Programm of the workshop The decomposition theorem Ulrich G¨ortz & Torsten Wedhorn

Introduction The theory of perverse sheaves and one of its crowning achievements, the decomposition theorem, are at the heart of a revolution which has taken place over the last thirty years in algebra, representation theory and algebraic geometry. The decomposition theorem is a powerful tool for investigating the topological properties of proper maps between algebraic varieties and is one of the deepest known facts relating their homological, Hodge-theoretic and arithmetic properties. By now there are three known approaches to the decomposition theorem: the original one, due to A. Beilinson, J. Bernstein, P. Deligne and O. Gabber ([BBD]), via the arithmetic properties of varieties over finite fields, the one of M. Saito ([Sa1], [Sa2], [Sa3]) via mixed Hodge modules, and the approach of De Cataldo and Migliorini, via classical Hodge theory. Each approach highlights different aspects of this important theorem. The goal of this workshop is the study of the decomposition theorem, of some applications, and of the proof of De Cataldo and Migliorini. Our main sources are [CM1] and [CM2].

Prerequisites Derived categories, “classical” Hodge theory.

Dates As dates we suggest: November 13, December 4 and January 22. One session each will take place at Essen and at Paderborn; the third place remains to be fixed.

1 1

Preliminaries (1st meeting) Hodge theory (60 minutes)

The goal of this talk is to recall some topological and Hodge theoretic facts on complex projective manifolds without proofs. The speaker should cover [CM2] 1.1–1.3 up to Example 1.3.1 using the formulations in [CM2] 5.2 (see also [CM1] 3.1). Start by recalling (very briefly) the notion of a pure Hodge structure and a polarization of pure Hodge structure ([De2] or [CM1] 3.1). Note that in the given references (in particular in [Vo]) there are often more 1

precise statements of the theorem (see also [CM2] 5.2). It is also suggested to express some of the more topological/complex analytic notions in terms of algebraic geometry (e.g. by replacing “family of projective manifolds” by “smooth projective morphism with connected fibers”). Recall briefly the notion of a mixed Hodge structure and Deligne’s theorem of the existence of functorial mixed Hodge structures on H • (X, Q) for all separated schemes X of finite type over C ([CM2] 5.2 and [De3]).

2

Intersection complexes (45 minutes)

Define the constructible derived category DY ([CM2] 1.5 (5)). Note that we assume the notion of a sheaf and the formalism of derived categories to be known. Recall briefly the functors f∗ , f! , f ∗ , f ! , some examples, adjointness properties, dualizing complex, Verdier duality D, and Df! = f∗ D and Df ! = f ∗ D ([CM2] 5.3). Define the intersection complex and the intersection cohomology as in [CM2] 2.1 and 1.5 and discuss its compatibility with Verdier duality. Explain [CM2] Example 2.2.1

3

Perverse sheaves (90 minutes)

Define perverse sheaves [CM2] 2.3 (we will always consider Q-coefficients and the middle perversity). Show that the intersection complexes are perverse sheaves. Define the perverse cohomology functors and describe kernel and cokernel of perverse sheaves in terms of perverse cohomology ([CM2] 2.5). Define the intermediate extension and explain the construction of the intersection complex using the intermediate extension ([CM2] 2.7 and [BBD]). Explain Example 2.7.2. Show that perverse sheaves form an abelian Artinian and Noetherian category whose simple objects are intersection complexes ([CM2] 2.3.6, see also [BBD]) and whose derived category is equivalent to DY (this last result by Bernstein should only be quoted). Explain the approach of MacPherson and Vilonen ([CM2] 5.7.1 and [MV]) for constructing perverse sheaves.

4

Perverse filtration (60 minutes)

Define the perverse spectral sequence [CM2] 1.5(7) and the perverse filtration [CM2] 2.4 (see also [CM1] §4). Explain the geometric construction in terms of restrictions to generic hyperplane sections (if possible in general but at least for affine varieties [CM3]). Prove the Lefschetz hyperplane theorem for the perverse cohomology [CM2] 2.6.1 and its consequence for intersection cohomology (Theorem 2.6.2).

2

2 1

The decomposition theorem (2nd meeting) Statement of the main theorem (75 minutes)

Statement of [CM2], Theorem 3.3.1 (see also [CM1] 2.1/2.2) and Examples [CM1] 2.4 (and 2.5 if time allows). For the examples recall briefly the notion of “semismallness” and “defect of semismallness” [CM1] 4.7.2 (see also next talk).

2

The decomposition theorem for semismall morphisms (60 minutes)

Explain some properties of semismall morphisms [CM2] Proposition 4.2.1 – Theorem 4.2.6. Reformulate the decomposition theorem for semi-small maps [CM2] Theorem 4.2.7 and Corollary 4.2.8. Explain the connection between the decomposition theorem for semismall maps and the nondegeneracy of intersection forms [CM2] Section 4.2.1.

3

Proof for semismall morphisms (60 minutes)

Give as much details as possible for the proof of the main theorem for semismall morphisms [CM2] 3.3.2/1 (see also [CM1] 2.6, §5 and §6).

4

Sketch of proof in the general case (60 minutes)

Sketch the proof in the general case [CM2] 3.3.2/2 (see also [CM1] 2.6, §5 and §6). Note that the Hodge theoretic nature of the perverse filtration follows from the geometric construction in terms of restrictions to generic hyperplane sections (talk 1.4)

3

Further examples and applications (3rd meeting)

These examples and applications are essentially independent of each other and the speaker is encouraged to choose from the material to make the talk as interesting as possible.

1

Toric varieties (60 minutes)

[CM2] Section 4.1 and the references therein.

2

Examples of semismall morphisms I: Springer resolution (60 minutes)

[CM2] Section 4.2.2 and the references therein.

3

3

Examples of semismall morphisms II: Hilbert schemes of points (60 minutes)

[CM2] Section 4.2.3 and the references therein.

4

Ngˆ o’s support theorem (75 minutes)

[CM2] 4.6, [N] Section 7.

Literatur [BBD] A.A. Beilinson, J.N. Bernstein, P. Deligne, Faisceaux pervers, Ast´erisque 100, Paris, Soc. Math. Fr. 1982. [CM1] M.A. de Cataldo, L. Migliorini, The Hodge Theory of Algebraic ´ maps, Ann. Scient. Ecole Norm. Sup., 4e s´erie, 38 (2005), 693–750. [CM2] M. De Cataldo, L. Migliorini, The decomposition theorem, perverse sheaves and the topology of algebraic maps, Bull. AMS 46 no. 4 (2009), 535–633. [CM3] M. de Cataldo, L. Migliorini, The perverse filtration and the Lefschetz Hyperplane Section Theorem, Annals of Math. 171 (2010), no. 3, 2089–2113. [De1]

P. Deligne, Th´eor`eme de Lefschetz et crit`eres de d´eg´en´erescence de suites spectrales, Publ. Math. IHES 35 (1969), 107–126.

[De2]

P. Deligne, Th´eorie de Hodge, II, Publ. Math. IHES 40 (1971), 5–57.

[De3]

P. Deligne, Th´eorie de Hodge, III, Publ. Math. IHES 44 (1974), 5–78.

[MV]

R. MacPherson, K. Vilonen, Elementary construction of perverse sheaves, Invent. Math. 84 (1986), no. 2, 403–435.

[N]

B. C. Ngˆ o, Le lemme fondamental pour les alg`ebres de Lie, arXiv:0801.0446v3

[Sa1]

M. Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), 849–995 (1989).

[Sa2]

M. Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), 221–333.

[Sa3]

M. Saito, Decomposition theorem for proper K¨ ahler morphisms, Tohoku Math. J. (2) 42, no. 2 (1990), 127–147.

4

[Vo]

C. Voisin, Hodge theory and complex algebraic geometry, I, II. Cambridge Studies in Advanced Mathematics, 76, 77. Cambridge University Press, Cambridge, 2003.

5