arXiv research

Holomorphic Koszul-Brylinski homology studied via Dolbeault cohomology.

The Hesse-Koszul flow converges to the Hesse-Einstein metric on compact Hessian manifolds.

Proves properties of Torelli Lie algebra for surfaces.

Constructs a geometric representation for holomorphic vector bundles.

Study convex projective manifolds with compact and convex ends.

Under certain integrability and geometric conditions, we prove division theorems for the exact sequences of holomorphic vector bundles and improve the results in the case of Koszul complex. By introducing a singular Hermitian structure on the trivial bundle, our results recover Skoda's division theorem for holomorphic …

Normal forms for Q-structures on graded manifolds explained.

Defines Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids.

Koszul duality for manifold modules proven.

Formula derived for holomorphic Poisson blow-ups.

The paper studies affine connections on singular warped products and their curvature.

The paper introduces new structures for left-symmetric algebroids.

In this note, we study the Koszul-Brylinski homology of holomorphic Poisson manifolds. We show that it is isomorphic to the cohomology of a certain smooth complex Lie algebroid with values in the Evens-Lu-Weinstein duality module. As a consequence, we prove that the Evens-Lu-Weinstein pairing on Koszul-Brylinski homolo…

Advances in Koszul modules and syzygies of algebraic varieties.

This paper constructs Koszul duals for Heegaard Floer Dehn surgery formulas.

Characterizes real left symmetric algebras with positive definite Koszul form and related Kähler-Einstein structures.

Study on deformation of affine structures on Lie groups using cohomology.

Paper proves Koszul duality for weighted A-infinity algebras.

Proves effective Chen ranks conjecture for Koszul modules.

The cohomology of affine structures on Lie groups is compared with that of Koszul-Vinberg algebras.

Study on deformations of pre-symplectic structures using an L-infinity algebra.

Paper constructs $L_{\infty}$-algebroids from homotopy Poisson structures.

Quantizes the relationship between Koszul and Schouten brackets in Poisson geometry.

Study submanifolds in Koszul-Vinberg geometry, a blend of Poisson and pseudo-Riemannian structures.

We examine functorial and homotopy properties of the exotic characteristic homomorphism in the category of Lie algebroids which was lastly obtained by the authors in [4]. This homomorphism depends on a triple (A,B,$\nabla$) where B $\subset$ A are regular Lie algebroids, both over the same regular foliated manifold (M,…

Constructs Koszul dual algebras for star-shaped diagrams in 3-manifolds.

Introduces a new operator generating higher Koszul brackets on differential forms.

This work is devoted to an intrinsic cohomology theory of Koszul-Vinberg algebras and their modules. Our results may be regarded as improvements of the attempt by Albert Nijenhuis in [NA]. The relationships between the cohomology theory developed here and some classical problems are pointed out, e.g. extensions of alge…

The Koszul-Tate resolution is described in the context of the geometry of jet spaces and differential equations. The application due to Barnich, Brandt, and Henneaux of this resolution to computing the horizontal cohomology is analyzed. Relations with the Vinogradov spectral sequence are discussed.

We deal with smooth real manifolds as well as complex analytic manifolds as well. It is well known that the concept of star product is powerful enough to produce all Poisson structures on real manifolds. According to [BdM] it is not known whether holomorphic star products exist on complex analytic manifolds. The main p…

Study cohomologies of complex bundles and sections.

We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This in particular gives a conceptual explanation of the appearance of graph cohomology of both the commutative and Lie types in computations of the cohomology of the ou…

The paper computes KV cochain differentials and their geometric implications.

Finite presentations for skein algebras linked to gauge field theory.

The abstract discusses cohomologies and deformations of Rota-Baxter operators on Lie algebroids and Koszul-Vinberg structures.

Study classifies 3D Hessian manifolds, proving their topology.

Constructs a new graded variety from algebraic data.

We show that if a generator of a differential Gerstenhaber algebra satisfies certain Cartan-type identities, then the corresponding Lie bracket is formal. Geometric examples include the shifted de Rham complex of a Poisson manifold and the subcomplex of differential forms on a symplectic manifold vanishing on a Lagrang…

Let M_0=G_0/H be a (pseudo)-Riemannian homogeneous spin manifold, with reductive decomposition g_0=h+m and let S(M_0) be the spin bundle defined by the spin representation Ad:H->\GL_R(S) of the stabilizer H. This article studies the superizations of M_0, i.e. its extensions to a homogeneous supermanifold M=G/H whose sh…

A generic degenerate Lagrangian system of even and odd variables on an arbitrary smooth manifold is examined in terms of the Grassmann-graded variational bicomplex. Its Euler-Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. However, non-tri…

Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k-rescaling of the rational cohomology ring of X. Assume H^*(X,Q) is a Koszul algebra. Then, the homotopy Lie algebra pi_*(Omega Y) tensor Q equals, up to k-rescaling, the graded rational Lie alge…

Researchers compute link surgery modules for 2-component L-space links.

The McCool group, denoted $PΣ_n$, is the group of pure symmetric automorphisms of a free group of rank $n$. The cohomology algebra $H^*(PΣ_n, \mathbb{Q})$ was determined by Jensen, McCammond and Meier. We prove that $H^*(PΣ_n, \mathbb{Q})$ is a non-Koszul algebra for $n \geq 4$, which answers a question of Cohen and Pr…

We give a combinatorial proof of the quasi-invertibility of $\widehat{CFDD}(\mathbb{I}_\mathcal{Z})$ in bordered Heegaard Floer homology, which implies a Koszul self-duality on the dg-algebra $\mathcal{A}(\mathcal{Z})$, for each pointed matched circle $\mathcal{Z}$. This is done by giving an explicit description of a r…

Study shows looping a 6-manifold over a 4-manifold results in a product of loops on spheres.

Diagrammatically describes algebra equivalent to perverse sheaves on isotropic Grassmannians.

Generalizes Lie bialgebroids to supermanifolds with homotopy Poisson structures.

Study of Batalin-Vilkovisky algebra on Poisson manifolds with diagonalizable modular symmetry.