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The dimension of automorphism groups of algebraic varieties with pseudo-effective log canonical divisors. (English) Zbl 1382.14003

Summary: Let \((X,D)\) be a log smooth pair of dimension \(n\), where \(D\) is a reduced effective divisor such that the log canonical divisor \(K_X + D\) is pseudo-effective. Let \(G\) be a connected algebraic subgroup of \(\mathrm{Aut}(X, D)\). We show that \(G\) is a semi-abelian variety of dimension \(\leq \min \{n-\bar {\kappa }(V), n\}\) with \(V:= X\setminus D\). In the dimension two, S. Iitaka claimed in his paper [Osaka J. Math. 16, 675–705 (1979; Zbl 0454.14016)] that \(\dim G\leq \bar {q}(V)\) for a log smooth surface pair with \(\bar{\kappa }(V) = 0\) and \(\bar {p}_g(V) = 1\). We (re-)prove and generalize this classical result for all surfaces with \(\bar{\kappa }=0\) without assuming Iitaka’s classification of logarithmic Iitaka surfaces or logarithmic \(K3\) surfaces.

MSC:

14C20 Divisors, linear systems, invertible sheaves
14J50 Automorphisms of surfaces and higher-dimensional varieties
14L10 Group varieties
14L30 Group actions on varieties or schemes (quotients)

Citations:

Zbl 0454.14016
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References:

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