arXiv research

The article examines twisted cohomologies on algebraic and analytic varieties.

By attaching a Lie algebra of germs of analytic vector fields to every point of a (real or complex) analytic variety V we construct the Nagano foliation of the variety. We prove that the Nagano foliation of V is a stratification. The treatment of the subject is totally coordinate free but relies on the Oka-Cartan-Serre…

The article proves a complex analytic inequality for stable Q-sheaves on Kähler varieties.

For a lattice $Γ$ of a simply connected solvable Lie group $G$, we describe the analytic germ in the variety of representations of $Γ$ at the trivial representation as an analytic germ which is linearly embedded in the analytic germ associated with the nilpotent Lie algebra determined by $G$. By this description, under…

Proves Kähler-Ricci shrinkers are complex analytic varieties.

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.…

A geometric construction of Sullivan's Stiefel-Whitney homology classes of a real analytic variety $X$ is given by means of the conormal cycle of an embedding of $X$ in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. W…

Study of twisted $L^2$-torsion on 3-manifold character varieties.

Introduces stability conditions for polarized varieties, linking to K-stability.

We prove that refined analytic torsion on a manifold with boundary is an analytic section of the determinant line bundle over the representation variety. As a fundamental application we establish a gluing formula for refined analytic torsion on connected components of the complex representation space which contain a un…

Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re(f) is Morse, we prove the existence of a stratified gradient-like vector field for Re(f) such that the unstable set of a critical point p on a stratum S of complex dimension s has real dimensi…

A mapping bends Teichmüller spaces into character varieties, preserving symplectic structure.

We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori.

For a G-invariant holomorphic 1-form with an isolated singular point on a germ of a complex-analytic G-variety with an isolated singular point (G is a finite group) one has notions of the equivariant homological index and of the (reduced) equivariant radial index as elements of the ring of complex representations of th…

Explains complex analytic invariants of vector fields and foliations.

We extend the holomorphic analytic torsion classes of Bismut and Köhler to arbitrary projective morphisms between smooth algebraic complex varieties. To this end, we propose an axiomatic definition and give a classification of the theories of generalized holomorphic analytic torsion classes for arbitrary projective mor…

Analyzes Kähler-Einstein metrics on families of Fano varieties.

Characterizes metabelian distributions and geodesics in sub-Riemannian manifolds.

Geodesics in 3D space with 1-2 analytic obstacles, proving geodesic independence.

The study shows boundedness of certain fibered varieties in algebraic geometry.

Complete, conformally flat metrics of constant positive scalar curvature on the complement of $k$ points in the $n$-sphere, $k \ge 2$, $n \ge 3$, were constructed by R\. Schoen [S2]. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop…

The paper examines bi-Lipschitz triviality of function germs on singular varieties.

Study non-existence of complex ball quotients in Torelli locus.

We discuss replica analytic continuation using several simple models in order to prove mathematically the validity of replica analysis, which is used in a wide range of fields related to large scale complex systems. While replica analysis consists of two analytical techniques, the replica trick (or replica analytic con…

In 1969, P. Deligne and D. Mumford compactified the moduli space of curves. Their compactification is a projective algebraic variety, and as such, it has an underlying analytic structure. Alternatively, the quotient of the augmented Teichmueller space by the action of the mapping class group gives a compactification of…

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

Study on metrics with singularities on spheres, showing moduli space structure.

We classify codimension two analytic submanifolds X of projective space having the property that any line through a general point p having contact to order two with X at p automatically has contact to order three. We give applications to the study of the Debarre--de Jong conjecture, and of n-dimensional varieties whose…

Let $X$ be a hyperkaehler manifold. Trianalytic subvarieties of $X$ are subvarieties which are complex analytic with respect to all complex structures induced by the hyperkaehler structure. Given a 2-dimensional complex torus $T$, the Hilbert scheme $T^{[n]}$ classifying zero-dimensional subschemes of $T$ admits a hype…

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

Let $X\subset P^N$ be a variety (respectively a patch of an analytic submanifold) and let $x\in X$ be a general point. We show that if the projective second fundamental form of $X$ at $x$ is isomorphic to the second fundamental form of a point of a Segre $P^n\times P^m$, $n,m\geq 2$, a Grassmaniann $G(2,n+2)$, $n\geq 4…

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…

For a smooth strictly pseudoconvex hypersurface in a complex manifold, we give a necessary and sufficient condition for being CR-diffeomorphic to a real-analytic CR manifold. Our condition amounts to a holomorphic extension property for the canonically associated function expressing $2$-jets of the formal Segre varieti…

Review of Gerber-Shiu function for practical actuarial science.

Proves properties of complex algebraic varieties and local systems.

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…

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

This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated characteristic classes, under morphisms of projective algebraic varieties. The formulas …

Study of Hermitian metrics on Lie algebroids over complex spaces.

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

We prove inequalities for Laplace eigenvalues of Kaehler manifolds generalising to higher eigenvalues the classical inequality for the first Laplace eigenvalue due to Bourguignon, Li, and Yau in 1994. We also obtain similar inequalities for analytic varieties in Kaehler manifolds.

Let K be an algebraically closed field endowed with a complete non-archimedean norm with valuation ring R. Let f:Y -> X be a map of K-affinoid varieties. In this paper we study the analytic structure of the image f(Y) in X; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are p…

New geometric conditions ensure compactness of $\bar{\partial}$-Neumann problem.

We show that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a non-trivial special test configuration preserving the stability threshold. When such a divisorial valuation exists, we show that the Fano variety degenerates to a uniquely determined twisted K-polystable Fan…

The paper studies coherent sheaves on subvarieties of Hopf manifolds.

Defines spectral varieties for non-simply connected manifolds and constructs conformal invariants.

The study of projective varieties with nef anticanonical divisors and log terminal singularities.

Study Fano fibrations and Kähler-Einstein metrics on their bases.