Аннотация:
Let $X$ be a smooth algebraic variety over a field $K$, and let $\Delta: X \to X \times X$ be the diagonal
embedding. Then the cohomology sheaves of the complex $\mathrm L\Delta^*\Delta_*\mathcal O_X$ are canonically identified with the sheaves of differential forms on $X$. In particular, there is a spectral sequence
from the Hodge cohomology of $X$ to the hypercohomology of the complex $\mathrm L\Delta^*\Delta_*\mathcal O_X$ . If the
characteristic of the base field $K$ is $0$ or larger then $\dim X$, the complex $\mathrm L\Delta^*\Delta_*\mathcal O_X$ is formal,
i.e. quasi-isomorphic to the direct sum of its cohomology sheaves. It follows that in this
case the above spectral sequence degenerates at the first page. It has been a longstanding
question whether this degeneration holds in any characteristic. I will explain a recent result
of Akhil Mathew showing that the analogous spectral sequence fails to degenerate for the
classifying stack of the finite group scheme $\mu_p$ over $\mathbb F_p$. This easily yields an example of a
smooth projective projective variety $X$ such that the spectral sequence does not degenerate.