1,205 research outputs found
Splitting vector bundles and A^1-fundamental groups of higher dimensional varieties
We study aspects of the A^1-homotopy classification problem in dimensions >=
3 and, to this end, we investigate the problem of computing A^1-homotopy groups
of some A^1-connected smooth varieties of dimension >=. Using these
computations, we construct pairs of A^1-connected smooth proper varieties all
of whose A^1-homotopy groups are abstractly isomorphic, yet which are not
A^1-weakly equivalent. The examples come from pairs of Zariski locally trivial
projective space bundles over projective spaces and are of the smallest
possible dimension.
Projectivizations of vector bundles give rise to A^1-fiber sequences, and
when the base of the fibration is an A^1-connected smooth variety, the
associated long exact sequence of A^1-homotopy groups can be analyzed in
detail. In the case of the projectivization of a rank 2 vector bundle, the
structure of the A^1-fundamental group depends on the splitting behavior of the
vector bundle via a certain obstruction class. For projective bundles of vector
bundles of rank >=, the A^1-fundamental group is insensitive to the splitting
behavior of the vector bundle, but the structure of higher A^1-homotopy groups
is influenced by an appropriately defined higher obstruction class.Comment: 38 pages; Significantly revised, comments still welcom
Birational invariants and A^1-connectedness
We study some aspects of the relationship between A^1-homotopy theory and
birational geometry. We study the so-called A^1-singular chain complex and
zeroth A^1-homology sheaf of smooth algebraic varieties over a field k. We
exhibit some ways in which these objects are similar to their counterparts in
classical topology and similar to their motivic counterparts (the (Voevodsky)
motive and zeroth Suslin homology sheaf). We show that if k is infinite the
zeroth A^1-homology sheaf is a birational invariant of smooth proper varieties,
and we explain how these sheaves control various cohomological invariants,
e.g., unramified \'etale cohomology. In particular, we deduce a number of
vanishing results for cohomology of A^1-connected varieties. Finally, we give a
partial converse to these vanishing statements by giving a characterization of
A^1-connectedness by means of vanishing of unramified invariants.Comment: 24 pages; To appear Crelle's journal. Part II of ArXiV v2 (regarding
the Luroth problem) is being reworked and will be uploaded separately; the
old version is still available at http://www-bcf.usc.edu/~aso
Electronic transport in a randomly amplifying and absorbing chain
We study localization properties of a one-dimensional disordered system
characterized by a random non-hermitean hamiltonian where both the randomness
and the non-hermiticity arises in the local site-potential; its real part being
ordered (fixed), and a random imaginary part implying the presence of either a
random absorption or amplification at each site. The transmittance (forward
scattering) decays exponentially in either case. In contrast to the disorder in
the real part of the potential (Anderson localization), the transmittance with
the disordered imaginary part may decay slower than that in the case of ordered
imaginary part.Comment: 7 LaTex pages plus 2 PS figures; e-mail: [email protected]
Phase Distribution in a Disordered Chain and the Emergence of a Two-parameter Scaling in the Quasi-ballistic to the Mildly Localized Regime
We study the phase distribution of the complex reflection coefficient in
different configurations as a disordered 1D system evolves in length, and its
effect on the distribution of the 4-probe resistance . The stationary () phase distribution is almost always strongly non-uniform and is in
general double-peaked with their separation decaying algebraically with growing
disorder strength to finally give rise to a single narrow peak at infinitely
strong disorder. Further in the length regime where the phase distribution
still evolves with length (i.e., in the quasi-ballistic to the mildly localized
regime), the phase distribution affects the distribution of the resistance in
such a way as to make the mean and the variance of diverge
independently with length with different exponents. As , these
two exponents become identical (unity). Obviously, these facts imply two
relevant parameters for scaling in the quasi-ballistic to the mildly localized
regime finally crossing over to one-parameter scaling in the strongly localized
regime.Comment: 12 LaTeX pages plus 3 EPS figure
Comparing Euler classes
We establish the equality of two definitions of an Euler class in algebraic
geometry: the first definition is as a "characteristic class" with values in
Chow-Witt theory, while the second definition is as an "obstruction class."
Along the way, we refine Morel's relative Hurewicz theorem in A^1-homotopy
theory, and show how to define (twisted) Chow-Witt groups for geometric
classifying spaces.Comment: 33 pages; Final version (before proofs). To appear Q. J. Mat
A^1-homotopy groups, excision, and solvable quotients
We study some properties of A^1-homotopy groups: geometric interpretations of
connectivity, excision results, and a re-interpretation of quotients by free
actions of connected solvable groups in terms of covering spaces in the sense
of A^1-homotopy theory. These concepts and results are well-suited to the study
of certain quotients via geometric invariant theory.
As a case study in the geometry of solvable group quotients, we investigate
A^1-homotopy groups of smooth toric varieties. We give simple combinatorial
conditions (in terms of fans) guaranteeing vanishing of low degree A^1-homotopy
groups of smooth (proper) toric varieties. Finally, in certain cases, we can
actually compute the "next" non-vanishing A^1-homotopy group (beyond
\pi_1^{A^1}) of a smooth toric variety. From this point of view, A^1-homotopy
theory, even with its exquisite sensitivity to algebro-geometric structure, is
almost "as tractable" (in low degrees) as ordinary homotopy for large classes
of interesting varieties.Comment: 48 pages, To appear Adv. Math, typographical and grammatical update
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