×

Formality of canonical symplectic complexes and Frobenius manifolds. (English) Zbl 0931.58002

It was shown by S. Barannikov and M. Kontsevich [Int. Math. Res. Not. 1998, No. 4, 201-215 (1998; Zbl 0914.58004)] that the formal moduli space of solutions to the Maurer-Cartan equations, modulo gauge equivalence associated to a very special class of differential Gerstenhaber-Batalin-Vilkovisky (dGBV) algebras, carries the natural structure of a Frobenius manifold.
To the author’s knowledge, only one example of such a special dGBV algebra was known: the one constructed out of the Dolbeault complex of an arbitrary Calabi-Yau manifold by Barannikov and Kontsevich [loc. cit.].
In this note, we produce another example of a special dGBV-algebra, namely, the one associated with an arbitrary symplectic manifold \((M,\omega)\) satisfying the hard Lefschetz condition which says that the cup product \[ [\omega^k]: H^{m-k}(M)\to H^{m+ k}(M) \] is an isomorphism for any \(k\leq m=1/2\dim M\). Applying then the machinery developed by S. Barannikov and M. Kontsevich [loc. cit.] and Yu. I. Manin to the moduli space of solutions of the associated Maurer-Cartan equation, we get a structure of a Frobenius manifold on the de Rham cohomology of \(M\).

MSC:

58A12 de Rham theory in global analysis
58H15 Deformations of general structures on manifolds
58J10 Differential complexes

Citations:

Zbl 0914.58004
PDFBibTeX XMLCite
Full Text: DOI arXiv