In this paper we will study a certain class of compact Kähler manifolds with trivial canonical bundle which contains all strict CalabiYau varieties as well as all hyperkähler manifolds. For the bigger class of manifolds with trivial rst Chern class c1(X) = 0∈H2(X,R) there exists
the following nice structure theorem, known as the BeauvilleBogomolov decomposition; see [Bea83]. Namely, each such manifoldX admits an étale covering X0 →X which decomposes
as X0 =T×Y i Yi× Y j Zj
whereT is a complex torus, the Yi are hyperkähler, and the Zj are simply connected strict
CalabiYau varieties of dimension at least 3.
Given a varietyX, the graded algebra H∗(OX) :=⊕dimX
i=0 Hi(X,OX)[−i]is considered an
important invariant; see, in particular, Abuaf [Abu15] who callsH∗(OX)the homological unit
of X and conjectures that it is stable under derived equivalences. In this paper, we want to
study varieties which have trivial canonical bundle and the property that the algebraH∗(OX) is generated by one element.
The main motivation are the following two observations. Let X be a compact Kähler
Observation 4.1.1. X is a strict CalabiYau manifold if and only if the canonical bundleωX
is trivial and H∗(OX) ∼= C[x]/x2 with degx = dimX. These conditions can be summarised
in terms of objects of the bounded derived category D(X) :=Db(Coh(X)) of coherent sheaves. Namely,X is a strict CalabiYau manifold if and only if OX ∈D(X) is a spherical object in the sense of Seidel and Thomas [ST01].
The above is a very simple reformulation of the standard denition of a strict CalabiYau manifold as a compact Kähler manifold with trivial canonical bundle such that Hi(OX) = 0
fori6= 0,dimX. The second observation is probably less well-known.
Observation 4.1.2. X is a hyperkähler manifold of dimension dimX= 2nif and only if ωX
is trivial and H∗(OX) ∼= C[x]/xn+1 with degx = 2. This is equivalent to the condition that
OX ∈D(X) is aPn-object in the sense of Huybrechts and Thomas [HT06].
Indeed, the structure sheaf of a hyperkähler manifold is one of the well-known examples of aPn-object; see [HT06, Ex. 1.3(ii)]. The fact that H∗(OX) also characterises the compact
hyperkähler manifolds follows from [HN11, Prop. A.1].
Inspired by this, we study the class of compact Kähler manifoldsXwith the property that
OX ∈D(X)is what we call a Pn[k]-object; see Denition 4.2.4. Concretely, this means:
(C1) The canonical bundle ωX is trivial,
(C2) There is an isomorphism ofC-algebras H∗(OX)∼=C[x]/xn+1 withdegx=k.
By Serre duality, such a manifold is of dimensiondimX=deg(xn) =n·k. For n= 1, we get exactly the strict CalabiYau manifolds while for k= 2we get the hyperkähler manifolds.
In this paper, we will study the case of highernand k. We construct examples and prove
some structure results. If OX is a Pn[k]-object with k >2, the manifold X is automatically
projective; see Lemma 4.3.10. Hence, we will call X a variety with Pn[k]-unit. The main
results of this paper can be summarised as
Theorem 4.1.3. Let n+ 1 =pν be a prime power. Then the following are equivalent:
1. There exists a variety with Pn[4]-unit,
2. There exists a variety with Pn[k]-unit for every even k,
3. There exists a strict Enriques variety of index n+ 1.
For n+ 1 arbitrary, the implications (iii)=⇒(ii)=⇒(i) are still true.
We do not know whether or not (i)=⇒(iii) is true in general ifn+ 1is not a prime power, but we will prove a slightly weaker statement that holds for arbitrary n+ 1; see Section4.5.2. In particular, the universal cover of a variety with Pn[4]-unit, withn+ 1 arbitrary, splits into
a product of two hyperkähler varieties; see Proposition 4.5.3.
Our notion of strict Enriques varieties is inspired by similar notions of higher dimensional analogues of Enriques surfaces due to Boissière, Nieper-Wiÿkirchen, and Sarti [BNS11] and Oguiso and Schröer [OS11]. There are known examples of strict Enriques varieties of index3 and 4. Hence, we get
Corollary 4.1.4. For n = 2 and n = 3 there are examples of varieties with Pn[k]-units for
The motivation for this work comes from questions concerning derived categories and the notions are inuenced by this. However, in this paper, with the exception Sections 4.6.5 and 4.6.6, all results and proofs are also formulated without using the language of derived categories.
The paper is organised as follows. In Section4.2.1, we x some notations and conventions. Sections4.2.2and4.2.3are a very brief introduction into derived categories and some types of objects that occur in these categories. In particular, we introduce the notion ofPn[k]-objects.
In Section4.3.1, we say a few words about compact hyperkähler manifolds. In Section4.3.2, we discuss automorphisms of BeauvilleBogomolov products and their action on cohomology. This is used in the following Section 4.3.3 in order to give a proof of Observation4.1.2. This proof is probably a bit easier than the one in [HN11, App. A]. More importantly, it allows us to introduce some of the notations and ideas which are used in the later sections. In Section4.3.4, we discuss a class of varieties which we call strict Enriques varieties. There are two dierent notions of Enriques varieties in the literature (see [BNS11] and [OS11]) and our notion is the intersection of these two; see Proposition 4.3.144. In Section 4.3.5, we quickly mention a generalisation; namely strict Enriques stacks.
We give the denition of a variety with a Pn[k]-unit together with some basic remarks
in Section 4.4.1. Section 4.4.2 provides two examples of varieties which look like promising candidates, but ultimately fail to have Pn[k]-units. In Section 4.4.3, we construct series of
varieties with Pn[k]-units out of strict Enriques varieties of index n+ 1. In particular, we
prove the implication (iii)=⇒(ii) of Theorem4.1.3.
In Section4.5.1, we make some basic observations concerning the fundamental group and the universal cover of varieties with Pn[k]-units. In Section 4.5.2, we specialise to the case
k = 4. We proof that the universal cover of a variety with Pn[4]-unit is the product of two
hyperkähler manifolds of dimension2n. Then we proceed to proof the implication (i)=⇒(iii) of Theorem4.1.3 forn+ 1a prime power.
Section4.6is a collection of some further observations and ideas. In Sections 4.6.1,4.6.2, and4.6.3, some further constructions leading to varieties with Pn[k]-units are discussed. We
talk briey about stacks withPn[k]-units in Section4.6.4. In Section 4.6.5, we prove that the
class of strict Enriques varieties is stable under derived equivalences, and in Section4.6.6 we study some derived autoequivalences of varieties withPn[k]-units. In the nal Section 4.6.7,
we contemplate a bit about varieties with Pn[k]-units as moduli spaces and constructions of
hyperkähler varieties.
Acknowledgements. The early stages of this work were done while the author was - nancially supported by the research grant KR 4541/1-1 of the DFG (German Research Foun- dation). He thanks Daniel Huybrechts, Marc Nieper-Wiÿkirchen, Sönke Rollenske, and Pawel Sosna for helpful discussions and comments. He also thanks the referee for helpful comments and suggestions.