  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.
