arXiv subjects
Publications and source records attributed to Peter O'Sullivan.
We study the behaviour of principal bundles under pullback along proper surjective morphisms of either schemes over an algebraically closed field of characteristic 0 or complex analytic spaces.
Let H be a complex Lie group acting holomorphically on a complex analytic space X such that the restriction to X_{\mathrm{red}} of every H-invariant regular function on X is constant. We prove that an H-equivariant holomorphic vector bundle E over X is $H$-finite, meaning f_1(E)= f_2(E) as H-equivariant bundles for two distinct polynomials f_1 and f_2 whose coefficients are nonnegative integers, if and only if the pullback of E along some H-equivariant finite étale covering of X is trivial as an H-equivariant bundle.
We consider essentially small rigid tensor categories (not necessarily abelian) which have a faithful tensor functor to a category of super vector spaces over a field of characteristic 0. It is shown how to construct for each such tensor category a super Tannakian hull, which is a universal faithful tensor functor to a super Tannakian category over a field of characteristic 0. The construction is analogous to the passage from an integral domain to its field of fractions.
Let $k$ be a field of characteristic $0$. We consider principal bundles over a $k$-scheme with reductive structure group (not necessarily of finite type). It is showm in particular that for $k$ algebraically closed there exists on any complete connected $k$-scheme a universal such bundle. As a consequence, an explicit description of principal bundles with reductive structure group over curves of genus $0$ or $1$ is obtained.
Let X be a smooth projective variety. Starting with a finite set of cycles on powers X^m of X, we consider the Q-vector subspaces of the Q-linear Chow groups of the X^m obtained by iterating the algebraic operations and pullback and push forward along those morphisms X^l -> X^m for which each component X^l -> X is a projection. It is shown that these Q-vector subspaces are all finite-dimensional, provided that the Q-linear Chow motive of X is a direct summand of that of an abelian variety.
It is shown that to every Q-linear cycle \barαmodulo numerical equivalence on an abelian variety A there is canonically associated a Q-linear cycle αmodulo rational equivalence on A lying above \barα. The assignment \barα-> αrespects the algebraic operations and pullback and push forward along homomorphisms of abelian varieties.
For $K$ a field, a Wedderburn $K$-linear category is a $K$-linear category $\sA$ whose radical $\sR$ is locally nilpotent and such that $\bar \sA:=\sA/\sR$ is semi-simple and remains so after any extension of scalars. We prove existence and uniqueness results for sections of the projection $\sA\to \bar\sA$, in the vein of the theorems of Wedderburn. There are two such results: one in the general case and one when $\sA$ has a monoidal structure for which $\sR$ is a monoidal ideal. The latter applies notably to Tannakian categories over a field of characteristic zero, and we get a generalisation of the Jacobson-Morozov theorem: the existence of a pro-reductive envelope $\Pred(G)$ associated to any affine group scheme $G$ over $K$. Other applications are given in this paper as well as in a forthcoming one on motives.