The Hirzebruch-Riemann-Roch theorem identifies for a holomorphic vector bundle $E$ the alternating sum of its abelian sheaf cohomology groups (hence for trivial bundle the index of the Dolbeault operator) with the Chern character of the bundle cupped with the Todd class of the underlying complex manifold.
Friedrich Hirzebruch, Topological methods in algebraic geometry, russian translation, Mir, Moscow, 1973.
Wikipedia, Hirzebruch-Riemann-Roch theorem
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