TL;DR
This paper investigates the Poisson structures and computes Poisson cohomology for two specific families of Fano threefolds, providing new methods and rederiving known results.
Contribution
It introduces a novel approach to compute Poisson cohomology for smooth cubic and del Pezzo quintic threefolds, reobtaining previous results with a different method.
Findings
Computed Poisson cohomology for the two Fano threefold families.
Reobtained earlier results by Loray, Pereira, and Touzet using a new approach.
Enhanced understanding of Poisson structures on Fano threefolds.
Abstract
We study the variety of Poisson structures and compute Poisson cohomology for two families of Fano threefolds - smooth cubic threefolds and the del Pezzo quintic threefold. Along the way we reobtain by a different method earlier results of Loray, Pereira and Touzet in the special case we are considering.
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