If c has trivial automorphism group and no invertible global. Pdf on connected automorphism groups of algebraic varieties. Pdf algebraic varieties with automorphism groups of maximal. Access full article top access to full text full pdf how to cite top.
Aug 11, 2018 let w be a quasiprojective variety over an algebraically closed field of characteristic zero. We show that if sautx is transitive on the smooth locus x reg, then it is infinitely transitive on x reg. In 1893, hurwitz applied the formula named after him to obtain an upper bound on the automorphism groups of algebraic curves over the complex numbers of genus at least 2, that is, on general type curves. The group scheme is locally of finite type and the tangent. Uma, lectures on the structure of algebraic groups and geometric applications, cmi lecture series in mathematics, vol. Let t be an algebraic torus and let c be a smooth a ne curve. In algebraic geometry, an algebraic group or group variety is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety. On connected automorphism groups of algebraic varieties michel brion abstract let x be a normal projective algebraic variety, g its largest connected automorphism group, and ag the albanese variety of g. Let g be a connected algebraic subgroup of \mathrmautx,d. Automorphism groups of affine varieties and vector fields. Is a finite group a subgroup of the automorphism group of a. The collection covers a wide range of topics and is. A lso, we obtain an optimal bound for the dimension of the. Algebraic varieties with automorphism groups of maximal rank article pdf available in mathematische annalen 3551 january 2012 with 22 reads how we measure reads.
Jordan groups and automorphism groups of algebraic. Let x be a normal projective algebraic variety, g its largest connected automorphism group, and ag the albanese variety of g. The paper intends to bring out relations between model theory, algebraic geometry, and symbolic dynamics. The main focus of this volume is on the problem of describing the automorphism groups of affine and projective varieties, a classical subject in algebraic geometry where, in both cases, the automorphism group is often infinite dimensional. Pdf jordan groups and automorphism groups of algebraic. Ams proceedings of the american mathematical society. Forms of automorphism groups of algebraic varieties.
Pdf flexible varieties and automorphism groups semantic. Then we show that very few connected algebraic semigroups can be realized as endomorphisms of some projective variety x, by. Abelian varieties as automorphism groups of smooth projective. Nonetheless, the problem of determining exactly which fano varieties are rational is far from solved. Jordan properties of automorphism groups of certain open algebraic. Since glr,z is the automorphism group of the algebraic torus c. Pdf algebraic varieties with automorphism groups of. Jordan property and automorphism groups of normal compact. Let w be a quasiprojective variety over an algebraically closed field of characteristic zero. We prove that if a contains no rational curves then the automorphism group g. A brief introduction to automorphisms of algebraic varieties. We study the automorphism groups of two families of varieties. On the automorphism group of certain algebraic varieties. Algebraic variety, automorphism of an encyclopedia of.
A note on automorphism groups of algebraic varieties. The first section is focused on jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational selfmaps of algebraic varieties. Jordan properties of automorphism groups of certain open. Sergei kovalenko, alexander perepechko, mikhail zaidenberg. Shramov, jordan property for groups of birational selfmaps, compos. Automorphism group algebraic variety these keywords were added by machine and not by the authors. In section 1 i discuss the makarlimanov and derksen invariants. On the birational automorphism groups of algebraic varieties. The aim of this paper is to prove the jordan property of the group autx for open. Of particular interest is the case of proalgebraic varieties over in. Automorphisms in birational and affine geometry levico. This process is experimental and the keywords may be updated as the learning algorithm improves.
Cusp singularities and discrete groups generated by re. Apr 19, 2018 abelian varieties as automorphism groups of smooth projective varieties. If autx and auty are isomorphic as ind groups, then x and y are isomorphic. In this symposium series, we aim to formulate a vision for future developments in complex, algebraic and arithmetic dynamics. For any projective algebraic variety x over a field k we set autkx to be the group of biregular maps of x to. We discussed arithmetic and algebraic features of dynamics on algebraic varieties. In this paper, we present these preliminary results which may have independent interest, with hopefully modest prerequisites. Algebraic varieties differ widely in how many birational automorphisms they have. The definition of a homomorphism depends on the type of algebraic structure. The automorphism group of the variety is typically huge in the examples i computed and might just include a copy of the group for no particular reason. This course introduces classical and new results on the algebraic structure of the identity component of the di.
The identity morphism identity mapping is called the trivial automorphism in some contexts. The appendix is the expanded version of my notes on open problems posted on the site of this workshop. On the birational automorphism groups of algebraic varieties masaki hanamura. There is a linear representation of this automorphism group on the vector space of global sections on of the pullback of from. Kyoto workshop on algebraic varieties and automorphism groups rims, kyoto university, july 7july 11, 2014 abstracts july 7 mon 11.
That means that there is a positive integer j j w such that every finite subgroup. On connected automorphism groups of algebraic varieties 45 ii. This is in some sense unfortunate, because the theory of algebraic groups even over the complex numbers, and still more over a nonalgebraically closed. For complete algebraic varieties over the field of complex numbers, the automorphism group is identical with the group of biholomorphic automorphisms. Jan 11, 2012 algebraic varieties with automorphism groups of maximal rank article pdf available in mathematische annalen 3551 january 2012 with 22 reads how we measure reads. To every such curve, we can associate a combinatorial object, a stable graph, which encode many properties of the curve. Characterization of affine toric varieties by their. More generally, we shall show that a connected and. We describe the automorphisms of cnc, deduce that the in. Introduction the subject matter of this note are automorphism groups of algebraic varieties.
In characteristic 0, we show that the dimension of ag is the rank of every maximal trivial direct summand of the tangent sheaf of x. Given an irreducible affine algebraic variety x of dimension n. In terms of category theory, an algebraic group is a group object in the category of algebraic varieties. An automorphism is simply a bijective homomorphism of an object with itself. Automorphism groups of configuration spaces and discriminant varieties vladimir lin and mikhail zaidenberg abstract. This question seems very pathological unless theres some natural map from the group to the automorphisms which ive missed. Variety means algebraic variety over k in the sense of serre so algebraic group.
Let x be an a ne toric variety di erent from the algebraic torus and let y be a normal a ne variety. On connected automorphism groups of algebraic varieties. X,y are smooth complete algebraic varieties, the connected automorphism group of x maps onto that of y under the homomorphism provided by the analogue of blanchards theorem. On connected automorphism groups of algebraic varieties michel brion institut fourier, b. The dimension of automorphism groups of algebraic varieties with. Automorphism groups of projective algebraic surfaces. We determine the isogeny class of ag in terms of the geometry of x. On automorphisms and endomorphisms of projective varieties.