2. Positive/mixed characteristic
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It was noted that there is no such map for $p=2$, so the question should be posed for $p>2$.
Problem 2.1.
[Iacopo Brivio] Does there exist a map $X \to \mathbb{Z}_p$ such that $(X,X_p)$ is log canonical (where $X_p$ is the closed fiber) and $X_p$ is irreducible but not $S_2$? -
Problem 2.2.
[Wanchun Shen] Is there an analogue of $\underline{\Omega}_X^\bullet$ in positive characteristic which makes the Hodge-to-de Rham spectral sequence degenerate at $E_1$? -
Problem 2.3.
[org.aimpl.user:wshen@math.harvard.edu] Is there an analog of the minimal exponent in positive characteristics? -
Problem 2.4.
[Rahul Ajit] Is there a notion of pseudo p-rationality? That is, $H^d_m(R) \rightarrow H^d_m(\mathbb{R} \pi_*\mathcal{O}_Y)$ is injective, where $\pi: Y \rightarrow X$ is a proper birational map and $d = \dim X$. (This definition assumes X is Cohen-Macaulay) -
Problem 2.5.
[Karl Schwede] If X is rational, this means something like $\mathcal{O}_X \rightarrow B$ where B is Cohen-Macaulay algebra, then $H^i_x(\mathcal{O}_X) \rightarrow H^i_x(B)$ is injective.- Is there an analog of this for Du Bois singularities? (where B is almost CM?)
- Is there a m-rational version of this?
- Is there a characteristic independent definition?
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Problem 2.6.
[Iacopo Brivio] Understand KSBA moduli spaces in char p (Hodge bundles, local freeness + commutes with base change, E_1 degeneration etc).
Cite this as: AimPL: Higher Du Bois and higher rational singularities, available at http://aimpl.org/higherdubois.