Incompressibility of algebraic varieties

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The main thrust of the proposal is to study numerical and discrete invariants of torsors and projective homogeneous varieties, such as essential and canonical dimensions, and their applications to the theory of algebraic groups over non-closed fields and related structures. The idea of building global information/objects out of compatible local data and the nature of the obstruction to doing so, are of great importance to mathematics and physics (manifolds are the quintessential example of a global object constructed by "patching together" local data). The algebraic version of manifolds (needed for example in applications to number theory, but also in far removed areas such as genetics) are called varieties, or more generally schemes. The theory of torsors provides the language and tools to measure the obstructions to patching local data together (say to construct a variety or scheme). The theory of essential dimension could be thought as a sort of "complexity theory" of torsors. Linked to this is the concept of canonical dimension and of compressible variety (a tool that allows us to know when certain type of constructions could not get simpler). Having introduced the central pieces of my research, I will proceed now to give a more detailed description of some of my current research and objectives. An algebraic variety is called incompressible, if all its rational endomorphisms are dominant. Results on incompressibility of varieties have numerous applications, especially to computation of essential dimension of algebraic structures. Incompressibility for several classes of varieties has been already established by the applicant. Most of the varieties (but not all of them) are projective homogeneous under an action of a semisimple affine algebraic group. The main thrust of the proposal is the study of incompressibility for the following classes of varieties (depending on a prime integer p): (a) Generic torsors over norm tori of separable p-primary extensions of fields of characteristic p. In characteristic different from p, the result has been already established. The proof makes use of Steenrod operations on Chow groups with coefficients in the finite field of p elements. Such operations are not available over fields of characteristic p. (b) Hypersurfaces Nrd = const, where Nrd is the reduced norm of a p-primary central simple algebra. The approach is based on study of motives of smooth compactifications of the variety. (c) Unitary grassmannians associated to a 2-primary division algebra endowed with an involution of unitary type. We expect to obtain an analogue of the results explained in the previous section concerning symplectic involutions. All of the above questions can be expressed in terms of the rationality of certain algebraic cycles. My objective is also to prove results on descent of rational cycles from the function field of a variety to the base field (joint with R. Fino). One of the varieties we are interested in is the variety Nrd = const given by the reduced norm of a degree p central division algebra (the result is already known in characteristic 0). Finally, we plan to study operations in connective K-theory with possible application to the above problems.