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Frobenius splitting of cotangent bundles of flag varieties. (English) Zbl 0959.14031

This article is yet a testimony to the power of Frobenius splittings. Let \(G\) be a semi-simple connected algebraic group over an algebraically closed field of characteristic \(p>0\). Moreover, let \(U\) be the unipotent radical of a Borel subgroup \(B\subseteq G\) and \(\mathfrak u\) the Lie algebra of \(U\). The authors establish a link between the \(G\)-invariant form \(\chi\) on the Steinberg module \(\text{St}= H^0(G/B, (p-1) \rho)\) and Frobenius splittings of the cotangent bundle \(T^\ast(G/B)\) of \(G/B\). They show that \(T^\ast (G/B)\) is Frobenius split and obtain the vanishing result \(H^i(G/B, S\mathfrak u^\ast\otimes\lambda)=0\), \(i>0\), where \(\lambda\) is any dominating weight and \(S\mathfrak u^\ast\) is the symmetric algebra of \(\mathfrak u^\ast\). This implies via standard techniques, that the subregular nilpotent variety is normal, Gorenstein and has rational singularities. The vanishing theorem is also proved in the parabolic case for \(P\)-regular dominant weights. Using the Koszul resolution the authors also obtain the Dolbeault vanishing theorem \[ H^i(G/B, \Omega^j_{G/B} \otimes \mathcal L(\lambda)) =0 \] for \(i>j\) and \(\lambda\in \mathcal C =\{ \lambda \mid (\lambda, \alpha^{\vee}) \geq -1\), for all \(\alpha \in R^+\}\), where \(R^+\) denotes the set of positive roots of the root system of \(G\). Another interesting consequence is an isomorphism between group cohomology \(H^i(G_1, H^0(G/B, \mu))^{[-1]}\) of the first Frobenius kernel of \(G\) and the space of sections of a homogeneous line bundle on \(T^\ast(G/B)\) conjectured by J. C. Janzen. Using the \(B\)-module structure of \(\text{St} \otimes \text{St}\) the authors obtain that \(T^\ast(G/B)\) carries a canonical Frobenius splitting. In particular \(H^0(G/B, S\mathfrak u^\ast \otimes \lambda)\) has a canonical splitting. The proofs work for all groups in a uniform manner.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14G20 Local ground fields in algebraic geometry
14G15 Finite ground fields in algebraic geometry
14M17 Homogeneous spaces and generalizations
14F17 Vanishing theorems in algebraic geometry
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