arXiv research

The study examines whether a specific type of hyperbolic manifolds remains unchanged under birational transformations.

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.

Study the intersection form on Kähler manifolds of dimension 4 and above.

Criterion found for blowing down in 6D symplectic geometry.

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…

The study classifies Fano varieties with specific pseudoindex.

Study bubbling Kahler metrics using algebraic geometry.

The paper connects Kähler-Ricci shrinkers to Fano fibrations in algebraic geometry.

Overview of algebraic geometry for almost complex manifolds.

Proves a generalized vanishing theorem for quasi-smooth stacks, with applications in K-theory and birational geometry.

Study collapsing Calabi-Yau metrics and flows on fiber spaces.

Proves common stellar subdivisions for all PL homeomorphic polyhedra.

Two types of nonvanishing results are presented for compact Kähler varieties.

Extends BCOV invariant to pairs of Calabi-Yau manifolds and pluricanonical divisors.

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.

We survey our recent papers (some being joint ones) about the relation between the geometry of a compact Kähler manifold and the existence of automorphisms of positive entropy on it. We also use the language of log minimal model program (LMMP) in biraitonal geometry, but not its more sophisticated technical part. We gi…

We prove an optimal result on the birational rigidity and K-stability of index $1$ hypersurfaces in $\mathbb{P}^{n+1}$ with ordinary singularities when $n\gg 0$ and also study the birational superrigidity and K-stability of certain weighted complete intersections. As an application, we show that birational superrigidit…

We study the formation of finite time singularities of the Kahler-Ricci flow in relation to high codimensional birational surgery in algebraic geometry. We show that the Kahler-Ricci flow on an n-dimensionl Kahler manifold contracts a complex submanifold $\mathbb{P}^m$ with normal bundle $\oplus_{j=1}^{n-m}\mathcal{O}_…

Program connects birational invariants with G-equivariant ones using Gromov-Witten theory.

We show a relation between the birational superrigidity of Fano manifold and its slope stability in the sense of Ross-Thomas.

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…

Study shows K-moduli spaces connect quartic surfaces to K3 surfaces, verifying predictions and classifying degenerations.

Researchers prove birational invariance of BCOV invariant using motivic integration.

We describe some relations between coefficients of irreducible components of the first Chern class [FP15] and birational germs introduced by Dloussky {Dl16] for intermediate Kato surfaces.

Let $M$ be a compact hyperkahler manifold with maximal holonomy (IHS). The group $H^2(M, R)$ is equipped with a quadratic form of signature $(3, b_2-3)$, called Bogomolov-Beauville-Fujiki (BBF) form. This form restricted to the rational Hodge lattice $H^{1,1}(M,Q)$, has signature $(1,k)$. This gives a hyperbolic Rieman…

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…

We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperkahler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the…

The purpose of this survey is to present analytic versions of the injectivity theorem and their applications. The proof of our injectivity theorems is based on a combination of the L^2-method for the dbar-equation and the theory of harmonic integrals. As applications, we obtain Nadel type vanishing theorems and extensi…

We introduce a weak concept of Morita equivalence, in the birational context, for Poisson modules on complex normal Poisson projective varieties. We show that Poisson modules, on projective varieties with mild singularities, are either rationally Morita equivalent to a flat partial holomorphic sheaf, or a sheaf with a …

In 1995 D. Joyce explicitly constructed a series of self-dual metrics with torus action on the connected sums of complex projective planes. In this paper we explicitly construct the twistor spaces of some of Joyce's self-dual metrics. Starting from a fiber space whose fibers are compact singular toric surfaces, we appl…

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…

Unified approach to invariants in equivariant geometry.

Homological stability fails for Cremona groups, rational varieties, and function fields.

We extend the framework of K-stability (Tian, Donaldson) to more general algebro-geometric setting, such as partial desingularisations of (fixed) singularities, (not necessarily flat) families over higher dimensional base and the classical birational geometry of surfaces. We also observe that "concavity" of the volume …

Study shows non-polyhedral structure in moduli spaces for n≥8.

We introduce an inductive argument for proving birational superrigidity and K-stability of singular Fano complete intersections of index one, using the same types of information from lower dimensions. In particular, we prove that a hypersurface in $\mathbb{P}^{n+1}$ of degree $n+1$ with only ordinary singularities of m…

We study the group of symplectic birational transformations of the plane. It is proved that this group is generated by $\mathrm{SL}(2,\mathbb{Z})$, 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 conjecture about integer sums of torus knot torsions.

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex proje…

Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.

We prove that every smooth Fano complete intersection of index $1$ and codimension $r$ in $\mathbb{P}^{n+r}$ is birationally superrigid and K-stable if $n\ge 10r$. We also propose a generalization of Tian's criterion of K-stability and, as an application, prove the K-stability of the complete intersection of a quadric …

This is the first of a series of papers, in which we study the plurigenera, the Kodaira dimension and more generally the Iitaka dimension on compact almost complex manifolds. Based on the Hodge theory on almost complex manifolds, we introduce the plurigenera, Kodaira dimension and Iitaka dimension on compact almost com…

Proves non-hyperbolicity of symplectic varieties with specific properties.