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Register a new userStability of Bott--Samelson Classes in Algebraic Cobordism
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In this paper, we construct stable Bott--Samelson classes in the projective limit of the algebraic cobordism rings of full flag varieties, upon an initial choice of a reduced word in a given dimension. Each stable Bott--Samelson class is represented by a bounded formal power series modulo symmetric functions in positive degree. We make some explicit computations for those power series in the case of infinitesimal cohomology. We also obtain a formula of the restriction of Bott--Samelson classes to smaller flag varieties.
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We prove that the $infty$-category of $mathrm{MGL}$-modules over any scheme is equivalent to the $infty$-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite $mathbb{P}^1$-loop spaces, we deduce that very effective $mathrm{MGL}$-modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers. Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that $Omega^infty_{mathbb{P}^1}mathrm{MGL}$ is the $mathbb{A}^1$-homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for $n>0$, $Omega^infty_{mathbb{P}^1} Sigma^n_{mathbb{P}^1} mathrm{MGL}$ is the $mathbb{A}^1$-homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension $-n$.
We introduce the notion of flag Bott-Samelson variety as a generalization of Bott-Samelson variety and flag variety. Using a birational morphism from an appropriate Bott-Samelson variety to a flag Bott-Samelson variety, we compute Newton-Okounkov bodies of flag Bott-Samelson varieties as generalized string polytopes, which are applied to give polyhedral expressions for irreducible decompositions of tensor products of $G$-modules. Furthermore, we show that flag Bott-Samelson varieties are degenerated into flag Bott manifolds with higher rank torus actions, and find the Duistermaat-Heckman measures of the moment map images of flag Bott-Samelson varieties with the torus action together with invariant closed $2$-forms.
We show that the motivic spectrum representing algebraic $K$-theory is a localization of the suspension spectrum of $mathbb{P}^infty$, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspension spectrum of $BGL$. In particular, working over $mathbb{C}$ and passing to spaces of $mathbb{C}$-valued points, we obtain new proofs of the topologic
We obtain geometric models for the infinite loop spaces of the motivic spectra $mathrm{MGL}$, $mathrm{MSL}$, and $mathbf{1}$ over a field. They are motivically equivalent to $mathbb{Z}times mathrm{Hilb}_infty^mathrm{lci}(mathbb{A}^infty)^+$, $mathbb{Z}times mathrm{Hilb}_infty^mathrm{or}(mathbb{A}^infty)^+$, and $mathbb{Z}times mathrm{Hilb}_infty^mathrm{fr}(mathbb{A}^infty)^+$, respectively, where $mathrm{Hilb}_d^mathrm{lci}(mathbb{A}^n)$ (resp. $mathrm{Hilb}_d^mathrm{or}(mathbb{A}^n)$, $mathrm{Hilb}_d^mathrm{fr}(mathbb{A}^n)$) is the Hilbert scheme of lci points (resp. oriented points, framed points) of degree $d$ in $mathbb{A}^n$, and $+$ is Quillens plus construction. Moreover, we show that the plus construction is redundant in positive characteristic.
In the framework of the problem of characterizing complete flag manifolds by their contractions, the complete flags of type $F_4$ and $G_2$ satisfy the property that any possible tower of Bott-Samelson varieties dominating them birationally deforms in a nontrivial moduli. In this paper we illustrate the fact that, at least in some cases, these deformations can be explained in terms of automorphisms of Schubert varieties, providing variations of certain isotropic structures on them. As a corollary, we provide a unified and completely algebraic proof of the characterization of complete flag manifolds in terms of their contractions.
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