Vanishing theorem for the cohomology of line bundles on Bott–Samelson varieties

Pasquier, Boris

doi:10.1016/j.jalgebra.2010.03.016

Cited by 21 publications

(24 citation statements)

References 12 publications

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“…In this setting, the Bott-Samelson variety Z i determined by the sequence i has a toric degeneration to the toric variety X(Σ(c(i))) (see [Pas10,§1] and [GK94]). Moreover, the line bundle L i,m over the Bott-Samelson variety Z i degenerates into the line bundle over the Bott manifold X(Σ(c(i))).…”

Section: Background On Grossberg-karshon Twisted Cubesmentioning

confidence: 99%

“…Grossberg and Karshon [GK94] constructed a one-parameter family of complex structures on a Bott-Samelson variety which makes the Bott-Samelson variety into a Bott manifold. This degeneration of complex structures can be interpreted as the toric degeneration of a Bott-Samelson variety to a Bott manifold by Pasquier [Pas10]. Indeed, there is a flat family X over C such that X(t) is isomorphic to the Bott-Samelson variety for all t ∈ C \ {0} and X(0) is the Bott manifold.…”

Section: Background On Grossberg-karshon Twisted Cubesmentioning

confidence: 99%

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Grossberg–Karshon Twisted Cubes and Hesitant Jumping Walk Avoidance

Lee¹

2020

Electron. J. Combin.

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Let $G$ be a complex simply-laced semisimple algebraic group of rank $r$ and $B$ a Borel subgroup. Let $\mathbf i \in [r]^n$ be a word and let $\boldsymbol{\ell} = (\ell_1,\dots,\ell_n)$ be a sequence of non-negative integers. Grossberg and Karshon introduced a virtual lattice polytope associated to $\mathbf i$ and $\boldsymbol{\ell}$ called a twisted cube, whose lattice points encode the character of a $B$-representation. More precisely, lattice points in the twisted cube, counted with sign according to a certain density function, yields the character of the generalized Demazure module determined by $\mathbf i$ and $\boldsymbol{\ell}$. In recent work, the author and Harada described precisely when the Grossberg–Karshon twisted cube is untwisted, i.e., the twisted cube is a closed convex polytope, in the situation when the integer sequence $\boldsymbol{\ell}$ comes from a weight $\lambda$ of $G$. However, not every integer sequence $\boldsymbol{\ell}$ comes from a weight of $G$. In the present paper, we interpret the untwistedness of Grossberg–Karshon twisted cubes associated with any word $\mathbf i$ and any integer sequence $\boldsymbol{\ell}$ using the combinatorics of $\mathbf i$ and $\boldsymbol{\ell}$. Indeed, we prove that the Grossberg–Karshon twisted cube is untwisted precisely when $\mathbf i$ is hesitant-jumping-$\boldsymbol{\ell}$-walk-avoiding.

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Section: Background On Grossberg-karshon Twisted Cubesmentioning

confidence: 99%

Section: Background On Grossberg-karshon Twisted Cubesmentioning

confidence: 99%

Grossberg–Karshon Twisted Cubes and Hesitant Jumping Walk Avoidance

Lee¹

2020

Electron. J. Combin.

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“…Finally, if {r σ | σ ∈ {+, −} n } is equal to the set of vertices of P i,m and its elements are all distinct, then the normal fan of P i,m is equal to Σ i [6, Theorem 6.2.1]. By the results of [24] as above, we know Σ i is smooth, so we may conclude P i,m is a smooth polytope. Moreover, if (i, m) satisfies the condition (P), then from [13] we know P i,m is a lattice polytope.…”

Section: A Functorial Desingularization Of Singularitiesmentioning

confidence: 99%

Singular string polytopes and functorial resolutions from Newton–Okounkov bodies

Harada¹,

Yang²

2018

Illinois J. Math.

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The main result of this note is that the toric degenerations of flag varieties associated to string polytopes and certain Bott-Samelson resolutions of flag varieties fit into a commutative diagram which gives a resolution of singularities of singular toric varieties corresponding to string polytopes. Our main tool is a result of Anderson which shows that the toric degenerations arising from Newton-Okounkov bodies are functorial in an appropriate sense. We also use results of Fujita which show that Newton-Okounkov bodies of Bott-Samelson varieties with respect to a certain valuation νmax coincide with generalized string polytopes, as well as previous results by the authors which explicitly describe the Newton-Okounkov bodies of Bott-Samelson varieties with respect to a different valuation ν min in terms of Grossberg-Karshon twisted cubes. A key step in our argument is that, under a technical condition, these Newton-Okounkov bodies coincide.

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“…[73]) first prove the Frobenius-split property for Bott-Samelson varieties and then deduce the Frobenius-split property for the generalized flag varieties. In [79], the author proves the vanishing theorem for the cohomology of line bundles on Bott-Samelson varieties. A standard monomial theory has been developed for Bott-Samelson varieties in [47]; see also [2].…”

Section: Bott-samelson Scheme Of Gmentioning

confidence: 99%