Topology Seminar/Graduate Student Seminar-Cohomology of Line Bundles with Rational Degree

Speaker: Harpreet Bedi (George Washington University)
Title: Cohomology of Line Bundles with Rational Degree
Place: Rome 771
Time: Thursday, August 24, 2017; 2:00-3:00 pm

We introduce the notion of a rational degree in Topology and Geometry. We begin by showing that the notion of degree in topology naturally extends from integers Z to Z[1/p] by simply taking direct limit. This idea also extends to geometry if we consider perfectoid fields and power series over it. We prove the following results, including rational degree analogues for two famous theorems.
  1. Theorem of Grothendieck on classification of vector bundles on projective lines.  We extend this theorem to projectivoid lines.
  2. Theorem of Serre on cohomology of line bundles on projective space. We consider line bundles of rational degree where degree is an element of Z[1/p] instead of Z.
  3.  Differential forms. We construct Cech complex for sheaf of differential forms and compute cohomology.

We also show various other results that are analogous to the classical results for standard integer degrees, e.g. Pic groups, Weierstrass Preparation Theorem, Maximum principle, Weil and Cartier Divisors and their equivalence on projective perfectoid, and we explain why the standard Riemann-Roch Theorem would not generalize.