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Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. (English) Zbl 0994.32021
Let \(\varphi\) be a plurisubharmonic function on a complex manifold \(X\). The complex singularity exponent \(c_K(\varphi)\) of \(\varphi\) on a compact set \(K\subset X\) is the supremum over \(c\geq 0\) such that \(\exp(-2c \varphi)\) is integrable on a neighborhood of \(K\). The notion plays an important role in complex analysis and algebraic geometry, and several other characteristics of singularities for analytic objects (holomorphic functions, coherent ideal sheaves, divisors, currents) are its particular cases.
The main results of the paper is lower semicontinuity of the map \(\varphi\mapsto c_K (\varphi)\), which means that if \(\varphi_j\to \varphi\) in \(L^1_{\text{loc}}(X)\) then \(\exp(-2 \subset\varphi_j) \to\exp(-2 \subset\varphi)\) in \(L^1\)-norm over a neighborhood of \(K\) for all positive \(c<c_K (\varphi)\).
As a consequence, a comparatively simple proof is given for the existence of Kähler-Einstein metrics on certain Fano orbifolds. In this way, the authors produce three new examples of rigid del Pezzo surfaces with quotient singularities which admit a Kähler-Einstein metric.

MSC:
32S05 Local complex singularities
14B05 Singularities in algebraic geometry
14J45 Fano varieties
32U05 Plurisubharmonic functions and generalizations
32U25 Lelong numbers
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