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1. In this note 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 A.Vaintrob and two of the authors, cf.[MSV] and [MS1]. Hopefully our result clarifies to some extent the constructions of op. cit.Recall that in [MSV] we discussed two kinds of sheaves on smooth complex algebraic (or analytic) varieties X. First, we defined the sheaf of conformal vertex superalgebras Ω ch X , called chiral de Rham algebra. These sheaves are canonically defined for an arbitrary X. Second, for some varieties X one can define a purely even counterpart of Ω ch X , a sheaf of graded vertex algebras O ch X , called a chiral structure sheaf, cf. op. cit., §5. For example, one can define O ch X for curves, and for flag spaces G/B. For an arbitrary X, there arises certain cohomological obstruction to the existence of O ch X . The infinitesimal incarnation of this obstruction is calculated in op. cit., §5, A.

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