For a holomorphic map of a compact complex manifold whose fixed points are isolated and satisfy , the holomorphic Lefschetz fixed-point formula for the structure sheaf saysThe left side uses sheaf cohomology of holomorphic functions; the right side uses the complex determinant, rather than its absolute value. This is a holomorphic form of the Atiyah-Bott fixed-point theorem.[ref]
A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications by M. F. Atiyah and R. Bott. section 4, holomorphic maps and their local contributions
On the complex projective line, , with , fixes zero and infinity. The two derivatives are and , so the local sum is . Globally and , giving the same answer. Individual local terms need not be integers.
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The Holomorphic Lefschetz fixed-point formula is an important result in complex geometry and algebraic geometry that relates fixed points of holomorphic maps to topological invariants of the underlying space. It is an extension of the classical Lefschetz fixed-point theorem which applies to smooth (differentiable) maps. ### Key Concepts 1.