Speaker
Description
For smooth projective varieties in characteristic zero, Hodge-to-de Rham
spectral sequence degenerates and Akizuki--Nakano vanishing theorem holds.
However, both statements can fail in positive and mixed characteristic.
Deligne and Illusie proved that, for a smooth variety $X$ in characteristic
$p$ admitting a $W_2$-lifting and satisfying $p>\dim X$, de Rham complex
decomposes and this implies the above degenerations and vanishings. More
recently, Petrov used the geometry of the de Rham stack to obtain such a
decomposition for smooth quasi-$F$-split varieties without any dimension
restriction.
When such a decomposition exists, the natural map to the de Rham complex
admits a retraction. In this talk, I will explain how, conversely, the
existence of such a retraction, which we call Hodge--Tate splitting, implies
a decomposition of the de Rham complex by using the geometry of de Rham
stacks. I will also discuss the corresponding logarithmic theory for SNC
divisors and an application to mixed characteristic.
This is joint work with Shou Yoshikawa.