Graded linearisations for linear algebraic group actions

Duration: 1 hour 8 mins

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Description:	Kirwan, F Friday 18th August 2017 - 11:30 to 12:30


Created:	2017-08-21 09:00
Collection:	Symplectic geometry - celebrating the work of Simon Donaldson
Publisher:	Isaac Newton Institute
Copyright:	Kirwan, F
Language:	eng (English)


Abstract:	Co-authors: Gergely Berczi (ETH Zurich), Brent Doran (ETH Zurich) In algebraic geometry it is often useful to be able to construct quotients of algebraic varieties by linear algebraic group actions; in particular moduli spaces (or stacks) can be constructed in this way. When the linear algebraic group is reductive, and we have a suitable linearisation for its action on a projective variety, we can use Mumford's geometric invariant theory (GIT) to construct and study such quotient varieties. The aim of this talk is to describe how Mumford's GIT can be extended effectively to actions of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action can be regarded as a graded linearisation in a natural way. The classical examples of moduli spaces which can be constructed using Mumford's GIT are the moduli spaces of stable curves and of (semi)stable bundles over a fixed curve. This more general construction can be used to construct moduli spaces of unstable objects, such as unstable curves (with suitable fixed discrete invariants) or unstable bundles (with fixed Harder-Narasimhan type).

Abstract:

Co-authors: Gergely Berczi (ETH Zurich), Brent Doran (ETH Zurich)
In algebraic geometry it is often useful to be able to construct quotients of algebraic varieties by linear algebraic group actions; in particular moduli spaces (or stacks) can be constructed in this way. When the linear algebraic group is reductive, and we have a suitable linearisation for its action on a projective variety, we can use Mumford's geometric invariant theory (GIT) to construct and study such quotient varieties. The aim of this talk is to describe how Mumford's GIT can be extended effectively to actions of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action can be regarded as a graded linearisation in a natural way.

The classical examples of moduli spaces which can be constructed using Mumford's GIT are the moduli spaces of stable curves and of (semi)stable bundles over a fixed curve. This more general construction can be used to construct moduli spaces of unstable objects, such as unstable curves (with suitable fixed discrete invariants) or unstable bundles (with fixed Harder-Narasimhan type).

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