arXiv research

Study on deformations of Lie groupoid morphisms and their properties.

Study on deformation cohomology for braided commutative structures.

Study canonical deformations of complex forms and their cohomology properties.

This research classifies deformations of Yang-Baxter operators using cohomology of $n$-Lie algebras.

The paper studies deformations of symplectic groupoids using cohomology and spectral sequences.

The paper studies deformations of Filippov algebroids using cohomology and DGLA.

The paper studies deformations of Lie ideals in Lie algebras.

Study on symplectic structures and their deformations.

Study of deformed Bott-Chern cohomology on complex manifolds.

Study of deformations of Virasoro symmetries using variational bihamiltonian cohomology.

We study deformations of Lie groupoids by means of the cohomology which controls them. This cohomology turns out to provide an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We prove several fundamental properties of the deformation cohomology including Morita invariance…

New cohomology $η$ for deformed Sasaki-Einstein manifolds derived from Dolbeault cohomology.

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

Study complex structure deformations on Lie algebras and Dolbeault cohomology.

In this paper, we establish a deformation theory for Dolbeault cohomology classes valued in holomorphic tensor bundles. We prove the extension equation which will play the role of Maurer-Cartan equation. Following the classical theory of Kodaira-Spencer-Kuranishi, we construct a canonical complete family of deformation…

New deformations of lattice cohomology help calculate knot invariants.

Maps geometric deformations to algebraic classes in Lie groupoids and algebroids.

In this note we define a notion of Courant pair as a Courant algebra over the Lie algebra of linear derivations on an associative algebra. We study formal deformations of Courant pairs by constructing a cohomology bicomplex with coefficients in a module from the cochain complexes defining Hochschild cohomology and Leib…

In this paper we introduce the notion of deformation cohomology for singular foliations and related objects (namely integrable differential forms and Nambu structures), and study it in the local case, i.e., in the neighborhood of a point.

Study YB operators and their deformations, finding integrable and nontrivial cases.

The paper explores moduli space of heterotic system using two deformation paths.

Analyzes complex structure deformations using cohomology contraction methods.

Study Witten deformation on noncompact manifolds with bounded geometry.

This study introduces a unified cohomology theory for braided algebras.

The study classifies nilpotent Lie foliations with cohomological obstructions.

We provide further techniques to study the Dolbeault and Bott-Chern cohomologies of deformations of solvmanifolds by means of finite-dimensional complexes. By these techniques, we can compute the Dolbeault and Bott-Chern cohomologies of some complex solvmanifolds, and we also get explicit examples, showing in particula…

We discuss a relation between deformed cohomologies of symmetry pseudo-groups and coverings of differential equations. Examples include the potential Khokhlov--Zabolotskaya equation and the Boyer--Finley equation.

We show that the Hochschild cohomology of the algebra obtained by formal deformation quantization on a symplectic manifold is isomorphic to the formal series with coefficients in the de Rham cohomology of the manifold. The cohomology class obtained by differentiating the star-product with respect to the deformation par…

Study on deformation theory of nearly G2 manifolds with obstructions.

While small deformations of Kähler manifolds are Kähler too, we prove that the cohomological property to be $\mathcal{C}^\infty$-pure-and-full is not a stable condition under small deformations. This property, that has been recently introduced and studied by T.-J. Li and W. Zhang in [Comparing tamed and compatible symp…

Every rack $Q$ provides a set-theoretic solution $c_Q$ of the Yang-Baxter equation. This article examines the deformation theory of $c_Q$ within the space of Yang-Baxter operators over a ring $\A$, a problem initiated by Freyd and Yetter in 1989. As our main result we classify deformations in the modular case, which ha…

We present some general results on properties of the bihamiltonian cohomologies associated to bihamiltonian structures of hydrodynamic type, and compute the third cohomology for the bihamiltonian structure of the dispersionless KdV hierarchy. The result of the computation enables us to prove the existence of bihamilton…

As is well-known, the Witten deformation of the De Rham complex computes the De Rham cohomology. In this paper we study the Witten deformation on a noncompact manifold and restrict it to differential forms which behave polynomially near infinity. Such polynomial differential forms naturally appear on manifolds with a c…

Let $A \Rightarrow M$ be a Lie algebroid. In this short note, we prove that a pull-back of $A$ along a fibration with homologically $k$-connected fibers, shares the same deformation cohomology of $A$ up to degree $k$.

The theory of multidimensional Poisson vertex algebras (mPVAs) provides a completely algebraic formalism to study the Hamiltonian structure of PDEs, for any number of dependent and independent variables. In this paper, we compute the cohomology of the PVAs associated with two-dimensional, two-components Poisson bracket…

We consider nilmanifolds with left-invariant complex structure and prove that small deformations of such structures are again left invariant if the Dolbeault-cohomology of the nilmanifold can be calculated using left-invariant forms. By a result of Console and Fino this is generically the case. Our main tool is an anal…

The paper studies holomorphic Poisson manifolds and their deformations under specific cohomology conditions.

Quantum cocycle invariants derived from Yang-Baxter cohomology.

We investigate the deformation theory of the simplest bihamiltonian structure of hydrodynamic type, that of the dispersionless KdV hierarchy. We prove that all of its deformations are quasi-trivial in the sense of B. Dubrovin and Y. Zhang, that is, trivial after allowing transformations where the first partial derivati…

We prove that, for some classes of complex nilmanifolds, the Bott-Chern cohomology is completely determined by the Lie algebra associated to the nilmanifold with the induced complex structure. We use these tools to compute the Bott-Chern and Aeppli cohomologies of the Iwasawa manifold and of its small deformations, com…

Study on deformations of special Lagrangians with boundary in Calabi-Yau manifolds.

Unique complex structures on specific Lie algebras.

Global homotopies upgrade classical map in differential geometry.

The paper studies deformations of symplectic forms and Lagrangian submanifolds.

We introduce a new cohomology for Lie algebroids, and prove that it provides a differential graded Lie algebra which ``controls'' deformations of the structure bracket of the algebroid. We also have a closer look at various special cases such as Lie algebras, Poisson manifolds, foliations, Lie algebra actions on manifo…

This thesis studies deformations of VB-algebroids and VB-groupoids in Lie algebroid and groupoid categories.

An algebra $A$ with identity $(a\circ b)\circ c-a\circ(b\circ c)=(a\circ c)\circ b-a\circ(c\circ b),$ is called right-symmetric. Cohomology and deformation theory for right-symmetric algebras are developed. Cohomologies of $gl_n$ and half-Witt algebras $W_n^{rsym}, p=0,$ $W_n^{rsym}({\bf m}), p>0,$ are calculated. In p…

Introduces a Morse complex on symplectic manifolds using gradient flows and proves its cohomology is independent of metrics and Morse functions.