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The \(\alpha\)-invariant on toric Fano manifolds. (English) Zbl 1088.32012
Let \(X\) be an \(n\)-dimensional compact complex manifolds with positive first Chern class \(c_1(X)\) and \(G\) a compact subgroup of \(Aut(X)\). Consider a \(G\)-invariant Kähler metric \(g=(g_{i\overline{j}})\) on \(X\) such that \(\omega ={\sqrt{-1}\over {2\pi}}\sum g_{i\overline{j}}dz_i\wedge d\overline{z}_{\overline {j}}\in c_1(X)\). Let \(P_G(X,\omega)\) be the set of all \(C^2\)-smooth \(G\)-invariant real-valued functions \(\varphi\) such that \(sup_X\varphi \)=0 and \(\omega+{\sqrt{-1}\over {2\pi}} \partial\overline{\partial}\varphi >0\). The \(\alpha_G(X)\) invariant is defined as the supremum of all \(\alpha >0\) such that there exists \(C(\alpha)\) with \(\int_Xe^{-\alpha\varphi}\omega^n\leq C(\alpha)\) for all \(\varphi \in P_G(X,\omega)\).
The author applies the Tian-Yau-Zelditch expansion of the Szegö kernel on polarized Kähler metrics to approximate almost plurisubharmonic functions and to obtain a formula to calculate the \(\alpha_G\)-invariants of all toric Fano manifolds precisely.

32Q20 Kähler-Einstein manifolds
14J45 Fano varieties
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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