arXiv research

Uniformizes varieties of general type with harmonic metrics.

Generalizes Toledo invariant to singular klt varieties, proving Milnor-Wood inequality.

Researchers propose a new approach to the Hopf problem for aspherical varieties.

We propose a definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type into semisimple Lie groups of Hermitian type. This definition allows to generalize the results known in the classical case of representations of complex hyperbolic lattices to this new setting: …

Complex projective manifolds without rational curves are quotients of Abelian varieties.

Study characteristic classes of a specific type of determinantal varieties.

We give a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z. This is done by applying the fixed point formula of Lefschetz type in Arakelov geometry to generalized flag varieties. Our proof involves the computation of the equivariant Ray-Singer torsion for all equivarian…

Theory and existence of extremal Kähler metrics on toric varieties.

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 paper examines when real matrix Schubert varieties are minimal submanifolds.

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

In this note we propose to show that the Kähler-Ricci flow fits naturally within the context of the Minimal Model Program for projective varieties. In particular we show that the flow detects, in finite time, the contraction theorem of any extremal ray and we analyze the singularities of the metric in the case of divis…

We develop a global Poincaré residue formula to study period integrals of families of complex manifolds. For any compact complex manifold $X$ equipped with a linear system $V^*$ of generically smooth CY hypersurfaces, the formula expresses period integrals in terms of a canonical global meromorphic top form on $X$. Two…

Abstract: Generalizes stability theories over toric varieties and Novikov type rings.

We study Riemannian geometry of canonical Kahler-Einstein currents on projective Calabi-Yau varieties and canonical models of general type with crepant singularities. We prove that the metric completion of the regular part of such a canonical current is a compact metric length space homeomorphic to the original project…

Classifies holomorphic parabolic geometries on complex manifolds.

Study Mabuchi solitons on toric Fano varieties, linking stability and energy.

We study the homotopy types of complements of arrangements of n transverse planes in R^4, obtaining a complete classification for n <= 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2-arrangement in R^4 is not determined by the cohomology ring, thereby …

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…

Study geometric criteria for cone structures on flag varieties.

The study bounds growth of Hodge numbers and computes $L^2$-Betti numbers for irregular varieties.

The study finds Kähler-Einstein metrics on certain Fano varieties of type AIII.

Paper introduces a new type of canonical metric for varieties with intermediate Kodaira dimension.

The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.

We study the algebraic properties of the generalized Futaki invariant of an almost Fano variety and prove that it is in fact a pushforward to a point of an appropriate equivariant Chow cohomology class of the variety. This allows us to use Bott-type formulae for calculating the invariant. We show this use on some examp…

The generic fiber of a Lagrangian fibration on an irreducible holomorphic symplectic manifold is an abelian variety. Associate a polarization type to such Lagrangian fibrations coming from polarizations on a generic fiber. We prove that this polarization type is constant in families of Lagrangian fibrations. Further, w…

The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.

Let X be a projective variety which is algebraic Lang hyperbolic. We show that Lang's conjecture holds (one direction only): X and all its subvarieties are of general type and the canonical divisor K_X is ample at smooth points and Kawamata log terminal points of X, provided that K_X is Q-Cartier, no Calabi-Yau variety…

We fix integers $k> 0$ and $n>0$. For a $k$-punctured Riemann surface $Σ\setminus \{ p_1,\ldots,p_k \}$ and a $k$-tuple $\boldsymbolμ=(μ^1,\ldots,μ^k)$ of partitions of $n$, we can define the character variety of type $\boldsymbolμ$. In this paper, we consider the case where $Σ=\mathbb{P}^1$ and $\boldsymbolμ$ is indiv…

The paper classifies certain singular projective varieties with specific properties.

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 prove rigidity of various types of holomorphic parabolic geometry on smooth complex projective varieties.

The paper shows how certain complex projective varieties can be broken down into simpler types.

Reductive quotients preserve klt singularities in algebraic geometry.

We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge-Ampère equations. This type of equations is precisely what is needed in order to construct Kähler-Einstein metrics over irreducible singular Kähler spaces with ample or trivial canonical sheaf and singular Kähle…

This paper formalizes manifolds in positive characteristic varieties.

Researchers describe character varieties for Hopf links, proving geometric properties.

Researchers extend period maps for Calabi-Yau types using modified Kato-Nakayama-Usui construction.

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…

Researchers characterize a specific type of projective variety based on its tangents.

Paper proves Kähler-Ricci solitons on a new type of variety.

New bounds found for certain types of algebraic varieties.

The authors study in detail new types of varieties with degenerate Gauss maps: varieties with multiple foci and their particular case, the so-called twisted cones. They prove an existence theorem for twisted cones and describe their structure.

Study of braid varieties and their Legendrian isotopy.

Review of instantons in noncommutative gauge theories across 4, 6, and 8 dimensions.

Kähler-Einstein metrics found on special types of symmetric varieties.

Study of Kähler-Ricci flow on toric Fano varieties with preserved symplectic condition.

Study two types of singular Kähler-Einstein metrics on complex varieties.