Serre–Tate theory for Calabi–Yau varieties

• Piotr Achinger and Maciej Zdanowicz

Abstract

Classical Serre–Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips the base of its universal deformation with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus.

In this paper, we construct canonical liftings modulo p2 of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Frobenius lifting on the moduli space of such varieties. The quite explicit construction uses Frobenius splittings and a relative version of Witt vectors of length two. If the variety has unobstructed deformations and bijective first higher Hasse–Witt operation, the Frobenius lifting gives rise to canonical coordinates. One of the key features of our liftings is that the crystalline Frobenius preserves the Hodge filtration.

We also extend Nygaard’s approach from K3 surfaces to higher dimensions, and show that no non-trivial families of such varieties exist over simply connected bases with no global one-forms.

Funding statement: The first author was supported by NCN SONATA grant number 2017/26/D/ST1/00913. The second author’s work was supported by Zsolt Patakfalvi’s Swiss National Science Foundation Grant No. 200021/169639. This work was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015–2019 Polish MNiSW fund.

A Finite height

In [59], Yobuko defines the notion of a quasi-F-splitting, more general than an F-splitting, and proves that a smooth quasi-F-split variety can be lifted modulo p2, generalizing the argument for smooth F-split varieties [25, Section 8.5]. He also shows that for Calabi–Yau varieties, being quasi-F-split is equivalent to the height of the associated Artin–Mazur formal group being finite. In this section, we give a somewhat different point of view on Yobuko’s construction. Based on this, we extend our construction of a canonical lifting associated to an F-splitting to quasi-F-split varieties, thus showing that there is a preferred lifting mod p2. In particular, we show that the smoothness assumption used by Yobuko is not necessary. Some of our results were observed earlier by Adrian Langer (unpublished).

It would be interesting to extend this construction to families and to generalize the Serre–Tate theory discussed in this paper to varieties with trivial canonical class of finite height in the spirit of [43].

A.1 The canonical lifting

Yobuko has defined a quasi-F-splitting of level m on an 𝐅p-scheme X as an additive map σ:Wm𝒪X𝒪X satisfying σ(1)=1 and which is F-linear in the sense that

σ(Fxy)=x0σ(y).

We have a short exact sequence

0Wm𝒪X𝑉Wm+1𝒪XRm𝒪X0.

Lemma A.1.1.

Suppose that σ:WmOXOX is a quasi-F-splitting of level m on an Fp-scheme X. Then the image of ker(σ) under V is an ideal in Wm+1OX.

Proof.

Suppose that xWm𝒪X satisfies σ(x)=0, and let yWm+1𝒪X. Then

yV(x)=V(F(y)x)andσ(F(y)x)=yσ(x)=0.

Corollary A.1.2.

Suppose that σ:WmOXOX is a quasi-F-splitting of level m on an Fp-scheme X. Then the quotient

𝒪X~=Wm+1𝒪X/V(kerσ)

defines a lifting of X over Z/p2Z, fitting inside a pushout diagram of exact sequences

Proof.

Since σ is surjective, the left square is a pushout. To prove that 𝒪X~ is flat, we need to check that the composition

𝒪X~𝒪X𝒪X~

equals multiplication by p. Let f~𝒪X~ be the image of

f=(f0,,fm)Wm+1𝒪X.

Then the image of f~ in 𝒪X is f0, which is the image under σ of

Ff=(f0p,,fmp)Wm𝒪X.

This in turn has image V(Ff)=pf in Wm𝒪X, which maps to pf~ under π. ∎

Definition A.1.3.

We call the lifting X~ constructed in Corollary A.1.2 the canonical lifting associated to the quasi-F-splitting σ.

Remark A.1.4.

Composing the map π:Wm+1𝒪X𝒪X~ with the Teichmüller lift [-]:𝒪XWm+1𝒪X yields a multiplicative section of 𝒪X~𝒪X. Consequently, the assertions of Lemma 5.2.2 hold for X~ as well.

A.2 The sheaf ℱm⁢𝒪X and Witt vectors mod p

Suppose that X is a smooth scheme over a perfect field k of characteristic p. Denote by C:ZΩX1ΩX1 the Cartier operator. The sheaf ZmΩX1F*mΩX1 consists of all local sections ω such that ω,C(ω),,Cm-1(ω) are closed, while the subsheaf BmΩX1 of ZmΩX1 consists of all local sections ω such that Cm-1(ω)BX1. The map

Dm:F*Wm𝒪XBmΩX1,(f0,,fm-1)f0pm-1-1df0++dfm-1

is Wm𝒪X-linear, and there is a short exact sequence of Wm𝒪X-modules

0Wm𝒪X𝐹F*Wm𝒪XDmBmΩX10.

In this situation, Yobuko defined a natural extension of 𝒪X-modules (denoted (em))

0𝒪Xm𝒪XBmΩX10

and showed that 𝒪X-linear splittings of this sequence correspond to quasi-F-splittings as defined previously. The exact sequence (em) is defined by the pullback diagram

In particular, if X is F-split, the bottom row is, and hence so is (em). Yobuko observes that (em) can also be defined as a pushout

It follows that quasi-F-splittings are precisely 𝒪X-linear splittings of (em).

We shall now elucidate the sheaf m and its 𝒪X-algebra structure. To this end, let X be an 𝐅p-scheme. Following [20, Section 3], we denote by

W¯m𝒪X=Wm𝒪X/pWm𝒪X

the mod-p reduction of the ring of Witt vectors of length m over 𝒪X. Recall from [20, Section 3] that the p-th power of the Teichmüller map

[-]p:𝒪XWm𝒪X

is additive modulo p, and therefore induces an 𝒪X-algebra structure

ρ=[-]pmodp:𝒪XW¯m𝒪X

on Wm𝒪X. In addition, the following triangle commutes:

Consequently, ρ is injective if X is reduced. We denote its cokernel by Bm, so that there is a short exact sequence of 𝒪X-modules

0𝒪XW¯m𝒪XBm0.

The formula

V(x)V(y)=V(FV(x)V(y))=pV(xy)

implies that the multiplication in W¯m𝒪X is highly degenerate.

Remark A.2.1.

The existence of ρ can be also seen as follows (cf. [47, Section 1.1]): the kernel VWm-1𝒪X of Rm-1:Wm𝒪X𝒪X has a natural divided power structure, and hence so does the kernel I of

Rm-1:W¯m𝒪X𝒪X.

It follows that fp=0 for every fI, and hence the absolute Frobenius of W¯m𝒪X factors naturally through 𝒪X.

Proposition A.2.2.

The following diagram is a pushout square:

Proof.

The commutativity of the diagram follows from the formula

F(f0,,fm-1)=(f0p,,fm-1p)
=[f0]p+VF(f1,,fm-1)
=[f0]p+p(f1,,fm-1)ρ(f0)modp.

Since the vertical arrows are surjective, it remains to check that the induced map

F:kerRm-1kerπ

is an isomorphism. This follows from

F(kerRm-1)=F(V(Wm-1𝒪X))=pWm-1𝒪X.

Corollary A.2.3.

Suppose that X is a smooth scheme over k. There are natural isomorphisms

W¯m𝒪Xm𝒪X𝑎𝑛𝑑BmBmΩX1

fitting inside a commutative diagram

We can therefore rephrase the definition of a quasi-F-splitting as follows.

Definition A.2.4.

Let X be an 𝐅p-scheme. A quasi-F-splitting of level m on X is an 𝒪X-linear splitting σ:W¯m𝒪X𝒪X of the map ρ:𝒪XW¯m𝒪X.

B Families with vanishing Kodaira–Spencer

Let k be a perfect field of characteristic p>0 and let S be a smooth k-scheme. Let f:XS be a smooth morphism. Applying Rf* to the short exact sequence on X

(B.1)0TX/STX/kf*TS/k0

yields an exact sequence on S

(B.2)0f*TX/Sf*TX/kf*f*TS/k𝛿R1f*TX/S.

The Kodaira–Spencer map of X/S is the map KSX/S:TS/kR1f*TX/S obtained as the composition of the adjunction map TS/kf*f*TS/k and δ:f*f*TS/kR1f*TX/S.

Theorem B.0.1 (compare [45, Lemma 3.5]).

Suppose KSX/S=0 and f*TX/S=0. Then there exists a canonical cartesian diagram

Proof.

The assumptions KSX/S=0 and f*TX/S=0 combined with the exactness of (B.2) show that the adjunction map TS/kf*f*TS/k factors uniquely through a map u:TS/kf*TX/k. By another adjunction, we obtain a map v:f*TS/kTX/k which splits (B.1). We check that v defines a 1-foliation (see [12, Chapter I, Section 1, p. 104]), i.e. that its image is closed under the Lie bracket and p-th iterates. The respective obstructions [12, Lemma 1.4, p. 105] are maps

2f*TS/kTX/SandFX*f*TS/kTX/S

2TS/kf*TX/SandFS*TS/kf*TX/S

which both vanish because the target is zero. Define Y to be the quotient by this 1-foliation. ∎

Remark B.0.2.

As the example S=𝐏k1 and X=𝐏(𝒪S𝒪S(1))S shows, the assumption that f*TX/S=0 is necessary. Note that in this example X/S still descends along FS/k Zariski-locally on S. To produce examples which do not descend even locally, one can take X/S a Brauer–Severi variety whose corresponding class in He´t2(S,𝐆m) is not divisible by p.

Of course, if KSX/S=0, then the adjunction map TS/kf*f*TS/k can locally be lifted to a map u:TS/kf*TX/k (since TS/k is locally free). However, for the above proof to work, we need the lifting u to be compatible with the restricted Lie algebra structures.

Acknowledgements

We would like to thank Bhargav Bhatt, Jędrzej Garnek, Adrian Langer, Daniel Litt, Arthur Ogus, Lenny Taelman, and Jakub Witaszek for helpful discussions. We thank the referees for many helpful comments regarding the manuscript. Part of this work was conducted during the first author’s stay at the MPIM and the Hausdorff Center for Mathematics in Bonn.

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