The automorphism group of a variety with torus action of complexity one (Q2876648)

Let \(X\) be a rational complete normal variety equipped with an effective action of an algebraic torus \(T\) of dimension \(\dim T=\dim X-1\). The automorphism group \(\Aut(X)\) is studied. Since \(X\) is a Mori dream space, \(\Aut(X)\) is a linear algebraic group containing \(T\). If \(T\) is not a maximal torus in \(\Aut(X)\), then \(X\) is a toric variety, in which case \(\Aut(X)\) can be described in terms of the so-called Demazure roots and the automorphisms of the defining fan [\textit{M. Demazure}, Ann. Sci. Éc. Norm. Supér. (4) 3, 507--588 (1970; Zbl 0223.14009)]; [\textit{T. Oda}, Convex bodies and algebraic geometry. An introduction to the theory of toric varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 15. Berlin etc.: Springer-Verlag (1988; Zbl 0628.52002)]; [\textit{D. A. Cox}, J. Algebr. Geom. 4, No. 1, 17--50 (1995; Zbl 0846.14032); erratum ibid. 23, No. 2, 393--398 (2014)]. Otherwise \(T\) is a maximal torus in \(\Aut(X)\) and the description of the identity component \(\Aut(X)^0\) is largely reduced to a description of its roots, i.e., the eigenweights of \(T\) in the Lie algebra of \(\Aut(X)\). The main result of the paper describes the roots of \(\Aut(X)\).NEWLINENEWLINEThe main tool used in this description is the Cox ring \(\mathcal{R}(X)\) of \(X\) [\textit{I. Arzhantsev} et al., Cox rings. Cambridge Studies in Advanced Mathematics 144. Cambridge: Cambridge University Press (2015; Zbl 1360.14001)]. Since \(X\) is a Mori dream space, \(\mathcal{R}(X)\) is finitely generated and \(\overline{X}=\text{Spec}\mathcal{R}(X)\) is an affine variety equipped with an action of a diagonalizable group \(H_X\) whose group of characters is \(\text{Cl}(X)\). Furthermore, \(\overline{X}\) contains an open \(H_X\)-stable subset \(\widehat{X}\) such that \(\text{codim}(\overline{X}\setminus\widehat{X})>1\) and there exists a good quotient \(\widehat{X}\to\widehat{X}/\!\!/H_X\simeq X\). It is proved that, for every Mori dream space \(X\), the automorphisms of \(X\) can be lifted to \(H_X\)-equivariant automorphisms of \(\widehat{X}\), i.e., automorphisms normalizing the \(H_X\)-action. Furthermore, the group of equivariant automorphisms \(\Aut(\widehat{X},H_X)\) is a subgroup of finite index in \(\Aut(\overline{X},H_X)\) and \(\Aut(\widehat{X},H_X)/H_X\simeq\Aut(X)\), \(\Aut(\overline{X},H_X)/H_X\simeq\Aut(U)\) for some open subset \(U\subseteq X\) with the complement of codimension \(>1\). This result reduces the description of the roots of \(\Aut(X)\) to those of \(\Aut(\overline{X},H_X)\) and hence to the homogeneous locally nilpotent derivations of \(\mathcal{R}(X)\). The latter can be done in terms of the combinatorial data defining \(\mathcal{R}(X)\) in the case under consideration [\textit{J. Hausen} and \textit{E. Herppich}, Lond. Math. Soc. Lect. Note Ser. 405, 414--428 (2013; Zbl 1290.13001); Adv. Math. 225, No. 2, 977--1012 (2010; Zbl 1248.14008)].NEWLINENEWLINEThe semisimple part \(\Aut(X)^{\text{ss}}\) of \(\Aut(X)\) is also described. The root system of \(\Aut(X)^{\text{ss}}\) splits into the ``vertical'' and ``horizontal'' parts. The vertical part, consisting of the roots whose root subgroups preserve generic \(T\)-orbit closures, splits into the simple components of type \(A\), as in the toric case. The horizontal part, consisting of the roots whose root subgroups act transversally to generic \(T\)-orbits, is of type \(A_k\) (\(k\leq3\)), \(A_1+A_1\), \(B_2\), or empty.NEWLINENEWLINEApplications include the description of almost homogeneous complete rational normal surfaces with Picard number 1 equipped with a 1-torus action (all of them are del Pezzo) and of almost homogeneous complete \(\mathbb{Q}\)-factorial threefolds with Picard number 1 and the reductive automorphism group of rank 2 (all of them are rational Fano varieties).