Geometry of Polarizations

Abstract

Let M be a real manifold of dimension 2n. Then an almost complex structure on M is a tensor field of type $\left( {1,1} \right)j:m \to {j_m}$ in End $\left( {{T_m}M} \right)$ for m in M such that $\left( i \right)j_m^2 = - 1$ and (ii)j is smooth. j thus defines a complex distribution $F:m \to {F_m} \subset T_m^c$ such that $T_m^c = {F_m} \oplus {\bar F_m}$. The almost complex manifold will be denoted (M,j). And the smooth functions fin A(M) which satisfy X\left( f \right) = 0 for all {V_F}\left( M \right) = \left\{ {X \in {V^C}\left( M \right)|{X_m} in {F_m},m in M} \right\} is the algebra of holomorphic functions.