A combinatorial description of the cup product for smooth complete toric varieties

Author: 
Date created: 
2018-12-11
Identifier: 
etd19980
Keywords: 
Cup product
Toric geometry
Tangent sheaf
Čech cohomology
Euler sequence
Deformation theory
Abstract: 

For any smooth variety X, there exists an associated vector space of first-order deformations. This vector space can be interpreted using sheaf cohomology; it is the first cohomology group H^1(X,T_X), where T_X is the tangent sheaf. One can ask when it is possible to "combine" two first-order deformations. The cup product takes elements of H^1(X,T_X) x H^1(X,T_X) and maps to the obstruction space H^2(X, T_X), and the vanishing of the cup product tells us precisely when this is possible. In this thesis we give a combinatorial description of the cup product (on the level of Čech cohomology) when X is a smooth, complete, toric variety with an associated fan Σ. We also give an example of a smooth, complete, toric 3-fold for which the cup product is nonvanishing.

Document type: 
Thesis
Rights: 
This thesis may be printed or downloaded for non-commercial research and scholarly purposes. Copyright remains with the author.
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Senior supervisor: 
Nathan Ilten
Department: 
Science: Department of Mathematics
Thesis type: 
(Thesis) M.Sc.
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