  We consider holomorphic twists of arbitrary supersymmetric theories in four
dimensions. Working in the BV formalism, we rederive classical results
characterizing the holomorphic twist of chiral and vector supermultiplets,
computing the twist explicitly as a family over the space of nilpotent
supercharges in minimal supersymmetry. The BV formalism allows one to work with
or without auxiliary fields, according to preference; for chiral superfields,
we show that the result of the twist is an identical BV theory, the holomorphic
$\beta\gamma$ system with superpotential, independent of whether or not
auxiliary fields are included. We compute the character of local operators in
this holomorphic theory, demonstrating agreement of the free local operators
with the usual index of free fields. The local operators with superpotential
are computed via a spectral sequence, and are shown to agree with functions on
a formal mapping space into the derived critical locus of the superpotential.
We consider the holomorphic theory on various geometries, including Hopf
manifolds and products of arbitrary pairs of Riemann surfaces, and offer some
general remarks on dimensional reductions of holomorphic theories along the
$(n-1)$-sphere to topological quantum mechanics. We also study an
infinite-dimensional enhancement of the flavor symmetry in this example, to a
recently-studied central extension of the derived holomorphic functions with
values in the original Lie algebra that generalizes the familiar Kac--Moody
enhancement in two-dimensional chiral theories.
