arXiv research

Study shows quantum behavior near infinity in metric asymptotics.

The present paper studies the structure of characteristic varieties of fundamental groups of graph manifolds. As a consequence, a simple proof of Papadima's question is provided on the characterization of algebraic links that have quasi-projective fundamental groups. The type of quasi-projective obstructions used here …

Study on Kähler-Einstein metrics on quasi-projective manifolds.

We give an elementary proof to the asymptotic expansion formula of Rochon-Zhang for the unique complete Kähler-Einstein metric of Cheng-Yau, Kobayashi, Tian-Yau and Bando on quasi-projective manifolds. The main tools are the solution formula for second order ODE's with constant coefficients and spectral theory for Lapl…

We prove two results relating 3-manifold groups to fundamental groups occurring in complex geometry. Let N be a compact, connected, orientable 3-manifold. If N has non-empty, toroidal boundary, and π_1(N) is a Kaehler group, then N is the product of a torus with an interval. On the other hand, if N has either empty or …

Even Artin groups generalize right-angled Artin groups by allowing the labels in the defining graph to be even. In this paper a complete characterization of quasi-projective even Artin groups is given in terms of their defining graphs. Also, it is shown that quasi-projective even Artin groups are realizable by K(pi,1) …

Study surfaces with free product fundamental groups, proving existence and properties.

Let $(M,ω)$ be a compact Kähler manifold with negative holomorphic sectional curvature. It was proved by Wu-Yau and Tosatti-Yang that $M$ is necessarily projective and has ample canonical bundle. In this paper, we show that any irreducible subvariety of $M$ is of general type. Moreover, we can extend the theorem to the…

Resolves conjectures on non-abelian Hodge loci for quasi-projective varieties.

Analyzes the moduli space of Higgs bundles to prove its quasi-projectivity.

Integral points are potentially dense in character varieties of quasi-projective varieties.

Stability results for complex Monge-Ampère equations in various classes.

Nilmanifolds are shown to be diffeomorphic to trivial bundles over tori.

Maps complex varieties into buildings with harmonic properties.

We discuss properties of complex algebraic orbifold groups, their characteristic varieties, and their abelian covers. In particular, we deal with the question of (quasi)-projectivity of orbifold groups. We also prove a structure theorem for the variety of characters of normal-crossing quasi-projective orbifold groups. …

The present paper describes a relation between the quotient of the fundamental group of a smooth quasi-projective variety by its second commutator and the existence of maps to orbifold curves. It extends previously studied cases when the target was a smooth curve. In the case when the quasi-projective variety is a comp…

Researchers create a projective space for quasimaps and study its connections to Calabi-Yau fibrations.

This paper has two purposes. First it partially extends the result in the author's previous work concerning the asymptotic expansion of the Tian-Yau metrics, by considering a slightly larger class of quasi-projective manifolds. This text is also intended to provide a quick introductory reference to the study of Ricci-f…

We define notions of higher order spectra of a complex quasi-projective manifold with an action of a finite group $G$ and with a $G$-equivariant automorphism of finite order, some of their refinements and give Macdonald type equations for them.

We consider the moduli space MN of flat unitary connections on an open Kaehler manifold U (complement of a divisor with normal crossings) with restrictions on their monodromy transformations. Using intersection and L2 cohomologies with degenerating coefficients we construct a natural symplectic form F on MN. When U is …

We define the C^*-action on moduli spaces of reductive representations of fundamental groups of quasi-compact Kaehler manifolds by solving Hermitian-Yang-Mills equation. As applications in algebraic geometry we show a non-abelian Hodge (p,q)-type theorem for families of quasi-projective manifolds. We also prove that an…

Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective threefold Y that admits a no…

We establish the correspondence between tame harmonic bundles and $μ_L$-stable parabolic Higgs bundles with trivial characteristic numbers. We also show the Bogomolov-Gieseker type inequality for $μ_L$-stable parabolic Higgs bundles. Then we show that any local system on a smooth quasi projective variety can be deforme…

In this paper, we construct a completion of the moduli space for polarized Calabi-Yau manifolds by using Ricci-flat Kähler-Einstein metrics and the Gromov-Hausdorff topology, which parameterizes certain Calabi-Yau varieties. We then study the algebro-geometric perperties and the Weil-Petersson geometry of such completi…

New algebraic-geometric method classifies superintegrable systems in any dimension.

In this work, we describe the asymptotic behavior of complete metrics with prescribed Ricci curvature on open Kahler manifolds that can be compactified by the addition of a smooth and ample divisor. First, we construct a explicit sequence of Kahler metrics with special approximating properties. Using those metrics as s…

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…

The paper finds non-isotopic exact Lagrangians in symplectic manifolds with $\mathbb{C}^*$-actions.

Let $K$ be a closed polydisc or ball in $\C^n$, and let $Y$ be a quasi projective algebraic manifold which is Zariski locally equivalent to $\C^p$, or a complement of an algebraic subvariety of codimension $\ge 2$ in such manifold. If $r$ is an integer satisfying $(n-r+1) (p-r+1)\geq 2$ then every holomorphic map from …

Let $L$ be a local system on a smooth quasi projective variety over $\cnum$. We see that $L$ is semisimple if and only if there exists a tame pure imaginary pluri-harmonic metric on $L$. Although it is a rather minor refinement of a result of Jost and Zuo, it is significant for the study of harmonic bundles and pure tw…

In this note, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space $\bar{\mathcal{M}}$ of smoothable Kahler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson current, which exists as a positive (1,1)-current on $\bar{\mathcal{M}}$ an…

The paper extends orientability results for DT invariants on Calabi-Yau 4-folds.

We introduce generalized Monge-Ampère capacities and use these to study complex Monge-Ampère equations whose right-hand side is smooth outside a divisor. We prove, in many cases, that there exists a unique normalized solution which is smooth outside the divisor.

Consider a smooth, projective family of canonically polarized varieties over a smooth, quasi-projective base manifold Y, all defined over the complex numbers. It has been conjectured that the family is necessarily isotrivial if Y is special in the sense of Campana. We prove the conjecture when Y is a surface or threefo…

Study torsion-free nilpotent fundamental groups of smooth varieties up to rank 7.

Proves cohomology theorems for tropical varieties.

Study on positivity of CM line bundles on moduli space of klt good minimal models with κ=1.

Study on how incomplete smooth metrics degenerate from asymptotically conical Ricci-flat Kähler metrics.

We survey the cohomology jumping loci and the Alexander-type invariants associated to a space, or to its fundamental group. Though most of the material is expository, we provide new examples and applications, which in turn raise several questions and conjectures. The jump loci of a space X come in two basic flavors: th…

We consider an extension of the results of S. Bando, R. Kobyashi, G. Tian, and S. T. Yau on the existence of Ricci-flat Kähler metrics on quasi-projective varieties Y=X\D with α[D]=c_1(X), α>1. The requirement that D admit a Kähler-Einstein metric is generalized to the condition that the link S in the normal bundle of …

Let $π: \mathcal{X}^* \rightarrow B^*$ be an algebraic family of compact Kähler manifolds of complex dimension $n$ with negative first Chern class over a punctured disc $B^*\in \mathbb{C}$. Let $g_t$ be the unique Kähler-Einstein metric on $\mathcal{X}_t= π^{-1}(t)$. We show that as $t\rightarrow 0$, $(\mathcal{X}_t, g…

We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized $\mathbb{Z}$-variation of Hodge structure $\mathbb{V}$ on a smooth complex quasi-projective variety $S$, are topologically tame. As an easy corollary of these results and of Peterzil-Starchenko's o-…

We reformulate the construction of Kontsevich's completion and use Lawson homology to define many new motivic invariants. We show that the dimensions of subspaces generated by algebraic cycles of the cohomology groups of two $K$-equivalent varieties are the same, which implies that several conjectures of algebraic cycl…

It was shown by Samelson and Wang that each compact Lie group K of even dimension admits left-invariant complex structures. When K has odd dimension it admits a left-invariant CR-structure of maximal dimension. This has been proved recently by Charbonnel and Khalgui who have also given a complete algebraic description …

The study proves rigidity for mixed Hodge structures and applies to curve families.

We consider a geometric property of the closest-points projection to a geodesic in Teichmüller space: the projection is called contracting if arbitrarily large balls away from the geodesic project to sets of bounded diameter. (This property always holds in negatively curved spaces.) It is shown here to hold if and only…

We realise Stroppel's extended arc algebra in the Fukaya-Seidel category of a natural Lefschetz fibration on the generic fiber of the adjoint quotient map on a type $A$ nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein-Gelfan'd-Gelf…

Develops quantization for non-compact complex manifolds with spectral gap.