A relative Yau-Tian-Donaldson conjecture and stability thresholds.(English)Zbl 1537.32082
Summary: Generalizing Fujita-Odaka invariant, we define a function \(\tilde{\delta}\) on a set ofgeneralized \(b\)-divisors over a smooth Fano variety. This allows us to provide a new characterization of uniform \(K\)-stability. A key role is played by a new Riemann-Zariski formalism for \(K\)-stability.
For any generalized \(b\)-divisor \(\mathbf{D}\), we introduce a (uniform) \(\mathbf{D}\)-log K-stability notion. We prove that the existence of a unique Kähler-Einstein metric with prescribed singularities implies this new \(K\)-stability notion when the prescribed singularities are given by the generalized \(b\)-divisor \(\mathbf{D}\).
We connect the existence of a unique Kähler-Einstein metric with prescribed singularities to a uniform \(\mathbf{D}\)-log Ding-stability notion which we introduce. We show that these conditions are satisfied exactly when \(\tilde{\delta}(\mathbf{D}) > 1\), extending to the \(\mathbf{D}\)-log setting the \(\delta\)-valuative criterion of Fujita-Odaka and Blum-Jonsson.
Finally we prove thestrong openness of the uniform \(\mathbf{D}\)-log Ding stability as a consequence of the strong continuity of \(\tilde{\delta}\).
For any generalized \(b\)-divisor \(\mathbf{D}\), we introduce a (uniform) \(\mathbf{D}\)-log K-stability notion. We prove that the existence of a unique Kähler-Einstein metric with prescribed singularities implies this new \(K\)-stability notion when the prescribed singularities are given by the generalized \(b\)-divisor \(\mathbf{D}\).
We connect the existence of a unique Kähler-Einstein metric with prescribed singularities to a uniform \(\mathbf{D}\)-log Ding-stability notion which we introduce. We show that these conditions are satisfied exactly when \(\tilde{\delta}(\mathbf{D}) > 1\), extending to the \(\mathbf{D}\)-log setting the \(\delta\)-valuative criterion of Fujita-Odaka and Blum-Jonsson.
Finally we prove thestrong openness of the uniform \(\mathbf{D}\)-log Ding stability as a consequence of the strong continuity of \(\tilde{\delta}\).
MSC:
| 32Q26 | Notions of stability for complex manifolds |
| 14J45 | Fano varieties |
| 32Q20 | Kähler-Einstein manifolds |
| 32U05 | Plurisubharmonic functions and generalizations |
Cite
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