# A “well known interpolation formula”

In the Atiyah-Bott paper on their Lefschetz theorem for elliptic complexes, they give a very nice elementary example of the Lefschetz theorem for the Dolbeault complex, by considering the automorphism of  ${\mathbb C}{\mathbb P}^n$ given by

$[z_0,\ldots, z_n] \mapsto [\gamma_0z_0, \ldots, \gamma_nz_n ]$

in homogeneous coordinates, where the $$\gamma_i$$ are distinct and nonzero complex numbers.  This has $$(n+1)$$ simple fixed points and applying the holomorphic Lefschetz theorem gives

$1 = \sum_{i=0}^n \frac{\gamma_i^n}{\prod_{j\neq i} (\gamma_i-\gamma_j)}.$

This is Example 2 on p. 460 of the second Atiyah-Bott paper.  They go on to describe this as a “well known interpolation formula”. Continue reading