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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Additional Information
- Alberto Della Vedova
- Affiliation: Department of Mathematics, Fine Hall, Princeton University, Princeton, New Jersey 08544 – and – Dipartimento di Matematica, Università degli Studi di Parma, Viale G. P. Usberti, 53/A, 43100 Parma Italy
- Email: della@math.princeton.edu
- Fabio Zuddas
- Affiliation: Dipartimento di Matematica, Università degli Studi di Parma, Viale G. P. Usberti, 53/A, 43100 Parma Italy
- Email: fabio.zuddas@unipr.it
- Received by editor(s): November 15, 2010
- Received by editor(s) in revised form: March 18, 2011
- Published electronically: June 8, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 6495-6511
- MSC (2010): Primary 32Q15
- DOI: https://doi.org/10.1090/S0002-9947-2012-05587-5
- MathSciNet review: 2958945