arXiv research

Study pseudoholomorphic maps using canonical connection.

New real algebraic maps with prescribed images and compositions are constructed locally like moment maps.

Canonical maps connect complex structures to Hitchin components.

This paper constructs real algebraic maps that are topologically special generic maps.

The author studies regions foliated by 1D families of functions and their applications.

We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endowed with a canonical proper symmetry. Our results generalize the constructions of Marle and Guillemin and Sternberg for canonical symmetries that have an associated momentum map. In these papers the momentum map played a cru…

The paper calculates the second variation of energy functions for families of canonically polarized manifolds.

We explore the intrinsic geometry of tangent bundles and properties of the mirror map.

Injectivity proven for measure homology of certain wild spaces.

We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the target is a point. We give some examples, such as Kodaira embeddings and fibrations. We prove the existence of a projective…

Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective threefold Y that admits a no…

The note proves positive currents induced by VKE with mixed singularities.

The study characterizes and analyzes spacelike surfaces with a canonical normal null direction in Minkowski 4-space.

We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to Fulton-MacPherson and Axelrod-Singer in the setting of smooth manifolds, as well as a simplicial variant of this compactification initiated by Kontsevich. Our constructions are elementary a…

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

We study a discrete dynamical system designed to find a 'most holomorphic' connection on a smooth complex vector bundle $E$. We examine the relation between the distance of the chern classes of $E$ from the $(p,p)$ axis of the Hodge diamond and singularity formation. Canonical connections and canonical metrics pulled b…

Solves complex Hessian equations in unstable cases, proving unique canonical solutions with singularities.

Study of endperiodic maps on infinite graphs, proving homotopy and eigenvalue properties.

Every action on a Poisson manifold by Poisson diffeomorphisms lifts to a Hamiltonian action on its symplectic groupoid which has a canonically defined momentum map. We study various properties of this momentum map as well as its use in reduction.

We prove that the mirror map is trivial for the canonical formal families of Calabi-Yau varieties constructed by Gross and the second author. In other words, the natural coordinate in a canonical Calabi-Yau family is a canonical coordinate in the sense of Hodge theory. This implies that the higher weight periods direct…

The paper shows how different geodesic flows on surfaces can be mapped to each other.

Every surface bundle with genus $g$ fiber has a canonical Heegaard splitting of genus $2g+1$. We classify the mapping class groups of such Heegaard splittings in the case when the surface bundle has a sufficiently complicated monodromy map.

This work improves understanding of symmetrizing Bregman divergences on positive definite matrices.

Uniformizes varieties with log-canonical singularities using ball quotients.

We study subelliptic biharmonic maps, i.e. smooth maps from a compact strictly pseudoconvex CR manifold M into a Riemannian manifold N which are critical points of a certain bienergy functional. We show that a map is subelliptic biharmonic if and only if its vertical lift to the (total space of the) canonical circle bu…

We prove that the mapping torus group $\FN \rtimes_α \Z$ of any automorphism $α$ of a free group $\FN$ of finite rank $n \geq 2$ is weakly hyperbolic relative to the canonical (up to conjugation) family $\mathcal H(α)$ of subgroups of $\FN$ which consists of (and contains representatives of all) conjugacy classes that …

Abstract: Generalizes supergravity c-map to quaternionic manifolds.

We define a canonical map from a certain space of laminations on a punctured surface into the quantized algebra of functions on a cluster variety. We show that this map satisfies a number of special properties conjectured by Fock and Goncharov. Our construction is based on the "quantum trace" map introduced by Bonahon …

For modelling of various physical processes, geodesic lines and almost geodesic curves serve as a useful tool. Trasformations or mappings between spaces (endowed with a metric or connection) which preserve such curves play an important role in physics, particularly in mechanics, and in geometry as well. Our aim is to c…

This paper studies the Fisher-Rao geometry on the parameter space of beta distributions. We derive the geodesic equations and the sectional curvature, and prove that it is negative. This leads to uniqueness for the Riemannian centroid in that space. We use this Riemannian structure to study canonical moments, an intrin…

A Reeb space is defined as the space of all the connected components of inverse images of a smooth map, which is a fundamental tool in studying smooth manifolds using generic smooth maps whose codimensions are not positive such as Morse functions, their higher dimensional versions including fold maps and general stable…

DICCA maps multi-view data into a shared latent space with interpretable components.

We give necessary and sufficient geometric conditions for a distribution (or a Pfaffian system) to be locally equivalent to the canonical contact system on Jn(R,Rm), the space of n-jets of maps from R into Rm. We study the geometry of that class of systems, in particular, the existence of corank one involutive subdistr…

In this article, we propose two algorithms for determining the Nielsen-Thurston classification of a mapping class $ψ$ on a surface $S$. We start with a finite generating set $X$ for the mapping class group and a word $ψ$ in $\langle X \rangle$. We show that if $ψ$ represents a reducible mapping class in $\Mod(S)$ then …

Let f:Σ_1 --> Σ_2 be an area preserving diffeomorphism between compact Riemann surfaces of constant curvature. The graph of f can be viewed as a Lagrangian submanifold in Σ_1\times Σ_2. This article discusses a canonical way to deform f along area preserving diffeomorphisms. This deformation process is realized through…

Geometrically classifies maps from R^0|2 to any manifold, unifying theories.

Study of homeomorphisms on infinite type surfaces with a classification theorem.

We analyse finite-time singularities of the Teichmüller harmonic map flow -- a natural gradient flow of the harmonic map energy -- and find a canonical way of flowing beyond them in order to construct global solutions in full generality. Moreover, we prove a no-loss-of-topology result at finite time, which completes th…

The paper bounds eigenvalues of specific operators on certain manifolds.

We use recent results by Bainbridge-Chen-Gendron-Grushevsky-Moeller on compactifications of strata of abelian differentials to give a comprehensive solution to the realizability problem for effective tropical canonical divisors in equicharacteristic zero. Given a pair $(Γ, D)$ consisting of a stable tropical curve $Γ$ …

New $L_\infty$ liftings derived from Chern-Simons classes for coherent sheaves.

Study of Pascal algebra matrices and their jet bundle map for vector bundles.

We classify all harmonic maps with finite uniton number from a Riemann surface into an arbitrary compact simple Lie group $G$, whether $G$ has trivial centre or not, in terms of certain pieces of the Bruhat decomposition of the group $Ω_\mathrm{alg}{G}$ of algebraic loops in $G$ and corresponding canonical elements. Th…

We study a geometrical condition (PHWC) which is weaker than horizontal weak conformality. In particular, we show that harmonic maps satisfying this condition, which will be called {\em pseudoharmonic morphisms}, include harmonic morphisms and can be described as pulling back certain germs to certain other germs. Final…

During the last thirty years, symplectic or Marsden--Weinstein reduction has been a major tool in the construction of new symplectic manifolds and in the study of mechanical systems with symmetry. This procedure has been traditionally associated to the canonical action of a Lie group on a symplectic manifold, in the pr…

Suppose that the 3-manifold M is given by integral surgery along a link L in S^3. In the following we construct a stable map from M to the plane, whose singular set is canonically oriented. We obtain upper bounds for the minimal numbers of crossings and non-simple singularities and of connected components of fibers of …

This paper investigates analytic properties of maps between hyperbolic surfaces, focusing on best Lipschitz maps and geodesic laminations.

Calculates Dehn twist actions on conformal blocks for modular categories.