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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.

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