## 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

### Keywords:

Calabi functional; test configuration; $$L$$-stability
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