Reduction cohomology of Riemann surfaces

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