arXiv research

Study classifies 3D Hessian manifolds, proving their topology.

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

Holomorphic Koszul-Brylinski homology studied via Dolbeault cohomology.

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.

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

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

Constructs a new graded variety from algebraic data.

Paper proves Koszul duality for weighted A-infinity algebras.

Proves effective Chen ranks conjecture for Koszul modules.

We study the deformation theory of pre-symplectic structures, i.e. closed two-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an $L_\infty$-algebra, which we call Koszul $L_\infty$-algebra. This $L_\infty$-algebra is a cousin of the Koszul…

Study reveals flatness of Hessian metrics with non-negative Ricci curvature on foliation leaves.

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

It is a classical fact in Poisson geometry that the cotangent bundle of a Poisson manifold has the structure of a Lie algebroid. Manifestations of this structure are the Lichnerowicz differential on multivector fields (calculating Poisson cohomology) and the Koszul bracket of differential forms. "Raising indices" by th…

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

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…

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…

Proves properties of Torelli Lie algebra for surfaces.

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.

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 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…

Normal forms for Q-structures on graded manifolds explained.

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…

Research connects geometric structures to knot theory and algebraic combinatorics.

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

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

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

The paper studies Kaehler and Kaehler-Einstein metrics on cohomogeneity one manifolds.

For each integer $k\geq 4$ we describe diagrammatically a positively graded Koszul algebra $\mathbb{D}_k$ such that the category of finite dimensional $\mathbb{D}_k$-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type ${\rm D}_k$ or ${\rm B}_{k-1}$, constructible with respect…

Generalizes Lie bialgebroids to supermanifolds with homotopy Poisson structures.

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

Recently S. Merkulov established a new link between differential geometry and homological algebra by giving descriptions of several differential geometric structures in terms of algebraic operads and props. In particular he described Nijenhuis structures as corresponding to representations of the cobar construction on …

We formulate a theory of pointed manifolds, accommodating both embeddings and Pontryagin-Thom collapse maps, so as to present a common generalization of Poincaré duality in topology and Koszul duality in $\mathcal{E}_n$-algebra.

The paper extends affine connection results to singular warped and twisted products.

The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study …