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

“Holomorphic Functional Calculus”

Writing up the Connes-Renault notes, which I mentioned in a previous post, leads to a number of interesting digressions. For instance, the notion of “holomorphic closure” is discussed at some length in these early notes. But what exactly is the relationship between “holomorphic closure”, “inverse closure”, “complete holomorphic closure” (= holomorphic closure when tensored with any matrix algebra), and so on? I was aware that there had been some progress in this area but had not really sorted it out in my mind. Here’s a summary (all these results are pretty old, so perhaps everyone knows this but me…)