Frobenius splitting methods in geometry and representation theory.

*(English)*Zbl 1072.14066
Progress in Mathematics 231. Boston, MA: Birkhäuser (ISBN 0-8176-4191-2/hbk). x, 250 p. (2005).

The theory of Frobenius splitting, introduced by V. B. Mehta and A. Ramanathan [Ann. Math. (2) 122, 27–40 (1985; Zbl 0601.14043)] and refined by S. Ramanan and A. Ramanathan [Invent. Math. 79, 217–224 (1985; Zbl 0553.14023)], is a powerful method in the study of flag varieties and representation theory of linear algebraic groups in positive characteristic. One can also recover from it many of the analogous results in characteristic 0 by reduction to positive characteristic. The present monography is the first exposition in book form of this theory and of its major applications. It addresses to mathematicians and graduate students having a basic knowledge of algebraic geometry and representation theory of algebraic groups as provided by standard texts.

The first chapter of the book is devoted to the general study of Frobenius split schemes. They are defined as follows : Let \(X\) be a separated scheme of finite type over an algebraically closed field \(k\) of characteristic \(p > 0\). Consider the absolute Frobenius morphism \(F = (\text{id}_X, F^{\#}) : X\rightarrow X\), where \(F^{\#} : {\mathcal O}_X\rightarrow F_{\ast}{\mathcal O}_X (={\mathcal O}_X)\) maps \(f\) to \(f^p\). \(X\) is Frobenius split if there exists a morphism of \({\mathcal O}_X\)-modules \(\phi : F_{\ast}{\mathcal O}_X\rightarrow {\mathcal O}_X\) which is a left inverse of \(F^{\#}\). This has remarkable consequences: \(X\) is reduced (because \(F^{\#}\) is injective), weakly normal and, if \(X\) is projective, \(\text{H}^i(X,L) = 0\), \( \forall i > 0\), for every ample line bundle \(L\) on \(X\) (because \(F^{\ast}L\simeq L^{\otimes p}\), hence \(\text{H}^i(X,L)\) injects into \(\text{H}^i(X,L^{\otimes p})\)). Moreover, (variants of) the Kodaira, Grauert-Riemenschneider and Kawamata-Viehweg vanishing theorems are quick consequences of the existence of a Frobenius splitting.

On the other hand, the existence of a Frobenius splitting is a quite restrictive condition: if \(X\) is smooth one can show (using the Cartier operator) that the dual of the locally free \({\mathcal O}_X\)-module \(F_{\ast}{\mathcal O}_X\) (if \(t_1,\dots ,t_n\) is a local system of parameters for \(X\) at a point \(x\), then the monomials \(t_1^{a_1}\dots t_n^{a_n}\) with \(0\leq a_i\leq p-1\) form a local \({\mathcal O}_X\)-basis for \(F_{\ast}{\mathcal O}_X\) at \(x\)) is isomorphic to \(F_{\ast}({\omega}_X^{\otimes 1-p})\) hence, if \(X\) is Frobenius split, \(\text{H}^0(X,{\omega}_X^{\otimes 1-p})\neq 0\). For example, the only smooth projective curves which are Frobenius split are the projective line and the elliptic curves of Hasse invariant 1. However, many varieties where a linear algebraic group acts with a dense orbit turn out to be Frobenius split: the flag varieties \(G/P\) (\(G\) a connected, simply-connected, semisimple algebraic group and \(P\subset G\) a parabolic subgroup) and their Schubert subvarieties (chapter 2), the cotangent bundles of flag varieties (chapter 5) and the equivariant embeddings of connected reductive groups, e.g., toric varieties (chapter 6).

The proofs of these results use a (simple) criterion of Mehta and Ramanathan asserting that, for \(X\) projective and smooth, a section \(\phi \in \text{H}^0(X,{\omega}_X^{\otimes 1-p})\) defines a Frobenius splitting of \(X\) iff, in the local expansion of \(\phi \) at a point \(x\in X\), the “monomial” \(t_1^{p-1}\dots t_n^{p-1}(dt_1\wedge \cdots \wedge dt_n)^{1-p}\) occurs with coefficient 1. The verification of this condition in specific situations, is a matter of geometry or representation theory. For example, in the case of Schubert varieties, one firstly proves that their Bott-Samelson-Demazure-Hansen desingularizations are Frobenius split using the concrete description of the canonical bundle of these desingularizations. Also, if \(X = G/P\), then \(\text{H}^0(X,{\omega}_X^{\otimes 1-p})\) is a \(G\)-module.

Once the Frobenius splitting of the above mentioned varieties has been established, many important geometric consequences can be quickly drawn: Schubert varieties have rational singularities and are projectively Cohen-Macaulay and projectively normal in any projective embedding defined by a complete linear system (chapter 3), the full and subregular nilpotent cones have rational singularities (chapter 5), the equivariant embeddings of reductive groups have rational singularities (chapter 6), etc.

The book also contains some remarkable applications of Frobenius splitting in the representation theory of semi-simple groups : the Demazure character formula (chapter 3), a proof of the Parthasarathy-Ranga Rao-Varadarajan-Kostant conjecture on the existence of certain components in the tensor product of two dual Weyl modules and the existence of good filtrations for such tensor products and for the coordinate rings of semisimple groups in positive characteristic (chapter 4), etc.

The final chapter of the book (the 7th) contains a result of a different nature : the punctual Hilbert schemes of a smooth, projective, Frobenius split surface are Frobenius split, too. The authors provide all the necessary preliminary results on symmetric products, punctual Hilbert schemes and the Hilbert-Chow morphism. A key point is the fact that, for a smooth surface \(X\), the Hilbert-Chow morphism \(\text{Hilb}^nX\rightarrow \text{Sym}^nX\) is crepant.

Each section of the book is complemented with exercises; each chapter ends with useful comments, and open problems are suggested throughout. The book leads clearly and rapidly an interested reader from basic results to the research level.

The first chapter of the book is devoted to the general study of Frobenius split schemes. They are defined as follows : Let \(X\) be a separated scheme of finite type over an algebraically closed field \(k\) of characteristic \(p > 0\). Consider the absolute Frobenius morphism \(F = (\text{id}_X, F^{\#}) : X\rightarrow X\), where \(F^{\#} : {\mathcal O}_X\rightarrow F_{\ast}{\mathcal O}_X (={\mathcal O}_X)\) maps \(f\) to \(f^p\). \(X\) is Frobenius split if there exists a morphism of \({\mathcal O}_X\)-modules \(\phi : F_{\ast}{\mathcal O}_X\rightarrow {\mathcal O}_X\) which is a left inverse of \(F^{\#}\). This has remarkable consequences: \(X\) is reduced (because \(F^{\#}\) is injective), weakly normal and, if \(X\) is projective, \(\text{H}^i(X,L) = 0\), \( \forall i > 0\), for every ample line bundle \(L\) on \(X\) (because \(F^{\ast}L\simeq L^{\otimes p}\), hence \(\text{H}^i(X,L)\) injects into \(\text{H}^i(X,L^{\otimes p})\)). Moreover, (variants of) the Kodaira, Grauert-Riemenschneider and Kawamata-Viehweg vanishing theorems are quick consequences of the existence of a Frobenius splitting.

On the other hand, the existence of a Frobenius splitting is a quite restrictive condition: if \(X\) is smooth one can show (using the Cartier operator) that the dual of the locally free \({\mathcal O}_X\)-module \(F_{\ast}{\mathcal O}_X\) (if \(t_1,\dots ,t_n\) is a local system of parameters for \(X\) at a point \(x\), then the monomials \(t_1^{a_1}\dots t_n^{a_n}\) with \(0\leq a_i\leq p-1\) form a local \({\mathcal O}_X\)-basis for \(F_{\ast}{\mathcal O}_X\) at \(x\)) is isomorphic to \(F_{\ast}({\omega}_X^{\otimes 1-p})\) hence, if \(X\) is Frobenius split, \(\text{H}^0(X,{\omega}_X^{\otimes 1-p})\neq 0\). For example, the only smooth projective curves which are Frobenius split are the projective line and the elliptic curves of Hasse invariant 1. However, many varieties where a linear algebraic group acts with a dense orbit turn out to be Frobenius split: the flag varieties \(G/P\) (\(G\) a connected, simply-connected, semisimple algebraic group and \(P\subset G\) a parabolic subgroup) and their Schubert subvarieties (chapter 2), the cotangent bundles of flag varieties (chapter 5) and the equivariant embeddings of connected reductive groups, e.g., toric varieties (chapter 6).

The proofs of these results use a (simple) criterion of Mehta and Ramanathan asserting that, for \(X\) projective and smooth, a section \(\phi \in \text{H}^0(X,{\omega}_X^{\otimes 1-p})\) defines a Frobenius splitting of \(X\) iff, in the local expansion of \(\phi \) at a point \(x\in X\), the “monomial” \(t_1^{p-1}\dots t_n^{p-1}(dt_1\wedge \cdots \wedge dt_n)^{1-p}\) occurs with coefficient 1. The verification of this condition in specific situations, is a matter of geometry or representation theory. For example, in the case of Schubert varieties, one firstly proves that their Bott-Samelson-Demazure-Hansen desingularizations are Frobenius split using the concrete description of the canonical bundle of these desingularizations. Also, if \(X = G/P\), then \(\text{H}^0(X,{\omega}_X^{\otimes 1-p})\) is a \(G\)-module.

Once the Frobenius splitting of the above mentioned varieties has been established, many important geometric consequences can be quickly drawn: Schubert varieties have rational singularities and are projectively Cohen-Macaulay and projectively normal in any projective embedding defined by a complete linear system (chapter 3), the full and subregular nilpotent cones have rational singularities (chapter 5), the equivariant embeddings of reductive groups have rational singularities (chapter 6), etc.

The book also contains some remarkable applications of Frobenius splitting in the representation theory of semi-simple groups : the Demazure character formula (chapter 3), a proof of the Parthasarathy-Ranga Rao-Varadarajan-Kostant conjecture on the existence of certain components in the tensor product of two dual Weyl modules and the existence of good filtrations for such tensor products and for the coordinate rings of semisimple groups in positive characteristic (chapter 4), etc.

The final chapter of the book (the 7th) contains a result of a different nature : the punctual Hilbert schemes of a smooth, projective, Frobenius split surface are Frobenius split, too. The authors provide all the necessary preliminary results on symmetric products, punctual Hilbert schemes and the Hilbert-Chow morphism. A key point is the fact that, for a smooth surface \(X\), the Hilbert-Chow morphism \(\text{Hilb}^nX\rightarrow \text{Sym}^nX\) is crepant.

Each section of the book is complemented with exercises; each chapter ends with useful comments, and open problems are suggested throughout. The book leads clearly and rapidly an interested reader from basic results to the research level.

Reviewer: Iustin Coandă (Bucureşti)

##### MSC:

14M15 | Grassmannians, Schubert varieties, flag manifolds |

20G15 | Linear algebraic groups over arbitrary fields |

20G05 | Representation theory for linear algebraic groups |

14C05 | Parametrization (Chow and Hilbert schemes) |

13A35 | Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure |