Reduction of generalized Calabi-Yau structures

Abstract

A generalized Calabi-Yau structure is a geometrical structure on a manifold which generalizes both the concept of the Calabi-Yau structure and that of the symplectic one. In view of a result of Lin and Tolman in generalized complex cases, we introduce in this paper the notion of a generalized moment map for a compact Lie group action on a generalized Calabi-Yau manifold and construct a reduced generalized Calabi-Yau structure on the reduced space. As an application, we show some relationship between generalized moment maps and the Bergman kernels, and prove the Duistermaat-Heckman formula for a torus action on a generalized Calabi-Yau manifold.

The Equivariant Cohomology Theory of Twisted Generalized Complex Manifolds

“…The same result, under the assumption that both the twisting three form H and the moment one form vanish, has already been proved by Nitta [NY06]. We will make use of the following fact established in the proof of [NY06, Thm.…”

“…Remark 6.2. Although [NY06] assume the generalized Calabi-Yau structure is untwisted, the proof of the above fact given by Nitta uses only the linear algebra of generalized complex structures and so applies to the twisted case as well.…”

“…Indeed, the Hamiltonian action on generalized Calabi-Yau manifolds has been studied in the very recent works of Nitta, [NY06], [NY07]. Under the assumption that both the twisting three form H and the moment one form α are zeroes, Nitta showed that the generalized Calabi-Yau structure descends to the generalized complex quotient.…”

Generalized reduction and pure spinors

“…More precisely, the main goal of this work is to find an explicit description of the pure spinor line bundle associated to the Dirac structure L red obtained by reducing a Dirac structure L under the procedure of [6]. In [23], Y. Nitta introduced a procedure to reduce pure spinors in the context of generalized Calabi-Yau structures. The motivation for this paper is to put the work of Y. Nitta into the broader perspective of [6] and to move toward a general procedure to reduce generalized Calabi Yau structures.…”

“…This pertubation method enables us to recover Nitta's reduction [23] as a particular instance of (0.2) when D is properly choosen. We are also able to obtain a new result on reduction of generalized Calabi-Yau structures in the presence of transversality conditions (see Proposition 4.8).…”