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

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.


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


Zbl 1328.14073
Full Text: DOI arXiv


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