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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Scalar curvature and asymptotic Chow stability of projective bundles and blowups


Authors: Alberto Della Vedova and Fabio Zuddas
Journal: Trans. Amer. Math. Soc. 364 (2012), 6495-6511
MSC (2010): Primary 32Q15
Published electronically: June 8, 2012
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05587-5
PII: S 0002-9947(2012)05587-5
Received by editor(s): November 15, 2010
Received by editor(s) in revised form: March 18, 2011
Published electronically: June 8, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.