Proves complex projective varieties are birational to cscK models.
problem Complex projective varieties and their cscK metrics.
method Birational models and Lefschetz pencils.
result Every complex projective variety is birational to a smooth projective manifold with cscK metric.
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.
Groups of birational transformations with specific properties are conjugate to groups of pseudo-automorphisms.
problem Classifying groups of birational transformations with fixed point properties.
method Importing ideas from geometric group theory, proving conjugacy to groups of pseudo-automorphisms.
result Groups of birational transformations with certain properties are conjugate to groups of pseudo-automorphisms.
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.
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.
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.
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 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 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.
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.
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.
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.
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.
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.
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.
Let M be a compact hyperkahler manifold with maximal holonomy (IHS). The group H2(M,R) is equipped with a quadratic form of signature (3,b2−3), called Bogomolov-Beauville-Fujiki (BBF) form. This form restricted to the rational Hodge lattice H1,1(M,Q), has signature (1,k). This gives a hyperbolic Rieman…
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…
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.
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.
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.
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.
Study shows K-moduli spaces connect quartic surfaces to K3 surfaces, verifying predictions and classifying degenerations.
problem Understanding the moduli spaces of quartic K3 surfaces and their birational models.
method Interpolates between GIT and Baily-Borel moduli spaces, describes wall crossings, and classifies degenerations.
result Verifies Laza-O'Grady's prediction and classifies Gorenstein canonical Fano degenerations of \(\mathbb{P}^3\).
Extends BCOV invariant to pairs of Calabi-Yau manifolds and pluricanonical divisors.
problem Extend BCOV invariant to new geometric pairs.
method Extend BCOV invariant to pairs (X,D), study blow-up behavior. result Results imply birational Calabi-Yau manifolds have the same BCOV invariant.
We prove that every algebraic curve X defined over the algebraic closure of the rationals is birational over the complex numbers to a Teichmuller curve.
We define a new notion of "b-stability" for a polarised algebraic variety, adapted to the existence problem for Kahler-Einstein metrics on Fano manifolds.
Paper defines new stability and metrics for complex spaces.
problem Stability and metrics for complex spaces with big cohomology classes.
method Introduces slope stability and Hermitian-Einstein metrics for big cohomology classes.
result Kobayashi Hitchin correspondence and Bogomolov Gieseker inequality proved.
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.
Study collapsing Calabi-Yau metrics and flows on fiber spaces.
problem Understanding the behavior of Calabi-Yau metrics and flows during collapsing.
method Analyzing the collapsing of Calabi-Yau metrics and Kähler-Ricci flows on fiber spaces.
result Identify the collapsed Gromov-Hausdorff limit and bounds for Hausdorff measure.
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.
Graph potentials link to topological QFTs, with computational methods.
problem Defining a topological quantum field theory using graph potentials.
method Using colored trivalent graphs and birational type to define a topological QFT.
result Graph potentials' birational type depends on the graph's homotopy type.
Study J-holomorphic curves from J-anti-invariant forms on almost complex 4-manifolds.
problem Understanding the relationship between J-anti-invariant forms and pseudoholomorphic curves. method Analyzing the zero set of closed J-anti-invariant 2-forms and their relation to J-holomorphic subvarieties. result The zero set of a closed J-anti-invariant 2-form supports a J-holomorphic subvariety in the canonical class, confirming a conjecture. Criterion found for blowing down in 6D symplectic geometry.
problem Blowing down criterion in 6D symplectic geometry.
method Criterion for blowing down in 6D symplectic geometry.
result Criterion established for blowing down in 6D symplectic geometry.