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Nonexistence of asymptotic GIT compactification. (English) Zbl 1306.14004

Consider \(X\) a smooth canonically polarized complex manifold and denote \(K_X\) its canonical bundle. It is known that \((X,K_X^r)\) is Chow stable for \(r>r_0\) for a certain \(r_0\). From general results of Geometric Invariant Theory (GIT), there exists a projective moduli space containing an open locus parametrizing smooth manifolds \((X,K_X^r)\). A natural and deep question is to understand what are the singular objects obtained by the GIT compactification. For instance one can ask if Chow semistable canonically varieties with the same \(r\) will appear as it is the case for moduli space of curves. This paper provides a surprising solution to this question by showing that it fails in general, the Chow semistable limit may be not asymptotically Chow semistable. Some simple examples are given including smooth hypersurfaces of large degree in \({\mathbb P}^3\). The paper is well written. The proof, which is interesting in itself, uses the relationship between various notions of stability: Chow stability, log-Chow stability, Kollár-Shepherd-Barron-Alexeev (KSBA) stability and \(K\)-stability.

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
53C55 Global differential geometry of Hermitian and Kählerian manifolds
14E30 Minimal model program (Mori theory, extremal rays)
14L24 Geometric invariant theory
32J05 Compactification of analytic spaces
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