Notes on GIT and symplectic reduction for bundles and varieties

By Richard P. Thomas. 2005

131 citations recorded by the source corpus.

These notes give an introduction to Geometric Invariant Theory and symplectic reduction, with lots of pictures and simple examples. We describe their applications to moduli of bundles and varieties, and their infinite dimensional analogues in gauge theory and the theory of special metrics on algebraic varieties. Donaldson's "quantisation" link between the infinite and finite dimensional situations is described, as are surprisingly strong connections between the bundle and variety cases.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/de4ebc8c06d5cd390cadcafa1cd1efe1aaf79b4b

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Uniform K-stability and asymptotics of energy functionals in Kähler geometry (2016, result): “zation of the Kempf-Ness/Hilbert-Mumford criterion is closely related to results of [Pau13], which they simplify to some extent. Our elementary argument is inspired by the discussion on pp.241{243 of [Tho06]. Theorem 5.4. Let fbe a function on Gwith log norm singularities. (i) For each 1-PS : C !G, ”
Blowing up Kähler manifolds with constant scalar curvature, II (2005, background): “To give further credit to this belief, we refer to the discussion of this issue in the preprint of Thomas [ 28 ], pages 27-28 and also to the recent preprint by Stoppa [27] where some partial result is obtained in this direction.”
Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature K\ (2010, methodology): “This was made more precise as the Yau-Tian-Donaldson conjecture (see [46]).”
Extremal metrics on blowups (2006, background): “Remark 2.4. Condition (i) and (ii) should be related to Mabuchi’s T-stability [24] and to Szekelyhidi’s relative K-stability [32] in the same way the analogue conditions for constant scalar curvature metrics are related to the asymptotic Chow semi-stability along the line of ideas described by Thoma”
Extremal metrics and K-stability (PhD thesis) (2006, background): “The destabilising test-configuration is an example of deformation to the normal cone of a subvariety, studied by Ross and Thomas [29] (see also Section 5.2), except we need to take into account the extremal C∗ -action as well.”
Smooth and singular Kahler-Einstein metrics (2014, methodology): “The interested reader is referred to the numerous existing surveys on related topics [10, 28, 33, 34, 40, 41, 62, 99, 115, 116, 117, 118, 119, 120, 125, 200, 226, 199, 234, 236, 245, 246, 247].”