[Paper Review] p-adic deformation of graph cycles

Xuanyu Pan|arXiv (Cornell University)|Oct 12, 2016

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper establishes a $p$-adic deformation theory for automorphisms of algebraic varieties, proving that if an automorphism acts as a Hodge-type operator on cohomology and the variety satisfies the infinitesimal Torelli theorem, then the automorphism lifts uniquely to a formal automorphism over the Witt ring. The key result generalizes classical deformation theory to the $p$-adic setting using crystalline cohomology and provides a criterion for the faithfulness of the automorphism group action on cohomology.

ABSTRACT

In this paper, we show that the infinitesimal Torelli theorem implies the existence of deformations of automorphisms. In the first part, we use Hodge theory and deformation theory to study the deformations of automorphisms of complex projective manifolds. In the second part, we use crystalline cohomology to explore the p-adic analogues of the first part, which generalizes a result of Berthelot and Ogus. The study of the deformations of automorphisms also provides criterions characterizing when the action of the automorphism group of a variety on its cohomology is faithful.

Motivation & Objective

• To extend the classical deformation theory of automorphisms from complex to $p$-adic settings using crystalline cohomology.
• To establish a criterion for lifting automorphisms over $W(k)$ based on their action on cohomology.
• To generalize results of Berthelot and Ogus on $p$-adic deformations to automorphisms of varieties.
• To provide a cohomological criterion for the faithfulness of the automorphism group action on $\ell$-adic and crystalline cohomology.
• To connect the preservation of Hodge filtration in cohomology with the existence of global $p$-adic deformations of automorphisms.

Proposed method

• Use of crystalline cohomology to study $p$-adic deformations of automorphisms, replacing de Rham cohomology in the complex case.
• Construction of obstruction maps via the Hodge filtration and the comparison isomorphism between de Rham and crystalline cohomology.
• Application of Grothendieck’s existence theorem to lift formal automorphisms on $\widehat{X}$ to algebraic automorphisms over $W(k)$.
• Identification of the map $\rho(g_n)_q$ as a component of the obstruction class in $\mathrm{H}^1(X_0, g_0^*T_{X_0/k}) \otimes_k (\pi^{n+1})$.
• Use of the injectivity of the map $\widehat{\Psi}_n$ to deduce that the obstruction vanishes when the Hodge filtration is preserved.
• Leveraging the fact that $\mathrm{H}_{\mathrm{cris}}^m(g_0)$ preserves the Hodge filtration to conclude that $\rho(g_n)_q = 0$, hence $ob(g_n) = 0$.

Experimental results

Research questions

• RQ1 Under what conditions does an automorphism of a variety over a perfect field of positive characteristic lift to an automorphism over the ring of Witt vectors $W(k)$?
• RQ2 How does the preservation of the Hodge filtration on crystalline cohomology relate to the existence of $p$-adic deformations of automorphisms?
• RQ3 Can the infinitesimal Torelli theorem be used to ensure the deformation of automorphisms in the $p$-adic setting?
• RQ4 What cohomological criterion guarantees that the action of the automorphism group on $\ell$-adic or crystalline cohomology is faithful?
• RQ5 How does the finiteness of the automorphism order interact with $p$-adic lifting and cohomological triviality?

Key findings

• An automorphism $g_0$ of a variety $X_0$ over a perfect field $k$ of positive characteristic lifts to an automorphism over $W(k)$ if $\mathrm{H}_{\mathrm{cris}}^m(g_0)$ preserves the Hodge filtration and $X_0$ satisfies the infinitesimal Torelli theorem of degree $m$.
• The obstruction to lifting $g_n$ modulo $W_{n+1}$ vanishes precisely when the induced map on cohomology preserves the Hodge filtration, as shown by the vanishing of $\rho(g_n)_q$.
• The formal automorphism $\lim_{\leftarrow} g_n$ on the formal scheme $\widehat{X}$ arises from a unique algebraic automorphism over $W(k)$ via Grothendieck’s existence theorem.
• If $\mathrm{H}_{\text{ét}}^m(g_0, \mathbb{Q}_l) = \mathrm{Id}$ for $l \neq \mathrm{char}(k)$, then $g_0$ lifts to an automorphism over $W(k)$, especially when $g_0$ has finite order.
• The faithfulness of the action of $\mathrm{Aut}(X_0)$ on $\mathrm{H}_{\text{ét}}^m(X_0, \mathbb{Q}_l)$ is implied by the faithfulness of the action on $\mathrm{H}_{\text{ét}}^m(X_K, \mathbb{Q}_l)$ for the generic fiber $X_K$, under the lifting condition.
• The result generalizes Berthelot and Ogus’s work on $p$-adic deformations by extending it from cycles to automorphisms, using the Hodge-type condition as a lifting criterion.