  We study quantization schemes on a K\"ahler manifold and relate several
interesting structures. We first construct Fedosov's star products on a
K\"ahler manifold $X$ as quantizations of Kapranov's $L_\infty$-algebra
structure. Then we investigate the Batalin-Vilkovisky (BV) quantizations
associated to these star products. A remarkable feature is that they are all
one-loop exact, meaning that the Feynman weights associated to graphs with two
or more loops all vanish. This leads to a succinct cochain level formula in de
Rham cohomology for the algebraic index.
