Special test configurations and $K$-stability of Fano varieties

By Chi Li. 2011

228 citations recorded by the source corpus.

For any flat projective family $(\mX,\mL)\rightarrow C$ such that the generic fibre $\mX_\eta$ is a klt Q-Fano variety and $\mL|_{\mX_\eta}\sim_{Q}-K_{X_{\eta}}$, we use the techniques from the minimal model program (MMP) to modify the total family. The end product is a family such that every fiber is a klt Q-Fano variety. Moreover, we can prove that the Donaldson-Futaki invariants of the appearing models decrease. When the family is a test configuration of a fixed Fano variety $(X,-K_X)$, this implies Tian's conjecture: given $X$ a Fano manifold, to test its K-(semi, poly)stability, we only need to test on the special test configurations.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b4b3dd485f98a9fa70c2dfeda7dd1f4f59123434

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In Applied Mathematics

1. Chi Li
2. Mattias Jonsson
3. Robert J. Berman
4. S. Boucksom
5. Yuchen Liu
6. H. Blum
7. Jiyuan Han
8. Y. Odaka

In Geometry and Topology

1. Chi Li
2. Mattias Jonsson
3. Robert J. Berman
4. S. Boucksom
5. Yuchen Liu
6. H. Blum
7. G. Tian
8. Y. Odaka

Citations in the corpus

• Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds (2013, background): “(3) Now we define the special degeneration following Tian [47] (See also [29]).”
• A generalization of the Ross--Thomas slope theory (2009, background): “Roughly speaking, the K-stability is positivity of the Dona ldson–Futaki invariants above but recentely it is pointed out by [15] that some pathol ogical test configurations of the following type should be “taken away” from our concern .”
• On the Yau‐Tian‐Donaldson Conjecture for Singular Fano Varieties (2017, background): “5 K-Polystability and Uniform K-Stability We recall the definitions of (log-)K-stability and uniform K-stability (see [9,18, 20, 41, 51]).”
• K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics (2012, methodology): “In general, KX /C does not extend as aQ-line bundle over the central fiber, but following [44] we say that a normal varietyX with aC∗-equivariant surjective morphism π to C is a special test configuration for the Fano variety X if X1 = X, the total space X is Q-Gorenstein and the central fiber is re”