  The first part of this paper provides a new formulation of chiral
differential operators (CDOs) in terms of global geometric quantities. The main
result is a recipe to define all sheaves of CDOs on a smooth cs-manifold; its
ingredients consist of an affine connection and an even 3-form that trivializes
the first Pontrjagin form. With the connection fixed, two suitable 3-forms
define isomorphic sheaves of CDOs if and only if their difference is exact.
Moreover, conformal structures are in one-to-one correspondence with even
1-forms that trivialize the first Chern form.
  Applying our work in the first part, we construct what may be called "chiral
Dolbeault complexes" of a complex manifold M, and analyze conditions under
which these differential vertex superalgebras admit compatible conformal
structures or extra gradings (fermion numbers). When M is compact, their
cohomology computes (in various cases) the Witten genus, the two-variable
elliptic genus and a spin-c version of the Witten genus. This part contains
some new results as well as provides a geometric formulation of certain known
facts from the study of holomorphic CDOs and sigma models.
