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Exploring Equivariant Chow Groups for Complexity One T-Varieties via Downgrading Techniques

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dc.contributor.advisor MALLICK, VIVEK
dc.contributor.author DIGHE, PAVANKUMAR
dc.date.accessioned 2024-04-08T10:15:37Z
dc.date.available 2024-04-08T10:15:37Z
dc.date.issued 2024-04
dc.identifier.citation 80 en_US
dc.identifier.uri http://dr.iiserpune.ac.in:8080/xmlui/handle/123456789/8620
dc.description.abstract A T-variety is an algebraic variety X with an effective torus action T. The number c(X) = dim(X)−dim(T) is called complexity of T-variety X. Altmann, Hausen and Süss have described these spaces in terms of pp-divisor and divisorial fans. This description of a T-variety involves a variety of dimension c(X) and some combinatorial data encoded in the form of pp-divisors, i.e., divisors where the coefficients come from the Grothendieck group associated with the semigroup of polyhedra having a common tail cone. In case of complete T-variety with c(X) = 1, Ilten and Süss have described these spaces in terms of marked fancy divisor. Through this combinatorial description of a T-variety, Ustøl Nødland has provided an description of the Chow group of X. In the first part of the thesis we study the computation of the equivariant Chow group of T-variety X with c(X) = 1. This computation involves the following steps. For a complete complexity 1 T-variety X, we compute combinatorial description of X×ETN as a T-variety, where ETN is Nd-dimensional space approximating the contractible space on which T acts freely. Subsequently, through the application of a downgrading technique,we introduce a structure of a T-variety of complexity 1 on the quotient space X×ETN/T . By using combinatorial criterion of completeness of a T-variety, we have proved that if X is complete then the quotient space X×ETN/T is complete. Once it has a complete, complexity 1 T-variety structure, one can use Ustøl’s result to compute Chow group of X×ETN/T . For an affine T-variety X with the action of a torus T, denoted temporarily, by T ↷ X. Assume that T′ is a subtorus of T. Then X is a T-variety with respect to the action of T′, T′ ↷ X. T′ ↷ X is called a downgrading of T ↷ X. The second part provides a combinatorial description of T′ ↷ X, in terms of a T/T′-invariant pp-divisor. We also describe the corresponding GIT fan. en_US
dc.description.sponsorship UGC India. en_US
dc.language.iso en en_US
dc.subject T-varieties en_US
dc.subject Chow Groups en_US
dc.subject Equivariant Chow Groups en_US
dc.subject Proper Polyhedral Divisors en_US
dc.subject Divisor en_US
dc.subject Downgrading en_US
dc.title Exploring Equivariant Chow Groups for Complexity One T-Varieties via Downgrading Techniques en_US
dc.type Thesis en_US
dc.description.embargo No Embargo en_US
dc.type.degree Ph.D en_US
dc.contributor.department Dept. of Mathematics en_US
dc.contributor.registration 20183615 en_US


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  • PhD THESES [590]
    Thesis submitted to IISER Pune in partial fulfilment of the requirements for the degree of Doctor of Philosophy

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