arXiv research

Mess showed that the genus 2 Torelli group $T_2$ is isomorphic to a free group of countably infinite rank by showing that genus 2 Torelli space is homotopy equivalent to an infinite wedge of circles. As an application of his computation, we compute the homotopy type of the zero locus of any classical genus 2 theta func…

Normal forms for Q-structures on graded manifolds explained.

A conformal geometry determines a distinguished, potentially singular, variant of the usual Yamabe problem, where the conformal factor can change sign. When a smooth solution does change sign, its zero locus is a smoothly embedded separating hypersurface that, in dimension three, is necessarily a Willmore energy minimi…

This article proves that the zero locus of a $\mathbb{Z}/2$ harmonic spinor on a 4 dimensional manifold is 2-rectifiable and has locally finite Minkowski content.

We characterize convex cocompact subgroups of the mapping class group of a surface in terms of uniform convergence actions on the zero locus of the limit set. We also construct subgroups that act as uniform convergence groups on their limit sets, but are not convex cocompact.

We characterize all LVMB manifolds X such that the holomorphic tangent bundle TX is spanned at the generic point by a family of global holomorphic vector fields, each of them having non-empty zero locus. We deduce that holomorphic connections on semi-stable holomorphic vector bundles over LVMB manifolds with this previ…

The classification of homogeneous compact Einstein manifolds in dimension six is an open problem. We consider the remaining open case, namely left-invariant Einstein metrics $g$ on $G = \mathrm{SU}(2) \times \mathrm{SU}(2) = S^3 \times S^3$. Einstein metrics are critical points of the total scalar curvature functional …

In this paper, we prove that the zero-locus of any global holomorphic log-one-form on a projective log-smooth pair $\left(X,D\right)$ of log-general type must be non-empty. Applying this result, we give an answer to the algebraic hyperbolicity part of Shafarevich's conjecture, with the generic fiber being Kawamata-log-…

Formula calculates homology groups of Milnor fibres for real hypersurface singularities.

We prove that the Euler form of a metric connection on real oriented vector bundle $E$ over a compact oriented manifold $M$ can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilisticall…

Proves non-positivity of Hirzebruch form on stable weights and connects to flat logarithmic connections.

We prove that a Gaussian ensemble of smooth random sections of a real vector bundle over compact manifold canonically defines a metric on the bundle together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet-Chern theorem stating that if the bundle and the manifold are oriented, then t…

This article describes various moduli spaces of pseudoholomorphic curves on the symplectization of a particular overtwisted contact structure on S^1 x S^2. This contact structure appears when one considers a closed self dual form on a 4-manifold as a symplectic form on the complement of its zero locus. The article is f…

Study parallel tractors and cotractors on almost Grassmannian structures.

For any compact oriented manifold $M$, we show that that the top degree multi-vector fields transverse to the zero section of $\wedge^{\text{top}}TM$ are classified, up to orientation preserving diffeomorphism, in terms of the topology of the arrangement of its zero locus and a finite number of numerical invariants. Th…

We use the symmetries of the tetrahedron, octahedron and icosahedron to construct local models for a $\mathbb{Z}/2$ harmonic 1-form or spinor in 3-dimensions near a singular point in its zero loci. The local models are $\mathbb{Z}/2$ harmonic 1-forms or spinors on $\mathbb{R}^3$ that are homogeneous with respect to res…

Real algebraic structures help classify overtwisted contact 3-spheres.

In this paper we construct an explicit representative for the Grothendieck fundamental class [Z] of a complex submanifold Z of a complex manifold X, under the assumption that Z is the zero locus of a real analytic section of a holomorphic vector bundle E. To this data we associate a super-connection A on the exterior a…

Study shows no hyperkähler fourfolds in specified conditions.

We describe the range of the Radon transform on the space $M$ of irreducible conics in $\CP^2$ in terms of natural differential operators associated to the $SO(3)$-structure on $M=SL(3, \R)/SO(3)$ and its complexification. Following \cite{moraru} we show that for any function $F$ in this range, the zero locus of $F$ is…

The paper constructs Gromov-Hausdorff metrics for Lorentzian spaces and calculates dimensions.

This study proves the local existence of a symplectic gradient flow on a flat torus.

The zero locus of a function f on a graph G is defined as the graph with vertex set consisting of all complete subgraphs of G, on which f changes sign and where x,y are connected if one is contained in the other. For d-graphs, finite simple graphs for which every unit sphere is a d-sphere, the zero locus of (f-c) is a …

Introduces gauge theory for string algebroids, solving Calabi system.

Study shows Futaki invariant vanishes on most Fano threefolds.

We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type opera…

Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, this interpretation allows one to pose questions that would not have been asked otherwi…

Extended signatures help distinguish non-concordant links.

The paper proves an isomorphism between $tt^*$ structures of Landau-Ginzburg and Calabi-Yau models.

The paper constructs non-convergent solutions to Vafa-Witten equations with specific harmonic 2-form limits.

The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial $A(x,y)$. Another "family version" of the volume conjecture depends on a quantization parameter, usually denoted $q$ or $\hbar$; this quan…

For curved projective manifolds we introduce a notion of a normal tractor frame field, based around any point. This leads to canonical systems of (redundant) coordinates that generalise the usual homogeneous coordinates on projective space. These give preferred local maps to the model projective space that encode geome…

Simplified computation of SFT invariants for Legendrian links.

Overview of algebraic geometry for almost complex manifolds.

Let X(Σ) be a smooth projective toric variety for a complex torus T_\C. In this paper, a real T_\C-invariant Poisson structure Π_Σis constructed on the complex manifold X(Σ), the symplectic leaves of which are the T_\C-orbits in X(Σ). It is shown that each leaf admits a Hamiltonian action by a sub-torus of the compact …

We show that the boundaries of thin strongly pseudoconvex Grauert tubes, with respect to the Guillemin-Stenzel Kähler metric canonically associated with the Poincaré metric on closed hyperbolic real-analytic surfaces, has nowhere vanishing Cartan CR-curvature. This result provides a wealth of examples of compact $3$-di…

The set of unrestricted homotopy classes $[M,S^n]$ where $M$ is a closed and connected spin $(n+1)$-manifold is called the $n$-th cohomotopy group $π^n(M)$ of $M$. Moreover it is known that $π^n(M) = H^n(M;\mathbb Z) \oplus \mathbb Z_2$ by methods from homotopy theory. We will provide a geometrical description of the $…

We introduce and compute a 2-parameter family deformation of the A-polynomial that encodes the color dependence of the superpolynomial and that, in suitable limits, reduces to various deformations of the A-polynomial studied in the literature. These special limits include the t-deformation which leads to the "refined A…

First BGG operators are a large class of overdetermined linear differential operators intrinsically associated to a parabolic geometry on a manifold. The corresponding equations include those controlling infinitesimal automorphisms, higher symmetries, and many other widely studied PDE of geometric origin. The machinery…

Study of symplectic Monge-Ampère equations using moment maps and contact structures.

In previous work with M.C. Fernandes, we found a Lie algebroid symmetry for the Einstein evolution equations of general relativity. The present work was motivated by the effort to explain the coisotropic structure of the constraint subset for the initial value problem by extending the notion of hamiltonian structure fr…

The paper develops a comprehensive theory of submanifolds in conformal geometries.