Some properties of symplectic and hyper-Kählerian structures. (English. Russian original) Zbl 0842.53019

Shmelev, A. S.

Russ. Acad. Sci., Dokl., Math. 49, No. 3, 511-514 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 336, No. 3, 304-306 (1994).

It is well known, that the Poincaré series for the moduli space of solutions of the gravitational field equations in empty space is a rational function, because its coefficients are polynomials. The author shows that a similar result holds for Kähler and hyper-Kählerian structures. The technique used by the author is as follows. Given the natural action of the group of germs of diffeomorphisms at a point on the space of jets of a geometric object (for instance, the Kähler form), the dimension of the foliation determined by the orbits is computed.

Reviewer: M.de León (Madrid)

MSC:

53B35	Local differential geometry of Hermitian and Kählerian structures
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
32Q15	Kähler manifolds
53C12	Foliations (differential geometric aspects)

Keywords:

Kähler structures; hyper-Kähler structures; moduli space of metrics