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Odaka, Yuji

The GIT stability of polarized varieties via discrepancy. (English) Zbl 1271.14067

Ann. Math. (2) 177, No. 2, 645-661 (2013).
The main result of the paper under review is to prove that if \(X\) is an equidimensional reduced projective scheme satisfying Serre’s \(S_2\) condition which is Gorenstein at all codimension \(1\) points and such that \(K_X\) is \(\mathbb Q\)-Cartier, \(L\) is an ample line bundle on \(X\) and \((X,L)\) is \(K\)-semistable then \(X\) has only semi-log-canonical singularities. If moreover \(L=-mK_X\) for some \(m>0\) (i.e. \(X\) is Fano), then \(X\) is log terminal (and in particular normal).
Recall that semi-log-canonical singularities are the broadest class of singularities that naturally arise in the minimal model program in the context of constructing a compactified moduli space for varieties of general type; in dimension \(1\) they correspond to nodal curves.
The notion of \(K\)-stability arises in the context of differential geometry and is related to the existence of Kähler metrics with constant scalar curvature. It is known that asymptotic Chow semi-stability or asymptotic Hilbert semi-stability imply \(K\)-semi-stability.
The author had previously shown [Osaka J. Math. 50, No. 1, 171–185 (2013; Zbl 1328.14073)] that semi-log canonical polarized varieties \((X,L)\) with \(K_X=0\) are \(K\)-semistable and semi-log canonical polarized varieties \((X,L)\) with \(L=mK_X\) for some \(m>0\) are \(K\)-stable.
Reviewer: Christopher Hacon (Salt Lake City)

Cited in 2 Reviews
Cited in 33 Documents

MSC:

14L24 Geometric invariant theory
14E30 Minimal model program (Mori theory, extremal rays)
32Q26 Notions of stability for complex manifolds

Keywords:

GIT stability; semi-log-canonical

Citations:

Zbl 1328.14073
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\textit{Y. Odaka}, Ann. Math. (2) 177, No. 2, 645--661 (2013; Zbl 1271.14067)
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