  Twilled L(ie)-R(inehart) algebas generalize, in the Lie-Rinehart context,
complex structures on smooth manifolds. An almost complex manifold determines
an almost twilled pre-LR algebra, which is a true twilled LR-algebra iff the
almost complex structure is integrable. We characterize twilled LR-structures
in terms of certain associated differential (bi)graded Lie and
G(erstenhaber)-algebras; in particular, the G-algebra arising from an almost
complex structure is a d(ifferential) G-algebra iff the almost complex
structure is integrable. Such G-algebras, endowed with a generator turning them
into a B(atalin)-V(ilkovisky) algebra, occur on the B-side of the mirror
conjecture. We generalize a result of Koszul to those dG-algebras which arise
from twilled LR-algebras. A special case thereof explains the relationship
between holomorphic volume forms and exact generators for the corresponding
dG-algebras and thus yields in particular a conceptual proof of the
Tian-Todorov lemma. We give a differential homological algebra interpretation
for twilled LR-algebras and by means of it we elucidate the notion of generator
in terms of homological duality for differential graded LR-algebras. Finally we
indicate how some of our results might be globalized by means of Lie groupoids.

Differential Geometry;Algebraic Geometry