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SUMMARY:Mike Roth (Queens University)
DTSTART:20251003T150000Z
DTEND:20251003T161500Z
DTSTAMP:20260423T005659Z
UID:CIRGET/145
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CIRGET/145/"
 >Reduced Čech complexes and computing higher direct images under toric fi
 brations.</a>\nby Mike Roth (Queens University) as part of CRM - Séminair
 e du CIRGET / Géométrie et Topologie\n\nLecture held in PK-5115.\n\nAbst
 ract\nLet $X$ be a topological space\, $F$ a sheaf of abelian groups on $X
 $\, and $\\{ U_{\\alpha}\\}_{\\alpha\\in I\\}$ an open cover of $X$.  Then
  one can form a Čech complex\, a complex of groups built from the values 
 of $F$ on the open sets and their intersections.    \n\nIf the higher coho
 mology of $F$ vanishes on all these open sets\, then it is a well-known th
 eorem of Leray that this complex computes the cohomology of $F$ on $X$.   
 For instance\, if $X$ is a manifold and the $U_{\\alpha}$ form a `good cov
 er’ (all the $U_{\\alpha}$ and their intersections are homeomorphic to $
 \\mathbb{R}^n$)\, then the Čech complex can be used to compute the topolo
 gical cohomology of $X$.\n\nFor special kinds of toric varieties — those
  whose fans are `simplicial’ -- it is known how to construct smaller (
 “reduced”) complexes which still correctly compute cohomology of sheav
 es.\n\nThis talk has three main goals : (1) To give an axiomatization of 
 ‘reduced Čech complexes’\, valid for any topological space\;  (2) To 
 extend the previous construction of reduced Čech complexes to all compact
  toric varieties (not just simplicial ones)\, and more generally to ’sem
 i-proper’ toric varieties\; (3) To use the previous method to give an al
 gorithm for computing higher direct images (roughly the `cohomology along 
 the fibres’) of line bundles for toric fibrations between smooth toric v
 arieties.\n\nNo previous knowledge of toric varieties is required.\nThis i
 s joint work with Sasha Zotine.\n
LOCATION:https://researchseminars.org/talk/CIRGET/145/
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