CM Stability and the Generalized Futaki Invariant I

Abstract

Based on the Cayley, Grothendieck, Knudsen Mumford theory of determinants we extend the CM polarization to the Hilbert scheme. We identify the weight of this refined line bundle with the generalized Futaki invariant of Donaldson. We are able to conclude that CM stability implies K-Stability. An application of the Grothendieck Riemann Roch Theorem shows that this refined sheaf is isomorphic to the CM polarization introduced by Tian in 1994 on any closed, simply connected base .

24 pages, fully rewritten, references added

References in corpus (5)

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Cited by in corpus (34)

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• Nonexistence of asymptotic GIT compactification
• Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds
• Extremal Kähler metrics
• Wall crossing for K-moduli spaces of plane curves
• Quasi-projectivity of the moduli space of smooth Kahler-Einstein Fano manifolds
• Hilbert-Mumford criterion for nodal curves
• Asymptotic Chow polystability in Kähler geometry
• Applications of the moduli continuity method to log K-stable pairs
• Compact moduli space of Kahler-Einstein Fano varieties
• Equivariant K-stability under finite group action
• On the limit of spectral measures associated to a test configuration
• K-stability of cubic fourfolds
• K-moduli of curves on a quadric surface and K3 surfaces
• The moduli space of Fano manifolds with Kähler-Ricci solitons
• Polystable log Calabi-Yau varieties and Gravitational instantons
• Ampleness of the CM line bundle on the moduli space of canonically polarized varieties
• Twisted cscK metrics and Kähler slope stability
• Canonical Kahler metrics and Arithmetics -- Generalising Faltings heights
• On the K-stability of complete intersections in polarized manifolds
• K-stability and birational models of moduli of quartic K3 surfaces
• Deligne pairings and the Knudsen-Mumford expansion
• On uniform K-stability of pairs
• Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures
• Worst Unstable Points of a Hilbert Scheme
• On the volume of some Fano K-moduli spaces
• On the continuity of Weil-Petersson volumes of the moduli space weighted points on the projective line
• Test Configurations for K-Stability and Geodesic Rays
• Multiplier ideal sheaves and geometric problems
• A note on Positivity of the CM line bundle