Positivity, vanishing theorems and rigidity of codimension one holomorphic foliations (Q2655882)

The present paper is a well written review paper and is essentially a summary of earlier papers of the author, which mainly study deformations of codimension one singular holomorphic foliations, as well as the geometric properties of the Kupka singular set of a foliation. New proofs of some results cited in earlier papers of the author are given. Let \(M\) be a complex manifold, let \(E\rightarrow M\) be a holomorphic vector bundle over \(M\) and let \(L\) be a holomorphic line bundle. The vector space of holomorphic sections of the vector bundle \(E\otimes L\), is in one to one correspondence with the space of meromorphic sections of \(E\) with a pole along a hypersurface \(D\subset M\) which is the zero locus of a holomorphic section of the line bundle \(L\). A codimension one holomorphic foliation with singularities on a manifold \(M\), may be defined by a holomorphic section \(\omega\) of \(T^{\ast}M\otimes L,\) satisfying the integrability condition \(\omega\wedge d\omega=0\) as a section of \(\bigwedge^{3}T^{\ast}M\otimes L^{\otimes2}\). Two sections \(\omega\) and \(\omega_{1}\) define the same foliation if there exists a nowhere vanishing holomorphic function \(\varphi\) on \(M\), such that \(\omega=\varphi\omega_{1}\); in this case the sections \(\omega\) and \(\omega_{1}\) are said equivalent. By \(\mathcal{F}(M,L)\) is denoted the set of equivalence classes of integrable, holomorphic sections of \(T^{\ast}M\otimes L\), the cotangent bundle of \(M\) twisted by a holomorpic line bundle \(L.\) If \(M\) is a compact complex manifold, it is well known that \(\mathcal{F}(M,L)\) is an algebraic variety with singularities in general. Let \(M\) be a compact complex manifold and let \(E\rightarrow M\) be a holomorphic vector bundle. \(E\) is said positive if there exists a hermitian metric \(<\cdot\) \(,\) \(\cdot>\) in \(E\) whose curvature tensor \(\pmb{\Theta}=(\Theta_{\sigma ij}^{\rho})\) has the property that the hermitian quadratic form \(\pmb{\Theta}(\zeta,\eta)=\underset{\rho,\sigma,i,j}{\sum} \Theta_{\sigma ij}^{\rho}\zeta^{\sigma}\overline{\zeta}^{\rho}\eta ^{i}\overline{\eta}^{j}\) is positive definite in the variables \(\zeta,\eta\). Let \(\omega\in\mathcal{F}(M,L).\) The Kupka singular set of the foliation \(\omega\) consists of the points: \(K_{\omega}=\{p\in M\) \(|\) \(\omega (p)=0;d\omega(p)\neq0\}\). The paper is organized as follows: In section 1, classical theorems are generalized, in particular Lefschtz's, Hartogs's and Zariski's theorems and the notion of positivity of holomorphic vector bundles in complex manifolds and its consequences are discussed. Section 3 is dedicated to the study of the Kupka singular set; the Zariski open set \(K(M,L)\) is introduced for those foliations having a single compact, connected component of the Kupka singular set. In section 4, it is proved that holomorphic foliations arising from the fibers of a generic rational map \(\phi:M\rightarrow\mathbb{P}^{1}\), describe some irreducible components of \(K(M,L)\). In section 5, some geometrical propeties of the Kupka set are described. In section 6, some other irreducible components of \(\mathcal{F} (M,L)\) are described: the logarithmic foliations. Finally in section 7, foliations on projective spaces are considered and several irreducible components are described.