arXiv research

New learning scheme solves high-dimensional semi-linear PDEs using sparse grids and Picard approximations.

In this paper, we study rational sections of the relative Picard scheme of a linear system on a smooth projective variety. We prove that if the linear system is basepoint-free and the locus of non-integral divisors has codimension at least two, then all rational sections of the relative Picard scheme come from restrict…

Study shows connections between Jacobian torsors and Fermat curves.

Develops a numerical scheme for solving path-dependent FBSDEs and PDEs.

Study splitting submanifolds in specific homogeneous spaces.

The aim of this work is to provide fast and accurate approximation schemes for the Monte Carlo pricing of derivatives in LIBOR market models. Standard methods can be applied to solve the stochastic differential equations of the successive LIBOR rates but the methods are generally slow. Our contribution is twofold. Firs…

Improved solver maintains positivity and accuracy across all time steps.

A new explicit scheme calculates XVA adjustments using neural networks and conditional expectations.

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

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

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

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…

The aim of this work is to provide fast and accurate approximation schemes for the Monte-Carlo pricing of derivatives in the Lévy LIBOR model of Eberlein and Özkan (2005). Standard methods can be applied to solve the stochastic differential equations of the successive LIBOR rates but the methods are generally slow. We …

New algorithms compute Volterra signature efficiently for time series analysis.

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 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.

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…

Computes Picard groups of complex parallelizable manifolds.

Mark and Paupert devised a general method for obtaining presentations for arithmetic non-cocompact lattices, $Γ$, in isometry groups of negatively curved symmetric spaces. The method involves a classical theorem of Macbeath applied to a $Γ$-invariant covering by horoballs of the negatively curved symmetric space upon w…

Study geometric properties of a complex hyperbolic group action.

Constructs explicit nontrivial cycles in Habiro cohomology of smooth varieties.

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 …

The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture fo…

The study classifies complex smooth Fano varieties with large pseudoindex.

In this paper we investigate codimension one Fano distributions on Fano manifolds with Picard number one. We classify Fano distributions of maximal index on complete intersections in weighted projective spaces, Fano contact manifolds, Grassmannians of lines and their linear sections, and describe their moduli spaces. A…

The goal of this article is to show that five explicitly given transformations, a rotation, two screw Heisenberg rotations, a vertical translation and an involution generate the Euclidean Picard modular groups with coefficient in the Euclidean ring of integers of a quadratic imaginary number field. We also obtain the r…

In this article we introduce algorithms which compute iterations of Gauss-Manin connections, Picard-Fuchs equations of Abelian integrals and mixed Hodge structure of affine varieties of dimension $n$ in terms of differential forms. In the case $n=1$ such computations have many applications in differential equations and…

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

Study classifies Calabi-Yau threefolds with non-Gorenstein involutions.

Study positive characteristic Fano 4-folds with nef tangent bundles.