arXiv research

New potentials found for sheaves on Calabi-Yau 4-folds.

Study real semi-stable degenerations and describe real loci via blow-ups.

Study homotopy groups in GIT quotients using transversality methods.

Study the boundary of Riemann surfaces with abelian automorphisms.

Study on algebraic fiber spaces and their anti-canonical divisors.

The study characterizes wobbly rank-2 bundles on Riemann surfaces using spectral curves.

A $k$-differential on a Riemann surface is a section of the $k$-th power of the canonical line bundle. Loci of $k$-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification of the moduli space of $k$-differentials. In this paper we give a complete description for the compa…

Smooth torus actions on moduli spaces of super stable curves and maps of genus zero.

Using stable log maps, we introduce log twisted differentials extending the notion of abelian differentials to the Deligne-Mumford boundary of stable curves. The moduli stack of log twisted differentials provides a compactification of the strata of abelian differentials. The open strata can have up to three connected c…

Conflict sets are loci of intersecting wavefronts emanating from $l$ different surfaces. We show that generically conflict sets are Legendrian: locally they admit the structure of wavefronts. Simple stable singularities for this problem in $\R^n$ occur when $0 \leq n-l \leq 4$. Other related sets, such as kite curves a…

Constructs a Morse-Bott function on symplectic Grassmannians.

Researchers create explicit representations for skein algebras of small surfaces, revealing their Azumaya loci.

In this note, we describe a procedure to construct generalized complex structures with an arbitrarily large number of type change loci on products of the circle with a connected sum of closed 3-manifolds. The loci need not be isotopic.

Study geodesics and shortest arcs on Lie groups with specific metrics.

In this paper, we study Fuchsian loci of ${\rm PSL}_n(\mathbb{R})$-Hitchin components. In particular, using the Bonahon-Dreyer parametrization of ${\rm PSL}_n(\mathbb{R})$-Hitchin components, we give an explicit parametrization of Fuchsian loci of a pair of pants.

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…

As an increasing number of genome-wide association studies reveal the limitations of attempting to explain phenotypic heritability by single genetic loci, there is growing interest for associating complex phenotypes with sets of genetic loci. While several methods for multi-locus mapping have been proposed, it is often…

We study the rational Picard group of the projectivized moduli space of holomorphic n-differentials on complex genus g stable curves. We define (n - 1) natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of n-differe…

Study Riemannian metrics on lens spaces, find cut loci and diameters.

Study Riemannian metrics on lens spaces, find cut loci and diameters.

We introduce the natural and fairly general notion of a subanalytic bundle (with a finite dimensional vector space $P$ of sections) on a subanalytic subset $X$ of a real analytic manifold $M$, and prove that when $M$ is compact, there is a Baire subset $U$ of sections in $P$ whose zero-loci in $X$ have tubular neighbou…

New method studies discriminantal loci of algebraic varieties.

New upper bound for geodesic complexity derived from cut locus decompositions.

Study curvature loci of 3-manifolds in R^6 and R^5.

Study geodesics and shortest arcs on Lie groups with specific metrics.

Generalizes results on Bieri-Neumann-Strebel-Renz invariants and tropical varieties.

Study geodesics on a special cylinder with arbitrary wind.

Let $X\subset \mathbb P^N$ be a scroll over a smooth curve $C$ and let $Ł=\mathcal O_{\mathbb P^N}(1)|_X$ denote the hyperplane bundle. The special geometry of $X$ implies that some sheaves related to the principal part bundles of $Ł$ are locally free. The inflectional loci of $X$ can be expressed in terms of these she…

Study Alexander invariants and cohomology jump loci in group extensions with trivial monodromy.

We show that some riemannian manifolds diffeomorphic to the sphere have the property that the cut loci of general points are smoothly embedded closed disks of codimension one. Ellipsoids with distinct axes are typical examples of such manifolds.

Study of Randers metrics on spheres with simple cut loci.

Constructs harmonic 1-forms on K3-fibred Calabi-Yau 3-folds.

We study the differential-geometric properties of the loci of fixed points of the elliptic isometries of the manifold of definite positive real matrices with the trace metric. We also give an explicit description of such loci and in particular we find their De Rham decomposition.

We prove a Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles $E$ and $F$ over a complex manifold under the condition that $E^*\ox F$ is Griffiths $k$-positive. We apply this result to find some homotopy groups of the Brill-Noether loci for a generic curve.

There is a remarkable type of field of two-planes special to four dimensions known as an Engel distributions. They are the only stable regular distributions besides the contact, quasi-contact and line fields. If an arbitrary two-plane field on a four-manifold is slightly perturbed then it will be Engel at generic point…

This paper is devoted to the classification of connected components of Prym eigenform loci in the strata H(2,2)^odd and H(1,1,2) in the Abelian differentials bundle in genus 3. These loci, discovered by McMullen are GL^+(2,R)-invariant submanifolds (of complex dimension 3) that project to the locus of Riemann surfaces …

Study of longest arcs and cut loci in deformed anti de-Sitter spaces.

The paper studies the locus in the rank 2 Higgs bundle moduli space corresponding to points which are critical for d of the Poisson commuting functions. These correspond to the Higgs field vanishing on a divisor of degree D. The degree D critical locus has an induced integrable system related to K(-D)-twisted Higgs bun…

New invariant $c_{sm}$ simplifies computing geometric invariants of recursive group orbits.

The study examines singularities and geometric properties of surfaces derived from frontals with specific singular points.

In this paper, we study the interplay between modules and sub-objects in holomorphic Poisson geometry. In particular, we define a new notion of "residue" for a Poisson module, analogous to the Poincaré residue of a meromorphic volume form. Of particular interest is the interaction between the residues of the canonical …

We give a new and detailed description of the structure of cut loci, with direct applications to the singular sets of some Hamilton-Jacobi equations. These sets may be non-triangulable, but a local description at all points except for a set of Hausdorff dimension $n-2$ is well known. We go further in this direction by …

The Gaiotto locus for Sp(2n) is shown to lie in the nilpotent cone.

Paper proposes BSP to find stable bimodules of cross-correlated features.

In this paper we study the invariant Carnot-Caratheodory metrics on $SU(2)\simeq S^3$, $SO(3)$ and $SL(2)$ induced by their Cartan decomposition and by the Killing form. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory dis…

The paper introduces new invariants to refine Alexander polynomials and bounds BNSR Σ-invariants.

The paper studies geometric loci and their invariants in complex dynamics.

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…