New K3 surfaces with two involutions and low Picard number constructed.
problem Finding K3 surfaces with specific properties and low Picard numbers.
method Construction of K3 surfaces over the rational numbers with low Picard numbers and two involutions.
result Explicit examples of K3 surfaces over the rational numbers with minimum Picard number 2 for various degrees.
New method finds 198,846 toric-colorable seeds of Picard number 5.
problem Enumerating toric-colorable seeds of Picard number 5.
method Binary matroid approach and dynamic programming algorithm.
result 198,846 mod 2 toric-colorable seeds of dimension four and Picard number five.
Study shows no hyperkähler fourfolds in specified conditions.
problem Identifying hyperkähler fourfolds in specific geometric settings.
method Classification and computation of Hodge numbers for fourfolds over rational homogeneous varieties.
result No hyperkähler fourfolds found in the specified conditions.
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…
Researchers characterize a specific type of projective variety based on its tangents.
problem Characterizing smooth projective horospherical varieties of Picard number one.
method Using methods of W-normal complete step prolongations and Lie algebra cohomology.
result A uniruled projective manifold of Picard number one is biholomorphic to the variety if its tangents match.
Method constructs fundamental domains for Picard modular groups.
problem Classify and understand torsion elements in Picard modular groups.
method Systematic construction of coarse fundamental domains.
result Classification of conjugacy classes of torsion elements.
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…
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 …
The paper classifies diffeomorphism classes of Calabi-Yau threefolds constructed by gluing methods.
problem Classifying diffeomorphism classes of Calabi-Yau threefolds constructed by gluing methods.
method Using differential-geometric gluing method and classifications of simply-connected 6-manifolds.
result Any two doubling Calabi-Yau threefolds with Picard number two are not diffeomorphic to each other when the underlying Fano threefolds are distinct families.
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.
problem Finding Kähler-Einstein metrics on smooth Fano symmetric varieties with specific properties.
method Used a combinatorial criterion for K-stability of Fano spherical varieties and computed algebraic moment polytopes and barycenters.
result Proved all smooth Fano symmetric varieties with Picard number one admit Kähler-Einstein metrics.
Study calculates Ricci bounds for special Fano manifolds.
problem Computing Ricci bounds for specific Fano manifolds.
method Using barycenter of moment polytopes with Duistermaat-Heckman measure.
result Greatest Ricci lower bounds can be arbitrarily close to zero.
The study classifies complex smooth Fano varieties with large pseudoindex.
problem Classifying Fano varieties with specific properties.
method Analyzing Fano varieties with large pseudoindex and Picard number greater than one.
result Classification of Fano varieties with pseudoindex at least n-2 and Picard number greater than one.
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…
Found a stable 3D shape with specific properties.
problem Finding K-stable Fano threefolds.
method Analyzing specific Fano threefolds with given properties.
result Identified a K-stable Fano threefold with Picard rank 3 and anti-canonical degree 28.
Algorithm identifies spheres with maximal Buchstaber number.
problem Characterizing (n−1)-dimensional PL spheres with specific vertex counts. method Computational algorithm for weak pseudo-manifolds, toric colorable seeds enumeration.
result Comprehensive characterization of (n−1)-spheres with maximal Buchstaber number. Survey connects singularity invariants to link pairings.
problem Understanding connections between singularities and link pairings.
method Use of Hermitian Variation Structures.
result Unified understanding of Picard--Lefschetz invariants and Blanchfield forms.
New examples show deletion type admissible pairs can be rigid under rational saturation.
problem Rigidity of admissible pairs of rational homogeneous spaces of Picard number one.
method Application of Mok's general criterion for non-subdiagram type admissible pairs.
result Examples of deletion type admissible pairs are rigid under rational saturation.
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 …
Study positive characteristic Fano 4-folds with nef tangent bundles.
problem Positive characteristic version of the Campana-Peternell conjecture for Fano 4-folds.
method Analyzes Fano 4-folds with nef tangent bundles in positive characteristic.
result Affirmative answer for Fano 4-folds with Picard number > 1 and nef tangent bundle.
A mG2-horospherical manifold is identified by its VMRT.
problem Recognizing mG2-horospherical manifolds of Picard number 1. method Using the method developed for symplectic Grassmannians, which involves constructing a flat Cartan connection and studying the positivity/negativity of vector bundles.
result The mG2-horospherical manifold ${f X}$ is the only smooth projective variety with the property of being recognized by its VMRT. 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…
Study splitting submanifolds in specific homogeneous spaces.
problem Classify splitting submanifolds in rational homogeneous spaces of Picard number one.
method Use global holomorphic vector fields and projection maps to analyze submanifolds.
result Proves submanifolds in certain spaces are rational or Hermitian symmetric.
Study generalizes Picard iteration for nonlinear PDEs, deriving bounds on error.
problem Generalize Picard iteration for nonlinear parabolic PDEs.
method Formulate Picard iteration as abstract state-transition model, derive generalization error bounds.
result Picard depth reduction reduces Picard truncation error without increasing estimation error.
Researchers found the global topology of the Eisenstein-Picard modular surface.
problem Understanding the global topology of the Eisenstein-Picard modular surface.
method Quotient space of the complex hyperbolic plane by the modular group.
result Determined the global topology of the Eisenstein-Picard modular surface as a 4-orbifold.
Characterizes solutions to Z-critical equations on surfaces using effective conditions.
problem Characterizing solutions to Z-critical equations on compact Kähler surfaces.
method Uses effective conditions and Picard number bounds to characterize solutions.
result Characterizes optimally destabilizing curves for Donaldson's J-equation and deformed Hermitian Yang-Mills equation.
A quaternionic version of Picard's theorem limits how many values a slice regular function can avoid.
problem How many values can a non-constant slice regular function of a quaternionic variable avoid?
method Investigates slice regular functions of quaternionic variables, extending the classical Picard theorem.
result A non-constant slice regular function of a quaternionic variable can avoid at most one value, similar to the classical Picard theorem.
An integral hyperbolic lattice is called reflective if its automorphism group is generated by reflections, up to finite index. Since 1981, it is known that their number is essentially finite. We show that K3 surfaces over C with reflective Picard lattices can be characterized in terms of compositions of their self-corr…
Picard modular groups are shown to be generated by complex reflections.
problem Understanding the structure of Picard modular groups using reflections.
method Using presentations from previous works to show generation by reflections.
result Picard modular groups mPU(2,1,Od) are generated by complex reflections. Classifies Fano varieties with large pseudoindex and non-free rational curves.
problem Classifying Fano varieties with specific properties.
method Extremal contractions and classification of varieties.
result Complete classification of Fano n-folds with pseudoindex at least n−2 and Picard number greater than one. 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…
We show that the pair (X,−KX) is K-unstable for a del Pezzo manifold X of degree five with dimension four or five. This disprove a conjecture of Odaka and Okada.
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
problem Short-time heat kernel asymptotics
method Picard-Lefschetz theory
result 1-Gevrey small-time expansion
We compute the Picard group of a stable b-symplectic manifold M by introducing a collection of discrete invariants Gr which classify M up to Morita equivalence.
Constructs stable bundles on K3 surfaces using monad construction.
problem Stability of bundles on K3 surfaces.
method Monad construction, Generalised Hoppe Criterion, computer aid.
result Examples of real stable bundles constructed on K3 surfaces.
Classifies Real primary Hopf surfaces and their associated groups.
problem Classifying Real primary Hopf surfaces and their associated groups.
method Complete classification up to Real biholomorphisms and equivariant diffeomorphisms.
result Detailed description of groups associated with Real primary Hopf surfaces.
Study finds Kähler-Einstein metrics on two Pasquier varieties.
problem Existence of Kähler-Einstein metrics on specific varieties.
method Analyzes Pasquier's two-orbits varieties to find metrics.
result New example of K-unstable Fano manifold with Picard number one.
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…
The goal of this paper is to study the geometry of cusped complex hyperbolic manifolds through their compactifications. We characterize toroidal compactifications with non-nef canonical divisor. We derive effective very ampleness results for toroidal compactifications of finite volume complex hyperbolic manifolds. We e…
Computes Picard groups of complex parallelizable manifolds.
problem Calculating Picard groups of specific compact complex manifolds.
method Analyzes tangent bundle triviality and uses lattice and Lie group properties.
result Computes Picard groups for certain compact 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.
problem Geometric properties of a specific modular group action.
method Explicit description of subgroups and conjugacy classes.
result Explicit description of a torsion-free subgroup of index 336.
Constructs explicit nontrivial cycles in Habiro cohomology of smooth varieties.
problem Describing and constructing nontrivial cycles in Habiro cohomology.
method Using either the Picard-Fuchs equation or push-forward of elements of the Habiro ring.
result Explicit classes for 1-parameter Calabi-Yau families and q-holonomic modules.
The loop space of the Riemann sphere consisting of all Ck or Sobolev Wk,p maps from the circle S1 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…
Study extends Nirenberg-Spencer's question to families of submanifolds.
problem Determine the germ of compact complex submanifolds in complex manifolds.
method Reformulate the question for families of submanifolds and their infinitesimal neighborhoods. Prove sufficient conditions for first-order neighborhoods and additional assumptions for submanifolds with nonzero vector fields.
result Affirmative answer to the reformulated question for certain submanifolds.
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…
A family of algebraic curves covering a projective variety X is called a web of curves on X if it has only finitely many members through a general point of X. A web of curves on X induces a web-structure, in the sense of local differential geometry, in a neighborhood of a general point of X. We study how the …