  We observe that an anti-symplectic manifold locally always admits a parity
structure. The parity structure can be viewed as a complex-like structure on
the manifold. This induces an odd metric and its Levi-Civita connection, and
thereby a new notion of an odd Kaehler geometry. Oversimplified, just to
capture the idea, the bosonic variables are ``holomorphic'', while the
fermionic variables are ``anti-holomorphic''. We find that an odd Kaehler
manifold in this new ``complex'' sense has a nilpotent odd Laplacian iff it is
Ricci-form-flat. The local cohomology of the odd Laplacian is derived. An odd
Calabi-Yau manifold has locally a canonical volume form. We suggest that an odd
Calabi-Yau manifold is the natural geometric notion to appear in covariant
BV-quantization. Finally, we give a vielbein formulation of anti-symplectic
manifolds.
