The Hard Lefschetz Theorem and the topology of semismall maps

Mark Andrea A. de Cataldo^{∗} and Luca Migliorini^{†}

Abstract

We introduce the notion of lef line bundles on a complex projective manifold. We prove that lef line bundles satisfy the Hard Lefschetz Theorem, the Lefschetz Decomposition and the Hodge-Riemann Bilinear Relations. We prove a generalization of the Grauert contractibility criterion which we call the Hodge Index Theorem for semismall maps. We study proper holomorphic semismall maps from complex manifolds and prove that, for constant coefficients, the Decomposition Theorem is equivalent to the non-degeneracy of certain intersection forms. We give a proof of the Decomposition Theorem for the complex direct image of the constant sheaf when the domain and the target are projective by proving that the forms in question are non-degenerate by virtue of our Hodge Index Theorem for semismall maps. A new feature uncovered by our proof is that the forms are definite.

Contents

1 Introduction 2

2 The Hard Lefschetz Theorem for lef line bundles 3 2.1 Semismall maps and lef line bundles . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The property HL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 The Hard Lefschetz Theorem and the signature of intersection forms …. 7

2.3.1 The use of pencils . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 The Hodge Index Theorem for semismall maps . . . . . . . . . . . . . . . . 9

3 The topology of semismall maps 11 3.1 Homological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 The bilinear forms associated with relevant strata . . . . . . . . . . . . . . . 13 3.3 Inductive study of semismall analytic maps . . . . . . . . . . . . . . . . . . 15 3.4 Signature and Decomposition Theorem in the projective case . . . . . . . . 17

∗Partially supported by N.S.F. Grant DMS 9701779.

†member of GNSAGA, supported by MURST funds

1 Introduction

Let X be a complex projective n-fold endowed with a line bundle L. The classical statement of the Hard Lefschetz Theorem is that if L is ample then, for every r ≥ 0, cupping with c_{1}(L)^{r} gives an isomorphism of H^{n−r}(X, ℚ) onto H^{n+r}(X, ℚ).

In this paper we introduce the notion of lef line bundles on projective manifolds. They are precisely the line bundles with the property that L^{⊗t} = f^{∗}A, where f : X → Y is a projective semismall morphism, t is some positive integer and A is an ample line bundle. Recall that f is semismall when the set of points in Y, whose fiber has dimension k, has codimension at least 2k. One has the following strict implications: ample ⇒ lef ⇒ semiample+big; the results we prove for lef line bundles fail for the last class.

Our first result is Theorem 2.3.1: if L is a line bundle such that a positive power is generated by its global sections, then the Hard Lefschetz Theorem holds for L if and only if L is lef. See also Proposition 2.2.7. A Lefschetz-type decomposition for H^{∗}(X, ℚ) ensues and we prove the Hodge-Riemann Bilinear relations, i.e. the signature of the intersection form on the associated primitive spaces has properties completely analogous to the ample case. We offer two proofs, see §2.3. and §2.3.1.. While the first is simpler, the second one characterizes monodromy invariants.

A second result is Theorem 2.4.1 where we prove a higher dimensional generalization of the Grauert criterion for the contractibility of a finite set of curves on a surface: if f : X → Y is a semismall map between 2m-dimensional projective varieties, X is nonsingular, y ∈ Y is a point and dim f^{−1}(y) = m, then the intersection form associated with the irreducible components of maximal dimension of f^{−1}(y) is (−1)^{m}-definite. We call this property the Hodge Index Theorem for semismall maps. It is proved using Theorem 2.3.1 and Deligne’s theory of Mixed Hodge Structures.

This result has a surprising consequence when applied to study the topology of semismall maps. Beilinson-Bernstein-Deligne, [2], proved the following fundamental result, called the Decomposition Theorem: if g : U → V is a proper morphism of algebraic varieties, then the direct image complex of an intersection cohomology complex of geometric origin on U, e.g. the intersection cohomology complex of U, is isomorphic, in the derived category, to a direct sum of shifted intersection cohomology complexes. An analogous result holds over any field. Even over the complex numbers, finite fields are necessary to their proof. Saito, [20], proved and extended these results in a Kähler setting using mixed Hodge modules. Borho-MacPherson, [4], have observed that, in the statement of the Decomposition Theorem, the shifts disappear when g is semismall, U is smooth and constant coefficients are taken.

Let f : X → Y be a proper semismall holomorphic map from a complex manifold. We propose a new approach to the Decomposition Theorem for such maps and for constant coefficients. Note that if f is not Kähler, see [20], then it is not known whether the theorem holds or not. The counterexamples to the Decomposition Theorem we are aware of are not for semismall maps.

We prove Theorem 3.3.3: the Decomposition Theorem is equivalent to the non-degene-racy of a collection of intersection forms that arise by studying a Mather-Thom-Whitney stratification of f. This result holds also for more general (i.e. non complex analytic) maps admitting stratifications with properties analogous to the complex analytic ones.

Using our Hodge Index Theorem for semismall maps we give a new proof of the Decomposition Theorem for constant coefficients and for X and Y projective, Theorem 3.4.1. This was actually our starting point and our main motivation to investigate lef bundles (l.e.f.= Lefschetz effettivamente funziona). A remarkable new feature we prove is that the intersection forms in questions are not only non-degenerate, but also definite.

We believe that the relation between the Decomposition Theorem and the intersection forms associated with the strata is illuminating. Furthermore, the direct proof of the non-degeneracy of these forms, with the new additional information on the signatures, neither relying on reduction to positive characteristic nor on Saito’s theory of mixed Hodge modules, sheds light on the geometry underlying the decomposition theorem and gives some indications on its possible extensions beyond the algebraic category.

Acknowledgments. Parts of this work have been done while the first author was visiting Hong Kong University in January 2000 and the Max-Planck Institut für Mathematik in the summer of 2000 and while the second author was visiting Harvard University in April 1999 and SUNY at Stony Brook in May 2000. We would like to thank D. Massey, A.J. Sommese and J.A. Wisniewski for useful correspondences.

2 The Hard Lefschetz Theorem for lef line bundles

In this section we introduce the notion of lef line bundle on a projective variety. It is a positivity notion weaker than ampleness but stronger than semiampleness and bigness combined. Lef line bundles satisfy many of the cohomological properties of ampleness. Our main result is that they also satisfy the Hard Lefschetz Theorem, the Lefschetz Decomposition and the Hodge Riemann Bilinear relations on the primitive spaces. These results are all false, in general, for line bundles which are simultaneously generated by their global sections and big. We study pencils of sections and give a second proof of the Hard Lefschetz Theorem. Finally we prove our Hodge Index Theorem for semismall maps.

2.1 Semismall maps and lef line bundles

Let f : X → Y be a proper holomorphic map. For every integer k define Y^{k} := {y ∈ Y | dim f^{−1}(y) = k}. The spaces Y^{k} are locally closed analytic subvarieties of Y whose disjoint union is Y. If a fiber is reducible, then it is understood that its dimension is the highest among the dimensions of its components.

Definition 2.1.1 We say that a proper holomorphic map f : X → Y of irreducible varieties is semismall if dim Y^{k} + 2k ≤ dim X for every k. Equivalently, f is semismall if and only if there is no irreducible subvariety T ⊆ X such that 2 dim T −dim f(T) > dim X.

Remark 2.1.2 A semismall map is necessarily generically finite.

From now on we shall assume that semismall maps are proper and surjective.

Definition 2.1.3 We say that a line bundle M on a complex projective variety X is lef if a positive multiple of M is generated by its global sections and the corresponding morphism onto the image is semismall.

Remark 2.1.4 If the map associated to a multiple of M generated by its global sections is semismall, then the map associated with any other multiple of M generated by its global sections is semismall as well. A lef line bundle is nef and big, but not conversely.

Proposition 2.1.5 (Weak Lefschetz theorem for lef line bundles) Let M be a lef line bundle on a smooth complex projective variety X. Assume that M admits a section s ∈ H^{0}(X, M) whose reduced zero locus is a smooth divisor Y. Denote by i : Y → X the inclusion. The restriction map i^{∗} : H^{r}(X) ⟶ H^{r}(Y) is an isomorphism for r < dimX − 1 and it is injective for r = dim X − 1.

Proof. The proof can be obtained by a use of the Leray spectral sequence coupled with the theorem on the cohomological dimension of constructible sheaves on affine varieties. See, for example, [19]. See also [12], Lemma 1.2. □

Remark 2.1.6 The Weak Lefschetz Theorem has been considerably strengthened by Goresky-MacPherson [15], II.1.1.

Proposition 2.1.7 (Bertini theorem for lef line bundles) Let M be a lef line bundle on a nonsingular complex projective variety X. Assume that M is generated by its global sections. Let W′ ⊆ |M| be the set of divisors Y in the linear system of M such that Y is smooth and M_{|Y} is lef. Then the set W′ contains a nonempty and Zariski open subset W ⊆ |M|.

Proof. The standard Bertini Theorem implies that a generic divisor D ∈ |M| is nonsingular. Let f : X → Y be the semismall map associated with |M| and Y^{k} be the locally closed subvarieties mentioned above. The set of divisors containing at least one among the closed subvarieties f^{−1}(\overline{Y^{k}}) is a finite union of linear proper subspaces of |M|. The conclusion follows. □

Remark 2.1.8 The Kodaira-Akizuki-Nakano Vanishing Theorem holds for lef line bundles. See [12], Theorem 2.4. See also [19].

Remark 2.1.9 The Kodaira-Akizuki-Nakano Vanishing Theorem for lef line bundles coupled with Serre Vanishing Theorem implies that if f : X → Y is a semismall holomorphic map from a complex projective manifold of dimension n onto a projective variety then R^{q}f_{∗}Ω_{X}^{p} = 0 for every p + q > n. This fact generalizes Grauert-Riemannschneider Vanishing Theorem and is false for semiample big line bundles.

Remark 2.1.10 One can define lef vector bundles in terms of the lefness of the tautological line bundle and prove that many of the standard properties of ample vector bundles hold for lef ones.

2.2 The property HL

Let X be a smooth, compact, oriented manifold of even real dimension 2n. We will use the notation H^{r}(X) for H^{r}(X, ℚ). The bilinear form on H^{∗}(X) := ⊕H^{r}(X) defined by (α, β) = ∫_{X} α ∧ β is non-degenerate by Poincaré duality. Let ω ∈ H^{2}(X). We define a bilinear form on H^{n−r}(X) by setting

Ψ(α, β) = (−1)^{\frac{(n−r)(n−r−1)}{2}} ∫_{X} ω^{r} ∧ α ∧ β,

for every 0 ≤ r ≤ n. The form Ψ is non-degenerate precisely when the linear map L^{r} = L_{ω}^{r} : H^{n−r}(X) ⟶ H^{n+r}(X), sending α to ω^{r} ∧ α, is an isomorphism.

Definition 2.2.1 We say that (X, ω) has property HL_{r} if the map L_{ω}^{r} : H^{n−r}(X) ⟶ H^{n+r}(X) given by the cup product with ω^{r} is an isomorphism. We say that (X, ω) has property HL if it has property HL_{r} for every 0 ≤ r ≤ n.

Note that property HL_{0} is automatic and that property HL_{n} is equivalent to ∫_{X} ω^{n} ≠ 0. Define H^{n−r}(X) ⊇ P^{n−r} = P_{ω}^{n−r} := Ker L_{ω}^{r+1} and call its elements primitive (with respect to ω). The following Lefschetz-type decomposition is immediate.

Proposition 2.2.2 Assume that (X, ω) has property HL. For every 0 ≤ r ≤ n, we have the following “primitive” decomposition

H^{n−r}(X) = P^{n−r} ⊕ L_{ω}(H^{n−r−2}(X)).

There is a direct sum decomposition

H^{n−r}(X) = ⊕L_{ω}^{i}P^{n−r−2i}.

The subspaces L_{ω}^{i}P^{n−r−2i} are pairwise orthogonal in H^{n−r}(X).

Remark 2.2.3 The projection of H^{n−r}(X) onto P^{n−r} is given by α → α−L_{ω}(L_{ω}^{r+2})^{−1}L_{ω}^{r+1}α, where (L_{ω}^{r+2})^{−1} denotes the inverse to (L_{ω}^{r+2}) : H^{n−r−2}(X) → H^{n+r+2}(X).

Definition 2.2.4 A rational Hodge structure of pure weight r is a rational vector space H with a bigraduation of H_{ℂ} = H ⊗ ℂ = ⊕H^{p,q} for p + q = r such that H^{p,q} = \overline{H^{q,p}}.

Definition 2.2.5 A polarization of the weight r Hodge structure H is a bilinear form Ψ on H, symmetric for r even, anti-symmetric for r odd, such that its ℂ-bilinear extension, to H_{ℂ}, still denoted by Ψ, satisfies :

a) the spaces H^{p,q} and H^{s,t} are Ψ-orthogonal whenever either p ≠ t, or q ≠ s;

b) i^{p−q}Ψ(α, \overline{α}) > 0, for every non-zero α ∈ H^{p,q}.

Let X be a nonsingular complex projective variety. For any ample line bundle M define L_{M} := L_{c_{1}(M)} : H^{n−r}(X) ⟶ H^{n−r+2}(X). Classical Hodge theory gives that

Ψ_{M}(α, β) = (−1)^{\frac{(n−r)(n−r−1)}{2}} ∫_{X} L_{M}^{r}(α) ∧ β is a polarization of the weight (n − r) Hodge structure P_{M}^{n−r} = KerL_{M}^{r+1} ⊆ H^{n−r}(X).

We say that (X, M) has property HL_{r} (HL, resp.) if (X, c_{1}(M)) has property HL_{r} (HL, resp.).

The HL property for a pair (X, M) with X projective and M nef implies that Ψ_{M} is a polarization. In fact, the first Chern class of such a line bundle can be written as a limit of rational Kähler classes and the following proposition applies.

Proposition 2.2.6 Let X be a compact connected complex Kähler manifold of dimension n and M be a line bundle such that (X, M) has property HL and c_{1}(M) = lim_{i→∞} ω_{i}, ω_{i} Kähler. The bilinear form Ψ_{M}(α, β) = (−1)^{\frac{(n−r)(n−r−1)}{2}} ∫_{X} L_{M}^{r}(α) ∧ β is a polarization of the weight (n − r) Hodge structure P_{M}^{n−r} = KerL_{M}^{r+1} ⊆ H^{n−r}(X), for every 0 ≤ r ≤ n.

Proof. The only thing that needs to be proved is the statement i^{p−q}Ψ_{M}(α, \overline{α}) > 0 for every non-zero α ∈ P_{M}^{n−r} ∩ H^{p,q}(X). Since the classes ω_{i} are Kähler, we have the decomposition H^{n−r}(X) = P_{ω_{i}}^{n−r}⊕L_{ω_{i}}(H^{n−r−2}(X)). Let π_{i} denote the projection onto P_{ω_{i}}^{n−r},

π_{i}(α) = α − L_{ω_{i}}(L_{ω_{i}}^{r+2})^{−1}L_{ω_{i}}^{r+1}α.

Since M satisfies the HL condition, the map L_{M}^{r+2} : H^{n−r−2}(X) → H^{n+r+2}(X) is invertible so that lim_{i→∞} (L_{ω_{i}}^{r+2})^{−1} = (L_{M}^{r+2})^{−1}. Identical considerations hold for the (p, q)-parts of these invertible maps. It follows that, if α ∈ P_{M}^{n−r}∩H^{p,q}(X), then lim_{i→∞} π_{i}(α) = α. Since the operators π_{i} are of type (0, 0), π_{i}(α) ∈ P_{ω_{i}}^{n−r}∩H^{p,q}(X). Therefore, i^{p−q}Ψ_{ω_{i}}(π_{i}(α), \overline{π_{i}(α)}) > 0. It follows that i^{p−q}Ψ_{M}(α, \overline{α}) ≥ 0. The HL property for M implies that Ψ_{M} is nondegenerate, therefore Ψ_{M} is a polarization of P_{M}^{n−r}. □

The following elementary fact highlights the connection between the HL property and lef line bundles.

Proposition 2.2.7 Let f : X → Y be a surjective projective morphism from a nonsingular projective variety X, A be a line bundle on Y and M := f^{∗}A. If M has property HL, then f is semismall.

Proof. If f is not semismall, then there exists an irreducible subvariety T ⊆ X such that 2 dim T − n > dim f(T). Let [T] ∈ H^{2(n−dim T)}(X) = H^{n−(2 dim T −n)}(X) be the fundamental class of T. The class c_{1}(M)^{2 dim T −n} can be represented by a ℚ-algebraic cycle that does not intersect T. It follows that c_{1}(M)^{(2 dim T −n)} · [T] = 0, i.e. M does not satisfy HL_{2 dim T −n}. □

Remark 2.2.8 A related fact, dealing with extremal ray contractions, can be found in [21].

2.3 The Hard Lefschetz Theorem and the signature of intersection forms

Our goal is to prove the following extension of the classical Hard Lefschetz Theorem which also constitutes a converse to Proposition 2.2.7. At the same time we prove that the Hodge-Riemann Bilinear Relations hold on the corresponding primitive spaces.

Theorem 2.3.1 Let X be a nonsingular complex projective variety and M be a lef line bundle on X. The pair (X, M) has property HL. In addition, Ψ_{M} is a polarization of P_{M}^{n−r} = Ker L_{M}^{r+1}.

Remark 2.3.2 Proposition 2.2.2 implies the decomposition of the singular cohomology of X into subspaces which are primitive with respect to M. It is immediate to check that dim_{ℂ} P_{M}^{l} = b_{l} − b_{l−2} and dim_{ℂ} P_{M}^{p+q} ∩ H^{p,q}(X) = h^{p,q}(X) − h^{p−1,q−1}(X).

The proof of Theorem 2.3.1 can be found below, after some preparatory remarks. A second proof, based on a variation of Deligne’s proof of the Hard Lefschetz Theorem, see [10], can be found at the end of §2.3.1.

We shall use induction on dim X. Note that the case n = 1 is classical, for M is then necessarily ample. The statement is invariant under taking non-zero positive powers of the line bundle M. Withouth loss of generality we assume that dim X ≥ 2 and that M is generated by its global sections. Let s ∈ H^{0}(X, M) be a section with smooth zero locus Y such that M_{|Y} is lef. Such a section exists by Proposition 2.1.7. Note that Y is necessarily connected by Bertini Theorem. Denote by i : Y → X the inclusion. Let L̂_{M}^{r} := (L_{M_{|Y}})^{r} : H^{n−1−r}(Y) → H^{n−1+r}(Y). The projection formula, coupled with Poincaré Duality, implies that L_{M}^{r} = i_{∗} ∘ L̂_{M}^{r−1} ∘ i^{∗}.

Lemma 2.3.3 If (Y, M_{|Y}) has property HL, then (X, M) has properties HL_{r} for r = 0 and 2 ≤ r ≤ n.

The pair (X, M) has property HL_{1} if and only if the restriction of the intersection form on H^{n−1}(Y) to the subspace i^{∗}H^{n−1}(X) is non-degenerate. If (Y, M_{|Y}) has property HL, then (X, M) has property HL if and only if the restriction of the intersection form on H^{n−1}(Y) to the subspace i^{∗}H^{n−1}(X) is non-degenerate.

Proof. The case r = 0 is trivially true. Let 2 ≤ r ≤ n. The first assertion follows from Proposition 2.1.5, the assumption that property HL holds for (Y, M_{|Y}) and the formula L_{M}^{r} = i_{∗} ∘ L̂_{M}^{r−1} ∘ i^{∗}. Let us prove the second statement. The map L_{M} : H^{n−1}(X) ⟶ H^{n+1}(X) is the composition of the injective map i^{∗} : H^{n−1}(X) ⟶ H^{n−1}(Y) and its surjective transpose i_{∗} : H^{n−1}(Y) ⟶ H^{n+1}(X). It is an isomorphism iff i^{∗}H^{n−1}(X) ∩ Ker i_{∗} = { 0 }, i.e. iff i^{∗}H^{n−1}(X) ∩ (i^{∗}H^{n−1}(X))^{⊥} = { 0 }. This latter statement is equivalent to the nondegeneracy of the restriction of the intersection form. The third assertion follows. □

Proof of Theorem 2.3.1. Assume that Theorem 2.3.1 holds in dimension dim X − 1. By Lemma 2.3.3 it is enough to show the statement of non-degeneracy on i^{∗}H^{n−1}(X). Consider i_{∗} : H^{n−1}(Y) → H^{n+1}(X). We have (i^{∗}H^{n−1}(X))^{⊥} = Ker i_{∗} ⊆ P_{M_{|Y}}^{n−1}(Y). By induction this last space is polarized by the intersection form. In particular, the intersection form is non-degenerate on (i^{∗}H^{n−1}(X))^{⊥} = Ker i_{∗} so that it is non-degenerate on i^{∗}H^{n−1}(X). It follows that (X, M) has property HL. We conclude by Proposition 2.2.6.

□

2.3.1 The use of pencils

Assume that dim X ≥ 2 and that M is generated by its global sections.

Choose a pencil l ⊆ |M| meeting the set W of Proposition 2.1.7. Two general elements u_{0} and u_{∞} ∈ l ∩ W meet transversally along a nonsingular submanifold A. Let Bl_{A} : X′ → X be the blowing-up of X along A. We obtain a nonsingular variety X′ and a proper and flat morphism f : X′ → l ≃ ℙ^{1}. Let U be the Zariski open and dense subset l ∩ W. The morphism f is smooth over U. For every u ∈ U, the fiber f^{−1}(u) is naturally isomorphic to the divisor Y_{u} corresponding to u ∈ U ⊆ l and, in particular, M_{|Y_{u}} is lef for every u ∈ U. We denote Y_{u_{0}} simply by Y and we denote the strict transform of Y on X′ by Y′. Clearly, Bl_{A}_{|Y′} : Y′ → Y is an isomorphism. Let j : U → ℙ^{1} denote the open imbedding and i′ : Y′ → X′ and i′′ : Y′ → f^{−1}(U) the closed imbeddings. The sheaves j^{∗}R^{q}f_{∗}ℚ_{X′} are local systems on U, for every q ≥ 0, corresponding to the π_{1}(U, u_{0})-module H^{q}(Y′). We drop the base point u_{0}.

Theorem 2.3.4 Let q ≤ n − 1. The natural map (Bl_{A} ∘ i′)^{q} : H^{q}(X) → H^{q}(Y′) is injective. The image (Bl_{A} ∘ i′)^{q}(H^{q}(X)) = H^{q}(Y′)^{π(U)}, the subspace of monodromy invariants. The π_{1}(U)-module H^{q}(Y′) is semisimple.

Proof. We have that Bl_{A} ∘ i′ = i ∘ Bl_{A}_{|Y′}. Since Bl_{A}_{|Y′} is an isomorphism, the first statement follows from Proposition 2.1.5. The subspace (Bl_{A} ∘ i′)^{q}(H^{q}(X)) ⊆ H^{q}(Y′) coincides with i^{′q}(H^{q}(X′)); see SGA VII,2, XVIII, Corollaire 5.1.6. The two subspaces of H^{q}(Y′), i^{′q}(H^{q}(X′)) and i^{′′q}(H^{q}(f^{−1}(U))) coincide; see [8], Corollaire 3.2.18. By the degeneration criterion of [7], we have the surjection H^{q}(f^{−1}(U)) → H^{0}(U, j^{∗}R^{q}f_{∗}ℚ_{X′}) ≃ H^{q}(Y′)^{π_{1}(U)}, where the last isomorphism is well-known, see [18], Lemma 5.3 for example. The restriction map i^{′∗} : H^{q}(X′) ⟶ H^{q}(Y′) factors as:

H^{q}(X′) ⟶ H^{q}(f^{−1}(U)) ⟶ H^{0}(U, j^{∗}R^{q}f_{∗}ℚ_{X′}) ⟶ H^{q}(Y′)^{π_{1}(U)} ⟶ H^{q}(Y′),

and the second statement follows. The last one follows from [8], Théorème 4.2.6 and footnote 1. □

Second Proof of Theorem 2.3.1. We prove the theorem by induction on dim X = n. As in the first proof, the statement is true when dim X = 1, we assume that the statement is true in dimension n − 1 and we replace, without loss of generality, M by a positive power which is generated by its global sections. We fix a pencil l as above. By Theorem 2.3.4, we can find a π_{1}(U)-invariant complement V to H^{n−1}(Y′)^{π_{1}(U)} in H^{n−1}(Y′). The Poincaré duality pairing on H^{n−1}(Y′) is π_{1}(U)- invariant, therefore it restricts to a non-degenerate pairing on H^{n−1}(Y′)^{π_{1}(U)}. This latter space is naturally identified with i^{∗}H^{n−1}(X). We conclude that (X, M) has the property HL by virtue of the third part of Proposition 2.3.3. Finally, we conclude by Proposition 2.2.6. □

2.4 The Hodge Index Theorem for semismall maps

Let us record the following consequence of Theorem 2.3.1. Coupled with Remark 2.4.5, it is a higher dimensional analogue of Grauert-Mumford contractibility test for curves on surfaces.

Theorem 2.4.1 (Hodge Index Theorem for semismall maps) Let f : X → Y be a semismall map from a nonsingular complex projective variety of even dimension n onto a projective variety Y and y ∈ Y be a point such that dim f^{−1}(y) = \frac{n}{2}. Denote by Z_{l}, 1 ≤ l ≤ r, the irreducible components of maximal dimension of f^{−1}(y). Then the cohomology classes [Z_{l}] ∈ H^{n}(X) are linearly independent and the symmetric matrix (−1)^{\frac{n}{2}} ||Z_{l} · Z_{m}|| is positive definite.

Proof. Let M = f^{∗}A be a lef line bundle, pull-back of an ample line bundle A on Y. The image of the cycle class map cl : H_{n}^{BM}(f^{−1}(y)) ⟶ H^{n}(X) belongs to the primitive space P_{M}^{n}(X) which is polarized by the intersection form by virtue of Theorem 2.3.1. We are left with showing that the map cl is injective.

Because of Poincarè Duality, the map cl is the transposed of the natural restriction map r : H^{n}(X) ⟶ H^{n}(f^{−1}(y)). We now prove that r is surjective. Let Y^{0} be an affine neighborhood of y ∈ Y such that Supp (R^{n}f_{∗}ℚ_{X})_{Y^{0}} = {y}. The theorem on the cohomological dimension of constructible sheaves on affine varieties, coupled with the assumption “f is semismall” gives that H^{p}(Y^{0}, (R^{q}f_{∗}ℚ_{X})_{|Y^{0}}) = {0}, for every p + q > n; see [12], Lemma 1.2 for example. The Leray Spectral sequence gives the surjection:

H^{n}(f^{−1}(Y^{0})) ⟶ (R^{n}f_{∗}ℚ_{X})_{y} ≃ H^{n}(f^{−1}(y)).

By [9], Proposition 8.2.6, we conclude that the map r is surjective as well. □

Remark 2.4.2 We sketch a second argument for the injectivity of cl. Choose a general pencil l of divisors on X as in §2.3.1 with the additional property that the natural restriction map H^{n−1}(X \ f^{−1}(y)) → H^{n−1}(X_{v}) is injective for a general member X_{v} of l; see [15], II.1.1. By Theorem 2.3.4, we have the isomorphism H^{n−1}(X) ≃ H^{n−1}(X_{v})^{π_{1}(V)}, where V is an appropriate Zariski dense open subset of l. This isomorphism factors as follows

H^{n−1}(X) →^{a} H^{n−1}(X \ f^{−1}(y)) ⟶^{b} H^{n−1}(X_{v})^{π_{1}(V)} ⊆ H^{n−1}(X_{v}).

Hence the injective maps b and a are isomorphisms. It follows that the natural cycle class map H_{n}^{BM}(f^{−1}(y)) ≃ H^{n}(X, X \ f^{−1}(y)) → H^{n}(X) is injective.

Corollary 2.4.3 Let f : X → Y and y be as above. The mixed Hodge Structure on H^{n−1}(X \ f^{−1}(y)) is pure of weight n − 1.

Proof. It follows from the surjectivity of the natural map H^{n−1}(X) → H^{n−1}(X \ f^{−1}(y)). □

Corollary 2.4.4 Let f : X → Y be a birational semismall map from a nonsingular quasi projective complex variety of even dimension n onto a quasi projective complex variety with an isolated singularity y ∈ Y such that f is an isomorphism outside y and dim f^{−1}(y) = \frac{n}{2}. Then the conclusions of Theorem 2.4.1 hold.

Proof. One finds a semismall projective completion f′′ : X′′ → Y′′ of f to which we apply Theorem 2.4.1. Since the bilinear form on the fibers in non-degenerate, the cycle classes of the fundamental classes of the fibers stay independent in H^{n}(X). □

Remark 2.4.5 The same proof shows that the statements of Theorem 2.4.1 and Corollary 2.4.4 holds unchanged if we consider any finite number of fibers over points as y above.

Remark 2.4.6 If (Y, y) is a germ of a normal complex space of dimension two, and f : X → Y is a resolution of singularities, then Grauert Contractibility Criterion, see [17] Theorem 4.4, implies that the form in question is non-degenerate and negative definite.

The following is a natural question. A positive answer would yield a proof of the Decomposition Theorem for semismall holomorphic maps from complex manifolds and for constant coefficients; see Theorem 3.3.3.

Question 2.4.7 Let f : V → W be a proper holomorphic semismall map from a complex manifold of even dimension n onto an analytic space Y. Assume that the fiber f^{−1}(w) over a point w ∈ W has dimension \frac{n}{2}. Is the intersection form on H_{n}^{BM}(f^{−1}(w)) non-degenerate? Is it (−1)^{\frac{n}{2}} -positive definite?

3 The topology of semismall maps

We now proceed to a study of holomorphic semismall maps from a complex manifold. First we need to prove Proposition 3.1.2, a simple splitting criterion in derived categories for which we could not find a reference. We study the topology of these maps by attaching one stratum at the time. In doing so a symmetric bilinear form emerges naturally; see Proposition 3.2.4 and Lemma 3.2.5. We then prove that the Decomposition Theorem for these maps and for constant coefficients is equivalent to the non-degeneration of these forms; Theorem 3.3.3. Finally, we give a proof of the Decomposition Theorem when the domain and target are projective, Theorem 3.4.1. A new feature that we discover is that the forms are definite by virtue of our Hodge Index Theorem for semismall maps.

3.1 Homological algebra

Let A be an abelian category with enough injectives, e.g. sheaves of abelian groups on a topological space, and C(A) be the associated category of complexes. Complexes and morphisms can be truncated. Given an integer t, we have two types of truncations: τ_{≤t}A and τ_{≥t}A. The former is defined as follows (τ_{≤t}A)^{i} := A^{i} for i ≤ t − 1, (τ_{≤t}A)^{t} := Ker(A^{t} → A^{t+1}), (τ_{≤t}A)^{i} := {0} for i > t. The latter is defined as follows (τ_{≥t}A)^{i} := {0} for i ≤ t − 1, (τ_{≥t}A)^{t} := Coker(A^{t−1} → A^{t}), (τ_{≥t}A)^{i} := A^{i} for i > t. Let h : A → B be a morphism of complexes. The truncations τ_{≤t}(h) : τ_{≤t}A → τ_{≤t}B and τ_{≥t}(h) : τ_{≥t}A → τ_{≥t}B are defined in the natural way. The operations of truncating complexes and morphisms of complexes induce functors in the derived category D(A).

If A is a complex acyclic in degrees l ≠ t for some integer t, i.e. if τ_{≤t}A ≃ τ_{≥t}A, then A ≃ H^{t}(A)[−t].

The cone construction for a morphism of complexes h : A → B gives rise, in a nonunique way, to a diagram of morphism of complexes A →^{h} B → M(h) →^{[1]} A[1]. A diagram

of morphisms X → Y → Z →^{[1]} X[1] in D(A) is called a distinguished triangle if it is isomorphic to a diagram arising from a cone.

A morphism h : A → B in D(A) gives rise to a distinguished triangle A →^{h} B → C → A[1]. If h = 0, then C ≃ A[1] ⊕ B and the induced morphism A[1] → A[1] is an isomorphism.

A morphism h : A → B in the derived category gives a collection of morphisms in cohomology H^{l}(h) : H^{l}(A) → H^{l}(B). A distinguished triangle A → B → C →^{[1]} A[1] gives rise to a cohomology long exact sequence:

… H^{l}(A) → H^{l}(B) → H^{l}(C) → H^{l+1}(A) …

A non-zero morphism h : A → B in the derived category may nonetheless induce the zero morphisms between all cohomology groups. However, we have the following simple and standard

Lemma 3.1.1 Let t be an integer and A and B be two complexes such that A ≃ τ_{≤t}A and B ≃ τ_{≥t}B. Then the natural map Hom_{D(A)}(A, B) → Hom_{A}(H^{t}(A), H^{t}(B)) is an isomorphism of abelian groups.

Proof. It is enough to replace B by an injective resolution placed in degrees no less than t. □

We shall need the following elementary splitting criterion.

Proposition 3.1.2 Let C ⟶^{u} A ⟶^{v} B ⟶^{[1]} C[1] be a distinguished triangle and t be an integer such that A ≃ τ_{≤t}A and C ≃ τ_{≥t}C. Then H^{t}(u) : H^{t}(C) → H^{t}(A) is an isomorphism iff

A ≃ τ_{≤t−1}B ⊕ H^{t}(A)[−t]

and the map v is the direct sum of the natural map τ_{≤t−1}B → B and the zero map.

Proof. Assume that H^{t}(u) is an isomorphism. Apply the functor Hom(A, −) to the

distinguished triangle τ_{≤t−1}B ⟶^{ν_{t−1}} τ_{≤t}B ⟶^{π} H^{t}(B)[−t] ⟶^{[1]} τ_{≤t−1}B [1] and we get the following exact sequence:

… → Hom^{−1}(τ_{≤t}A, H^{t}(B)[−t]) → Hom^{0}(τ_{≤t}A, τ_{≤t−1}B) →
Hom^{0}(τ_{≤t}A, τ_{≤t}B) → Hom^{0}(τ_{≤t}A, H^{t}(B)[−t])) → …

Since H^{t}(B)[−t] is concentrated in degree t, Hom^{−1}(τ_{≤t}A, H^{t}(B)[−t]) = {0}. The morphism H^{t}(v) = 0, for H^{t}(u) is surjective. It follows that there exist a unique lifting v′ of τ_{≤t}(v), i.e. there exists a unique v′ : A → τ_{≤t−1}B such that τ_{≤t}(v) = ν_{t−1} ∘ v′. We complete v′ to a distinguished triangle:

τ_{≤t}A ⟶^{v′} τ_{≤t−1}B ⟶^{v′′} M(v′) ⟶^{[1]} τ_{≤t}A [1] .

By degree considerations, the morphism H^{l}(v′) = 0 for l ≥ t. Since v′ is a lifting of τ_{≤t}(v), the morphism H^{l}(v′) is an isomorphism for l ≤ t − 1 and it is the zero map for l ≥ t. This implies that M(v′) ≃ H^{t}(A)[−t + 1] and that H^{t−1}(v′′) = 0. By virtue of Lemma 3.1.1, we get that v′′ = 0. The desired splitting follows. The converse can be read off the long exact cohomology sequence. □

3.2 The bilinear forms associated with relevant strata

Let f : X → Y be a proper holomorphic semismall map with X nonsingular connected of dimension n. Let us summarize the results from stratification theory (cf. [15], Ch. 1) that we shall need in the sequel. They are based essentially on Thom First Isotopy Lemma.

There exists a collection of disjoint locally closed and connected analytic subvarieties Y_{i} ⊆ Y such that:

a) Y = ∐_{i} Y_{i} is a Whitney stratification of Y.

b) Y_{i} ∩ \overline{Y_{j}} ≠ ∅ iff Y_{i} ⊆ \overline{Y_{j}}.

c) the induced maps f_{i} : f^{−1}(Y_{i}) → Y_{i} are stratified submersions; in particular they are topologically locally trivial fibrations.

We call such data a stratification of the map f.

Definition 3.2.1 A stratum Y_{i} is said to be relevant if 2 dim f^{−1}(Y_{i}) − dim Y_{i} = n. Let I′ ⊆ I be the set of indices labeling relevant strata.

Let i ∈ I be any index and d_{i} := dim Y_{i}. Define L_{i} := (R^{n−d_{i}}f_{∗}ℚ_{X})_{|Y_{i}}. It is a local system on Y_{i}.

Remark 3.2.2 If Y_{i} is not relevant, then L_{i} is the zero sheaf. If Y_{i} is relevant, then the stalks (L_{i}^{∗})_{y_{i}} ≃ H_{n−d_{i}}^{BM}(f^{−1}(y_{i})) of the dual local system are generated exactly by the fundamental classes of the irreducible and reduced components of maximal dimension of the fiber over y_{i}.

The following is elementary and holds also when the stratum is not relevant when we consider the local system dual to the one generated by the components of maximal dimension of the fibers.

Lemma 3.2.3 The local system L_{i} splits as a direct sum L_{i} ≃ ⨁_{j=1}^{m_{i}} L_{ij} of irreducible local sub-systems.

Proof. It is enough to show the statement for L_{i}^{∗}, i ∈ I′. Going around a loop in Y_{i} has the effect of permuting the elements of the basis of Remark 3.2.2. The associated monodromy representation factors through a finite symmetric group so that it splits into a sum of irreducibles. □

Let S := Y_{i}, d := dim S and L_{S} := L_{i}. We now proceed to associating with S a symmetric bilinear form on the local system L_{S}^{∗}.

Let s ∈ S and choose a small-enough euclidean neighborhood U of s in Y such that a) S′ := S ∩ U is contractible and b) the restriction i^{∗} : H^{n−d}(f^{−1}(U)) → H^{n−d}(f^{−1}(s)) is an isomorphism.

Let F_{1}, …, F_{r} be the irreducible and reduced components of maximal dimension of f^{−1}(S′). By virtue of a) above and of the topological triviality over S′, the intersections f_{j} := f^{−1}(s) ∩ F_{j} are exactly the irreducible and reduced components of maximal dimension of f^{−1}(s). The analogous statement is true for every point s′ ∈ S′ and the components for the point s can be canonically identified with the ones of s′. The specialization morphism i_{s}^{!} : H_{n+d}^{BM}(f^{−1}(S′)) → H_{n−d}^{BM}(f^{−1}(s)), associated with the regular imbedding i_{s} : {s} → S′, sends the fundamental class of a component F_{l} to the fundamental class of the corresponding f_{l} and it is an isomorphism; see [13], Ch. 10. We have the following sequence of maps:

H_{n−d}^{BM}(f^{−1}(s)) ⟶^{(i_{s}^{!})^{−1}} H_{n+d}^{BM}(f^{−1}(S′)) ⟶^{(∩μ_{f^{−1}(U)})^{−1}} H^{n−d}(f^{−1}(U), f^{−1}(U \ S′))

→^{nat} H^{n−d}(f^{−1}(U)) ⟶^{i^{∗}} H^{n−d}(f^{−1}(s)) ⟶^{κ} H_{n−d}^{BM}(f^{−1}(s))^{∗}.

The second map is the inverse to the isomorphism given by capping with the fundamental class μ_{f^{−1}(U)} (cf. [16], IX.4). The third map is the natural map in relative cohomology. The fourth map is an isomorphism by virtue of condition b) above. The map κ is an isomorphism by the compactness of f^{−1}(s).

We denote the composition, which is independent of the choice of U:

ρ_{S,s} : H_{n−d}^{BM}(f^{−1}(s)) ⟶ H_{n−d}^{BM}(f^{−1}(s))^{∗}.

We have that

ρ_{S,s}(f_{h})(f_{k}) = deg F_{h} · f_{k},

where the refined intersection product takes place in f^{−1}(U) and has values in H_{0}^{BM}(f^{−1}(s)).

Since the map f is locally topologically trivial along S, the maps ρ_{S,s} define a map of local systems

ρ_{S} : L_{S}^{∗} ⟶ L_{S}.

We record the following fact for future use.

Proposition 3.2.4 If S is not relevant, then ρ_{S} is the zero map between trivial local systems. Let s ∈ S be a point. The map ρ_{S,s} is an isomorphism iff the natural map r_{k} : H^{k}(f^{−1}(U)) → H^{k}(f^{−1}(U \ S′)) is an isomorphism for every k ≤ n − d − 1, iff the natural map s_{k} : H^{k}(f^{−1}(U \ S′)) → H^{k+1}(f^{−1}(U), f^{−1}(U \ S′)) is an isomorphism for every k ≥ n − d.

Proof. The domain and the range of ρ_{S,s} are dual to each other. The statement follows from the relative cohomology sequence for the pair (f^{−1}(U), f^{−1}(U \S′)), the isomorphisms H^{k}(f^{−1}(U)) ≃ H^{k}(f^{−1}(s)), H^{k}(f^{−1}(U), f^{−1}(U \ S′)) ≃ H_{2n−k}^{BM}(f^{−1}(S′)) and the fact that dim f^{−1}(s) ≤ \frac{n−d}{2}, dim f^{−1}(S) ≤ \frac{n+d}{2}. □

Since f_{i} : f^{−1}(S) → S is a stratified submersion, given any point s ∈ S, we can choose an analytic normal slice N(s) to S at s such that f^{−1}(N(s)) is a locally closed complex submanifold of X of dimension n − d. We now use this fact to express the map ρ_{S,s} in terms of the refined intersection pairing on f^{−1}(N(s)).

Lemma 3.2.5 If s ∈ S, then ρ_{S,s}(f_{h})(f_{k}) = deg f_{h} · f_{k}, where the refined intersection product on the r.h.s. takes place in f^{−1}(N(s)) and has values in H_{0}^{BM}(f^{−1}(s)). In particular, the map ρ_{S} : L_{S}^{∗} ⟶ L_{S} is symmetric.

Proof. Since f_{i} : f^{−1}(S) → S is a stratified submersion and N(s) is a normal slice to S at s, F_{j} meets f^{−1}(N(s)) transversally at the general point of f_{j}. It follows that the refined intersection product f^{−1}(N(s)) · F_{j} is the fundamental class of f_{j} in H_{n−d}^{BM}(f^{−1}(s)). The result follows by applying [13], 8.1.1.a) to the maps f^{−1}(s) → f^{−1}(N(s)) → f^{−1}(U). □

3.3 Inductive study of semismall analytic maps

Let f : X → Y and {Y_{j}}, i ∈ I, S be as in section 3.2. We assume, for simplicity, the Y_{i} to be connected and I to be finite. There is no loss of generality, for strata of the same dimension do not interfere with each other from the point of view of the analysis that follows and could be treated simultaneously. As usual, we define a partial order on the index set I by setting i ≺ j iff Y_{i} ⊆ \overline{Y_{j}}. We fix a total order I = \{i_{1} < … < i_{ι}\} which is compatible with the aforementioned partial order and define the open sets U_{≥i} := ∐_{j≥i} Y_{j}. Similarly, U_{>i} := ∐_{j>i} Y_{j}. Let α_{i} : U_{>i} → U_{≥i} be the open imbedding. We can define the intermediate extension of a complex of sheaves K^{•} on U_{>i} to a complex of sheaves on U_{≥i} by setting

α_{i}_{!∗}K^{•} = τ_{≤−dimY_{i}−1}Rα_{i}_{∗}K^{•}.

See [2]. The construction is general and can be iterated so that one can form the intermediate extension of a complex of sheaves on any Y_{i} to a complex on \overline{Y_{i}} ∩ U_{>j} for j < i. In particular, let L be a local system on Y_{i}. The intermediate extension of L[dim Y_{i}] to \overline{Y}_{i} ∩ U_{>j} for j < i is called the intersection cohomology complex associated with L and is denoted by IC_{\overline{Y}_{i}∩U_{>j}}(L).

Definition 3.3.1 Let f : X → Y be a proper holomorphic semismall map from a nonsingular connected complex manifold X of dimension n. We say that the Decomposition Theorem holds for f if there is an isomorphism

Rf_{∗}ℚ_{X}[n] ≃ ⨁_{k∈I′} IC\overline{Y_{k}}(L_{k}) ≃ ⨁_{k∈I′} ⨁_{m=1}^{m_{k}} IC\overline{Y_{k}}(L_{km}),

where the L_{km} are as in Lemma 3.2.3.

Remark 3.3.2 The Decomposition Theorem holds, in the sense defined above, for X, Y and f algebraic (cf. [2]) and for f a Kähler morphism (cf. [20]). In both cases, a far more general statement holds. As observed in [4], §1.7, in the case of semismall maps these results can be expressed in the convenient form of Definition 3.3.1.

We now proceed to show that the non-degeneracy of the forms ρ_{S} associated with the strata Y_{i} implies the Decomposition Theorem.

Recall that I′ ⊆ I is the subset labeling relevant strata. For ease of notation set

V := U_{>i},  V′ := U_{≥i},  S := Y_{i}

and let

V ⟶^{α} V′ ←^{β} S

be the corresponding open and closed imbeddings.

Theorem 3.3.3 Assume that the Decomposition Theorem holds over V. The map ρ_{S} is an isomorphism iff the Decomposition Theorem holds over V′ and the corresponding isomorphism restricts to the given one over V.

Proof. Denote by g the map f_{|} : f^{−1}(V) → V. By cohomology and base change, (Rf_{∗}ℚ_{X}[n])_{|V} ≃ Rg_{∗}ℚ_{f^{−1}(V)}[n]. Similarly for V′. Clearly, we have (Rf_{∗}ℚ_{X}[n])_{|V} ≃ α^{∗}[(Rf_{∗}ℚ_{X}[n])_{|V′}]. There is a distinguished “attaching” triangle, see [3], 5.14:

β_{∗}β^{!}(Rf_{∗}ℚ_{X}[n]_{|V′}) ⟶^{u} (Rf_{∗}ℚ_{X}[n])_{|V′} ⟶^{v} Rα_{∗}(Rf_{∗}ℚ_{X}[n]_{|V}) ⟶^{w [1]} β_{∗}β^{!}(Rf_{∗}ℚ_{X}[n]_{|V′})[1].

On the open set V the complex β_{∗}β^{!}(Rf_{∗}ℚ_{X}[n])_{|V′} is isomorphic to zero and the map v restricts to an isomorphism. Recalling the notation in §3.2, the long exact sequence of cohomology sheaves is, stalk-wise along the points of S, the long exact sequence for the cohomology of the pair (f^{−1}(U), f^{−1}(U \ S′). In addition the map H^{−d}(u) is identified, stalk-wise along the points of S, with the map ρ_{S,s}. The statement follows from Proposition 3.2.4 which allows us to apply proposition 3.1.2. □

Remark 3.3.4 In the algebraic and Kähler case, the results [2] and [20], coupled with Theorem 3.3.3, imply that the forms ρ_{S} are non-degenerate for every i ∈ I; see also [5], Theorem 8.9.14. To our knowledge these results have no implications as to the sign of the intersection forms. Surprisingly, in the projective case we can determine that these forms are definite; see §3.4.

Remark 3.3.5 If one can prove that intersection forms ρ_{S} are non-degenerate, then the Decomposition Theorem follows. Two non-trivial examples are: Hilbert schemes of points on a surface and moduli spaces of semistable sheaves on surfaces; see [11] and [1].

The following statement is known as a consequence of the characterizing properties of intersection cohomology sheaves; see [14], §6.2 and [4], §1.8. Recall that a map is small if all but one the inequalities in the definition of semismall are strict. Note that in this case there is only one relevant stratum.

Corollary 3.3.6 Let f : X → Y be a proper holomorphic small map from a connected complex manifold. Then the Decomposition Theorem holds for f.

Proof. All the maps ρ_{S} are identically zero and are isomorphisms. □

3.4 Signature and Decomposition Theorem in the projective case

In this section we use Theorem 2.4.1, the previous inductive analysis and a Bertini-type argument to give a proof of the following theorem

Theorem 3.4.1 Let f : X → Y be a semismall map from a nonsingular complex projective variety of dimension n onto a complex projective variety. The Decomposition Theorem holds for f, i.e. there is a canonical isomorphism

Rf_{∗}ℚ_{X}[n] ≃ ⨁_{k∈I′} IC\overline{Y_{k}}(L_{k}) ≃ ⨁_{k∈I′} ⨁_{m=1}^{m_{k}} IC\overline{Y_{k}}(L_{km}).

For every relevant stratum S of dimension d the associated intersection form is nondegenerate and (−1)^{\frac{n−d}{2}} -definite.

Proof. By virtue of Proposition 3.3.3 we are reduced to checking that the intersection form associated with a relevant d-dimensional stratum S is non-degenerate and (−1)^{\frac{n−d}{2}} - definite. If d = 0, then the conclusion follows from Proposition 2.4.1. Let d > 0. Let A be a very ample divisor on Y. The line bundle M := f^{∗}A is lef and generated by its global sections. By virtue of Proposition 2.1.7, we can choose d general sections H_{1}, …, H_{d} in the linear system |A| such that their common zero locus H has the property that f^{−1}(H) is nonsingular of dimension n − d, f^{−1}(H) → H is semismall, H meets S at a non-empty finite set of points s_{1}, …, s_{r} so that, for at least one index 1 ≤ l ≤ r, a small neighborhood of a point s_{l} in H is a normal slice to S at s_{l}. By virtue of Theorem 2.4.1 the intersection form of f^{−1}(s_{l}) ⊆ f^{−1}(H) has the required properties at the point s_{l}, and therefore at every point s ∈ S. We conclude by applying Lemma 3.2.5.

□

Remark 3.4.2 Theorem 3.4.1 can be applied even when the spaces are not complete, in the presence of a suitable completion of the morphism: one for which the domain is completed to a projective manifold, the target to a projective variety and the map to a semismall one. In general this may not be possible, but it can be done in several instances, e.g. the Springer resolution of the nilpotent cone of a complex semisimple Lie algebra, the Hilbert scheme of points on an algebraic surface mapping on the corresponding symmetric product, isolated singularities (see below), certain contraction of holomorphic symplectic varieties …

Corollary 3.4.3 Let f : X → Y be a birational semismall map from a nonsingular quasi projective complex variety of dimension n onto a quasi projective complex variety Y with isolated singularities. Assume that f is an isomorphism outside the isolated singularities. The Decomposition Theorem holds for f.

Proof. We can reduce the statement to the complete projective case: see Corollary 2.4.4 and Remark 2.4.5. □

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Authors’ addresses:

Mark Andrea A. de Cataldo, Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794, USA. e-mail: mde@math.sunysb.edu

Luca Migliorini, Dipartimento di Matematica, Università di Trento, Via Sommarive, 14, 38050 Povo (Tn), ITALY. e-mail: luca@alpha.science.unitn.it