Buildings and Schubert schemes (Q2832144)

The topic of this book is resolutions of singularities of Schubert varieties. Much of the material is already contained in earlier publications of the author: [Adv. Math. 71, No. 2, 186--231 (1988; Zbl 0688.14046)], and mainly the author's Thèse [Géométrie des groupes semi-simples, résolutions equivariantes, et Lieu singulier de leurs cellules de Schubert. Montpellier (1981)]. Howevere it is not easy to find access to the latter. As a technical clarification, let us mention that in the book under review, and in this review, the term resolution is understood in the weak sense that we do not require that a resolution is an isomorphism over all of the smooth locus.NEWLINENEWLINETo describe the key questions and results of the book in a specific case, we set up some notation and at the same time recall some basic definitions. Let \(n\geq 1\), let \(k\) be a field, and let \(G=\mathrm{GL}_{n, k}\) be the general linear group over \(k\). Let us consider the flag variety \(\text{Flag}\), i.e., the projective variety of flags NEWLINE\[NEWLINE {\mathcal F}_1 \subset \cdots \subset {\mathcal F}_{n-1} \subset {\mathcal F}_n = k^n NEWLINE\]NEWLINE with \(\dim_k {\mathcal F}_i = i\) for all \(i\). Denoting by \(e_1, \dots, e_n\) the standard basis of \(k^n\), we have the standard flag given by \({\mathcal S}^{(i)} = \langle e_1, \dots, e_i\rangle_k\), \(i=1, \dots, n\). The \textit{Schubert cells} in \(\text{Flag}\) are the orbits under the action by the group \(B\) of upper triangular matrices in \(\mathrm{GL}_{n,k}\), the so-called standard Borel subgroup. It is well known that the set of orbits is in bijection with the symmetric group \(\underline{S}_n\) on \(n\) letters (the Weyl group of \(G\)). Explicitly, given a permutation \(w\in\underline{S}_n\), the corresponding Schubert cell is NEWLINE\[NEWLINE C_w = \{ ({\mathcal F}_i)_i;\;\dim {\mathcal F}_i\cap {\mathcal S}_j = m_{ij} \}, NEWLINE\]NEWLINE where \(M= (m_{ij})_{i,j}\) is the \textit{relative position} matrix attached to \(w\), defined by NEWLINE\[NEWLINE m_{ij} = \#\{ k \leq i;\;w(k) \leq j \}. NEWLINE\]NEWLINE It is known that as a variety \(C_w\) is isomorphic to affine space over \(k\), of dimension the number of inversions of the permutation \(w\).NEWLINENEWLINEA \textit{Schubert variety} in \(\text{Flag}\) is the closure of a Schubert cell (with the reduced scheme structure). It is easy to see that as a set the closure of \(C_w\) is NEWLINE\[NEWLINE S_w = \overline{C_w} = \{ ({\mathcal F}_i)_i;\;\dim {\mathcal F}_i\cap {\mathcal S}_j \geq m_{ij} \}. NEWLINE\]NEWLINENEWLINENEWLINEIn general, \(S_w\) is a singular variety, and it is an interesting question to describe the singularities, the singular locus, and resolutions of singularities of \(S_w\). The best known type of resolution is the Demazure resolution [\textit{M. Demazure}, Ann. Sci. Éc. Norm. Supér. (4) 7, 53--88 (1974; Zbl 0312.14009)] also called the Bott-Samelson resolution [\textit{R. Bott} and \textit{H. Samelson}, Am. J. Math. 80, 964--1029 (1961; Zbl 0101.39702)] or Hansen resolution [\textit{H. C. Hansen}, Math. Scand. 33, 269--274 (1974; Zbl 0301.14019)]. In the book under review, the construction of resolutions is approached somewhat differently (the relation to the usual construction of the Demazure resolution is discussed in Chapter 15, see the comments below).NEWLINENEWLINEFor \(\mathrm{GL}_n\), one can obtain a resolution not depending on any choices in the following way: Given the numbers \(m_{ij}\) attached to the permutation \(w\) as above, consider the following variety: NEWLINE\[NEWLINE\begin{aligned} \widehat{S}_w := \{ ({\mathcal H}_{ij})_{i,j};\;& {\mathcal H}_{ij} \subseteq k^n,\;\dim_k {\mathcal H}_{ij} = m_{ij},\\ & \forall i \leq i', j\leq j': {\mathcal H}_{ij}\subseteq {\mathcal H}_{i'j'},\\ & \forall j: {\mathcal H}_{nj} = {\mathcal S}_j \} \end{aligned}NEWLINE\]NEWLINE Clearly this is a closed subvariety of a product of Grassmann varieties, and mapping \(({\mathcal H}_{ij})_{i,j}\) to \(({\mathcal H}_{in})_i\) we obtain a flag in \(k^n\) which lies in \(S_w\) because \({\mathcal H}_{in}\cap {\mathcal S}_j\) contains \({\mathcal H}_{ij}\) and hence has dimension \(\geq \dim {\mathcal H}_{ij} = m_{ij}\). On the other hand it is easy to write \(\widehat{S}_w\) as a successive fibration of Grassmann varieties which shows that it is smooth. Furthermore, over \(C_w\) we have an obvious section to the map \(\widehat{S}_w\to S_w\) by mapping \(({\mathcal F}_i)_i\) to \(({\mathcal F}_i\cap {\mathcal S}_j)_{i,j}\). This resolution and its variants for varieties of partial flags are the theme of Chapter 3 (Def.~3.2), after some basic notions have been discussed in Chapters 1 and 2.NEWLINENEWLINEBased on this resolution, a combinatorial description of the singular locus of a Schubert variety for \(\mathrm{GL}_n\) is given (Thm.~4.38). Because of the differences in the approach, and in notation, it is not immediately apparent (to the reviewer, at least), how this description compares with other combinatorial characterizations of the singular locus (e.g., in terms of pattern avoidance of permutations, see the papers cited in the next paragraph).NEWLINENEWLINETo complement the bibliography in the book, we list in chronological order the following papers (among many others) about combinatorial descriptions of singular loci of Schubert varieties: [\textit{V. Lakshmibai} and \textit{C. S. Seshadri}, Bull. Am. Math. Soc., New Ser. 11, 363--366 (1984; Zbl 0549.14016)], relying on standard monomial theory. The state of the art at around the year 2000 is given in the book [\textit{S. Billey} and \textit{V. Lakshmibai}, Singular loci of Schubert varieties. Boston, MA: Birkhäuser (2000; Zbl 0959.14032)]. Afterwards, the following authors have obtained more precise information about the singular locus of Schubert varieties: [\textit{L. Manivel}, Int. Math. Res. Not. 2001, No. 16, 849--871 (2001; Zbl 1023.14022)], [\textit{S. C. Billey} and \textit{G. S. Warrington}, Trans. Am. Math. Soc. 355, No. 10, 3915--3945 (2003; Zbl 1037.14020)], [\textit{C. Kassel} et al., J. Algebra 269, No. 1, 74--108 (2003; Zbl 1032.14012)], [\textit{S. Gaussent}, Commun. Algebra 31, No. 7, 3111--3133 (2003; Zbl 1065.14060)]. Some of these results extend to the cases of other reductive groups; see the book by Billey and Lakshmibai and the references given there.NEWLINENEWLINEWhile the above definition of resolution is easy to write down, it does not generalize well, in this form, to other reductive algebraic groups, because there is no direct analog of the relative position matrix. Instead, as the author proposes, one can use the theory of buildings and generalized galleries in order to arrive at a construction of resolutions of singularities of Schubert varieties which has the construction described above as a special case, but applies to general reductive groups.NEWLINENEWLINEThis is carried out in the following chapters of the book. In Chapter 5 (The Flag Complex) the definition of building is recalled, and the example of the general linear group is discussed. Here we also find the definition of \textit{generalized} gallery (Def.~5.1) which is of key importance in the sequel. To describe it, recall the usual notion of (non-generalized) gallery: a sequence of chambers such that every chamber in the sequence shares a codimension \(1\) facet with the next one (including the possibility that it equals the next one). In a generalized gallery, the simplices need not have full dimension (and, for the connecting facets, codimension \(1\), resp.). Rather, a generalized gallery is defined as a sequence of facets \((F_1, \dots, F_n)\) (or \((F_0, \dots, F_n)\)), such that \(F_{2i} \supset F_{2i+1}\) for all \(i\), and \(F_{2i+1} \subset F_{2i+2}\) for all \(i\). Chapter 6 is about gallery varieties. Given a generalized gallery as above, for each facet \(F_i\) we have the partial flag variety \(\text{Flag}_i\) of the same type. The associated gallery variety is the closed subvariety of the product \(\prod_i \text{Flag}_i\) of partial flag varieties consisting of tuple \((\mathcal D_i)_i\), where the relative position between \(\mathcal D_i\) and \(\mathcal D_{i+1}\) is the one specified by \(F_i\) and \(F_{i+1}\) (here the notion of relative position is to be understood in terms of the Weyl group of the underlying group, or double cosets thereof). In Chapter 7 it is shown that the resolution defined in terms of the relative position matrix as explained above can be obtained as a gallery variety.NEWLINENEWLINEIn the following three chapters, the combinatorial ingredients required to handle general reductive groups are discussed. Chapter 8 is about the Coxeter complex, Chapter 9 about \textit{minimal} generalized galleries in a Coxeter complex, and Chapter 10 about the notion of minimal generalized galleries in the building of a reductive group.NEWLINENEWLINEWith these tools at hand, the author generalizes the theory which was previously set up for \(\mathrm{GL}_n\) over a field to the case of a reductive group scheme over an arbitrary base scheme. To start with, in Chapter 11 basic notions about parabolic subgroups and their relative position are established; for a large part, these topics are already discussed in [\textit{M. Demazure} (ed.) and \textit{A. Grothendieck} (ed.), Schémas en groupes. III: Structure des schémas en groupes réductifs. Exposés XIX à XXVI. Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3), dirigé par Michel Demazure et Alexander Grothendieck. Revised reprint. Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0212.52810)], see also [\textit{M. Demazure} (ed.) and \textit{A. Grothendieck} (ed.), Séminaire de géométrie algébrique du Bois Marie 1962-64. Schémas en groupes (SGA 3). Tome III: Structure des schémas en groupes réductifs. New annotated edition of the 1970 original published bei Springer. Paris: Société Mathématique de France (2011; Zbl 1241.14003)]). In Chapters 12 and 13, Schubert schemes and gallery schemes are defined in this context. Using these, the author defines a smooth resolution of Schubert schemes in Chapter 14. The main result (Theorem 14.18) is phrased as follows: Consider the universal Schubert scheme \(\overline{\Sigma}\) over the scheme \(\text{Relpos}\) of relative positions (i.e., we consider all Schubert varieties simultaneously; in the case of a split group scheme, the scheme of relative positions is just a disjoint union of copies of the base scheme). As above, the Schubert scheme is defined using the operation of schematic closure. Then the scheme of minimal galleries \(\mathcal C\) is a smooth resolution of the fiber product \(\Gamma^m \times_{\text{Relpos}} \overline{\Sigma}\), where \(\Gamma^m\) denotes the scheme of types of minimal galleries. For Schubert varieties in the flag variety of a split group, we can read this as saying that for each Weyl group element \(w\) (i.e., a point in the relative position scheme), we get a resolution of singularities once we choose a reduced expression for this element (in other words, a minimal gallery connecting the base chamber with the chamber corresponding to \(w\)).NEWLINENEWLINENext, the construction via generalized gallery schemes is compared to the more common point of view of Demazure resolutions written as contracted products. It is shown that both approaches are basically equivalent (Prop.~15.4). Finally, Chapter 16 discusses questions of functoriality and compatibility with base change.NEWLINENEWLINEDemazure resolutions have been studied from many points of view. Let us mention as examples the papers by \textit{S. Gaussent} [Indag. Math., New Ser. 12, No. 4, 453--468 (2001; Zbl 1065.14065)] and by \textit{S. Gaussent} and \textit{P. Littelmann} [Duke Math. J. 127, No. 1, 35--88 (2005; Zbl 1078.22007)]. In the latter one, several concepts treated in the book at hand are developed in the context of a Kac-Moody group, i.e., for affine Grassmannians and affine flag varieties, with the Bruhat-Tits building replacing the Tits building.NEWLINENEWLINEThe heavy notation used in the book is not always easy to digest. There is an index (unfortunately it is more difficult than necessary to find the desired information there, because the index is not sorted alphabetically, but according to chapters).