Contents/conteúdo

Mathematics Department Técnico Técnico

Summer Lectures in Geometry  RSS

Sessions

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.


For detailed overviews of each course see https://camgsd.tecnico.ulisboa.pt/encontros/slg/.

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