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On the Mixed Hodge Structures of the Intersection Cohomology Stalks Of On the mixed Hodge structures of the intersection cohomology stalks of complex hypersurfaces TAKAHIRO SAITO Abstract. We consider a hypersurface in Cn with an isolated singular point at the origin, and study the mixed Hodge structure of the stalk of its intersection cohomology complex at the origin. In particular we express the dimension of each graded piece of the weight filtration of this mixed Hodge structure in terms of the numbers of the Jordan blocks in the Milnor monodromy. Contents 1. Introduction 1 Acknowledgments 4 2. Intersection cohomology 4 3. Limit mixed Hodge structures and mixed Hodge modules 6 3.1. Limit mixed Hodge structures 6 3.2. Mixed Hodge modules 9 4. Milnor monodromies and IC stalks 11 4.1. Milnor monodromies 11 4.2. IC stalks and Milnor monodromies 14 5. Main theorem 16 6. Proof of Proposition 5.8 24 References 26 arXiv:1602.02976v2 [math.AG] 10 Feb 2017 1. Introduction In this paper, we reveal a new relationship between the mixed Hodge structures of the stalks of the intersection cohomology complexes (we call them IC stalks for short) and the Milnor monodromies, by using the results on motivic Milnor fibers shown by Matsui-Takeuchi [11] and on motivic nearby fibers by Stapledon [19]. Date: October 8, 2018. 2010 Mathematics Subject Classification. 32C38, 32S35, 32S40, 32S55, 32S60. 1 2 TAKAHIROSAITO For a natural number n 2, let f C[x1,...,xn] be a non-constant polynomial of n variables with≥ coefficients∈ in C such that f(0) = 0. As- sume that f is convenient and non-degenerate at 0 (see Definitions 4.7 and 4.9). We denote by V the hypersurface x Cn f(x)=0 in Cn. Then, it is well-known that 0 V is a smooth{ ∈ or isolated| singular} ∈ point of V . We denote by ICV := ICV (Q) the intersection cohomology complex of V with rational coefficients. This is the underlying perverse H sheaf of the mixed Hodge module ICV . By Morihiko Saito’s theory, the stalk (ICV )0 of ICV at 0 is a complex of mixed Hodge structures. How- ever, to the best of our knowledge, the mixed Hodge structure of (ICV )0 has not been fully studied yet. For a complex C of mixed Hodge struc- W k tures and integers r, k Z, we denote by grr H (C) the r-th graded k ∈ piece of H (C) with respect to the weight filtration W•. For an inte- W k ger w Z, we say that C has a pure weight w if grr H (C) = 0 for any r, k∈ Z with r = k + w. The purity of the weights of IC stalks are important.∈ Kazhdan-Lustzig6 computed the Kazhdan-Lustzig poly- nomials by using the purity of IC stalks of Schubert varieties in flag varieties in [9]. Denef-Loeser proved that if f is quasi-homogeneous, (ICV )0[ (n 1)](=: (ICV )0) has a pure weight 0 in [3] (see Proposi- tion 4.14).− By− using this result, they computed the dimensions of the intersection cohomologyg groups of complete toric varieties. W k In general, (ICV )0 has mixed weights 0, that is grr H ((ICV )0)=0 ≤ W k for r>k. In this paper, we will describe the dimensions of grr H ((ICV )0) very explicitly.g We denote by N0 the dimension of the invariantg sub- space of the (n 1)-st Milnor monodromy Φ0,n−1 of f at 0. Assumingg that n 3, for− any k Z, we have ≥ ∈ 1 if k =0, k dim H ((ICV )0)= N if k = n 2, and 0 − 0 otherwise g n−2 (see Proposition 4.12). Thus, if H ((ICV )0) does not have a pure weight, the dimension N0 is decomposed into those of the graded pieces W n−2 W n−2 grr H ((ICV )0). We shall describe dimg grr H ((ICV )0) in terms of the numbers of the Jordan blocks for the eigenvalue 1 in the (n 1)-st − Milnor monodromyg Φ0,n−1. For a natural number s gZ≥0, we denote 1 ∈ by Js the number of the Jordan blocks in Φ0,n−1 for the eigenvalue 1 with size s. Then our main result is the following. ON THE MHS OF IC STALKS OF COMPLEX HYPERSURFACES 3 Theorem 1.1 (Theorem 5.1). Assume that n 3 and f is convenient and non-degenerate at 0. Then for any r Z,≥ we have ∈ W 0 1 if r =0, dim grr H ((ICV )0)= 0 if r =0, 6 and g W n−2 1 dim grr H ((ICV )0)= Jn−r−1. For the case where n = 2, see Theorem 5.9. In particular, we obtain g the following result on the purity of the IC stalk (ICV )0. Corollary 1.2 (Corollary 5.2). In the situation of Theorem 1.1, the following conditions are equivalent. g (i) The IC stalk (ICV )0 has a pure weight 0. (ii) There is no Jordan block for the eigenvalue 1 with size > 1 of the (n 1)-st Milnorg monodromy Φ0,n−1. − Therefore, the result on the purity of the IC stalks for quasi-homogeneous polynomials by Denef-Loeser (Proposition 4.14) is a special case of our result. Moreover, as a corollary of the above theorem, we obtain a result on the mixed Hodge structures of the cohomology groups of the link of the isolated singular point 0 in V . We denote by L the link of 0 in V , that is, the intersection of V and a sufficiently small sphere centered at 0. Then L is a (2n 3)-dimensional orientable compact real manifold and each cohomology− group Hk(L; Q) of L has a canonical mixed Hodge structure. For any k n 2, the k-th cohomology group Hk(L; Q) ≤ − k is isomorphic (as mixed Hodge structures) to H ((ICV )0). Assuming that n 3, by the Poincar´eduality, we have Hk(L; Q) = 0 for any ≥ k =0, n 2, n 1, 2n 3. Then we obtain the followingg result. 6 − − − Corollary 1.3 (Corollary 5.5). In the situation of Theorem 1.1, for any r Z, we have ∈ W 0 W 2n−3 1 if r =0, dim grr H (L; Q) = dim gr2(n−1)−rH (L; Q)= 0 if r =0, 6 and W n−2 W n−1 1 dim grr H (L; Q) = dim gr2(n−1)−rH (L; Q)= Jn−r−1. We prove Theorem 1.1 by combining the results of Stapledon [19] with that of Matsui-Takeuchi [11]. For this purpose, we deform the hypersurface V to a suitable one which can be compactified in Pn nicely. See Section 6 for the details. 4 TAKAHIROSAITO Acknowledgments. The author would like to express his hearty grat- itude to Professor Kiyoshi Takeuchi for drawing the author’s attention to this problem and the several discussions for this work. His thanks goes also to Yuichi Ike for many discussions and answering to his ques- tions on sheaf theory, to Tatsuki Kuwagaki for answering to his ques- tions on algebraic varieties and toric varieties, and to Tomohiro Asano for discussions on matters in Section 6. 2. Intersection cohomology In this section, we introduce some basic notations and recall in- tersection cohomology theory. We follow the notations of [8] and [7] (see also [4] and [18]). Let X be an algebraic variety over C and K a field. We denote by KX the constant sheaf on X with stalk K b b and by D (KX ) or D (X) the bounded derived category of sheaves of b KX -modules on X. For F D (X) and an integer d Z, we de- note by F [d] the shifted complex∈ of F by the degree d∈. Moreover, we denote by τ ≥dF, τ ≤dF the truncated complexes of F . We denote b b by Dc (X) the full triangulated subcategory of D (X) consisting of complexes whose cohomology sheaves are constructible. For a mor- phism f : X Y of algebraic varieties, one can define Grothendieck’s → L −1 ! six operations Rf∗, Rf!, f , f , and R om as functors of de- rived categories of sheaves. Let g ⊗: X CH be a morphism of al- → gebraic varieties. Denote by C∗ the universal covering of C∗ and by p: C∗ x C Im(x) > 0 C∗, x exp(2π√ 1x) the covering map. Consider≃ { ∈ the| diagram: }f → 7→ − f ′ −1 ∗ π / ∗ (X g (0)) C∗ C C \ × π p f f g−1(0) / Xo ? _ X g−1(0) / C∗ , i j \ g where the maps i and j are the inclusion maps and the square on b b −1 the right is Cartesian. For F Dc (X), we define ψg(F ) Dc (g (0)) as i−1R(j π) (j π)−1F Db∈(g−1(0)) and φ (F ) Db(g∈−1(0)) as the ◦ ∗ ◦ ∈ c g ∈ c mapping cone of the morphism F g−1(0) ψg(F ). Then we obtain the b b −|1 → functors ψg,φg : Dc (X) Dc (g (0)). We call ψg and φg the nearby cycle and the vanishing cycle→ functors, respectively. In this paper, for a K-vector space H, we denote by H∗ the dual K-vector space of H. If X is a purely n-dimensional smooth variety, there exist canonical isomorphisms Hk(X; K) (H2n−k(X; K))∗ ≃ c ON THE MHS OF IC STALKS OF COMPLEX HYPERSURFACES 5 for any k Z by the Poincar´eduality.
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