Nicolas Addington (University of Oregon)
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PDL C-38
Pre-talk: Cohomology of hypersurfaces
Title: In the main talk I'll discuss some hard questions about the cohomology of a smooth hypersurface X in CP^{n+1}, so in the pre-talk I'll discuss what's easy, or at least classical. The Lefschetz hyperplane theorem gives everything but the middle cohomology H^n(X,Z). The Hirzebruch-Riemann-Roch formula gives the Hodge numbers and thus the rank of H^n(X,Z). Griffiths' residue calculus gives the action of the automorphism group on H^{p,q}(X), and thus on H^n(X,C), although not on H^n(X,Z). (It also gives information about the variation of Hodge structure as X deforms, but there may not be time for this.)
Title: In the main talk I'll discuss some hard questions about the cohomology of a smooth hypersurface X in CP^{n+1}, so in the pre-talk I'll discuss what's easy, or at least classical. The Lefschetz hyperplane theorem gives everything but the middle cohomology H^n(X,Z). The Hirzebruch-Riemann-Roch formula gives the Hodge numbers and thus the rank of H^n(X,Z). Griffiths' residue calculus gives the action of the automorphism group on H^{p,q}(X), and thus on H^n(X,C), although not on H^n(X,Z). (It also gives information about the variation of Hodge structure as X deforms, but there may not be time for this.)
Title: Integral cohomology of some hypersurfaces, and applications
Abstract: Abstractly we know all about the integral cohomology of a smooth hypersurface in complex projective space, but if we really want to compute in examples, it is surprisingly hard to get our hands on, especially the "transcendental part" not generated by Poincaré duals of subvarieties. I'll explain that for some hypersurfaces with big automorphism groups, the middle cohomology is generated by the Aut(X)-orbit of the Poincaré dual of the set of real points. Then I'll give three applications: one to the Torelli problem for Calabi-Yau threefolds, answering a question of Aspinwall, Morrison, and Szendrői, joint with Ben Tighe; one to intermediate Jacobians of cubic threefolds, in progress with Benson Farb; and one to a question that Max Lieblich asked me a few years ago, on surfaces of maximal Picard number.