×

Log homogeneous varieties. (English) Zbl 1221.14053

Ferrer Santos, Walter (ed.) et al., Actas del XVI coloquio Latinoamericano de álgebra. Madrid: Revista Matemática Iberoamericana (ISBN 978-84-611-7907-7/pbk). Biblioteca de la Revista Matemática Iberoamericana, 1-39 (2007).
Let \(X\) be a nonsingular algebraic variety and \(G\) a connected algebraic group, both defined over an algebraically closed field \(k\) of characteristic zero. Assume that \(G\) operates faithfully on \(X\) as an algebraic transformation group with an open orbit \(U\) such that \(D_0:=X\backslash U\) is a divisor with only simple normal crossings. If the global vector fields on \(X\) associated to the Lie algebra \(\mathfrak g\) generate the logarithmic tangent bundle \(T_X(-\log D_0)\) then \(X\) is called \(\log\) homogeneous (with respect to \(G\)). This is equivalent to the condition that \(G\) operates transitively on the strata \(D_i\backslash D_{i+1}\) with \(D_{i+1}:=\mathrm{Sing} D_i\) for all \(i\geq 0\).
The author proves several fundamental structure theorems for log homogeneous varieties and analyzes in particular two important \(G\)-equivariant morphisms of \(X\), namely the universal map \(\alpha\) onto the abelian Albanese variety \({\mathcal A}(X)\) and the Tits morphism \(\tau\), an extension of the canonical surjection \(U=G/H\rightarrow G/N_G(H^0)\) to \(X\) and associated to a finite dimensional \(G\)-invariant subspace of the global sections of the line bundle \(-K_X-D_0\). Let \(X\) be complete. Then the \(\alpha\)-fibers and also \(\tau(X)\) are spherical (almost homogeneous and rational) varieties, but \(\tau(X)\) may have singularities. It is shown that \(X\) is \(\log\)-parallelizable (i.e. the bundle \(T_X(-\log D_0)\) is trivial) if and only if \(G\) is an extension of an abelian variety by an algebraic torus. (For the classification of compact log parallelizable weakly Kähler manifolds [see J. Winkelmann, Osaka J. Math. 41, No. 2, 473–484 (2004; Zbl 1058.32011)].
The irreducible components of the \(\tau\)-fibers are \(\log\) parallelizable and the \(G\)-equivariant product map \(\alpha\times\tau\) is affine and surjective; the irreducible components of its fibers are toric varieties. Moreover any closed \(G\)-orbit in \(X\) is mapped isomorphically onto the unique closed \(G\)-orbit in \({\mathcal A}(X)\times\tau(X)\). The classification of complete log homogeneous algebraic varieties is reduced to a problem concerning automorphism groups of spherical varieties, and it is solved under additional assumptions. Finally the author introduces the subclass of strongly log homogeneous varieties which is closely related to actions of reductive groups. The characterization of this subclass is used for the classification of complete log homogeneous surfaces.
The methods of the paper are purely algebraic. They involve results from the theory of algebraic transformation groups and some basic notations from logarithmic birational geometry and holomorphic transformation groups.
For the entire collection see [Zbl 1184.13002].

MSC:

14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations
32M12 Almost homogeneous manifolds and spaces
32M10 Homogeneous complex manifolds

Citations:

Zbl 1058.32011
PDFBibTeX XMLCite
Full Text: arXiv