Holomorphic geometric structures on compact complex manifolds (Q2757192)

Let \(M\) be a compact complex manifold of dimension \(n\). Let \(D^r({\mathbb C}^n)\) denote the group of \(r\)-jets of local biholomorphic mappings of \({\mathbb C}^n\) fixing the origin. Let \(R^r(M)\) be the \textsl{bundle of \(r\)-frames on \(M\)} (or of \(r\)-jets of local mappings of \({\mathbb C}^n\) into \(M\)), which is a principal fiber bundle over \(M\) with structural group \(D^r(M)\). Let \(Z\) be a Zariski open subset of a projective algebraic manifold equipped with an action of \(D^r({\mathbb C}^n)\). According to \textit{M. Gromov} [Géométrie Différentielle, Trav. Cours 33, 65-139 (1988; Zbl 0652.53023)], a geometric structure of type \(Z\) and of order \(r\) on \(M\) is a holomorphic mapping \(\varphi : R^r(M)\to Z\) which is \(D^r({\mathbb C}^n)\)-equivariant, namely satisfies \(\varphi(s\cdot g) = g^{-1} \cdot \varphi(s)\) for all \(s\in R^r(M)\) and all \(g\in D^r({\mathbb C}^n)\). Also, \(\varphi\) is called \textsl{of affine algebraic type} if \(Z\) is affine. Such a geometric structure is called \textsl{rigid} if there exists an integer \(l\) such that every \((r+l+1)\)-jet of local biholomorphism of \(M\) which preserves the \((l+1)\)-th jet of \(\varphi\) is uniquely determined by its \((r+l)\)-th order part. For instance, an affine holomorphic connection or a holomorphic Riemannian metric are rigid geometric structures. A \textsl{local isometry of \(\varphi\)} is a local biholomorphism between two open subsets of \(M\) which preserves the structure \(\varphi\). Let \(\text{Is}^{\text{loc}}(\varphi)\) denote the pseudogroup of such local isometries. The geometric structure \(\varphi\) is called locally homogeneous if \(\text{Is}^{\text{loc}} (\varphi)\) acts transitively on \(M\). In this case, there is only one local model form for \(\varphi\). NEWLINENEWLINENEWLINEThe author conjectures that on an arbitrary compact complex manifold, all holomorphic geometric structures of affine algebraic type are locally homogeneous. It is easy to see that a holomorphic Riemannian metric on a compact complex surface is a homogeneous geometric structure, because its curvature (a holomorphic function) must be constant. In his first theorem, the author verifies the conjecture under the additional assumption that \(M\) is Kähler and its first Chern class vanishes. Also, using a decomposition theorem due to \textit{A. Beauville} [J. Differ. Geom. 18, 755-782 (1983; Zbl 0537.53056)], with the same assumptions, he proves that if there exists a rigid geometric structure on \(M\) of affine algebraic type, then \(M\) has a finite cover which is a complex torus. This result generalizes a theorem by \textit{M. Inoue, S. Kobayashi} and \textit{T. Ochiai} [J. Fac. Sci. Univ. Tokyo, Sect. I A 27, 247-264 (1980; Zbl 0467.32014)] about affine connections. In his third result, without any assumption on \(M\), the author establishes that if there exists a holomorphic geometric structure on \(M\), then \(p+d\geq n\), where \(p\) is the generic dimension of the orbits of \(\text{Is}^{\text{loc}} (\varphi)\) and \(d\) is the algebraic dimension of \(M\). This generalizes a previous result by \textit{F. A. Bogomolov} [Math. USSR Izv. 13, 499-555 (1979; Zbl 0439.14002)] in the case \(d=0\).