10.29.1 The constructionCA83ED725CCA

Let $K$ be a number field of degree $s+2t$, with $s\geq 1$ real embeddings $\sigma _1,\ldots ,\sigma _ s$ and $t\geq 1$ pairs of complex conjugate embeddings $\sigma _{s+k},\overline{\sigma }_{s+k}$, $1\leq k\leq t$. Write $\mathcal O_ K$ for its ring of integers and $\mathcal O_ K^{*,+}$ for the group of totally positive units, that is, of units $\varepsilon $ with $\sigma _ j(\varepsilon )>0$ for $1\leq j\leq s$. Let $\mathbb H\subset \mathbb C$ be the upper half-plane. Write points of $\mathbb H^ s\times \mathbb C^ t$ as $(w,z)$, with $w_ j=x_ j+\mathrm i y_ j$, and put

An element $a\in \mathcal O_ K$ acts by the translation $(w,z)\mapsto (w+\sigma _{\mathbb R}(a),z+\sigma _{\mathbb C}(a))$, and $\varepsilon \in \mathcal O_ K^{*,+}$ acts by

A subgroup $U\subset \mathcal O_ K^{*,+}$ is admissible if its image

is a lattice in $\mathbb R^ s$. By Dirichlet's unit theorem, when $t=1$ every subgroup of finite index in $\mathcal O_ K^{*,+}$ is admissible. For admissible $U$, Oeljeklaus and Toma showed that the semidirect product $\Gamma =\mathcal O_ K\rtimes U$ acts freely and properly discontinuously on $\mathbb H^ s\times \mathbb C^ t$ with compact quotient. The compact complex manifold

is the Oeljeklaus–Toma manifold of type $(s,t)$. The same construction works if $\mathcal O_ K$ is replaced by any lattice $L\subset K$ of rank $s+2t$ that is stable under multiplication by $U$; for $s=t=1$ this produces the Inoue surfaces $S_ M$. The action is still free and cocompact: a point fixed by $(a,\varepsilon )$ with $\varepsilon \neq 1$ would have real coordinates $w_ j$, since $\sigma _ j(\varepsilon )\neq 1$ by injectivity of $\sigma _ j$, and $(\sigma _{\mathbb R},\sigma _{\mathbb C})(L)$ is again a lattice in $\mathbb R^ s\times \mathbb C^ t$. All results of this section other than corollary 10.29.7 hold, with the same proofs, in this generality, with $L$ in place of $\mathcal O_ K$; the only property of $\mathcal O_ K$ they use is that it spans $K$ over $\mathbb Q$. Corollary 10.29.7 also holds for every Inoue surface $S_ M$, since these contain no curves. The sources are discussed in section 25.18.

The functions $y_ j=\operatorname {Im}w_ j$ are invariant under the translations and are multiplied by $\sigma _ j(\varepsilon )$ under the unit $\varepsilon $. Hence the closed real $(1,1)$-form

on $\mathbb H^ s\times \mathbb C^ t$ is $\Gamma $-invariant and descends to a closed semipositive form on $X$, still denoted $\theta $. It has rank $s$ everywhere, and its kernel is spanned by the vectors $\partial /\partial z_ k$. The potential $-\sum _ j\log y_ j$ itself is not invariant, since a unit adds $-\sum _ j\log \sigma _ j(\varepsilon )$ to it.