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Lower bounds on the Calabi functional. (English) Zbl 1149.53042

The Calabi functional is the \(L^2\) norm of the scalar curvature of Kähler metrics, running over a fixed Kähler class on a compact Kähler manifold. The author establishes an analogue of the Atiyah-Bott result for this Calabi functional. Assume that \(X\) is a smooth complex projective variety, let \(L\to X\) be a fixed ample bundle, and consider all the Kähler metrics \(\omega\) in the class \(c_1(L)\). The author defines a numerical invariant \(\Psi(\chi)\) using the \(\mathbb{C}^*\) action on the vector spaces \(H^0(X_0,\mathcal{L}^k)\), related to the generalized Futaki invariant. Consider \(S(\omega){{\omega^n}\over {n!}}={1\over 2}\rho\wedge {{\omega^{n-1}}\over {(n-1)!}}\), where \(\rho\) is the Ricci form representing \(c_1(X)\). Then \[ \inf_\omega\| S(\omega)\| _{L^2}\geq \sup_\chi \Psi(\chi). \]

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
32Q15 Kähler manifolds
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