arXiv research

Paper describes holomorphic polyvector fields on toric varieties.

Study BV operators on holomorphic polyvector fields on toric varieties.

Classifies holomorphic parabolic geometries on complex manifolds.

Given a singular Schubert variety Z in a compact Hermitian symmetric space it is a longstanding question to determine when Z is homologous to a smooth variety Y. We identify those Schubert varieties for which there exist first-order obstructions to the existence of Y. This extends (independent) work of M. Walters, R. B…

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

We present a simple proof of a precise version of the localization theorem in equivariant cohomology. As an application, we describe the cohomology algebra of any compact symplectic variety with a multiplicity-free action of a compact Lie group. This applies in particular to smooth, projective spherical varieties.

The paper studies stability of CR structures on compact manifolds.

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

We prove a criterion for the existence of harmonic metrics on Higgs bundles that are defined on smooth loci of klt varieties. As one application, we resolve the quasi-etale uniformisation problem for minimal varieties of general type to obtain a complete numerical characterisation of singular quotients of the unit ball…

Gauduchon's theorem extended to singular spaces with smoothing.

Strong formal properties for toric and homogeneous Kähler manifolds.

For every fibration $f : X \to B$ with $X$ a compact Kähler manifold, $B$ a smooth projective curve, and a general fiber of $f$ an abelian variety, we prove that $f$ has an algebraic approximation.

Paper proves structure for compact Kähler manifolds with pseudo-effective tangent bundles.

Proves tropical Hodge theory for smooth projective varieties, conditional on Laplacian regularity.

We give a numerical characterization of the Kahler cone of a possibly singular compact analytic variety which is embedded in a smooth ambient space.

We show that if a holomorphic $n$ dimensional compact torus action on a compact connected complex manifold of complex dimension $n$ has a fixed point then the manifold is equivariantly biholomorphic to a smooth toric variety.

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 …

Three-dimensional smooth, compact toric varieties (SCTV), when viewed as real six-dimensional manifolds, can admit G-structures rendering them suitable for internal manifolds in supersymmetric flux compactifications. We develop techniques which allow us to systematically construct G-structures on SCTV and read off thei…

Consider a sequence of minimal varieties M_i in a Riemannian manifold N such that the boundary measures are uniformly bounded on compact sets. Let Z be the set of points at which the areas of the M_i blow up. We prove that Z behaves in some ways like a minimal variety without boundary: in particular, it satisfies the s…

We initiate the study of holomorphically convex groups: groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers. If $G$ is a holomorphically convex group of cohomological dimension two, we show that $G$ is isomorphic to the fundamental group …

New SKT manifolds created using toric geometry.

Constructs Morse homology for complex algebraic varieties.

Let $X$ be a smooth manifold belonging to one of these three collections: acyclic manifolds (compact or not, possibly with boundary), compact connected manifolds (possibly with boundary) with nonzero Euler characteristic, integral homology spheres. We prove that $Diff(X)$ is Jordan. This means that there exists a const…

We use gauge theoretic and algebraic methods to examine sufficient conditions for smooth points on the moduli space of flat connections on a compact manifold and on the character variety of a finitely generated and presented group. We give a complete proof of the slice theorem for the action of the group of gauge trans…

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…

Given a compact Riemann surface $X$ and a semisimple affine algebraic group $G$ defined over $\mathbb C$, there are moduli spaces of Higgs bundles and of connections associated to $(X,\, G)$. We compute the Brauer group of the smooth locus of these varieties.

We study Lagrangian subalgebras of a semisimple Lie algebra with respect to the imaginary part of the Killing form. We show that the variety $\Lagr$ of Lagrangian subalgebras carries a natural Poisson structure $Π$. We determine the irreducible components of $\Lagr$, and we show that each irreducible component is a smo…

We propose a way to define and compute invariants of general smooth 4-manifolds based on topological twists of non-Lagrangian 4d N=2 and N=3 theories in which the problem is reduced to a fairly standard computation in topological A-model, albeit with rather unusual targets, such as compact and non-compact Gepner models…

Smooth maps bound Betti numbers of zero sets.

A group morphism is constructed, which can be realized as the induced morphism of fundamental groups from a holomorphic map between compact Kahler manifolds, but can not be realized by a holomorphic map between smooth projective varieties. And it is also proved that there exists no such example between abelian groups.

We characterise the virtually abelian groups which are fundamental groups of compact Kähler manifolds and of smooth projective varieties. We show that a virtually abelian group is Kähler if and only if it is projective. In particular, this allows to describe the Kähler condition for such groups in terms of integral sym…

The equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group. We prove that toric manifolds (i.e. compact smooth toric varieties) are isomorphic as varieties if and only if their equivariant cohomology algebras are…

Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomog…

We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted $\barpartial$-Laplacian on compact Kähler manifolds. This situation typically occurs for a sequence of Fano manifolds with anticanonical Kähler class. We apply it to show that, if an almost smooth Fano-Ricci limit space admits a Kähler…

The traceless $SU(2)$ character variety $R(S^2,\{a_i,b_i\}_{i=1}^n)$ of a $2n$-punctured 2-sphere is the symplectic reduction of a Hamiltonian $n$-torus action on the $SU(2)$ character variety of a closed surface of genus $n$. It is stratified with a finite singular stratum and a top smooth symplectic stratum of dimens…

Let G be a complex reductive group and K a maximal compact subgroup. If X is a smooth projective G-variety, with a fixed (not necessarily integral) K-invariant Kaehler form, then the K-action is Hamiltonian. Let M be the zero fiber of the corresponding moment map. It is well known that the quotient M/K is a complex spa…

In this paper we prove geometric residue theorems for bundle maps over a compact manifold. The theory developed associates residues to the singularity submanifolds of the map for any invariant polynomial. The theory is then applied to a variety of settings: smooth maps between equidimensional manifolds, CR-singularitie…

The paper proves the existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.

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

The study classifies complex smooth Fano varieties with large pseudoindex.

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

A natural family of affine cubic surfaces arises from SL(2)-characters of the 4-holed sphere and the 1-holed torus. The ideal locus is a tritangent plane which is generic in the sense that the cubic curve at infinity consists of three lines pairwise intersecting in three double points. We show that every affine cubic s…

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

We prove that if $K$ is a compact subset of an affine variety O = P^n - D (where D is a projective hypersuface), and if K is a compact subset of a closed analytic subvariety V \subset O, then the projective hull K^ of K has the property that K^ \cap O is contained in V. If V is smooth and 1-dimensional, then K^ \cap O …

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 …

For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We show that this twisted complex is a functorial invariant of the isotopy class of the tangle, and that it provides a factoriz…

The paper trivializes moment maps for various geometric structures.

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…