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Register a new userDeformation of Bott-Samelson varieties and variations of isotropy structures
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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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We compute the Newton--Okounkov bodies of line bundles on a Bott--Samelson resolution of the complete flag variety of $GL_n$ for a geometric valuation coming from a flag of translated Schubert subvarieties. The Bott--Samelson resolution corresponds to the decomposition $(s_1)(s_2s_1)(s_3s_2s_1)(ldots)(s_{n-1}ldots s_1)$ of the longest element in the Weyl group, and the Schubert subvarieties correspond to the terminal subwords in this decomposition. We prove that the resulting Newton--Okounkov polytopes for semiample line bundles satisfy the additivity property with respect to the Minkowski sum. In particular, they are Minkowski sums of Newton--Okounkov polytopes of line bundles on the complete flag varieties for $GL_2$,ldots, $GL_{n}$.
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 use the toric degeneration of Bott-Samelson varieties and the description of cohomolgy of line bundles on toric varieties to deduce vanishings results for the cohomology of lines bundles on Bott-Samelson varieties.
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.
Let $G$ be a connected and simply connected complex semisimple Lie group. For a collection of homogeneous $G$-spaces $G/Q$, we construct a finite atlas ${mathcal{A}}_{rm BS}(G/Q)$ on $G/Q$, called the Bott-Samelson atlas, and we prove that all of its coordinate functions are positive with respect to the Lusztig positive structure on $G/Q$. We also show that the standard Poisson structure $pi_{G/Q}$ on $G/Q$ is presented, in each of the coordinate charts of ${mathcal{A}}_{rm BS}(G/Q)$, as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl-Yakimov, making $(G/Q, pi_{G/Q}, {mathcal{A}}_{rm BS}(G/Q))$ into a Poisson-Ore variety. Examples of $G/Q$ include $G$ itself, $G/T$, $G/B$, and $G/N$, where $T subset G$ is a maximal torus, $B subset G$ a Borel subgroup, and $N$ the uniradical of $B$.
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