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.