Lectures on the deformations of log-canonical Poisson brackets

Lecture 1: Poisson bracket, examples, Schouten bracket, Poisson cohomology

Lecture 2: Interpretation of 0-th, 1-st, 2-nd and 3-rd Poisson cohomology. Computation for {x,y} =1 and {x,y}=x.

Lecture 3: Computation of Poisson cohomology for {x,y}=x (continued). Homotopy formula. Computation for {x,y} = xy. Computation of Poisson cohomology for any log-canonical bracket.

Lecture 4: T-log-symplectic log-canonical brackets. Examples coming from "action data". Finite-dimensionality of 2-nd Poisson cohomology for T-log-symplectic brackets.

Lecture 5, part 1 and part 2: Smoothing diagrams. Chain vs cycle dychotomy. Discussion of "no cycles" condition.

Lecture 6: No cycles => unobstructedness of formal Poisson deformations. Algebraicity of the formal deformations. Classification of (strongly symmetric) Poisson CGL extensions