Atiyah–Bott fixed-point theorem

Remarks

The cohomology groups are finite-dimensional because the complex is elliptic, and the fixed points are finitely many because simple fixed points are isolated and $X$ is compact. The idea of the proof is to write $L(T)$ as an alternating sum of traces of operators of the form $T_i e^{-t \Delta_i}$, where $\Delta_i$ are the Laplacians of the complex. These traces are integrals of a kernel over the graph of $f$ restricted to the diagonal, and the transversality condition $\det(1 - df_x) \ne 0$ localizes the integral at the fixed points as $t \to 0$.

For the de Rham complex with $\varphi_i = \Lambda^i (df)^*$, the numerator at $x$ is $\det(1 - df_x)$, so each fixed point contributes $\sgn \det(1 - df_x) = \pm 1$. This recovers the Lefschetz fixed-point theorem for maps with simple fixed points. For the Dolbeault complex of a compact complex manifold and a holomorphic map $f$, it gives the holomorphic Lefschetz formula $$\sum_q (-1)^q \tr\big(f^* \mid H^{0,q}(X)\big) = \sum_{f(x) = x} \frac{1}{\det_\C(1 - \partial f_x)}.$$

The hypothesis that the fixed points are simple cannot be dropped from the formula as stated. For the identity map every point is fixed, and the right side is not defined.

Going deeper

For a student

The theorem is a fixed-point counterpart of the Atiyah–Singer index theorem: for $f = \id$ and no fixed-point condition, $L(T)$ would be the index of the complex. Applied to the Dolbeault complex, it computes traces of group elements on holomorphic cohomology, which leads to the Weyl character formula for compact Lie groups acting on flag manifolds. Applied to the spin complex and the signature complex, it gives fixed-point formulas for the $\hat{A}$-genus and the signature.