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Nikita A. Karpenko

Publications and source records attributed to Nikita A. Karpenko.

12 recordsLinked to original sources
arXiv2014

A numerical invariant for linear representations of finite groups

We study the notion of essential dimension for a linear representation of a finite group. In characteristic zero we relate it to the canonical dimension of certain products of Weil transfers of generalized Severi-Brauer varieties. We then proceed to compute the canonical dimension of a broad class of varieties of this type, extending earlier results of the first author. As a consequence, we prove analogues of classical theorems of R. Brauer and O. Schilling about the Schur index, where the Schur index of a representation is replaced by its essential dimension. In the last section we show that essential dimension of representations can behave in rather unexpected ways in the modular setting.

math.RT↗
arXiv2012

On standard norm varieties

Let $p$ be a prime integer and $F$ a field of characteristic 0. Let $X$ be the {\em norm variety} of a symbol in the Galois cohomology group $H^{n+1}(F,μ_p^{\otimes n})$ (for some $n\geq1$), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field $F(X)$ has the following property: for any equidimensional variety $Y$, the change of field homomorphism $\CH(Y)\to\CH(Y_{F(X)})$ of Chow groups with coefficients in integers localized at $p$ is surjective in codimensions $< (\dim X)/(p-1)$. One of the main ingredients of the proof is a computation of Chow groups of a (generalized) Rost motive (a variant of the main result not relying on this is given in Appendix). Another important ingredient is {\em $A$-triviality} of $X$, the property saying that the degree homomorphism on $\CH_0(X_L)$ is injective for any field extension $L/F$ with $X(L)\ne\emptyset$. The proof involves the theory of rational correspondences reviewed in Appendix.

math.AG↗
arXiv2012

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.

math.AG↗
arXiv2011

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.

math.AG↗
arXiv2011

Upper motives of algebraic groups and incompressibility of Severi-Brauer varieties

Let G be a semisimple affine algebraic group of inner type over a field F. We write C for the class of all finite direct products of projective G-homogeneous F-varieties. We determine the structure of the Chow motives with coefficients in a finite field of the varieties in C. More precisely, it is known that the motive of any variety in C decomposes (in a unique way) into a sum of indecomposable motives, and we describe the indecomposable summands which appear in the decompositions. In the case where G is the group of automorphisms of a given central simple F-algebra A, for any variety in the class C (which includes the generalized Severi-Brauer varieties of the algebra A) we determine its canonical dimension at any prime p. In particular, we find out which varieties in C are p-incompressible. If A is a division algebra of degree p^n for some n, then the list of p-incompressible varieties includes the generalized Severi-Brauer variety X(p^m; A) of ideals of reduced dimension p^m for m=0,1,...,n.

math.AG↗
arXiv2011

Incompressibility of quadratic Weil transfer of generalized Severi-Brauer varieties

Let X be the variety obtained by the Weil transfer with respect to a quadratic separable field extension of a generalized Severi-Brauer variety. We study (and, in some cases, determine) the canonical dimension, incompressibility, and motivic indecomposability of X. We determine the canonical 2-dimension of X (in the general case).

math.AG↗
arXiv2011

Isotropy of orthogonal involutions

An orthogonal involution on a central simple algebra becoming isotropic over any splitting field of the algebra, becomes isotropic over a finite odd degree extension of the base field (provided that the characteristic of the base field is not 2). The proof makes use of a structure theorem for Chow motives with finite coefficients of projective homogeneous varieties, of incompressibility of certain generalized Severi-Brauer varieties, and of Steenrod operations.

math.AG↗
arXiv2009

Upper motives of outer algebraic groups

Let G be a semisimple affine algebraic group over a field F. Assuming that G becomes of inner type over some finite field extension of F of degree a power of a prime p, we investigate the structure of the Chow motives with coefficients in a finite field of characteristic p of the projective G-homogeneous varieties. The complete motivic decomposition of any such variety contains one specific summand, which is the most understandable among the others and which we call the upper indecomposable summand of the variety. We show that every indecomposable motivic summand of any projective G-homogeneous variety is isomorphic to a shift of the upper summand of some (other) projective G-homogeneous variety. This result is already known (and has applications) in the case of G of inner type and is new for G of outer type (over F).

math.AG↗
arXiv2009

Hyperbolicity of orthogonal involutions

We show that a non-hyperbolic orthogonal involution on a central simple algebra over a field of characteristic different from 2 remains non-hyperbolic over some splitting field of the algebra.

math.AG↗
arXiv2006

Rational surfaces and canonical dimension of PGL_6

The "canonical dimension" of an algebraic group over a field by definition is the maximum of the canonical dimensions of principal homogenous spaces under that group. Over a field of characteristic zero, we prove that the canonical dimension of the projective linear group PGL_6 is 3. We give two distinct proofs, both of which rely on the birational classification of rational surfaces over a nonclosed field. One of the proofs involves taking a novel look at del Pezzo surfaces of degree 6.

math.AG↗
arXiv2003

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.

math.AG↗