Perverse sheaves.
(Faisceaux pervers.)

*(French)*Zbl 0536.14011
Astérisque 100, 172 p. (1982).

The success of the ’intersection homology’ introduced by M. Goresky and R. MacPherson [Topology 19, 135–165 (1980; Zbl 0448.55004)] in giving a good approach to deep problems in algebraic geometry has been both striking and somewhat mysterious. The paper under review approaches these notions from a general viewpoint: This is needed as the final result – a relative version of the hard Lefschetz theorem – is deduced from arguments in finite characteristic.

The first hundred pages consist of abstract, largely formal considerations. First one considers a triangulated category \(D\) with a \(t\)-structure, which consists of full subcategories \(D^{\leq n}\) and \(D^{\geq n}\) satisfying certain axioms. This is a rich structure, and several results involving homology and functors between such categories are obtained. These lead on to a study of gluing, in a context where one has 3 pairs of adjoint functors satisfying various exactness conditions. This context is provided by categories of sheaves over a space, a closed subspace and its complement, and the categories so constructed include those of sheaves whose cohomology on one of the subspaces vanishes below (or above) a certain dimension. Given a stratified space, and an integer valued function on the index set of strata (a perversity) an iteration of the construction yields categories of ’perverse sheaves’.

If \(X\) is an algebraic variety over \(\mathbb C\) one considers Whitney stratifications of \(X\) by subvarieties, and a perversity \(p\) depending only on the dimension of the subvariety. Taking the limit (as the stratification gets finer) gives a category of sheaves depending only on \(X\) and \(p\), and locally defined. For finite characteristic, regular stratifications are not available: a more technical definition is needed, and one works first with the coefficient group \(\mathbb Z/\ell^ n\), then (taking a limit) with \(\mathbb Z_{\ell}\); tensoring gives a category of perverse sheaves with coefficients \(\mathbb Q_{\ell}\) (or \(\bar{\mathbb Q}_{\ell})\) which has better properties, and some time is spent obtaining formal properties.

The results of geometric significance hold when we restrict to the middle perversity. Verdier duality gives an involution of this category of perverse sheaves (corresponding to the duality studied by Goresky and MacPherson): and the existence of such an involution is a useful guide to guessing which results should hold for perverse sheaves. However, the key result is one due to M. Artin (in SGA 4 =Sémin. Géométrie Algébrique 4. Lecture Notes in Mathematics. 305 (1973; Zbl 0245.00002)) which translates in the present context to give right exactness of direct images for affine morphisms. After various results about induced mappings, this leads to the result that the category of perverse sheaves is Noetherian and Artinian. Chapter 4 concludes with a partial result on the exactness of the ‘nearby cycles’ functor, leading to an estimation of certain Betti numbers (a variant on the theme that the middle dimensional homology group is the largest and the most interesting) which is needed for the main purity theorem.

A sheaf over a variety in finite characteristic is “pure” (of weight \(w\)) if for each fixed point of the Frobenius \(F_{q^ n}\) the eigenvalues of \(F_{q^ n}\) on the stalk have absolute value \(q^{\frac{1}{2}nw}\). It is “mixed” if it has a finite filtration with pure quotients; a complex of sheaves is mixed if each cohomology sheaf is (in practice this condition always holds). Deligne’s earlier result [P. Deligne, Publ. Math., Inst. Hautes Étud. Sci. 52, 137–252 (1980; Zbl 0456.14014)] is that the functor \(Rf_{!}\) preserves mixed complexes of weights \(\leq w\). For affine embeddings of perverse sheaves, this also holds for weights \(\geq w\), whence follows the key result that a simple mixed perverse sheaf is pure: and a pure perverse sheaf a direct sum of simple ones: This leads to a calculus of weights of perverse sheaves with respect to various functors. These permit a relativisation of the proof of hard Lefschetz: If \(f:X\to Y\) is a projective morphism, \(F\) a pure perverse sheaf on \(X\), \(L\in H^ 2(X, \mathbb Q_{\ell}(1))\) the first Chern class of a relatively ample invertible sheaf, then \(L^ i: {}^ pH^{- i}f_*F(-i)\to {}^ pH^ if_*F(i)\) is an isomorphism.

The paper concludes with a comparatively brief discussion of how to deduce results in characteristic 0 from those in finite characteristic, with particular reference to hard Lefschetz and the invariant cycle theorem (local and global).

For the entire collection see [Zbl 0515.00020].

The first hundred pages consist of abstract, largely formal considerations. First one considers a triangulated category \(D\) with a \(t\)-structure, which consists of full subcategories \(D^{\leq n}\) and \(D^{\geq n}\) satisfying certain axioms. This is a rich structure, and several results involving homology and functors between such categories are obtained. These lead on to a study of gluing, in a context where one has 3 pairs of adjoint functors satisfying various exactness conditions. This context is provided by categories of sheaves over a space, a closed subspace and its complement, and the categories so constructed include those of sheaves whose cohomology on one of the subspaces vanishes below (or above) a certain dimension. Given a stratified space, and an integer valued function on the index set of strata (a perversity) an iteration of the construction yields categories of ’perverse sheaves’.

If \(X\) is an algebraic variety over \(\mathbb C\) one considers Whitney stratifications of \(X\) by subvarieties, and a perversity \(p\) depending only on the dimension of the subvariety. Taking the limit (as the stratification gets finer) gives a category of sheaves depending only on \(X\) and \(p\), and locally defined. For finite characteristic, regular stratifications are not available: a more technical definition is needed, and one works first with the coefficient group \(\mathbb Z/\ell^ n\), then (taking a limit) with \(\mathbb Z_{\ell}\); tensoring gives a category of perverse sheaves with coefficients \(\mathbb Q_{\ell}\) (or \(\bar{\mathbb Q}_{\ell})\) which has better properties, and some time is spent obtaining formal properties.

The results of geometric significance hold when we restrict to the middle perversity. Verdier duality gives an involution of this category of perverse sheaves (corresponding to the duality studied by Goresky and MacPherson): and the existence of such an involution is a useful guide to guessing which results should hold for perverse sheaves. However, the key result is one due to M. Artin (in SGA 4 =Sémin. Géométrie Algébrique 4. Lecture Notes in Mathematics. 305 (1973; Zbl 0245.00002)) which translates in the present context to give right exactness of direct images for affine morphisms. After various results about induced mappings, this leads to the result that the category of perverse sheaves is Noetherian and Artinian. Chapter 4 concludes with a partial result on the exactness of the ‘nearby cycles’ functor, leading to an estimation of certain Betti numbers (a variant on the theme that the middle dimensional homology group is the largest and the most interesting) which is needed for the main purity theorem.

A sheaf over a variety in finite characteristic is “pure” (of weight \(w\)) if for each fixed point of the Frobenius \(F_{q^ n}\) the eigenvalues of \(F_{q^ n}\) on the stalk have absolute value \(q^{\frac{1}{2}nw}\). It is “mixed” if it has a finite filtration with pure quotients; a complex of sheaves is mixed if each cohomology sheaf is (in practice this condition always holds). Deligne’s earlier result [P. Deligne, Publ. Math., Inst. Hautes Étud. Sci. 52, 137–252 (1980; Zbl 0456.14014)] is that the functor \(Rf_{!}\) preserves mixed complexes of weights \(\leq w\). For affine embeddings of perverse sheaves, this also holds for weights \(\geq w\), whence follows the key result that a simple mixed perverse sheaf is pure: and a pure perverse sheaf a direct sum of simple ones: This leads to a calculus of weights of perverse sheaves with respect to various functors. These permit a relativisation of the proof of hard Lefschetz: If \(f:X\to Y\) is a projective morphism, \(F\) a pure perverse sheaf on \(X\), \(L\in H^ 2(X, \mathbb Q_{\ell}(1))\) the first Chern class of a relatively ample invertible sheaf, then \(L^ i: {}^ pH^{- i}f_*F(-i)\to {}^ pH^ if_*F(i)\) is an isomorphism.

The paper concludes with a comparatively brief discussion of how to deduce results in characteristic 0 from those in finite characteristic, with particular reference to hard Lefschetz and the invariant cycle theorem (local and global).

For the entire collection see [Zbl 0515.00020].

Reviewer: C. T. C. Wall

##### MSC:

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14G15 | Finite ground fields in algebraic geometry |

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |