## 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

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 *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

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

*X*as an additive map

*F*-linear in the sense that

We have a short exact sequence

### Lemma A.1.1.

*Suppose that *

### Proof.

Suppose that

### Corollary A.1.2.

*Suppose that *

*defines a lifting of X over *

### Proof.

Since σ is surjective, the left square is a pushout. To prove that

equals multiplication by *p*. Let

Then the image of

This in turn has image

### Definition A.1.3.

We call the lifting *canonical lifting* associated to the quasi-*F*-splitting σ.

### Remark A.1.4.

Composing the map

### A.2 The sheaf ℱ m 𝒪 X and Witt vectors mod *p*

Suppose that *X* is a smooth scheme over a perfect field *k* of characteristic *p*. Denote by

is

In this situation, Yobuko defined a natural extension of

and showed that *F*-splittings as defined previously. The exact sequence

In particular, if *X* is *F*-split, the bottom row is, and hence so is

It follows that quasi-*F*-splittings are precisely

We shall now elucidate the sheaf *X* be an

the mod-*p* reduction of the ring of Witt vectors of length *m* over *p*-th power of the Teichmüller map

is additive modulo *p*, and therefore induces an

on

Consequently, ρ is injective if *X* is reduced. We denote its cokernel by

The formula

implies that the multiplication in

### Remark A.2.1.

The existence of ρ can be also seen as follows (cf. [47, Section 1.1]): the kernel *I* of

It follows that

### Proposition A.2.2.

*The following diagram is a pushout square:*

### Proof.

The commutativity of the diagram follows from the formula

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

is an isomorphism. This follows from

### Corollary A.2.3.

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

*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 *quasi- F-splitting of level m* on

*X*is an

## B Families with vanishing Kodaira–Spencer

Let *k* be a perfect field of characteristic *S* be a smooth *k*-scheme. Let *X*

yields an exact sequence on *S*

The *Kodaira–Spencer map* of

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

*Suppose *

## Proof.

The assumptions *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

by adjunction correspond to maps

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*. To produce examples which do not descend even locally, one can take *p*.

Of course, if *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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**Received:**2020-11-09

**Revised:**2021-06-08

**Published Online:**2021-09-03

**Published in Print:**2021-11-01

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