The mixed-characteristic lifting problem for smooth projective varieties (Grothendieck, SGA 1, Exposé III)

SGA 1, Exposé III ('Morphismes lisses : propriétés de prolongement') proves that prolonging a smooth scheme across an infinitesimal thickening of its base is controlled by a canonical obstruction class in $H^2$ of the sheaf of infinitesimal automorphisms, with the isomorphism classes of prolongations forming a torsor under $H^1$ when the obstruction vanishes. At the end of the Exposé Grothendieck records what was then open: the general algebraic mechanism of these classes ('on ne sait à peu près rien'), their computation for abelian schemes (families of abelian varieties) over Artinian rings, the classification of lifts and of their dependence on coefficient-field choices in equal characteristic, and — the core — the lifting problem in unequal characteristic, where it was unknown whether a smooth projective variety over a field of characteristic $p$ must lift over the Witt ring or a Cohen $p$-ring at all, and whether the first obstruction class in $H^2(X_0,\Theta_{X_0/k})$ could be nonzero. The mechanism was supplied by Schlessinger's functors of Artin rings and Illusie's cotangent complex; the abelian case was settled (Serre–Tate theory, Norman–Oort); non-liftable smooth projective varieties exist (Serre 1961; Mumford's unpublished surface; numerous surface and Calabi–Yau examples through 2022; new obstruction types in 2021, 2023 and 2026); curves, abelian varieties, K3 and Enriques surfaces lift, in some classes only over ramified extensions. What survives is the problem in its original generality: no criterion is known that, given a smooth projective variety over a field of characteristic $p$, decides whether it lifts to characteristic zero, and the classification of liftable varieties is open.

Background

SGA 1 (Séminaire de Géométrie Algébrique du Bois Marie 1960–61, published as Lecture Notes in Mathematics 224) develops the theory of the étale fundamental group (the scheme-theoretic analogue of the topological fundamental group, built from the finite étale covers of a scheme); its Exposé III, 'Morphismes lisses : propriétés de prolongement' (a chapter by Grothendieck, with the volume co-credited to M. Raynaud, who added updating footnotes to the re-edition), establishes the prolongation properties of smooth morphisms that the later Exposés need. A morphism of schemes $f\colon X\to S$ is smooth if it is flat, locally of finite presentation, and has geometrically regular fibers; equivalently it is formally smooth, meaning that infinitesimal thickenings of the base can always be lifted locally on $X$. The converse direction — prolonging a given scheme that is smooth over a smaller base to an infinitesimally thickened base — is not formal, and its failure is measured by cohomology: if $S_n=(S,\mathcal I^{n+1})$ for an ideal $\mathcal I$ of a noetherian scheme $S$ and $X_n$ is a smooth $S_n$-scheme with special fiber $X_0$, then Théorème III.6.3 attaches a canonical obstruction class in $H^2(X_0,\mathcal G)$, where $\mathcal G$ is the sheaf of infinitesimal automorphisms — concretely $H^i(X_0,\mathcal G)\cong H^i(X_0,\Theta_{X_0/S_0})\otimes\mathrm{gr}^{n+1}_\mathcal I(S)$, with $\Theta=\mathcal{Hom}(\Omega^1,\mathcal O)$ the tangent sheaf, $\Omega^1$ denoting the sheaf of Kähler differentials — whose vanishing is necessary and sufficient for a smooth prolongation $X_{n+1}$ to exist; when it vanishes, the isomorphism classes of prolongations form a principal homogeneous space (torsor) under $H^1(X_0,\mathcal G)$. For proper $X_0$ with $H^2(X_0,\mathcal O)=0$, formal lifts algebraize, so a smooth projective $X_0$ with $H^2(X_0,\Theta)=H^2(X_0,\mathcal O)=0$ lifts over every complete local ring with residue field $k$ (Théorème III.7.3); in particular every smooth projective curve lifts (Corollaire III.7.4), which is the input for the transcendental study of the fundamental group in the rest of SGA 1.

The picture splits according to the characteristic of the complete local noetherian ring $A$ with residue field $k$. In equal characteristic, Cohen's structure theorem provides a coefficient field $k\to A$, so $X_0\otimes_k A$ is a lift; but the lift depends on the chosen embedding, and two embeddings can yield non-isomorphic schemes once $H^1(X_0,\Theta)\ne 0$ — Grothendieck asks exactly which lifts arise this way and anticipates that the answer is governed by the Kodaira–Spencer bracket (the pairing measuring to first order how the fibers of a smooth morphism vary along the base). In unequal (mixed) characteristic — the ring of $p$-typical Witt vectors $W(k)$, the universal complete discrete valuation ring of characteristic $0$ with perfect residue field $k$, and more generally Cohen $p$-rings, from which every complete local ring with residue field $k$ receives a map, so that lifting over a Cohen ring is the strongest form of the problem — there is no section $A\to k$ and everything is obstructed. At the time of writing it was unknown whether every smooth projective $X_0$ lifts (Grothendieck conjectured that non-liftable examples exist, with $X_0$ a surface), and every smooth $X_0$ over a field of characteristic $p>0$ carries a first obstruction class in $H^2(X_0,\Theta_{X_0/k})$ to lifting over a Cohen $p$-ring, whose possible nonvanishing was open. The Exposé closes by asking for the general algebraic mechanism of the obstruction classes — 'on ne sait pour l'instant à peu près rien' ('at present one knows almost nothing') — including in the simplest cases such as abelian schemes over Artinian rings, expressing the hope that 'il se trouvera des gens pour chiader la question' (that people will be found to grind away at the question), tied to the 'theory of moduli' of algebraic structures.

Problem Statement

The problem, as recorded at the end of SGA 1, Exposé III (§§6.6–7), is to build a complete prolongation theory for smooth schemes over complete local noetherian rings, in three linked facets.

(a) Mechanism. Develop the general algebraic theory of the obstruction classes of Théorème III.6.3 — the dependence of the classes in $H^2(X_0,\Theta_{X_0/k})$ on the thickening, their relation to the deformation theory of $X_0$, and explicit computations in basic cases, notably abelian schemes (smooth proper group schemes with connected geometric fibers, i.e. families of abelian varieties) over Artinian rings.

(b) Unequal characteristic — the surviving core. Let $k$ be a field of characteristic $p>0$ and let $A$ be a complete local noetherian ring of unequal characteristic with residue field $k$ — most strongly a Cohen $p$-ring (for $k$ perfect, the Witt ring $W(k)$), since a lift over it implies a lift over every such $A$. The open question is an effective criterion deciding liftability for given $X_0$ and $A$, equivalently a determination of which smooth projective $k$-schemes $X_0$ are the special fiber of a smooth $A$-scheme, formal or projective — that is, a classification of the smooth projective $k$-schemes that lift and those that do not. Two further questions of the same facet concern (i) whether liftability is typical — whether the generic smooth projective variety over a given field of characteristic $p$ lifts, the universal form of the question being already settled negatively by the known non-liftable examples — and (ii) whether the higher obstruction classes in the tower can be controlled by finitely many invariants.

(c) Non-uniqueness. Classify lifts up to isomorphism: the torsor structure under the $H^1$ groups, and in equal characteristic precisely which lifts arise from which coefficient-field embeddings $k\to A$, and when two embeddings give isomorphic lifted schemes.

Facets (a) and (c) are settled background: (a) by the standard machinery of Schlessinger functors and the cotangent complex (Schlessinger 1968; Illusie 1971/72), and (c) by Kodaira–Spencer theory together with the Serre–Tate and Norman–Oort analysis of lifts (Katz 1981; Norman–Oort 1980). The decisive open part (b) in its general form is to find an effective criterion deciding, for a smooth projective variety over a field of characteristic $p$, whether it lifts to characteristic zero — equivalently, to classify the smooth projective $k$-schemes that lift and those that do not. No such criterion is known, and the classification is open.

References

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