Riemannian holonomy and algebraic geometry (Q2467593)

Riemannian manifolds \((M^n,g)\) equipped with some additional geometric structure (read: tensor field) occur in many situations in differential geometry and have interesting properties. In case the Riemannian geometry is compatible with the geometric structure the corresponding tensor field is parallel with respect to the Levi-Civita connection and the restricted holonomy group \(\text{Hol}(M^n)\) reduces to a subgroup of \(\text{SO}(n)\) preserving the structure. Berger's theorem classifies all possible holonomy groups of simply connected, irreducible and non-symmetric Riemannian manifolds. In the non-generic case (i.e.\ \(\text{Hol}(M^{n})\neq\text{SO}(n)\)) the admissible classes are Kähler manifolds (\(\text{Hol}(M^{2n})=\text{U}(n)\)), Calabi-Yau manifolds (\(\text{Hol}(M^{2n})=\text{SU}(n)\)), hyper-Kähler manifolds (\(\text{Hol}(M^{4n})=\text{Sp}(n)\)), quaternionic-Kähler manifolds (\(\text{Hol}(M^{4n})=\text{Sp}(n)\text{Sp}(1)\)), parallel \(\text{G}_2\)-manifolds (\(\text{Hol}(M^{7})=\text{G}_2\)) and parallel \(\text{Spin}(7)\)-manifolds (\(\text{Hol}(M^{8})=\text{Spin}(7)\)). The author devotes his survey article to review certain of the above structures in the light of algebraic geometry studying special algebraic varieties. The three considered classes are the compact, simply-connected Riemannian manifolds with holonomy \[ \text{SU}(n),\quad \text{Sp}(n),\quad \text{Sp}(n)\text{Sp}(1). \] As an introduction, the author presents basic facts on Riemannian geometry including the splitting theorem of de Rham, the theorem of Berger and the holonomy principle. The main part of the article is divided into three sections, each of which is devoted to one of these classes and completed by an overview on current research in the related field of algebraic geometry. The author starts with the case \(\text{Hol}(M^{2n})=\text{SU}(n)\), \(n\geq3\) of Calabi-Yau manifolds. He reminds the reader that these are Ricci-flat Kähler manifolds and states Bochner's principle. As one main consequence of the latter, he proves that a compact Kähler manifold of dimension \(2n\geq6\) with holonomy \(\text{SU}(n)\) is projective. Finally, he presents a structural theorem for compact, simply connected Calabi-Yau manifolds. These are certain products whose factors are either projective Calabi-Yau or irreducible hyper-Kähler manifolds. He proceeds with \(\text{Hol}(M^{4n})=\text{Sp}(n)\), the hyper-Kähler case. After recalling basic properties, he introduces the two standard series of \(4n\)-dimensional, irreducible, simply connected Kähler symplectic manifolds: the Douady space \(S^{[2n]}\) based on a K3 surface \(S\) and \(K_{2n}\) coming from a complex torus \(T\) of dimension \(4\). Eventually he provides an overview on attempts to construct additional examples of compact hyper-Kähler manifolds and presents further geometric properties of them. In the last section he exposes the case \(\text{Hol}(M^{4n})=\text{Sp}(n)\text{Sp}(1)\), \(n\geq2\). After constructing the twistor space \(Z\) of \(M^{4n}\) he investigates its properties. This culminates in the theorem of LeBrun and Salamon: If \(M^{4n}\) has positive scalar curvature then \(Z\) is a Fano contact manifold admitting a Kähler-Einstein metric. This enables the author to carry the classification problem of quaternionic-Kähler manifolds over to algebraic geometry. Studying Fano contact manifolds he devotes the last part to the conjecture that these are homogeneous spaces.