# EP/K005545/1 - Geometry and arithmetics through the theory of algebraic cycles

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Fellow - Pure Maths and Mathematical Statistics, University of Cambridge

Start Date

04/2013

End Date

03/2018

Value

£401,912

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Summary and Description of the grant

The basic objects of algebraic geometry are algebraic varieties. These are defined locally as the zero locus of polynomial equations. The main goal of algebraic geometry is to classify varieties. An approach consists in attaching invariants to varieties. Some invariants are of an arithmetic nature, e.g. the gcd of the degrees of closed points on X. Some are of a topological nature, e.g. the singular cohomology of the underlying topological space of X. Some are of a geometric nature, e.g. the Chow groups of X. A codimension-n algebraic cycle on X is a formal sum of irreducible subvarieties of codimension n and the Chow group CH^n(X) is the abelian group with basis the irreducible subvarieties of codimension n in X modulo a certain equivalence relation called rational equivalence. Roughly, rational equivalence is the finest equivalence relation on algebraic cycles that makes it possible to define unambiguously an intersection product on cycles. Moreover, the aforementioned invariants for X are encoded (or at least expected to be) in the Chow groups X. Therefore, in some sense, algebraic cycles are the finest invariants for algebraic varieties, and the theory of algebraic cycles lies at the very heart of geometry, topology and number theory.

I will integrate methods from K-theory, Galois cohomology and number theory to derive new results in the theory of algebraic cycles on varieties defined over finitely generated fields or other fields of arithmetic interest. Conversely, I will use the theory of algebraic cycles to derive new results of arithmetic

interest. In addition, the outcome of such results will shed new light on the geometry of such varieties. Thus, by its very nature, my research proposal on the theory of algebraic cycles is intradisciplinary within the mathematical sciences.

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Name: Geometry and arithmetics through the theory of algebraic cycles - EP/K005545/1

Start Date: 2013-04-01T00:00:00+00:00

End Date: 2018-03-31T00:00:00+00:00

Organization: University of Cambridge

Description: The basic objects of algebraic geometry are algebraic varieties. These are defined locally as the zero locus of polynomial equations. The main goal of algebraic geometry is to classify varieties. An approach consists in attaching invariants to varieties. ...