arXiv research

Rectifies zero loci of certain spinors in 4D manifolds.

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…

Paper calculates virtual dimension of spinor zero loci on 3-manifolds.

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

We describe and study the loci equidistant from finitely many points in the so-called complex hyperbolic geometry, i.e., in the geometry of a holomorphic $2$-ball $\Bbb B$. In particular, we show that the bisectors (= the loci equidistant from $2$ points) containing the (smooth real algebraic) curve equidistant from gi…

Study Fuchsian loci in ${ m PSL}_n(\mathbb{R})$-Hitchin components of a pair of pants.

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

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

We study topology change in (2+1)D gravity coupling with non-Abelian SO(2,1) Higgs field from the point of view of Morse theory. It is shown that the Higgs potential can be identified as a Morse function. The critical points of the latter (i.e. loci of change of the spacetime topology) coincide with zeros of the Higgs …

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.

Study shows no hyperkähler fourfolds in specified conditions.

Study homotopy groups in GIT quotients using transversality methods.

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…

Classifies Teichmüller curves in specific hyperelliptic components of meromorphic differentials.

Study shows finiteness of Teichmueller curves in specific Prym loci.

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

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

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.

Models for 3D harmonic 1-forms and spinors near singular points.

The paper studies conjugate loci on ellipsoids and Liouville manifolds.

Study elliptic isometries on a matrix manifold with specific metrics.

Surveying recent results on the geometry of Jacobian loci.

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

Study cohomology jump loci in 3-manifolds, focusing on Alexander polynomial.

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

Study geodesics on a special cylinder with arbitrary wind.

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

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…

Let $M$ be a compact oriented 3-dimensional smooth manifold. In this paper, we construct a moduli space consisting of pairs $(Σ, ψ)$ where $Σ$ is a $C^1$-embedding simple closed curve in $M$, $ψ$ is a $\mathbb{Z}/2$-harmonic spinor vanishing only on $Σ$, and $\|ψ\|_{L^2_1}\neq 0$. We prove that when $Σ$ is $C^2$, a nei…

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

This paper classifies components of meromorphic differential strata.

The paper examines critical points in Higgs bundle moduli spaces.

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 on algebraic fiber spaces and their anti-canonical divisors.

Study of Randers metrics on spheres with simple cut loci.

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

Classifies connected components of meromorphic differentials with residue conditions.

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 …

Uniformizing maps and period maps are topologically tame, leading to algebraicity of Hodge loci.

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.

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

We study the moduli space of SU(3) structure manifolds X that form the internal compact spaces in four-dimensional N=1/2 domain wall solutions of heterotic supergravity with flux. Together with the direction perpendicular to the four-dimensional domain wall, X forms a non-compact 7-manifold Y with torsionful G2 structu…

The study characterizes a complex curve of residueless meromorphic differentials on elliptic curves.

Study the boundary of Riemann surfaces with abelian automorphisms.