Study on Klt varieties with trivial canonical class, focusing on holonomy and stability.
problem Holonomy group of singular Kähler-Einstein metrics on klt varieties with numerically trivial canonical divisor.
method Investigation of holonomy group properties, finiteness of connected components, Bochner principle for holomorphic tensors, and connections between irreducibility of holonomy representations and stability of the tangent sheaf.
result Refinement of known decompositions for tangent sheaves of varieties with trivial canonical divisor, showing that up to finite quasi-étale covers, varieties with strongly stable tangent sheaves are either Calabi-Yau or irreducible holomorphic symplectic.
The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, and irreducible, simply-connected Calabi-Yau-- and holomorphic-symplectic manifolds. The decomposition of the simply…
Counterexample disproves log canonical Beauville--Bogomolov decomposition.
problem Disproving the log canonical Beauville--Bogomolov decomposition.
method Constructing a specific log canonical, K-trivial variety with non-birational fibers.
result Provides a counterexample to the Beauville--Bogomolov decomposition in the log canonical setting.
Study complex solvmanifolds with trivial canonical bundle and hypercomplex geometry.
problem Characterize and produce examples of complex solvmanifolds with trivial canonical bundle.
method Characterize invariant trivializing sections using Koszul 1-form, provide algebraic obstructions, and exhibit specific examples.
result New examples of complex solvmanifolds with trivial canonical bundle and algebraic obstructions for triviality.
Decomposes complex manifolds with trivial canonical bundle into homogeneous structures.
problem Decomposing complex manifolds with trivial canonical bundle into homogeneous structures.
method Using MMP and foliation theory, we prove a decomposition theorem and deduce properties of holomorphic geometric structures.
result Holomorphic geometric structures on X are locally homogeneous away from an analytic subset of complex codimension at least two. The paper shows how compact Kähler manifolds with a special bundle can be broken down into simpler parts.
problem Understanding compact Kähler manifolds with a specific type of anti-canonical bundle.
method Introduced a new approach to extend a strategy from smooth projective varieties to singular Kähler spaces, using a flatness criterion for pseudo-effective sheaves.
result Compact Kähler manifolds with a nef anti-canonical bundle admit a locally trivial fibration with rational connected fibers and a Calabi-Yau base.
Compute Dolbeault and Bott-Chern cohomologies of complex solvmanifolds.
problem Compute cohomologies of complex solvmanifolds.
method Build finite-dimensional double subcomplexes and decompose them into indecomposable ones.
result Characterize the ∂∂ˉ-Lemma property and compute triple ABC-Massey products. In this note we prove that Thompson's group F cannot be the fundamental group of a symplectic 4-manifold with trivial canonical class by showing that its Hausmann-Weinberger invariant q(F) is strictly positive.
Calculates Dehn twist actions on conformal blocks for modular categories.
problem Order of Dehn twists on spaces of conformal blocks.
method Quantum representations of mapping class groups, ribbon twists, monoidal powers.
result Order of Dehn twists generalizes previous results for small quantum group.
Classifies 6D homogeneous spaces with holomorphically trivial canonical bundle.
problem Classifying 6D homogeneous spaces with specific geometric properties.
method Analyzing Lie algebras and homogeneous spaces, studying connections and instantons.
result Compact non-Kähler homogeneous spaces are unique in their solutions to the Hull-Strominger system.
We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the h…
We generalize the idea of unknotting knots to Seifert surfaces. We define an operation called ribbon twist which serves as the equivalent of a crossing change for knots. A Seifert surface is considered untwisted, the equivalent to unknotted, if it is isotopic to a standardly embedded n-fold punctured torus. A Seifert s…
Study symplectic 4-orbifolds with vanishing canonical class, finding new structures and resolutions.
problem Investigate symplectic 4-orbifolds with vanishing canonical class.
method Introduced a cyclic orbifold covering and a successive symplectic blowing-down procedure.
result Minimal resolution of symplectic 4-orbifolds yields symplectic Calabi-Yau manifolds.
The article classifies six-dimensional solvmanifolds with non-invariant trivializing sections of their canonical bundle.
problem Classifying six-dimensional solvmanifolds with non-invariant trivializing sections of their canonical bundle.
method Complete classification of six-dimensional solvable strongly unimodular Lie algebras admitting complex structures, identifying those with non-invariant holomorphic sections of their canonical bundle.
result Construction of a new six-dimensional solvmanifold with non-invariant holomorphic sections of its canonical bundle.
The paper studies finiteness of canonical quotients in Dehn quandles of surfaces.
problem Finiteness of canonical quotients in Dehn quandles of surfaces.
method Analyzing the Dehn quandle structure and using quandle quotients.
result For surfaces of positive genus, the n-quandle of their Dehn quandles is finite for specific values of n. The classical McKay correspondence establishes an explicit link from the representation theory of a finite subgroup G of SU(2) and the geometry of the minimal resolution of the quotient of the affine plane by G. In this paper we discuss a possible generalization of the McKay correspondence to the case when G is replace…
We give an algebraic characterisation for the triviality of the canonical bundle of a complex supermanifold in terms of a certain Batalin-Vilkovisky superalgebra structure. As an application, we study the Calabi-Yau case, in which an explicit formula in terms of the Levi-Civita connection is achieved. Our methods inclu…
Compact complex manifolds with trivial canonical bundle and ∂∂ˉ-Lemma have smooth deformations and a surjective Albanese map.
problem Understanding the structure and deformations of compact complex manifolds with specific properties.
method Analyzing the Kuranishi space, showing smooth deformations, and studying the Albanese map.
result Compact complex manifolds with trivial canonical bundle and ∂∂ˉ-Lemma have smooth deformations and a surjective Albanese map. Smooth complex surfaces with triple intersections using differential geometry.
problem Smooth complex surfaces with trivial canonical bundle and triple intersections.
method Explicit construction of local smoothings and solutions to nonlinear elliptic PDEs.
result Existence of smoothings for d-semistable SNC complex surfaces with trivial canonical bundle. In this paper we use the Lubotzky alternative for finitely generated linear groups to determine which 4-manifolds admitting a free circle action can be endowed with a symplectic structure with trivial canonical class. The content of this paper partly overlaps with the content of the unpublished preprint "Symplectic 4-m…
Let (M,I,J,K) be a compact hypercomplex manifold admitting an HKT-metric. Assume that the canonical bundle of (M,I) is trivial as a holomorphic line bundle. We show that the holonomy of Obata connection on M is contained in SL(n,H). In Appendix we apply these arguments to compact nilmanifolds equipped with abelian hype…
It is shown that the cotangent bundle of a matched pair Lie group is itself a matched pair Lie group. The trivialization of the cotangent bundle of a matched pair Lie group are presented. On the trivialized space, the canonical symplectic two-form and canonical Poisson bracket are explicitly written. Various symplectic…
In this note we show that if a compact Kahler manifold with trivial canonical bundle is the total space of a holomorphic fibration without singular fibers, then the fibration is a holomorphic fiber bundle. In the algebraic case, the fibration becomes trivial after a finite base change.
We identify the canonical contact structure on the link of a simple elliptic or cusp singularity by drawing a Legendrian handlebody diagram of one of its Stein fillings. We also show that the canonical contact structure on the link of a numerically Gorenstein surface singularity is trivial considered as a real plane bu…
The paper addresses deformations of Kähler spaces with vanishing first Chern class.
problem Deformations of Kähler spaces with specific properties.
method Analyzes locally trivial deformation spaces and uses cohomological vanishing conditions.
result Shows that under certain conditions, deformations of Kähler spaces are projective varieties.
The paper classifies certain singular projective varieties with specific properties.
problem Classifying projective klt pairs with nef anti-log canonical divisors.
method Establishes a structure theorem using locally trivial rationally connected fibrations.
result Projective klt pairs can be decomposed into rationally connected and Calabi-Yau varieties.
We review some constructions and properties of complex manifolds admitting pluriclosed and balanced metrics. We prove that for a 6-dimensional solvmanifold endowed with an invariant complex structure J having holomorphically trivial canonical bundle the pluriclosed flow has a long time solution for every invariant init…
New groups can't be fundamental groups of symplectic Calabi-Yau manifolds.
problem Proving certain groups cannot be fundamental groups of symplectic Calabi-Yau manifolds.
method Using the method of Friedl and Vidussi, they showed these groups cannot be fundamental groups of SCY manifolds.
result The Lodha--Moore groups and their n-adic generalizations cannot be fundamental groups of any SCY manifold. We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong Kähler with torsion (SKT), generalized Gauduchon, balanced and strongly Gauduchon metrics is studied.…
We construct the first combinatorial 1-cocycle with values in the Z[x,x−1]-module of isotopy classes of singular long knots in 3-space with a signed planar double point, and which represents a non trivial cohomology class in the topological moduli space of long knots. It can be interpreted as an invaria…
Study on non-Kähler Calabi-Yau manifolds and their geometric structures.
problem Holomorphic geometric structures on non-Kähler compact complex manifolds.
method Beauville-Bogomolov decomposition and weak Bochner principle.
result Rigidity of Vaisman Calabi-Yau manifolds implies they are Kodaira manifolds.
On compact Kähler manifolds, we classify regular holomorphic foliations of codimension 1 whose canonical bundle is numerically trivial.
New solutions found for complex structures on specific manifolds.
problem Constructing smooth solutions to the Hull-Strominger system.
method Using fibrations over K3 orbisurfaces.
result Proved existence of solutions for certain manifolds.
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…
Study projective klt pairs with nef anti-canonical divisor and their properties.
problem Characterize properties of projective klt pairs with nef anti-log canonical divisors.
method Analyze maximally rationally connected fibration and use numerical dimension analogy.
result Numerical dimension of anti-log canonical divisor on X matches that on a general fiber. The study explores polarized deformations of SKT Calabi-Yau manifolds using Aeppli classes.
problem Understanding polarized deformations of SKT Calabi-Yau manifolds.
method Introducing small deformations polarized by Aeppli classes and investigating their properties.
result Existence of primitive elements in Bott-Chern classes and metrics comparison.
Develops spectral triples for quantum projective space.
problem Noncommutative geometry of quantum projective space.
method Introduces Kähler structures and spectral triples for quantum projective space.
result Produces an even spectral triple for quantum projective space.
Here we are fixing an output of a trivial calculation based on Konsevich's differential 2-form for the Chern class of polygon bundle. As a result an interesting combinatorics and arithmetics jumps right out of a jukebox. The calculation gives very simple rational combinatorial characteristics (we call it "curvature") o…
We describe which knots can be obtained as cycles in the canonical book representation of K_n, the complete graph on n vertices. We show that the canonical book representation of K_n contains a Hamiltonian cycle that is a composite knot if and only if n>11 and we show that when p and q are relatively prime, the (p,q) t…
We show that the Craighero-Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero-Gattazzo surface is the …
Extending work of Chen, we prove the Weinstein conjecture in dimension three for strongly fillable contact structures with either non-vanishing first Chern class or with strong and exact filling having non-trivial canonical bundle. This implies the Weinstein conjecture for certain Stein fillable contact structures obta…
Modulo trivial exceptions, we show that smoothly nontrivial symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particul…
We study a canonical spanning surface obtained from a knot or link diagram depending on a given Kauffman state, and give a sufficient condition for the surface to be essential. By using the essential surface, we can see the triviality and splittability of a knot or link from its diagrams. This has been done on the exte…
It is well-known that a knot in a contact manifold (M,C) transverse to a trivialized contact structure possesses the natural framing given by the first of the trivialization vectors along the knot. If the Euler class eC∈H2(M) of C is nonzero, then C is nontrvivializable and the natural framing of transvers…
The paper studies quadratic Poisson structures on Lie algebras, finding a 10-parametric family.
problem Compatibility of quadratic Poisson structures with linear structures on Lie algebras.
method Developed general theory and studied families of functions in involution.
result Found a 10-parametric family of quadratic Poisson structures on $\gl(3)^*$.
Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.
problem Existence of non-trivial holomorphic vector fields on compact complex manifolds.
method Vanishing result for measure preserving holomorphic vector fields, Gibbs stability, and log terminal singularities.
result No non-trivial holomorphic vector fields on compact complex manifolds with big anti-canonical line bundle.
We study the special algebraic properties of alternating 3-forms in 6 and 7 dimensions and introduce a diffeomorphism-invariant functional on the space of differential 3-forms on a closed manifold M in these dimensions. Restricting the functional to closed forms in a fixed cohomology class, we find that a critical poin…
For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic anti-canonical hypersurfaces along the three rational curves can be deformed to smooth t…