arXiv research

Proofs for Gauss map properties on surfaces with finite geometric type.

The paper calculates the Kobayashi-Royden metric on a punctured sphere and finds rational coefficients.

The paper establishes Schwarz type lemmas for holomorphic maps between pseudo-Hermitian and Hermitian manifolds.

A quaternionic version of Picard's theorem limits how many values a slice regular function can avoid.

Compactness results for Hermitian manifolds help understand Type IIB flow.

The Type IIA flow converges on symplectic manifolds, with singularity models identified.

The paper presents presentations for Euclidean Picard modular groups.

Unified approach to totally ramified values in various surface theories.

Paper generalizes Bloch-Ros principle to various surface classes.

Formalizes integral curves on Banach manifolds in Lean.

New theorem for surfaces in hyperbolic and de Sitter spaces.

Constructs explicit nontrivial cycles in Habiro cohomology of smooth varieties.

We establish a Lehto--Virtanen-type theorem and a rescaling principle for an isolated essential singularity of a holomorphic curve in a complex space, which are useful for establishing a big Picard-type theorem and a big Brody-type one for holomorphic curves.

New examples show deletion type admissible pairs can be rigid under rational saturation.

We introduce logarithmic Picard algebroids, a natural class of Lie algebroids adapted to a simple normal crossings divisor on a smooth projective variety. We show that such algebroids are classified by a subspace of the de Rham cohomology of the divisor complement determined by its mixed Hodge structure. We then solve …

Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.

Study generalizes Picard iteration for nonlinear PDEs, deriving bounds on error.

Researchers found the global topology of the Eisenstein-Picard modular surface.

Picard modular groups are shown to be generated by complex reflections.

New method finds 198,846 toric-colorable seeds of Picard number 5.

We determine the number of cusps of minimal Picard modular surfaces. The proof also counts cusps of other Picard modular surfaces of arithmetic interest. Consequently, for each N > 0 there are finitely many commensurability classes of nonuniform arithmetic lattices in SU(2, 1) that contain an N-cusped surface. We also …

We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the first part of the paper, we show that, under mild restrictions, the moduli spaces of smoo…

Method constructs fundamental domains for Picard modular groups.

We study deformation of spherical $CR$ circle bundles over Riemann surfaces of genus > 1. There is a one to one correspondence between such deformation space and the so-called universal Picard variety. Our differential-geometric proof of the structure and dimension of the unramified universal Picard variety has its own…

Heat kernel resurgent structure from Picard-Lefschetz theory

We compute the Picard group of a stable b-symplectic manifold $M$ by introducing a collection of discrete invariants $\mathfrak{Gr}$ which classify $M$ up to Morita equivalence.

Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomog…

Constructs stable bundles on K3 surfaces using monad construction.

Study extends Nirenberg-Spencer's question to families of submanifolds.

Study shows no hyperkähler fourfolds in specified conditions.

Researchers characterize a specific type of projective variety based on its tangents.

Classifies Real primary Hopf surfaces and their associated groups.

Geometric structures modeled on rational homogeneous manifolds are studied to characterize rational homogeneous manifolds and to prove their deformation rigidity. To generalize these characterizations and deformation rigidity results to quasihomogeneous varieties, we first study horospherical varieties and geometric st…

The paper classifies diffeomorphism classes of Calabi-Yau threefolds constructed by gluing methods.

We generalize Demailly's construction of projective jet bundles and strictly negatively curved pseudometrics on them to the logarithmic case. We establish this logarithmic generalization explicitly via coordinates, just as Noguchi's generalization of the jets used by Green-Griffiths. As a first application, we give a m…

This note is a report on the observation that some singular varieties admit Calabi--Yau coverings. As an application, we construct 18 new Calabi--Yau 3-folds with Picard number one that have some interesting properties.

Kähler-Einstein metrics found on special types of symmetric varieties.

Found a stable 3D shape with specific properties.

New K3 surfaces with two involutions and low Picard number constructed.

For a Poisson manifold $M$ we develop systematic methods to compute its Picard group $Pic(M)$, i.e., its group of self Morita equivalences. We establish a precise relationship between $Pic(M)$ and the group of gauge transformations up to Poisson diffeomorphisms showing, in particular, that their connected components of…

Survey connects singularity invariants to link pairings.

Computes Picard groups of complex parallelizable manifolds.

Study geometric properties of a complex hyperbolic group action.

Study calculates Ricci bounds for special Fano manifolds.

The loop space of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps from the circle $S^1$ to the sphere is an infinite dimensional complex manifold. We compute the Picard group of holomorphic line bundles on this loop space as an infinite dimensional complex Lie group with Lie algebra the first Dolbe…

We investigate a method of construction of Calabi--Yau manifolds, that is, by smoothing normal crossing varieties. We develop some theories for calculating the Picard groups of the Calabi--Yau manifolds obtained in this method. Some applications are included, such as construction of new examples of Calabi--Yau 3-folds …

New method speeds up diffusion models inference to sub-linear time.

The study classifies complex smooth Fano varieties with large pseudoindex.