Isotropy of unitary involutions

We prove the so-called Unitary Isotropy Theorem, a result on isotropy of a unitary involution. The analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal, symplectic, and unitary involutions are formal consequences of this theorem. A component of the proof is a detailed study of the quasi-split unitary grassmannians.Comment: final version, to appear in Acta Mat

Ellipsoidal flows in relativistic hydrodynamics of finite systems

A new class of 3D anisotropic analytic solutions of relativistic hydrodynamics with constant pressure is found. We analyse, in particular, solutions corresponding to ellipsoidally symmetric expansion of finite systems into vacuum. They can be utilized for relativistic description of the system evolution in thermodynamic region near the softest point and at the final stage of the hydrodynamic expansion in A+A collisions. The solutions can be used also for testing of numerical hydrodynamic codes solving relativistic hydrodynamic equations for anisotropic expansion of finite systemsComment: 7 pages, talk given at RHIC school 2004 (Budapest, december 2004

Unitary grassmannians

We study projective homogeneous varieties under an action of a projective unitary group (of outer type). We are especially interested in the case of (unitary) grassmannians of totally isotropic subspaces of a hermitian form over a field, the main result saying that these grassmannians are 2-incompressible if the hermitian form is generic. Applications to orthogonal grassmannians are provided.Comment: 25 page

Holes in I^n

Let F be an arbitrary field of characteristic not 2. We write W(F) for the Witt ring of F, consisting of the isomorphism classes of all anisotropic quadratic forms over F. For any element x of W(F), dimension dim x is defined as the dimension of a quadratic form representing x. The elements of all even dimensions form an ideal denoted I(F). The filtration of the ring W(F) by the powers I(F)^n of this ideal plays a fundamental role in the algebraic theory of quadratic forms. The Milnor conjectures, recently proved by Voevodsky and Orlov-Vishik-Voevodsky, describe the successive quotients I(F)^n/I(F)^{n+1} of this filtration, identifying them with Galois cohomology groups and with the Milnor K-groups modulo 2 of the field F. In the present article we give a complete answer to a different old-standing question concerning I(F)^n, asking about the possible values of dim x for x in I(F)^n. More precisely, for any positive integer n, we prove that the set dim I^n of all dim x for all x in I(F)^n and all F consisists of 2^{n+1}-2^i, i=1,2,...,n+1 together with all even integers greater or equal to 2^{n+1}. Previously available partial informations on dim I^n include the classical Arason-Pfister theorem, saying that no integer between 0 and 2^n lies in dim I^n, as well as a recent Vishik's theorem, saying the same on the integers between 2^n and 2^n+2^{n-1} (the case n=3 is due to Pfister, n=4 to Hoffmann). Our proof is based on computations in Chow groups of powers of projective quadrics (involving the Steenrod operations); the method developed can be also applied to other types of algebraic varieties.Comment: 29 page

Incompressibility of orthogonal grassmannians

We prove the following conjecture due to Bryant Mathews (2008). Let Q be the orthogonal grassmannian of totally isotropic i-planes of a non-degenerate quadratic form q over an arbitrary field (where i is an integer in the interval [1, (\dim q)/2]). If the degree of each closed point on Q is divisible by 2^i and the Witt index of q over the function field of Q is equal to i, then the variety Q is 2-incompressible.Comment: 5 page