Kontsevich's graph complex, GRT, and the deformation complex of the sheaf of polyvector fields

Abstract

Kontsevich's graph complex, GRT, and the deformation complex of the sheaf of polyvector fields V.A. Dolgushev, C.L. Rogers, and T.H. WillwacherTo the memory of Boris Vasilievich Fedosov AbstractWe generalize Kontsevich's construction [38] of L ∞ -derivations of polyvector fields from the affine space to an arbitrary smooth algebraic variety. More precisely, we construct a map (in the homotopy category) from Kontsevich's graph complex to the deformation complex of the sheaf of polyvector fields on a smooth algebraic variety. We show that the action of Deligne-Drinfeld elements of the Grothendieck-Teichmüller Lie algebra on the cohomology of the sheaf of polyvector fields coincides with the action of odd components of the Chern character. Using this result, we deduce that theÂ-genus in the Calaque-Van den Bergh formula [12] for the isomorphism between harmonic and Hochschild structures can be replaced by a generalizedÂ-genus.

Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus

“…None of the solutions Q(P) known so far contains any 2-cycles (or "eyes" • ⇄ •). 7 We now report a classification of Poisson bi-vector symmetriesṖ = Q(P) which are given by those Kontsevich graphs Q = O r(γ) on k internal vertices that can be obtained at 5 k 9 by orienting k-vertex connected graphs γ without double edges. By construction, this extra assumption keeps only those Kontsevich graphs which may not contain eyes.…”

“…The definition of insertion γ 1 • i γ 2 of the entire graph γ 1 into vertices of γ 2 and the construction of Lie bracket [·, ·] of graphs and differential d in the non-oriented graph complex, referring to a sign convention, are as follows (cf. [8] and [7,11,12]); these definitions apply to sums of graphs by linearity. Definition 1.…”

The orientation morphism: from graph cocycles to deformations of Poisson structures

“…Having studied the natural differential graded Lie algebra (dgLa) structure on the space of graded skew-symmetric endomorphisms End * , * skew (T poly (M) [1]), we observe that its construction goes in parallel with the dgLa structure on the vector space k Gra i edge i #Vert=:k 1 S k of finite non-oriented graphs with wedge ordering of edges (and without leaves). Referring to [8,9,10,15] (and references therein), as well as to [6,7,14] with explicit examples of calculations in the graph complex, we summarize the set of analogous objects and structures in Table 1 below. Table 1.…”

“…The existence of this formula with some vanishing right-hand side is implied in[10,15,8] where it is stated that there is an action of the graph complex on Poisson structures (or Maurer-Cartan elements of T poly (M )). The precise right-hand side is all but written in[9]; still to the best of our knowledge, the exact formula is presented here and on p. 7 below for the first time.…”

Open problems in the Kontsevich graph construction of Poisson bracket symmetries

“…In the papers [21] (see also [22] and [4,8,28] for a pedagogical review), Kontsevich introduced the graph complex -one of the many -with parity-even vertices, with a wedge ordering of parity-odd edges, and the differential d = [•−•, ·] produced by the graded commutator of graph insertions into vertices. This direction of research was furthered by Willwacher et al [11,14,29]: in particular, in [30] a generating function counts the numbers of nonzero (i.e. not equal to minus itself) unoriented graphs with respect to their bi-grading by the vertex-edge numbers.…”