15 research outputs found
Groups of order p^3 are mixed Tate
A natural place to study the Chow ring of the classifying space BG, for G a
linear algebraic group, is Voevodsky's triangulated category of motives, inside
which Morel and Voevodsky, and Totaro have defined motives M(BG) and M^c(BG),
respectively. We show that, for any group G of order p^3 over a field of
characteristic not p which contains a primitive p^2-th root of unity, the
motive M(BG) is a mixed Tate motive. We also show that, for a finite group G
over a field of characteristic zero, M(BG) is a mixed Tate motive if and only
M^c(BG) is a mixed Tate motive.Comment: 17 page
The categorical DT/PT correspondence and quasi-BPS categories for local surfaces
We construct semiorthogonal decompositions of Donaldson-Thomas (DT)
categories for reduced curve classes on local surfaces into products of
quasi-BPS categories and Pandharipande-Thomas (PT) categories, giving a
categorical analogue of the numerical DT/PT correspondence for Calabi-Yau
3-folds. The main ingredient is a categorical wall-crossing formula for DT/PT
quivers (which appear as Ext-quivers in the DT/PT wall-crossing) proved in our
previous paper. We also study quasi-BPS categories of points on local surfaces
and propose conjectural computations of their K-theory analogous to formulas
already known for the three dimensional affine space.Comment: 42 page
Categorical and K-theoretic Donaldson-Thomas theory of (part II)
Quasi-BPS categories appear as summands in semiorthogonal decompositions of
DT categories for Hilbert schemes of points in the three dimensional affine
space and in the categorical Hall algebra of the two dimensional affine space.
In this paper, we prove several properties of quasi-BPS categories analogous to
BPS sheaves in cohomological DT theory.
We first prove a categorical analogue of Davison's support lemma, namely that
complexes in the quasi-BPS categories for coprime length and weight are
supported over the small diagonal in the symmetric product of the three
dimensional affine space. The categorical support lemma is used to determine
the torsion-free generator of the torus equivariant K-theory of the quasi-BPS
category of coprime length and weight.
We next construct a bialgebra structure on the torsion free equivariant
K-theory of quasi-BPS categories for a fixed ratio of length and weight. We
define the K-theoretic BPS space as the space of primitive elements with
respect to the coproduct. We show that all localized equivariant K-theoretic
BPS spaces are one dimensional, which is a K-theoretic analogue of the
computation of (numerical) BPS invariants of the three dimensional affine
space.Comment: 45 pages, to appear in Forum Math. Sigm
The local categorical DT/PT correspondence
In this paper, we prove the categorical wall-crossing formula for certain
quivers containing the three loop quiver, which we call DT/PT quivers. These
quivers appear as Ext-quivers for the wall-crossing of DT/PT moduli spaces on
Calabi-Yau 3-folds. The resulting formula is a semiorthogonal decomposition
which involves quasi-BPS categories studied in our previous papers, and we
regard it as a categorical analogue of the numerical DT/PT correspondence. As
an application, we prove a categorical DT/PT correspondence for sheaves
supported on reduced plane curves in the affine three dimensional space.Comment: 35 page
Deformed dimensional reduction
Since its first use by Behrend, Bryan, and Szendr\H{o}i in the computation of
motivic Donaldson-Thomas (DT) invariants of ,
dimensional reduction has proved to be an important tool in motivic and
cohomological DT theory. Inspired by a conjecture of Cazzaniga, Morrison, Pym,
and Szendr\H{o}i on motivic DT invariants, work of Dobrovolska, Ginzburg, and
Travkin on exponential sums, and work of Orlov and Hirano on equivalences of
categories of singularities, we generalize the dimensional reduction theorem in
motivic and cohomological DT theory and use it to prove versions of the
Cazzaniga-Morrison-Pym-Szendr\H{o}i conjecture in these settings.Comment: 40 page
On Rationally Ergodic and Rationally Weakly Mixing Rank-One Transformations
We study the notions of weak rational ergodicity and rational weak mixing as
defined by Jon Aaronson. We prove that various families of infinite
measure-preserving rank-one transformations possess (or do not posses) these
properties, and consider their relation to other notions of mixing in infinite
measure.Comment: Some typos corrected, a reference added, arguments expande
Ergodicity and Conservativity of products of infinite transformations and their inverses
We construct a class of rank-one infinite measure-preserving transformations
such that for each transformation in the class, the cartesian product
of the transformation with itself is ergodic, but the product
of the transformation with its inverse is not ergodic. We also
prove that the product of any rank-one transformation with its inverse is
conservative, while there are infinite measure-preserving conservative ergodic
Markov shifts whose product with their inverse is not conservative.Comment: Added references and revised some arguments; removed old section 6;
main results unchange