Bruno De Oliveira

Hyperbolicity of nodal hypersurfaces

Abstract We show that a nodal hypersurface X in ℙ3 of degree d with a sufficiently large number l of nodes, , is algebraically quasi-hyperbolic, i.e. X can only have finitely many rational and elliptic curves. Our results use the theory of symmetric differentials and algebraic foliations and give a very striking example of the jumping of the number of symmetric differentials in families.

Local structure of closed symmetric 2-differentials

In this article we provide a description of the local structure of closed symmetric 2-differentials on a complex surface. The main technical result of the article is that a sum of two local singular analytic functions which both are constant along two different smooth holomorphic foliations can be locally holomorphic only if both functions have at most meromorphic singularities. As a corollary we proof that locally a product of two singular closed differentials on a surface is holomorphic only if the singularities of the differentials are at most exponents of local meromorphic functions.

Vanishing theorems of negative vector bundles on projective varieties and the convexity of coverings

We give a new proof of the vanishing of H1(X, V ) for negative vector bundles V on normal projective varieties X satisfying rank V < dim X. Our proof is geometric, it uses a topological characterization of the affine bundles associated with non-trivial cocycles α ∈ H1(X, V ) of negative vector bundles. Following the same circle of ideas, we use the analytic characteristics of affine bundles to obtain convexity properties of coverings of projective varieties. We suggest a weakened version of the Shafarevich conjecture: the universal covering X̃ of a projective manifold X is holomorphically convex modulo the pre-image ρ−1(Z) of a subvariety Z ⊂ X. We prove this conjecture for projective varieties X whose pullback map ρ∗ identifies a nontrivial extension of a negative vector bundle V by O with the trivial extension.

Singularities on complete algebraic varieties

We prove that any finite set of n-dimensional isolated algebraic singularities can be afforded on a simply connected projective variety.