Schubert varieties are arithmetically Cohen-Macaulay.

*(English)*Zbl 0541.14039Let G be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic. Let B be a Borel subgroup and \(Q\supset B\) a parabolic subgroup of G. The closure of a B-orbit in G/Q is called a Schubert variety. We prove in this paper the following theorems:

Theorem 1. Let X and Y be unions of Schubert varieties in G/Q taken with the reduced subscheme structure. Then their scheme theoretic intersection \(X\cap Y\) is reduced. - Theorem 2. Let X be a union of Schubert varieties in G/Q taken with the reduced subscheme structure and L a line bundle on G/Q with \(H^ 0(G/Q,L)\neq 0.\) Then \(H^ i(X,L)=0\) for \(i>0\) and \(H^ 0(G/Q,L)\to H^ 0(X,L),\) the restriction map, is surjective. - Theorem 3. Let X be a Schubert variety in G/Q. Then X is Cohen-Macaulay and in the projective embedding given by any ample line bundle on G/Q it is arithmetically Cohen-Macaulay.

The proofs proceed by reducing to characteristic p\(>0\). The key point is the fact that the p-th power map \({\mathcal O}_{G/Q}\to F_*{\mathcal O}_{G/Q}\) where F is the absolute Frobenius morphism, has a splitting which gives compatible splitting of the corresponding map for any Schubert variety in G/Q. This is a further development of the methods of the papers ”Frobenius splitting and cohomology vanishing for Schubert varieties” by V. B. Mehta and the author [preprint (to appear)] and ”Projective normality of flag varieties and Schubert varieties” by S. Ramanan and the author [Invent. Math. (to appear)].

Theorem 1. Let X and Y be unions of Schubert varieties in G/Q taken with the reduced subscheme structure. Then their scheme theoretic intersection \(X\cap Y\) is reduced. - Theorem 2. Let X be a union of Schubert varieties in G/Q taken with the reduced subscheme structure and L a line bundle on G/Q with \(H^ 0(G/Q,L)\neq 0.\) Then \(H^ i(X,L)=0\) for \(i>0\) and \(H^ 0(G/Q,L)\to H^ 0(X,L),\) the restriction map, is surjective. - Theorem 3. Let X be a Schubert variety in G/Q. Then X is Cohen-Macaulay and in the projective embedding given by any ample line bundle on G/Q it is arithmetically Cohen-Macaulay.

The proofs proceed by reducing to characteristic p\(>0\). The key point is the fact that the p-th power map \({\mathcal O}_{G/Q}\to F_*{\mathcal O}_{G/Q}\) where F is the absolute Frobenius morphism, has a splitting which gives compatible splitting of the corresponding map for any Schubert variety in G/Q. This is a further development of the methods of the papers ”Frobenius splitting and cohomology vanishing for Schubert varieties” by V. B. Mehta and the author [preprint (to appear)] and ”Projective normality of flag varieties and Schubert varieties” by S. Ramanan and the author [Invent. Math. (to appear)].

##### MSC:

14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

**OpenURL**

##### References:

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