arXiv research

Existence of metrics on non-Kähler varieties, generalizing previous work.

Investigates admissible metrics on compact Kähler varieties and their stability.

Paper defines new stability and metrics for complex spaces.

We show that a general $n$-dimensional polarized abelian variety $(A,L)$ of a given polarization type and satisfying $h^0(A, L) \geq \dfrac{8^n}{2} \cdot \dfrac{n^n}{n !}$ is projectively normal. In the process, we also obtain a sharp lower bound for the volume of a purely one-dimensional complex analytic subvariety i…

Establishes Hermite-Einstein metrics on complex spaces with singularities.

Let $X_{\mathbb{C}}$ be a smooth real affine variety with compact real points $X_{\mathbb{R}}$. We show that $X_{\mathbb{C}}$ is diffeomorphic to the normal bundle of $X_{\mathbb{R}}$ provided that $X_{\mathbb{C}}$ admits a complete Riemannian metric of nonnegative sectional curvature which is also invariant under the …

The paper studies fundamental groups of compact Kahler varieties with nef anti-canonical bundles.

Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces proved.

Constructs Morse homology for complex algebraic varieties.

The paper proves the existence of singular cscK metrics on smoothable varieties.

Let $Y$ be a compact Kähler normal space and $α\in H^{1,1}(Y,\mathbb{R})$ a Kähler class. We study metric properties of the space $\mathcal{H}_α$ of Kähler metrics in $α$ using Mabuchi geodesics. We extend several results by Calabi, Chen, Darvas previously established when the underlying space is smooth. As an applicat…

The paper proves inequalities for orbifold second Chern classes in Fujiki's class.

Formula for sections on complex manifolds with non-isolated components.

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

We study the Gauss map and the dual variety of a real-analytic immersion of a connected compact real-analytic manifold into a sphere or into a hyperbolic space. The dual variety is defined to be the set of all normal directions of the immersion. First, we show that the image of the Gauss map characterizes the manifold.…

Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings. We study the variety of unitary representations of the fundamental group of U with certain restrictions related to the divisor. We show that the possible singularities of this variety as well as of the corresponding modul…

The paper proves a theorem and characterizes connections over normal varieties.

Constructs moduli spaces for Calabi-Yau cones and Sasaki-Einstein manifolds.

New connections on symmetric spaces with invariant properties.

Studying the (long-term) behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampére equations. The purpose of this article, the second of a series on this subject, is to develop a viscosity theory for degenerate complex Mong…

Solves Monge-Ampère equations on compact Hessian manifolds using the Perron method.

The paper proves a Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles.

The paper addresses deformations of Kähler spaces with vanishing first Chern class.

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. …

In this paper we deal with singular varieties of bounded mean curvature in the viscosity sense. They contain all varifolds of bounded generalized mean curvature. In the first part we investigate the second-order properties of these varieties, obtaining results that are new also in the varifold's setting. In particular …

We study the rate of growth of normalized Hodge numbers along a tower of abelian covers of a smooth projective variety with semismall Albanese map. These bounds are in some cases optimal. Moreover, we compute the $L^2$-Betti numbers of irregular varieties that satisfy the weak generic Nakano vanishing theorem e.g., var…

Given a 3 manifold M with torus boundary and an ideal triangulation, Yoshida and Tillmann give different methods to construct surfaces embedded in M from ideal points of the deformation variety. Yoshida builds a surface from twisted squares whereas Tillmann produces a spun-normal surface. We investigate the relation be…

Logarithmic connections on principal bundles over normal varieties are studied.

Galois action on manifold structures of complex varieties is abelian.

We survey Lagrangian fibrations of holomorphic symplectic varieties, both compact and non-compact, whose fibres are Jacobians and Prym varieties.

Paper describes holomorphic polyvector fields on toric varieties.

New proof shows almost all surface group actions are dense.

We construct normal rationally connected varieties (of arbitrarily large dimension) not containing any smooth rational curves.

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

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

A natural oriented (2k+2)-chain in CP^{2k+1} with boundary twice RP^{2k+1}, its complex shade, is constructed. Via intersection numbers with the shade, a new invariant, the shade number of k-dimensional subvarieties with normal vector fields along their real part, is introduced. For an even-dimensional real variety, th…

Proves Torelli group action is ergodic on Lie group character varieties.

We investigate a method of construction of Calabi--Yau manifolds, that is, by smoothing normal crossing varieties. We develop some theories for calculating the Picard groups of the Calabi--Yau manifolds obtained in this method. Some applications are included, such as construction of new examples of Calabi--Yau 3-folds …

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

Segre varieties' hyperplane sections are unstable under certain conditions.

Characterizes structures preserved by groups in high-dimensional spaces.

Compact character varieties of punctured spheres are proven.

Proves properties of complex algebraic varieties and local systems.

3D hyperbolic manifolds map one-to-one to their boundary character varieties.

This expository monograph cuts a short path from the common, elementary background in geometry (linear algebra, vector bundles, and algebraic ideals) to the most advanced theorems about involutive exterior differential systems: (1) The incidence correspondence of the characteristic variety, (2) Guillemin normal form an…

Let $S$ be a smooth rational curve on a complex manifold $M$. It is called ample if its normal bundle is positive. We assume that $M$ is covered by smooth holomorphic deformations of $S$. The basic example of such a manifold is a twistor space of a hyperkahler or a 4-dimensional anti-selfdual Riemannian manifold $X$ (n…

Compact parameterization improves Bayesian neural network performance.

Let $(X,ω)$ be a compact normal Kähler space, with Hodge metric $ω$. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical Kähler metrics, we introduce a geodesic metric structure on $\mathcal H_ω(X)$, the space of Kähler potentials, whose completion is the fin…