Descent of line bundles to GIT quotients of flag varieties by maximal torus

Preprint English OPEN
Kumar, Shrawan (2007)
  • Subject: 14L24 | 57S25 | Mathematics - Algebraic Geometry | Mathematics - Group Theory | 14L30
    arxiv: Mathematics::Algebraic Geometry | Mathematics::Symplectic Geometry

Let L be a homogeneous ample line bundle on any flag variety G/P and let T be a maximal torus of G. We prove a general necessary and sufficient condition for L to descend as a line bundle on the GIT quotient of G/P by T. We use this result to explicitly determine exactly which L descend to the GIT quotient for any simple complex algebraic group G and any parabolic subgroup P.
  • References (7)

    L(Dℓ;α1,...,αℓ) = (Zα1 + Zθ + L′(α3,...,αℓ))∩ (Zα1 + Zα2 + Zθ + L′(α4,...,αℓ))∩ (L′(α3,α2,α1,−θ) + L′(α5,...,αℓ)) ∩ · · · ∩(L′(αℓ−5,...,α1,−θ) + L′(αℓ−3,αℓ−2,αℓ−1,αℓ)) ∩(L′(αℓ−4,...,α1,−θ) + Zαℓ−2 + Zαℓ−1 + Zαℓ) ∩(L′(αℓ−3,...,α1,−θ) + Zαℓ−1 + Zαℓ)]W = [L′(α1,...,αℓ)]W.

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