Essential dimension is an invariant of algebraic objects, roughly speaking given by the number of parameters needed to describe these objects. It was introduced by J.Buhler and Z.Reichstein [Compos. Math. 106, No. 2, 159–179 (1997; Zbl 0905.12003)] in the context of Tschirnhaus transformations and has later been generalized and studied in various contexts. Essential dimension has been a very active field of research in the past 15 years. The goal of the paper under review, a proceedings article for the International Congress of Mathematics 2010, is to provide a survey on all this research.
Let us first recall the definition, in the form given below due to A.Merkurjev; see [G. Berhuy and G. Favi, Doc. Math., J. DMV 8, 279–330 (2003; Zbl 1101.14324)]. Let \(k\) be a field and \(\mathcal{F}: \mathrm{Fields}_k \to \mathrm{Sets}\) be a functor. Usually \(\mathcal{F}(K)\) will be the set of isomorphism classes of algebraic objects over \(K\) of some type, e.g. \(n\)-dimensional quadratic forms, smooth curves of genus \(g\), central simple algebras of degree \(n\) or alike. Then for a field extension \(K/k\) and an object \(a\in \mathcal{F}(K)\) the essential dimension of \(a\) is defined as \[ \mathrm{ed}(a):= \min_{K_0} \mathrm{trdeg}_{k} K_0, \] where \(K_0\) runs over all fields of definition of \(a\), i.e., intermediate fields \(K_0\) of \(K/k\) such that \(a\) lies in the image of the map \(\mathcal{F}(K_0)\to \mathcal{F}(K)\) induced by the inclusion \(K_0\hookrightarrow K\). The essential dimension of the functor \(\mathcal{F}\) is defined as the maximal essential dimension \(\mathrm{ed}(a)\) of all objects \(a\) over field extensions \(K\) of \(k\).
Most research in essential dimension concentrates on the case where \(\mathcal{F}\) is the Galois-cohomology functor \(H^1(-,G)\) for an (affine) algebraic group \(G\). We write \(\mathrm{ed}(G)\) for \(\mathrm{ed } H^1(-,G)\) and call it the essential dimension of \(G\). Several connections with classical problems are outlined in the paper. For instance the problem of computing the essential dimension of symmetric groups \(S_n\) has its starting point in the work of F. Klein [Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade. Leipzig 1884; reprint Birkhäuser, Basel (1993; Zbl 0803.01037)] on quintic equations, and for projective linear groups \(\mathrm{PGL}_n\) it goes back to the work of C. Procesi [Atti Accad. Naz. Lincei, Ser. 8, 239–255 (1967; Zbl 0204.04802)] on universal division algebras.
Essential dimension of an (affine) algebraic group \(G\) is then described in terms of generically free \(G\)-actions on varieties. The action of \(G\) on \(X\) is said to be generically free, if a \(G\)-invariant dense open subset \(U\) of \(X\) exists, where all stabilizers are trivial. Let \(G\) act linearly and generically freely on a vector space \(V\). Then \(\mathrm{ed}(G)\) is the least value of \(\dim X - \dim G\), where \(X\) runs over all generically free \(G\)-varieties admitting a \(G\)-equivariant dominant rational map \(V-\!\to X\). In particular, \(\dim V - \dim G\) gives an upper bound on \(\mathrm{ed}(G)\) for any generically free linear representation \(V\) of \(G\).
Several methods are presented to give lower bounds on \(\mathrm{ed}(G)\) for an algebraic group \(G\). These include lower bounds given by the degree \(d\) of a non-trivial cohomological invariant \(H^1(-,G) \to H^d(-,\mu_n)\) (this method was invented by J.-P. Serre see [Z. Reichstein, Transform. Groups 5, No. 3, 265–304 (2000; Zbl 0981.20033)]), and a fixed point method which bounds the essential dimension of a connected algebraic group \(G\) by \(\mathrm{rank}(A) - \mathrm{rank}(C^0_G(A))\) for a finite abelian subgroup \(A\) of \(G\) (whose order is prime to \(\mathrm{char}(k)\)), see e.g. [Ph.Gille and Z.Reichstein, Comment. Math. Helv. 84, No. 1, 189–212 (2009; Zbl 1173.11022)]. Sample applications are given to obtain lower bounds for \(\mathrm{ed}(G)\) with \(G=O_n\), \(\mu_n^r\), \(\mathrm{PGL}_{p^r}\), and for \(G=\mathrm{SO}_n\), \(\mathrm{Spin}_n\) and various exceptional simple algebraic groups, respectively.
Then central extensions of the form \(1\to \mu_p^r \to G \to \bar{G} \to 1\) are discussed. For such extensions a lower bound on \(\mathrm{ed}(G)\) can be given either by a sum of indices of certain central simple algebras attached to this sequence or by a formula depending on the dimensions of linear representations of \(G\). This method has its origin in the paper of M. Florence [Invent. Math. 171, No. 1, 175–189 (2008; Zbl 1136.14035)] and was vastly generalized in sequel by P. Brosnan, Z.Reichstein and A. Vistoli [Ann. Math. (2) 171, No. 1, 533–544 (2010; Zbl 1252.11034)] and by N.Karpenko and A.Merkurjev [Invent. Math. 172, No. 3, 491–508 (2008; Zbl 1200.12002)]. Applications include the computation of the precise value of \(\mathrm{ed}(G)\) for spin groups and finite \(p\)-groups. Several other applications have since been worked out, in particular algebraic tori by M.MacDonald, A.Meyer, Z.Reichstein and the reviewer in the preprint [“Essential dimension of algebraic tori”, http://www.math.uni-bielefeld.de/LAG/man/399.pdf] or the group \(\mathrm{Sim}(A,\sigma)\) of similitudes of a division algebra with involution, where \(A\) has \(p\)-power degree by the reviewer [“Essential dimension of involutions and subalgebras”, http://www.math.uni-bielefeld.de/LAG/man/403.pdf].
Next the author discusses two types of problems which he calls of type 1 and of type 2 respectively. Type 1 problems are those problems in the structure theory of algebraic objects which are considered up to prime to \(p\) extensions. Type 2 problems are those problems which remain after solving the corresponding type 1 problem. The author observes that most existing methods apply for type 1 problems only, whereas many long-standing open problems are of type 2. The problem to decide if some central simple algebra is a crossed product and the problem of computing the torsion index of an algebraic group are provided as examples.
Then finite groups of low essential dimension are discussed. The classifications of groups of essential dimension 0, 1 and 2 are given. The first case is trivial, the second on due to J.Buhler and Z.Reichstein [Compos. Math. 106, No. 2, 159–179 (1997; Zbl 0905.12003)] and the last and most difficult case (considered only over an algebraically closed field of characteristic 0) is due to A.Duncan [Math. Res. Lett. 17, No. 2, 263–266 (2010; Zbl 1262.14057)].
The paper concludes with a discussion of open problems on essential dimension.
For the entire collection see [Zbl 1220.00032].