\(K\)-destabilizing test configurations with smooth central fiber. (English) Zbl 1244.53083
Bielawski, Roger (ed.) et al., Variational problems in differential geometry. Proceedings of the workshop, Leeds, UK, March 30–April 2, 2009. Cambridge: Cambridge University Press (ISBN 978-0-521-28274-1/pbk). London Mathematical Society Lecture Note Series 394, 24-36 (2012).
This note is a continuation of the authors’ earlier works. They give examples of special degenerations with positive Futaki invariant and smooth limit. If \(X\subset {\mathbb P}^N\times {\mathbb C}\) is a subscheme obtained from a polarized normal variety \((V,A)\) and a representation \(\rho:{\mathbb C}^*\rightarrow \text{GL}(N+1)\), the pair \((X,L)\) with the relatively ample polarization \(L\) induced by the hyperplane bundle gives a test configuration.
A test configuration \((X,L)\) is destabilizing if the Futaki invariant of the induced action on the central fiber \((f^{-1}(0),L|_{f^{-1}(0)})\) is greater than zero. Here, \(f:X\rightarrow {\mathbb C}\) is a flat \({\mathbb C}^*\)-equivariant map such that \(L|_{f^{-1}(0)}\) is ample on \(f^{-1}(0)\) and \((f^{-1}(1),L|_{f^{-1}(1)})\simeq (V,A^r)\) for some \(r>0\).
Suppose the Futaki invariant \(F(X,L)\) is \(>0\). Considering the central fiber \(X_0\subset{\mathbb P}^N \) invariant under \(\rho\), they apply the resolution of singularities by Kollár to get a smooth manifold \(\tilde P\) with \({\mathbb C}^*\)-action and an equivariant map \(\beta:\tilde P\rightarrow {\mathbb P}^N\). The strict transform \(\tilde X_0\) of \(X_0\) is invariant and smooth in which the strict transform \(\tilde X_1\) of the fiber \(X_1\subset X\) degenerates to \(X_0\) under a \({\mathbb C}^*\)-action on \(\tilde P\), that is smooth. Hence this machinery gives an invariant family \(\tilde X\subset {\mathbb P}^N\times {\mathbb C}\) with an equivariant birational morphism \(\pi:\tilde X\rightarrow X\).
This will contribute to the conjecture that “A polarized manifold \((M,A)\) admits a Kähler metric of constant scalar curvature in the class \(c_1(A)\) if and only if it is \(K\)-polystable.”
For the entire collection see [Zbl 1230.53006].
A test configuration \((X,L)\) is destabilizing if the Futaki invariant of the induced action on the central fiber \((f^{-1}(0),L|_{f^{-1}(0)})\) is greater than zero. Here, \(f:X\rightarrow {\mathbb C}\) is a flat \({\mathbb C}^*\)-equivariant map such that \(L|_{f^{-1}(0)}\) is ample on \(f^{-1}(0)\) and \((f^{-1}(1),L|_{f^{-1}(1)})\simeq (V,A^r)\) for some \(r>0\).
Suppose the Futaki invariant \(F(X,L)\) is \(>0\). Considering the central fiber \(X_0\subset{\mathbb P}^N \) invariant under \(\rho\), they apply the resolution of singularities by Kollár to get a smooth manifold \(\tilde P\) with \({\mathbb C}^*\)-action and an equivariant map \(\beta:\tilde P\rightarrow {\mathbb P}^N\). The strict transform \(\tilde X_0\) of \(X_0\) is invariant and smooth in which the strict transform \(\tilde X_1\) of the fiber \(X_1\subset X\) degenerates to \(X_0\) under a \({\mathbb C}^*\)-action on \(\tilde P\), that is smooth. Hence this machinery gives an invariant family \(\tilde X\subset {\mathbb P}^N\times {\mathbb C}\) with an equivariant birational morphism \(\pi:\tilde X\rightarrow X\).
This will contribute to the conjecture that “A polarized manifold \((M,A)\) admits a Kähler metric of constant scalar curvature in the class \(c_1(A)\) if and only if it is \(K\)-polystable.”
For the entire collection see [Zbl 1230.53006].
Reviewer: Mohamed Darwish (Jeddah)
MSC:
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
