Liftability of smooth projective varieties from characteristic p to characteristic zero: criteria and obstructions (the general question raised by Serre's 1961 non-liftable examples)

Serre's 1961 PNAS note constructed the first smooth projective varieties over algebraically closed fields of characteristic p that lift over no characteristic-zero ring, answering negatively the two lifting questions it poses; the note itself poses no further questions. The problem considered here is the general question its examples left open, posed in the subsequent literature (Bernasconi–Brivio–Kawakami–Witaszek, Section 1): under which conditions does a smooth projective variety in characteristic p admit a lifting to characteristic zero — over some complete noetherian local ring of characteristic zero, or over the Witt ring W(k) or its length-two truncation W2(k) — by sufficient conditions guaranteeing a lift, obstructions for non-lifting, and decisions for the natural classes where liftability is unknown. The recognized frontier cases are the folklore conjecture that every globally F-split normal projective variety lifts to a flat projective scheme over W(k) (open in dimension at least 3 after the surface case was proved in 2022) and the conjecture that ordinary varieties with trivial canonical class have unobstructed deformations over W(k) (open in general). The literature from 1961 through 2026 — including the descendants of Serre's note, the Deligne–Illusie W2-theory, and the 2021–2026 wave of new obstructions (étale fundamental groups, étale cohomology rings) and new liftable classes (F-split surfaces, Fano threefolds) — leaves the problem open: no general criterion is known, and both obstruction and positive results continue to accumulate.

Background

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $X_0$ be a smooth projective variety over $k$. A lifting of $X_0$ to characteristic zero is a proper flat scheme $X$ over a complete noetherian local ring $A$ of characteristic zero with residue field $k$, together with an isomorphism between the special fiber of $X$ over $A$ and $X_0$ (flatness means the structure varies without jumps, so all fibers share their homological shape). The standard refinements ask for a lift over the ring $W(k)$ of Witt vectors of $k$ — the complete discrete valuation ring with residue field $k$ and uniformizer $p$, the universal characteristic-zero thickening of $k$ — or over its length-two truncation $W_2(k) = W(k)/p^2W(k)$; allowing ramified finite extensions of $W(k)$ gives an intermediate notion. These notions genuinely differ: by a complement in Serre's note, every ring over which his examples lift is annihilated by a power of $p$ (so has positive characteristic, never zero), and they lift over no characteristic-zero ring. Deciding which characteristic-$p$ varieties lift fixes how far characteristic-zero tools reach into positive characteristic, since every characteristic-zero variety can be reduced modulo $p$: for instance, a smooth projective variety of dimension at most $p$ over a perfect field that lifts over $W_2(k)$ has degenerate Hodge–de Rham spectral sequence (the filtration of de Rham cohomology by sheaf cohomology of differential forms degenerates) and satisfies Kodaira vanishing (the higher cohomology of the dual of an ample line bundle vanishes; Deligne–Illusie).

In a note in PNAS, Serre posed the question in two precise forms — does a given smooth projective $X_0$ lift over a given characteristic-zero $A$, and does it lift over some characteristic-zero ring — and answered both negatively. His construction takes a finite group $G$ and a homomorphism $G$ to $PGL_n(k)$, the group of automorphisms of projective space, i.e. invertible matrices modulo scalars (a projective representation), satisfying hypothesis (D): the fixed-point set of every nontrivial element has codimension at least 4. A classical Bertini-type argument then gives a smooth $G$-stable complete intersection $Y_0$ in $P^{n-1}_k$ (a variety cut out by the expected number of homogeneous equations) of dimension at least 3 avoiding all fixed loci; the $G$-action on $Y_0$ is then free and the quotient $X_0 = Y_0/G$ is smooth projective. The key Lemma of the note shows that if $X_0$ lifts over $A$, then the projective representation lifts to a homomorphism $G$ to $PGL_n(A)$: the proof lifts the étale $G$-cover $Y_0$ over $X_0$ (using that the étale fundamental group — the profinite group classifying finite étale covers, analogous to the topological fundamental group — is invariant under infinitesimal deformation, a theorem of Grothendieck quoted in the note) together with the hyperplane line bundle. Serre then embeds an elementary abelian $p$-group (a product of copies of the cyclic group of order $p$) of order greater than $p^5$, for $p$ at least 7, into $PGL_5(k)$ via unipotent matrices (identity plus nilpotent; these have order $p$ in characteristic $p$), whereas such a subgroup cannot exist in $PGL_5$ over a field of characteristic zero, where commuting elements of order $p$ are simultaneously diagonalizable modulo scalars, bounding the rank of an elementary abelian $p$-subgroup. Hence $X_0$ lifts over no characteristic-zero ring, even over an algebraically closed residue field. A second complement notes that $X_0$ does not lift even as a cycle in the sense of Shimura's reduction of cycles. The note poses no further questions; the problem its examples left open, posed in the subsequent literature (Bernasconi–Brivio–Kawakami–Witaszek, Section 1), is the general question of under which conditions an arbitrary smooth projective variety in characteristic $p$ admits a lifting to characteristic zero.

The subsequent literature built both directions. On the negative side, Raynaud's 1978 counterexamples to Kodaira vanishing are not liftable to $W_2(k)$ (as documented by Achinger and Zdanowicz), and non-liftable smooth projective varieties now exist in essentially every class without Frobenius positivity: rational simply-connected varieties with well-behaved cohomology (Achinger–Zdanowicz), Calabi–Yau varieties — smooth projective with trivial canonical class, the canonical bundle being the top exterior power of the cotangent bundle — in every characteristic $p$ at least 5 (Achinger–Zdanowicz), varieties with trivial dualizing sheaf (for smooth varieties, the same as the canonical bundle) in every characteristic (Rössler–Schröer), and, for every prime $p$, a smooth projective surface that cannot even be rationally dominated by any liftable variety (van Dobben de Bruyn), which in particular does not itself lift. New obstruction types continue to appear: non-$p$-prime-to-$p$ discrete finite generation of the étale fundamental group (Esnault–Srinivas–Stix), étale cohomology rings (the positive-characteristic analogue of singular cohomology) admitting no rational form (Hu–Zhang), and families of abelian varieties over curves that do not lift to $W_2(k)$ (Cheng). On the positive side, curves (by a theorem of Grothendieck quoted already in Serre's note), abelian varieties (projective varieties carrying an algebraic group structure), K3 surfaces (simply connected surfaces with trivial canonical class) and Enriques surfaces (quotients of K3 surfaces by a fixed-point-free involution) lift to characteristic zero (documented in Liedtke's survey); every globally F-split variety — one whose absolute Frobenius map (raising local functions to their $p$-th powers) $O_X$ to $F$-pushforward $O_X$ admits an $O_X$-linear splitting — lifts over $W_2(k)$ (recorded by Achinger and Zdanowicz); every globally F-split surface lifts over $W(k)$ (Bernasconi–Brivio–Kawakami–Witaszek); and smooth Fano threefolds (threefolds whose anti-canonical class is ample) lift over $W(k)$ (Kawakami–Tanaka). Liftability is a birational invariant for smooth surfaces but not in dimension at least 3 (Liedtke–Satriano).

The recognized open frontier consists of: (a) the folklore conjecture — stated as Conjecture 1.2 in Bernasconi–Brivio–Kawakami–Witaszek and crystallized in Section 1.7 of Achinger–Zdanowicz, which records that no non-liftable globally F-split variety is known — that every globally F-split normal projective variety lifts to a flat projective scheme over $W(k)$: proved in dimension two, open in dimension at least 3; and (b) Conjecture 1.1.1 of Achinger–Zdanowicz: a smooth projective variety with trivial canonical class whose Hasse–Witt operation (the Frobenius action on $H^d(X, O_X)$, $d$ the dimension) is bijective — the ordinary case — has unobstructed deformations over $W(k)$: all known non-liftable examples with trivial canonical class are supersingular (vanishing Hasse–Witt), so the ordinary case is open in general (for ordinary abelian varieties the conjecture holds in all dimensions by classical Serre–Tate theory, and for ordinary K3 surfaces by work of Nygaard and Ogus, as recalled by Achinger–Zdanowicz). Beyond these, no effective criterion is known that decides, for an explicitly given smooth projective variety in characteristic $p$, whether it lifts to characteristic zero: the Deligne–Illusie description of the gerbe (a stack encoding all lifts) of $W_2$-liftings is structural, not a decision procedure, and for lifts over arbitrary characteristic-zero rings the obstruction theory is incomplete. Adjacent problems — lifting group actions on curves (the local lifting problem, surveyed by Obus), lifting automorphisms of a fixed variety (Esnault–Oguiso; Xie–Zheng), and lifting the Frobenius endomorphism itself — are related but distinct lineages, not the core.

Problem Statement

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $X_0$ be a smooth projective $k$-variety. Say that $X_0$ lifts to characteristic zero if there exist a complete noetherian local ring $A$ of characteristic zero with residue field $k$ and a proper flat $A$-scheme $X$ whose special fiber is isomorphic to $X_0$ (Serre's question (2)); the standard variants ask for a flat projective lift over the Witt ring $W(k)$ of $k$ (the complete discrete valuation ring of characteristic zero with residue field $k$ and uniformizer $p$), over its length-two truncation $W_2(k) = W(k)/p^2W(k)$, or over a ramified finite extension of $W(k)$ (an intermediate requirement between lifting over $W(k)$ and lifting over some characteristic-zero ring). These lift notions are genuinely different: a flat projective lift over a complete noetherian local ring is in particular a proper flat lift, but the converse — extending or algebraizing an ample line bundle from the special fiber — is not automatic, and unobstructedness of the deformation functor over $W(k)$ (the conclusion sought in item (2b) below) is strictly stronger than the mere existence of a flat lift. Serre (1961) proved that not every $X_0$ lifts, in the strongest form: his quotient examples lift over no characteristic-zero ring at all. The problem, in the framing given to it in Bernasconi–Brivio–Kawakami–Witaszek (Section 1), is to characterize the liftable ones: (1) (Criteria) Give conditions that decide liftability of a given smooth projective variety in characteristic $p$ — sufficient conditions guaranteeing a lift (over $W(k)$, a ramified extension, or some characteristic-zero ring) and obstructions certifying non-liftability. (2) (Frontier classes) Decide liftability and unobstructedness for the natural classes where they are currently unknown, in particular: (a) the folklore conjecture (Conjecture 1.2 of Bernasconi–Brivio–Kawakami–Witaszek; Section 1.7 of Achinger–Zdanowicz) that every globally F-split normal projective variety (one whose absolute Frobenius map $O_X \to F_*O_X$, raising local functions to their $p$-th powers, admits an $O_X$-linear splitting) lifts to a flat projective scheme over $W(k)$, open in dimension at least 3; (b) Conjecture 1.1.1 of Achinger–Zdanowicz: a smooth projective variety with trivial canonical class (vanishing canonical bundle) and bijective Hasse–Witt action (the Frobenius action) on $H^d(X, O_X)$ ($d$ the dimension) has unobstructed deformations over $W(k)$ — that is, its formal deformation functor over $W(k)$ is unobstructed, which is strictly stronger than the existence of a flat lift — open in general (it holds for ordinary abelian varieties in every dimension by classical Serre–Tate theory and for ordinary K3 surfaces by the Nygaard–Ogus theory recalled by Achinger–Zdanowicz). The overarching objective is the characterization of liftability for smooth projective varieties in positive characteristic. The lift notions (lifts over some characteristic-zero ring, over $W(k)$, and over $W_2(k)$) are strictly different: by a complement in Serre's note, every ring over which his examples lift is annihilated by a power of $p$, so they lift over no characteristic-zero ring. Among the natural classes, curves, abelian varieties, K3 and Enriques surfaces are known to lift to characteristic zero, while globally F-split varieties in dimension at least 3, and ordinary varieties with trivial canonical class, are recorded in the cited literature as open.

References

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