m at h . A G ] 1 8 Ju n 20 06 CM Stability And The Generalised Futaki Invariant I Sean
نویسندگان
چکیده
Based on the Cayley, Grothendieck, Knudsen Mumford theory of determinants we extend the CM polarisation to the Hilbert scheme. The Baum Fulton Macpherson GRR Theorem enables us to show that on any flat, proper, local complete intersection family the restriction of the extension and the original CM sheaf are isomorphic (under a mild hypothesis on the base). As a consequence the CM stability implies the K Stability in the sense of Donaldson. When the CM polarisation is ample on the base of the family, CM stability and K stability are equivalent. §0 Introduction In order to establish a relationship between Kähler Einstein metrics on a fixed Fano manifold X and the stability of an associated projective model the second author was led to consider a family of projective varieties parametrized by a base B:
منابع مشابه
ar X iv : m at h / 06 06 50 5 v 3 [ m at h . D G ] 1 9 A pr 2 00 8 CM STABILITY AND THE GENERALIZED FUTAKI INVARIANT II SEAN
The Mabuchi K-energy map is exhibited as a singular metric on the refined CM polarization of any equivariant family X p → S. Consequently we show that the generalized Futaki invariant is the leading term in the asymptotics of the reduced K-energy of the generic fiber of the map p. Properness of the K-energy implies that the generalized Futaki invariant is strictly negative.
متن کاملar X iv : m at h / 06 06 50 5 v 4 [ m at h . D G ] 2 3 A pr 2 00 8 CM STABILITY AND THE GENERALIZED FUTAKI INVARIANT II
The Mabuchi K-energy map is exhibited as a singular metric on the refined CM polarization of any equivariant family X p → S. Consequently we show that the generalized Futaki invariant is the leading term in the asymptotics of the reduced K-energy of the generic fiber of the map p. Properness of the K-energy implies that the generalized Futaki invariant is strictly negative.
متن کامل0 Ju n 20 06 CM Stability And The Generalised Futaki Invariant II Sean
Let (X,L) be a smooth polarized complex variety of dimension n. In this paper we identify the leading terms of the (reduced) K-energy map with the weight F1(λ) provided that (X,L) moves in a good family X f → B. Consequently, we deduce that the properness of the K-Energy map implies the K-Stability of the variety. §0 Resumé of Results Let (X, L) be a compact polarised manifold. Assume L is very...
متن کاملm at h . A G ] 1 9 A pr 2 00 8 CM STABILITY AND THE GENERALIZED FUTAKI INVARIANT I
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 t...
متن کاملAnalysis of Geometric Stability
We identify the difference between the CM polarisation and the Chow polarisation on the “Hilbert scheme”. As a consequence, we give a numerical criterion for the CM stability as in Mumfords’ G.I.T.. Also, we write down an explicit formula for the generalised futaki invariant interms of weights and multiplicities of the associated degeneration. Research supported in part by an NSF Postdoctoral F...
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