Scalar curvature and asymptotic Chow stability of projective bundles and blowups

Della Vedova, Alberto; Zuddas, Fabio

doi:10.1090/S0002-9947-2012-05587-5

Scalar curvature and asymptotic Chow stability of projective bundles and blowups
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by Alberto Della Vedova and Fabio Zuddas PDF

Trans. Amer. Math. Soc. 364 (2012), 6495-6511 Request permission

Abstract:

The holomorphic invariants introduced by Futaki as obstruction to asymptotic Chow semistability are studied by an algebraic-geometric point of view and are shown to be the Mumford weights of suitable line bundles on the Hilbert scheme of $\mathbb P^n$.

These invariants are calculated in two special cases. The first is a projective bundle $\mathbb P(E)$ over a curve of genus $g \geq 2$, and it is shown that it is asymptotically Chow polystable (with every polarization) if and only if the bundle $E$ is slope polystable. This proves a conjecture of Morrison with the extra assumption that the involved polarization is sufficiently divisible. Moreover it implies that $\mathbb P(E)$ is asymptotically Chow polystable (with every polarization) if and only if it admits a constant scalar curvature Kähler metric. The second case is a manifold blown up at points, and new examples of asymptotically Chow unstable constant scalar curvature Kähler classes are given.

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