This is a report on joint work with Fyodor Malikov and Vadim Schechtman. We compute the cohomological obstruction to the existence of certain sheaves of vertex algebras on smooth varieties. These sheaves have been introduced and studied in the previous work by Malikov, Schechtman and Vaintrob, and are canonically defined for an arbitrary X. One can try to define a purely even counterpart of W^ch_X, a sheaf of graded vertex algebras \CO^ch_X, called a chiral structure sheaf. The obstraction to its existence turns out to admit a very simple expression in terms of characteristic classes of X, namely it is expressed in terms of the second component of Chern character of the tangent bundle of X. The obstruction to the existence of a globally defined Virasoro field L(z); it is given by the first Chern class c₁(X)/2. It provides a geometric criterion for a manifold to admit a BUá6ñ-structure: those are recisely the manifolds which admit the above mentioned sheaf \CO^ch_X and for which the conformal anomaly vanishes. If such a manifold is Calabi-Yau (i.e. has the trivial canonical bundle) then \CO^ch_X is a sheaf of conformal vertex algebras.

From a different viewpoint, one can regard the above result as a geometric interpretation of the second component of the Chern character.