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Special test configuration and \(K\)-stability of Fano varieties. (English) Zbl 1301.14026

The authors show that if \(f:(\mathcal X, \mathcal L)\to C\) is a flat projective morphism over a smooth curve such that \(\mathcal X\) is normal, \(\mathcal L\) is \(f\)-ample, and \(\mathcal L \sim _{\mathbb Q,C}-K_{\mathcal X}\) holds over a non-empty open subset \(C^*\subset C\), then there exists a finite morphism of smooth curves \(\phi: C'\to C\) and a flat projective morphism \(f':(\mathcal X ', \mathcal L')\to C'\) such that \(\mathcal L'\) is \(f\)-ample, \(\mathcal L' \sim _{\mathbb Q,C'}-K_{\mathcal X'}\) holds over \(C'\) and \[ (\mathcal X ', \mathcal L')|_{\phi ^{-1}(C^*)}\cong (\mathcal X, \mathcal L) |_{C^*}\times _C C'. \] The authors then show that the Donaldson-Futaki invariants decrease via this process (and the inequality is strict unless we can assume \(C^*=C\)) and they show that this implies Tian’s conjecture:
Theorem. To test the \(K\)-(semi, poly)-stability of a Fano manifold, it suffices to test special test configurations.

MSC:

14J45 Fano varieties
14E30 Minimal model program (Mori theory, extremal rays)
14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
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