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Register a new userClemens-Schmid exact sequence in characteristic p
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For a semistable family of varieties over a curve in characteristic $p$, we prove the existence of a Clemens-Schmid type long exact sequence for the $p$-adic cohomology. The cohomology groups appearing in such a long exact sequence are defined locally
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This is a sequel to our work in tropical Hodge theory. Our aim here is to prove a tropical analogue of the Clemens-Schmid exact sequence in asymptotic Hodge theory. As an application of this result, we prove the tropical Hodge conjecture for smooth projective tropical varieties which are rationally triangulable. This provides a partial answer to a question of Kontsevich who suggested the validity of the tropical Hodge conjecture could be used as a test for the validity of the Hodge conjecture.
We give counterexamples to the degeneration of the HKR spectral sequence in characteristic $p$, both in the untwisted and twisted settings. We also prove that the de Rham--$mathrm{HP}$ and crystalline--$mathrm{TP}$ spectral sequences need not degenerate.
In this paper we show an Arakelov inequality for semi-stable families of algebraic curves of genus $ggeq 1$ over characteristic $p$ with nontrivial Kodaira-Spencer maps. We apply this inequality to obtain an upper bound of the number of algebraic curves of $p-$rank zero in a semi-stable family over characteristic $p$ with nontrivial Kodaira-Spencer map in terms of the genus of a general closed fiber, the genus of the base curve and the number of singular fibres. An extension of the above results to smooth families of Abelian varieties over $k$ with $W_2$-lifting assumption is also included.
In this paper we focus on pairs consisting of the affine $N$-space and multiideals with a positive exponent. We introduce a method lifting to characteristic 0 which is a kind of the inversion of modulo p reduction. By making use of it, we prove that Mustata-Nakamuras conjecture and some uniform bound of divisors computing log canonical thresholds descend from characteristic 0 to certain classes of pairs in positive characteristic. We also pose a problem whose affirmative answer gives the descent of the statements to the whole set of pairs in positive characteristic.
We prove a Hochschild-Kostant-Rosenberg decomposition theorem for smooth proper schemes $X$ in characteristic $p$ when $dim Xleq p$. The best known previous result of this kind, due to Yekutieli, required $dim X<p$. Yekutielis result follows from the observation that the denominators appearing in the classical proof of HKR do not divide $p$ when $dim X<p$. Our extension to $dim X=p$ requires a homological fact: the Hochschild homology of a smooth proper scheme is self-dual.
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