  We study the algebraic conditions leading to the chain property of complexes
for vertex operator algebra $n$-point functions with differential being defined
through reduction formulas. The notion of the reduction cohomology of Riemann
surfaces is introduced. Algebraic, geometrical, and cohomological meanings of
reduction formulas is clarified. A counterpart of the Bott-Segal theorem for
Riemann surfaces in terms of the reductions cohomology is proven. It is shown
that the reduction cohomology is given by the cohomology of $n$-point
connections over the vertex operator algebra bundle defined on a genus $g$
Riemann surface $\Sigma^{(g)}$. The reduction cohomology for a vertex operator
algebra with formal parameters identified with local coordinates around marked
points on $\Sigma^{(g)}$ is found in terms of the space of analytical
continuations of solutions to Knizhnik-Zamolodchikov equations. For the
reduction cohomology, the Euler-Poincare formula is derived. Examples for
various genera and vertex operator cluster algebras are provided.
