry for almost complex manifolds without genericity.
method Reviewing results based on pseudoholomorphic maps and intersection theory.
result Introduction of birational morphism between almost complex manifolds.
Quantum trace maps abelian character varieties to SL2 character varieties.
problem Classifying irreducible representations of Chekhov-Fock algebras.
method Non-commutative deformation of algebraic morphisms.
result Induces birational morphism between torus and SL2 character variety.
The study examines whether a specific type of hyperbolic manifolds remains unchanged under birational transformations.
problem Birational invariance of balanced hyperbolic manifolds.
method Analysis of the class of balanced hyperbolic manifolds.
result The class of balanced hyperbolic manifolds is birationally invariant.
Study on birational rigidity and stability of hypersurfaces and complete intersections, proving non-locally closed property.
problem Birational rigidity and stability of hypersurfaces and complete intersections.
method Optimal results on birational rigidity and K-stability, proving non-locally closed property.
result Birational superrigidity is not a locally closed property.
Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension 1. We import ideas from geometric group theory to prove that a group of birational transformations that satisfies a fixed point property on CAT(0) cubical complexes, for example a discrete countable group with Kazhdan Pro…
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.
Study connects Chern class components to Kato surface germs.
problem Understanding intermediate Kato surfaces through Chern class components.
method Relates coefficients of Chern classes to birational germs.
result Established relations between Chern class components and Kato surface germs.
Program connects birational invariants with G-equivariant ones using Gromov-Witten theory.
problem Establishing a connection between birational invariants and G-equivariant ones.
method Gromov-Witten theory, Chen-Ruan cohomology, and equivariant atoms.
result New interpretations and applications of classical invariants.
We show a relation between the birational superrigidity of Fano manifold and its slope stability in the sense of Ross-Thomas.
Researchers prove birational invariance of BCOV invariant using motivic integration.
problem Proving the birational invariance of BCOV invariant for Calabi-Yau manifolds and varieties.
method Motivic integration theory applied to Calabi-Yau varieties with Kawamata log terminal singularities.
result Birational Calabi-Yau manifolds have the same BCOV invariant.
In this paper we study the birational geometry of HyperKaehler manifolds by combining the method of minimal model program and the traditional approach of symplectic geometry.
In this paper we will survey some recent developments in the last decade or so on variation of Geometric Invariant Theory and its applications to Birational Geometry such as the weak Factorization Theorems of nonsingular projective varieties and more generally projective varieties with finite quotient singularities. Al…
This is a survey on symplectic birational geometry. In arbitrary dimension, this subject is centered around the notion of uniruledness. In low dimensions, we will also discuss Kodaira dimension and minimality.
We study degenerate complex Monge-Ampère equations of the form (ω+ddcφ)n=etφμ where ω is a big semi-positive form on a compact Kähler manifold X of dimension n, t∈R+, and μ=fωn is a positive measure with density f∈Lp(X,ωn), p>1. We prove the existence and unicity of bou…
A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally invariant. These vector spaces so metrized will be referred to as the pseudonormed spaces…
Unique birational structure proven on Inoue surfaces.
problem Proving uniqueness of birational structures on Inoue surfaces.
method Generalizing a result by Bruno Klingler, proving uniqueness of structures.
result The natural (Aff2(C),C2)-structure on an Inoue surface is the unique (Bir(P2),P2(C))-structure. A symplectic manifold (M,ω) is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and blow-down. This theorem follows from a general Relative/Absolute correspondence for a symple…
We describe a birational map between subvarieties in the character varieties of mutative 3-manifolds. By studying the birational map, one can decide in certain circumstances whether a mutation surface is detected by an ideal point of the character variety.
Building on the work of the fourth author in math.AG/9904074, we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blowings up and blowings down with sm…
Constructs BCOV invariant for Calabi-Yau pairs.
problem No specific problem stated; focuses on construction.
method Constructs BCOV invariant for Calabi-Yau pairs, covering classical and equivariant cases.
result Expected well-behaved under birational equivalence.
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.
Study the intersection form on Kähler manifolds of dimension 4 and above.
problem Understand the intersection form on higher-dimensional Kähler manifolds.
method Investigate fundamental properties and applications to birational geometry.
result Present open problems in the relationship between birational invariants and topological invariants.
A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hypekahler manifold M, showing that it is commensurable to an arithmetic subgroup in SO(3, b_2-3). A Teichmuller space of M is a space of complex structures on M up to is…
The study proves Fano complete intersections' rigidity and stability under certain conditions.
problem The rigidity and stability of Fano complete intersections.
method Birational superrigidity and K-stability criteria.
result Fano complete intersections of index 1 and codimension r in P^(n+r) are birationally superrigid and K-stable for n ≥ 10r.
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.
Homological stability fails for Cremona groups, rational varieties, and function fields.
problem Homological stability in Cremona groups fails in both possible ways.
method Explained the failure of homological stability for Cremona groups.
result Homological stability fails for Cremona groups in both possible ways.
New proof for Fano manifolds, showing rigidity and stability.
problem Proving rigidity and stability of Fano manifolds.
method Birational superrigidity and K-stability approach.
result Projectively normal Fano manifolds of index 1 are birationally superrigid and K-stable.
New invariant distinguishes real algebraic surfaces.
problem Distinguishing real algebraic surfaces up to birational diffeomorphism.
method Introducing real (logarithmic)-Kodaira dimension.
result Constructs infinite families of non-birationally diffeomorphic surfaces.
Proves conjecture about integer sums of torus knot torsions.
problem Integrality of sums of (g-1)st powers of adjoint Reidemeister torsions for torus knots.
method Introduced Verlinde numbers from modular S-matrix, proved integrality through recursion formulas.
result Proven integrality of sums of (g-1)st powers of adjoint Reidemeister torsions for all torus knots and non-negative g.
Proves common stellar subdivisions for all PL homeomorphic polyhedra.
problem Proving common stellar subdivisions for all PL homeomorphic polyhedra.
method Proved weighted strong factorization conjecture and iterated blowups.
result Every two PL homeomorphic polyhedra have a common stellar subdivision.
Study on deformations of Lie groupoid morphisms and their properties.
problem Understanding the deformation theory of Lie groupoid morphisms.
method Established deformation theory, cohomology, and properties of morphisms.
result Invariance and stability properties of morphisms, Morita invariance of cohomology, and simultaneous deformations.
Introduces conformal Riemannian morphisms generalizing various Riemannian concepts.
problem None explicitly stated; generalizing Riemannian concepts.
method Introduces conformal Riemannian morphisms and proves properties.
result Every injective conformal Riemannian morphism is an injective conformal immersion, and similar results for surjective and bijective cases.
Defines and partially characterizes p symphonic morphisms.
problem No specific problem stated; focuses on definition and characterization.
method Defines p symphonic morphisms and partially characterizes them.
result Characterization of p symphonic morphisms.
The paper explores new algebraic structures and morphisms in graded settings.
problem Understanding new algebraic structures and morphisms in graded settings.
method Introducing and analyzing L∞-, P∞-, and S∞-algebras, and thick morphisms in a Z2imesZ-graded context. result Shifted S∞-thick morphisms induce L∞-morphisms of shifted S∞-structures. Algorithms compute the topology of hyperelliptic curves in 2D and 3D.
problem Computing the topology of hyperelliptic curves in higher dimensions.
method Birational mapping of the plane or space to compute the topology of the curve.
result Algorithms implemented in { t Maple} for computing the topology of hyperelliptic curves.
Proves non-hyperbolicity of symplectic varieties with specific properties.
problem Non-hyperbolicity of primitive symplectic varieties with certain conditions.
method Uses ergodicity, birational contractions, and cycle spaces.
result Vanishing Kobayashi pseudometric for symplectic varieties with Lagrangian fibrations.
Proves K-stability and superrigidity of certain singular Fano hypersurfaces.
problem Proving K-stability and superrigidity of singular Fano hypersurfaces.
method Inductive argument using information from lower dimensions and adjunction type results for local volumes of singularities.
result Proves birational superrigidity and K-stability of singular Fano hypersurfaces with specific conditions.
Study bubbling Kahler metrics using algebraic geometry.
problem Analyzing the degeneration of Kahler metrics with Euclidean volume growth.
method Algebraic construction of birational modifications to simplify degenerations, comparing with analytic constructions.
result Provide a framework to compare algebraic and analytic approaches to bubbling phenomena.
We study the behavior of the modular class of a Lie algebroid under general Lie algebroid morphisms by introducing the relative modular class. We investigate the modular classes of pull-back morphisms and of base-preserving morphisms associated to Lie algebroid extensions. We also define generalized morphisms, includin…
We show that thick morphisms (or microformal morphisms) between smooth (super)manifolds, introduced by us before, are classical limits of `quantum thick morphisms' defined here as particular oscillatory integral operators on functions.
Let $\g\_2$ be the Hochschild complex of cochains on $C^\infty(\RM^n)$ and $\g\_1$ be the space of multivector fields on $\RM^n$. In this paper we prove that given any G_∞-structure ({\rm i.e.} Gerstenhaber algebra up to homotopy structure) on $\g\_2$, and any C_∞-morphism φ ({\rm i.e.} morphism of co…