20/06/2001, 11:30 — 12:30 — Amphitheatre Pa2, Mathematics Building
Bruno de Oliveira, University of Pennsylvania
Introduction to the Hartshorne Conjecture - I
In the early 60's Hartshorne studied the subvarieties that are
the generalization of ample divisors for higher codimensions.
Motivated by his study Hartshorne proposed the following
conjecture: Let \(X\) be a smooth projective variety, \(A\) and
\(B\) be two smooth subvarieties of \(X\) with ample normal bundle
and such that \(\dim A + \dim B \geq \dim X\). Then \(A\)
intersects \(B\). We will use this problem to illustrate the
interplay of complex, differential and algebraic geometry. We will
always target a diverse audience. To that effect we review notions
of algebraic geometry : line bundles, vector bundles,
\(P^n\)-bundles and the ampleness property. From complex
differential geometry: Kahler manifolds, Hermitean metrics,
connections, curvature, the positivity of vector bundles and
vanishing theorems. From several complex variables: strongly
q-convex spaces, the finiteness of cohomology groups and cycle
spaces of complex manifolds. The notions and results mentioned
above will then be applied to explain the reason of the conjecture,
why the conjecture might not be true and to prove special cases. In
particular we will do the case where the ambient variety \(X\) is a
hypersurface in \(P^n\) (done by Barlet), a \(P^ 2\)-bundle over a
surface (done by Barlet, Schneider and Peternel) and a
\(P^1\)-bundle over a threefold.