A Fr\"olicher-type inequality for generalized complex manifolds
  We prove a Fr\"olicher-type inequality for a compact generalized complex manifold $M$, and show that the equality holds if and only if $M$ satisfies the generalized $\partial\bar{\partial}$-Lemma. In particular, this gives a unified proof of analogous results in the complex and symplectic cases. 
Kwokwai Chan and Yat-Hin Suen
math.DG math.SG