Small resolutions of Schubert varieties in symplectic and orthogonal Grassmannians
Recall [\textit{M. Goresky} and \textit{R. MacPherson}, Invent. Math. 72, 77-129 (1983; Zbl 0529.55007)] that a resolution \(p:\widetilde{X}\to X\) of an irreducible complex variety \(X\) is said to be small if, for each \(i>0\), one has \(\text{codim}_X \{x\in X\mid\dim p^{-1}(x)\geq i\}>2i\). Let \(G=Sp(2n,\mathbb{C})\) or \(SO(2n,\mathbb{C})\), and let \(P=\mathbb{P}_n\). Recall [\textit{V. Lakshmibai} and \textit{C. S. Seshadri}, Proc. Indian Acad. Sci., Sect. A 87, No. 2, 1-54 (1978; Zbl 0447.14011)] that the Schubert varieties in \(Sp(2n,\mathbb{C})/P\) are indexed by \(\bigcup_{0\leq r\leq n} I_{n,r}\) where \(I_{n,r}=\{(\lambda_,\dots,\lambda_r)\mid 1\leq\lambda_1<\cdots<\lambda_r\leq n\}\). Similarly, the Schubert varieties in \(SO(2n)/\mathbb{P}_n\) are labelled by the set \(\bigcup_{\substack{ 0\leq r\leq n\\ (n-r)\text{ even}}} I_{n,r}\). The main results of this paper are: Theorem 1.1. Let \(\lambda=(\lambda_1,\dots,\lambda_r)\in I_{n,r}\). (i) The Schubert variety \(X(\lambda)\subset Sp(2n)/\mathbb{P}_n\) has a small resolution if \(\lambda_r\leq n-r\). (ii) Assume \(n-r\) is even so that \(\lambda\) gives rise to a Schubert variety \(X(\lambda)\) in \(SO(2n)/\mathbb{P}_n\). \(X(\lambda)\) has a small resolution if: (a) \(\lambda_r<n-r\); or (b) for \(r\geq 2\), \(\lambda_r=n\), \(\lambda_{r-1}\leq n-r\). Theorem 1.2. Let \(\lambda=(n)\), \(n\geq 3\), and let \(Q\) be any parabolic subgroup contained in \(\mathbb{P}_n\subset Sp(2n)\). Let \(X(\Lambda)\) be the inverse image of \(X(\lambda)\subset Sp(2n)/\mathbb{P}_n\) under the projection \(Sp(2n)/Q\to Sp(2n)/\mathbb{P}_n\). Then \(X(\Lambda)\) does not admit any small resolution.
scientific article; zbMATH DE number 716438