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Super-rigid affine Fano varieties. (English) Zbl 1408.14052
Motivated by the study of automorphsm groups of affine varieties and the study of birational super-rigidity of Fano varieties, this paper studies affine Fano varieties and their super-rigidity.
This paper defines affine Fano varieties as the following: $$X\setminus S_X$$ is an affine Fano variety with completion $$X$$ and boundary $$S_X$$ if $$X$$ is a $$\mathbb{Q}$$-factorial projective normal variety of Picard rank $$1$$ and $$S_X$$ is a prime divisor on $$X$$ such that $$(X, S_X)$$ is purely log terminal and $$-(K_X+S_X)$$ is ample. Then super-rigidity for affine Fano varieties is defined as an analogue of birational super-rigidity of Fano varieties.
As the first result, this paper provides a sufficient condition for affine Fano varieties to be super-rigid, that is, $$X\setminus S_X$$ is super-rigid if for any mobile linear system $$\mathcal{M}_X$$ on $$X$$ and a positive rational number $$\lambda$$ such that $$K_X+S_X+\lambda \mathcal{M}_X\sim_{\mathbb{Q}} 0$$ the pair $$(X, S_X+\lambda \mathcal{M}_X)$$ is log canonical along $$S_X$$ This is an analogue of the Noether-Fano inequality.
Another sufficient condition is provided by using the alpha-invariants, which says that $$X\setminus S_X$$ is super-rigid if $$\alpha(S_X, \text{Diff}_{S_X}(0))\geq 1$$.
With those conditions, many examples and non-examples are provided.

##### MSC:
 14E07 Birational automorphisms, Cremona group and generalizations 14J45 Fano varieties 14J50 Automorphisms of surfaces and higher-dimensional varieties 14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) 14R20 Group actions on affine varieties 14R25 Affine fibrations 14C20 Divisors, linear systems, invertible sheaves 14E05 Rational and birational maps 14J17 Singularities of surfaces or higher-dimensional varieties 14J70 Hypersurfaces and algebraic geometry
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