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CAMBRIDGE TRACTS IN MATHEMATICSGeneral Editors

B. BOLLOBAS, F. KIRWAN, P. SARNAK, C.T.C. WALL

132 Mixed Hodge Structures and Singularities

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Cambridge University Press978-0-521-62060-4 - Mixed Hodge Structures and SingularitiesValentine S. KulikovFrontmatterMore information

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Valentine S. KulikovMoscow State University of Printing

Mixed Hodge Structures and Singularities

CAMBRIDGEUNIVERSITY PRESS






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Mixed Hodge structures and singularities/Valentine S. Kulikov.p. cm. – (Cambridge tracts in mathematics ; 132)

Includes bibliographical references and index. ISBN 0-521-62060-0 (hb)

1. Hodge theory. 2. Singularities (Mathematics) I. Title. II. Series. QA564.K85 1998 516.3’5-dc21 97-11978 CIP

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Contents

Introduction page xi

I The Gauss-Manin connection 11 Milnor fibration, Picard-Lefschetz monodromy transforma-

tion, topological Gauss-Manin connection 11.1 Milnor fibration 11.2 Cohomological Milnor fibration 11.3 Topological Gauss-Manin connection 21.4 Picard-Lefschetz monodromy transformation 2

2 Connections, locally constant sheaves and systems of lineardifferential equations 32.1 Connection as a covariant differentiation 32.2 Equivalent definition: a covariant derivative along a

vector field 42.3 Local calculation of connections. Relation to differen-

tial equations 52.4 The integrable connections. The De Rham complex 62.5 Local systems and integrable connections 72.6 Dual local systems and connections 8

3 De Rham cohomology 103.1 The Poincare lemma 103.2 Relative De Rham cohomology 113.3 De Rham cohomology for smooth Stein morphisms 113.4 Coherence theorem 123.5 On the absence of torsion in the De Rham cohomology

sheaves 123.6 Relation between . ig^(/*Qy) and f*Mp(Q'f) 13

4 Gauss-Manin connection on relative De Rham cohomology 14






vi Contents

4.1 Identification of sheaves of sections of cohomologicalfibration and of relative De Rham cohomology 15

4.2 Calculation of the connection on a relative De Rhamcohomology sheaf 16

4.3 The division lemma. The connections on the sheaves3&&K(X/S) for p ^ n - \ 17

4.4 The sheaf '3V = / * Q J / 5 / d ( / * Q ^ ) 194.5 Meromorphic connections 204.6 The Gauss-Manin connection as a connecting

homomorphism 215 Brieskorn lattices 23

5.1 Brieskorn lattice "38 245.2 Calculation of the Gauss-Manin connection V on '38 255.3 Increasing filtration on 38^ 255.4 A practical method of calculation of the Gauss-Manin

connection 275.5 Calculation of the Gauss-Manin connection of quasi-

homogeneous isolated singularities 286 Absence of torsion in sheaves 38^ of isolated

singularities 306.1 The presence of a connection implies the absence of

torsion 306.2 A theorem of Malgrange 316.3 Connection on a pair (E9 F) 326.4 Sheaves 3%^& are locally free 32

7 Singular points of systems of linear differential equations 337.1 Differential equations of Fuchsian type 337.2 Systems of linear differential equations and

connections 347.3 Decomposition of a fundamental matrix Y(t) 357.4 Regular singular points 367.5 Simple singular points 367.6 Simple singular points are regular 377.7 Connections with regular singularities 397.8 Residue and limit monodromy 41

8 Regularity of the Gauss-Manin connection 428.1 The period matrix and the Picard-Fuchs equation 428.2 The regularity theorem follows from Malgrange's

theorem 448.3 The regularity theorem and connections with

logarithmic poles 449 The monodromy theorem 46

9.1 Two parts of the monodromy theorem 46






Contents vii

9.2 Eigenvalues of monodromy 479.3 The size of Jordan blocks 499.4 Consequences of the monodromy theorem.

Decomposition of integrals into series 4910 Gauss-Manin connection of a non-isolated hypersurface

singularity 5110.1 De Rham cohomology sheaves 5110.2 Coherence 5210.3 Relation between J%p(f*Q'f) and / * J ^ ( Q / ) 5310.4 A general method of extension of a singular

connection over the whole disk 5310.5 The sheaves ^£_i ) and the Gauss-Manin connection

5410.6 The sheaves $$\§\ and the Gauss-Manin connectionft^X)f0) 5610.7 A generalization of diagram (5.3.4) 57

II Limit mixed Hodge structure on the vanishingcohomology of an isolated hypersurface singularity 60

1 Mixed Hodge structures. Definitions. Deligne's theorem 601.1 Pure Hodge structure 601.2 Polarised HSs 611.3 Mixed Hodge structure 611.4 Deligne's theorem 62

2 The limit MHS according to Schmid 622.1 Variation of HS: geometric case 622.2 Variation of HS: definition 632.3 Classifying spaces and period mappings 632.4 The canonical Milnor fibre 642.5 The Schmid limit Hodge filtration F's 672.6 An interpretation of Fs in terms of the canonical

extension of 3@ 692.7 The weight filtration of a nilpotent operator 702.8 Schmid's theorem 73

3 The limit MHS according to Steenbrink 733.1 The limit MHS for projective families: the case of

unipotent monodromy 743.2 The limit MHS for projective families: the general

case 753.3 Brie skorn construction 773.4 Limit MHS on a vanishing cohomology 783.5 The weight filtration on Hn(XOQ). Symmetry of Hodge

numbers 79






viii Contents

4 Hodge theory of a smooth hypersurface according toGriffiths-Deligne 824.1 The Gysin exact sequence 824.2 Hodge theory for a complement U = X\Y. Hodge

filtration and pole order filtration 834.3 De Rham complex of the sheaf B[Y]x a n d the

cohomology of a hypersurface Y 854.4 The case of a smooth hypersurface Y in a projective

space X = Pn+l 864.5 Generalization to the case of a hypersurface with

singularities 875 The Gauss-Manin system of an isolated singularity 88

5.1 Hodge theory of a smooth hypersurface in the relativecase 89

5.2 The Gauss-Manin differential system 905.3 Interpretation of the complex DRz/s (B[T]z) in terms

of the morphism / : X —> S 915.4 Connection between the differential system J¥x and

the Brieskorn lattice i?f(0) 946 Decomposition of a meromorphic connection into a direct

sum of the root subspaces of the operator tdt. The V-filtration and the canonical lattice 956.1 'Block' decomposition 956.2 Decomposition of a meromorphic connection ^S into

a direct sum of the root subspaces 966.3 The order function a and the V' -filtration 986.4 Identification of the zero fibre of the canonical

extension 3§ and the canonical fibre of the fibration H 996.5 The decomposition of sections co £ <J& into a sum of

elementary sections 1006.6 Transfer of automorphisms from the Milnor lattice H

to the meromorphic connection ^S 1017 The limit Hodge filtration according to Varchenko and to

Scherk-Steenbrink 1037.1 Motivation of Scherk- Steenbrink's construction of the

Hodge filtration 1037.2 The definition of the limit Hodge filtration F s s

according to Scherk- Steenbrink 1067.3 The Scherk-Steenbrink theorem 1087.4 Varchenko's theorem about the operator of

multiplication by / in Q / 1107.5 The definition of the limit Hodge filtration F' on

Hn(Xoo) according to Varchenko 111






Contents ix

7.6 Comparison of the filtrations Fss and FVa 1117.7 Supplement on the connection between the Gauss -

Manin differential system Mx and its meromorphicconnec t ion^ 112

Spectrum of a hypersurface singularity 1158.1 The definition of the spectrum of an isolated singularity 1158.2 The spectral pairs Spp(f) 1178.3 Properties of the spectrum 1188.4 The spectra of a quasihomogeneous and a semi-

quasihomogeneous singularity 1198.5 Calculation of the spectrum of an isolated singularity

in terms of a Newton diagram 1228.6 Calculation of the geometric genus of a hypersurface

singularity in terms of the spectrum 1278.7 Spectrum of the join of isolated singularities 1278.8 Spectra of simple, uni- and bimodal singularities 1298.9 Semicontinuity of the spectrum. Stability of spectrum

for //-const deformations 1308.10 Spectrum of a non-isolated singularity 1328.11 Relation between the spectrum of a singularity with a

one-dimensional critical set and spectra of isolatedsingularities of its Iomdin series 134

III The period map of a //-const deformation of an isolatedhypersurface singularity associated with Brieskornlattices and MHSs 139

1 Gluing of Milnor fibrations and meromorphic connectionsof a /^-const deformation of a singularity 1391.1 Milnor fibrations 1401.2 Cohomological fibration 1411.3 Canonical extension of the sheaf 3$ and the

meromorphic connection 1422 Differentiation of geometric sections and their root

components wrt a parameter 1442.1 Geometric sections and their root components 1442.2 Formulae for derivatives of geometric sections and

their root components wrt a parameter 1462.3 Decomposition of the root components of geometric

sections into Taylor series for upper diagonal deforma-tions of quasihomogeneous singularities 148

2.4 The sheaves Grpig?(0) 150






x Contents

3 The period map 1513.1 Identification of meromorphic connections in a

//-const family of singularities 1513.2 The period map defined by the embedding of

Brieskorn lattices 1523.3 Example: the period map for E\2 singularities 1543.4 The period map for hyperbolic singularities Tp^r 1563.5 The period map for simply-elliptic singularities 1593.6 The period map defined by MHS on the vanishing

cohomology 1634 The infinitesimal Torelli theorem 165

4.1 The V' -filtration on Jacobian algebra. The necessarycondition for //-const deformation 165

4.2 Calculation of the tangent map of the period map. Thehorizontally of the MHS-period map 167

4.3 The infinitesimal Torelli theorem 1694.4 The period map in the case of quasihomogeneous

singularities 1715 The Picard-Fuchs singularity and Hertling's invariants 172

5.1 The Picard-Fuchs singularity PFS(f) according toVarchenko 172

5.2 The Hertling invariant Hen (f) 1745.3 The Hertling invariants Her2(f) and Her^if) 1115.4 Hertling's results 179

References 181Index 185






Introduction

The aim of this book is to introduce and at the same time to survey someof the topics of singularity theory which study singularities by means ofdifferential forms. Here differential forms associated with a singularity arethe main subject as well as the main tool of investigation. Differentialforms provide the main discrete invariants of a singularity as well ascontinuous invariants, i.e. they make it possible to study moduli ofsingularities of a given type.

A singularity is a local object. It is a germ of an algebraic variety, or ananalytic space, or a holomorphic function. However, the majority of theideas and methods, used in the theory under consideration, originated inthe 'global' algebraic geometry. Therefore we first give a very brief andschematic description of these ideas.

The idea of using differential forms and their integrals to define numeri-cal invariants of algebraic varieties goes back to the classic writers ofalgebraic geometry. It will be important for us that holomorphic andalgebraic forms can be used to calculate the singular cohomology of asmooth algebraic variety over C. Developing the ideas of Atiyah and Hodge(1955), Grothendieck (1966) showed that H\Xan, C) ~ Hl

DR(X/C), whereH^iX/C) is the De Rham cohomology. Grothendieck defined Hl

DR(X/C)as the hypercohomology W(X9 Q'x) of the complex of sheaves of holo-morphic differential forms on X. The comparison theorem enables us tocalculate the cohomology of the complement to a hypersurface in projectivespace by means of the cohomology classes generated by rational differentialforms.

Algebraic differential forms have also proved to be useful in the study ofthe monodromy of a family of complex varieties, using the Gauss-Maninconnection. The monodromy transformation is the transformation of fibres






xii Introduction

(or their homotopic invariants) of a locally trivial fibration correspondingto a loop in the base. This notion appears when studying the multivaluedanalytic function, where it corresponds to the notion of the covering orsliding transformation. Very often the monodromy transformation appearsin the following situation. Let / : X —> S be a proper holomorphic map ofan analytic space to the disk in the complex plane. Let Xt be the fibref-\t), t€S,S' = S\{0} and X' = f~l(Sf). By reducing the radius of S,if necessary, we can make the fibration / ' : X' —> Sf a locally trivial C°°-fibration. The monodromy transformation T associated with the loop in *S"surrounding 0 is called the monodromy transformation of the family / Theaction of the monodromy Ton the vector space H*(Xt) is obtained by theparallel displacement of the cohomology classes in the fibres of the locallyconstant fibration H= \JteS>Hl(Xu C) = R'f+Cx'. Grothendieck alsodefined the relative De Rham cohomology sheaves J$l

DR(X/S) ~Uff^(Q'XfS). If / : X —• S is a smooth proper morphism of algebraicvarieties/C, then from Grothendieck's theorem it follows the existenceof the canonical isomorphism of coherent analytic sheavesJgl

DR(X/S)m ~ #7* a n (C) ®c t^s*. The presence of the locally constantsheaf H = i?'/*(C) in ^l

DR(X/S) defines a topological connection on thesheaf J¥l

DR(X/S). Katz and Oda (1968) gave an algebraic definition of thecanonical connection (the Gauss-Manin connection) on the sheavesJ$l

DR(X/S) such that the sheaves of its horizontal sections are 7?'/*(C).They calculated this connection explicitly and showed that it reduces to thedefinition orginally given by Manin (1958) for the case in which X/S is analgebraic curve over the field of functions.

The family of varieties Xt defined by the morphism /degenerates at thepoint 0 G S, and the Gauss-Manin connection has a singularity at the point0. This singularity is regular. The notion of a regular connection generalizesthe classical notion of a differential equation with a regular singular point,and is the subject of Deligne's book [Dl]. Katz gave an algebraic proof ofthe regularity of the Gauss-Manin connection (1970). Analytic proofs weregiven by Griffiths (1971) and by Deligne [Dl]. The regularity theorem isrelated to the monodromy theorem. When the space X as well as all fibresXu t ^ 0, is smooth, the monodromy theorem states that T is quasi-unipotent on H*(Xt, Q), i.e. there exist positive integers A: and TV such that(Tk — \)N = 0. There are several proofs of this basic theorem (Clemens1969; Katz 1971; Griffiths & Schmid 1975). Many of the characteristicfeatures of the degenerate family / : X —> S become apparent in the proper-ties of the monodromy. The monodromy of the family / is closelyconnected with the mixed Hodge structure (MHS) on the cohomology






Introduction xiii

H*(X0) and H*(Xt). The Hodge structure and the period map of a familyof algebraic varieties defined by the Hodge structure on the cohomology offibres H*(Xt) are the second most important notions used in this book.

The concept of a pure Hodge structure is a formalization of the structureof the cohomology groups of a compact Kahler manifold. From the theorydeveloped by Hodge in the 1930s it follows that Hn(X, C) =(&p+q=nH

p'q, HP>i = Hq'p, where the cohomology vector space is identi-fied with the vector space of harmonic forms, and harmonic forms, andconsequently the cohomology, can be decomposed into the direct sum ofcomponents of the type (/?, q). Under a variation of a variety X'm a familyXt the variation of subspaces Hp>q(Xt) is not complex analytic. For thisreason it is more convenient to study the Hodge filtrationFpHn(X, C) = © r ^p / / r w ~ r , which does depend analytically on theparameter t. One can give an equivalent definition of the Hodge structurein terms of the Hodge filtration. Using the isomorphismHn(X, C) ~ H^R(X)9 we can express the Hodge filtration in terms of thestupid (obvious) filtration a^pQ'x = {0 —> Q% —> Q^+ 1 —•...} on theDe Rham complex. One of Griffiths's discoveries [Gr] was that the Hodgefiltration is related to the pole order filtration: if co has a pole of order notgreater than k + 1 along a non-singular projective variety X, then theresidue Resa> is a sum of forms of the type (/?, q) with q ^ k. This gives apurely algebraic definition of the Hodge filtration.

The theory of MHS developed by Deligne [D2, D3] is used more andmore. The definition of the MHS includes, besides a decreasing Hodgefiltration F\ an increasing weight filtration W. Deligne showed that thecohomology of any algebraic variety (possibly non-proper and singular)has a natural MHS. The MHS also appears in investigations of degenera-tions of algebraic varieties. Let a morphism f:X—>S define a family ofnon-singular projective varieties over the punctured disk S\ and letATo = / ~ 1 ( 0 ) be the degenerate fibre. Schmid [Sm] and Steenbrink [SI]investigated the question of what happens with the Hodge structure onHn(Xt, Z), when t is limited to the point 0. The limit object appears to bea MHS. The Hodge filtration of the limit MHS is in a sense the limit of theHodge filtration on Hn(Xt), and the weight filtration is related to themonodromy.

The study of period maps goes back to the investigations of Abel andJacobi on integrals of algebraic functions. A tempestuous development ofthe theory of periods of integrals begins after Griffiths's papers (1968).Griffiths studied the properties of integrals in terms of the notions of theperiod matrix space and the period map which he introduced. Let us






xiv Introduction

consider a family Xt of non-singular projective varieties depending on aparameter t € S, and defined by a proper morphism / : X —> S. Using theconnection on the fibration H= \J tesHn(Xt), we can displace the Hodgestructures on Hn(Xt) to the cohomology space H = Hn (Xto) of a fixedfibre. Considering the Hodge filtration FpHn{Xt) only, we can associate aflag F'(t) in Hto every point t e S9

H = F\t) D F\t) D . . . D Fn+\i) = {0},

and hence obtain a point F(t) of the manifold of flags of given type. Weobtain the period map O: S —• J^. In coordinates the period map is givenby the periods of integrals (we have to choose a basis in the homologyspace consisting of continuous families of cycles, and to define thesubspaces Fp(f) by bases of differential forms). In fact the definition of Ois not correct, since as t moves round a loop in S9 the identification ofHn(Xto) with itself need not be the identity. So we have to consider eithera map O: S —» ^ of the universal cover of S, or a map O of S to a quotientof .^"by some group.

Griffiths found that to first order, Fp is deformed only into the subspaceFp~l. In terms of the Gauss-Manin connection this can be interpreted asV F ^ c Q ^ F ^ " 1 (the horizontality theorem). In fact, Griffiths consid-ered polarized Hodge structures on the primitive cohomology groupsPn{Xt). He constructed the period matrix space D of all possible polarizedHodge structures of given type, which is a submanifold of the flag manifolddistinguished by the Hodge-Riemann bilinear relations. It turns out that Dis an open homogeneous complex manifold, there is a naturally definedproperly discontinuous group T of analytic automorphisms of D such thatM = D/T is an analytic space, and then we obtain the period mapO: S -> M.

The period map can be used for the description of the moduli ofalgebraic varieties. Here there are problems about Torelli-type theorems.The global Torelli problem is the question of whether the period mapO: ^S —> M of the moduli space of algebraic varieties of a given type isan embedding, i.e. whether the period matrix uniquely characterizes thepolarized algebraic variety. The affirmative answer to this question in thecase of algebraic curves is the usual Torelli theorem. The local Torelliproblem is one of deciding when the Hodge structure on H*(Xt, C)separates points in the local moduli space (Kuranishi space) of Xt. Theinfinitesimal Torelli problem is one of deciding when the tangent map dOto the period map of the universal family is a monomorphism. The criterionfor the infinitesimal Torelli theorem to hold, which was obtained by






Introduction xv

Griffiths (1968), stimulated the appearance of many papers on this theme.We are not able in this introduction to go in detail into the problemstouched upon above, and we refer the reader to one of the surveys on thetheory of Hodge structures and periods of integrals, e.g. to [K-Ku] and[B-Z], where one can find other references.

The aim of this book is to transfer the ideas and notions, describedabove, to the local situation - to the case of isolated singularities ofholomorphic functions / : (Cw+1, 0) —> (C, 0). Again we have a morphism/ : X —> 5, but now the fibres are local analytic hypersurfaces in an openset X C Cn+l. In Chapter I we introduce the main personage of this book -the meromorphic Gauss-Manin connection ^S of a singularity / a n d theBrieskorn lattice 3%^ in it. We prove the regularity of the singularity ofthe Gauss-Manin connection and the monodromy theorem. The discussionin Chapter I is based on the classical papers of Brieskorn [Br] andMalgrange [M]. In Chapter II we consider the limit MHS appearing on thevanishing cohomology Hn{Xu C) of an isolated singularity / In the mainwe follow the development as it occurred historically. Here the maincontribution comes from the papers of Steenbrink, Varchenko and Scherk.Initially the MHS on the vanishing cohomology Hn(Xt) was constructedby Steenbrink [S3] following a suggestion of Delinge. He used an embed-ding of the morphism / : X —> S in a projective family Y —• S and thelimit MHS in the case of a degeneration of projective varieties [SI]. ThenVarchenko [V2, V3] proposed and accomplished the direct introduction ofthe limit Hodge filtration F' on Hn(Xoo, C) (by means of asymptotics ofintegrals), without using an embedding to a projective family. Followingthis idea Scherk and Steenbrink [Sc-S] introduced the filtration F' in adifferent way. They showed how the filtration F' is obtained from theembedding of the Brieskorn lattice ^ ( 0 ) in the meromorphic connection.Finally, in Chapter III we consider the period map of a //-const deformationfy(x) of isolated singularities parametrized by points ye Y of a non-singular manifold. The basis of this chapter is the papers of Saito [Sa8],Karpishpan [Ka2] and Hertling [Hel, He2]. First we consider the periodmap defined via embedding of Brieskorn lattices but not the one definedvia the limit MHS on Hn(Xt, C) as it should be according to the ideologypresented at the beginning of this introduction. Instead it is connected withthe following. Firstly, the limit MHS is determined by the embedding ofBrieskorn lattice, and, secondly, the limit MHS on Hn(Xt, C) is a rougherinvariant than the embedding ^ ( 0 ) c ~S is (because the filtration Fm isdefined not by the embedding 3%^ c *J& but by the embedding of adjointobjects Gr'vJ^^ c Grv^M). We give examples of the explicit calculation






xvi Introduction

of the period maps for the deformations of unimodular singularities andprove the horizontality theorem and the infinitesimal Torelli theorem.

Now we give a detailed description of the contents of this book. In §1 ofChapter I we recall the definition of the Milnor fibration [Mi] / : X —> S ofan isolated singularity / : (Cn + 1, 0) -» (C, 0). The restriction of /over thepunctured disk S' is a smooth locally trivial fibration. We introduce thecohomological fibration H= \J tes'Hn(Xt, C) defining the topologicalGauss-Manin connection Vtop on the sheaf of sections 3@ =H®0S' = Rnf*CX' ® @s'- In §2 we develop elements of the theory ofconnections on locally free sheaves, the presentation of which one can findin Deligne's book [Dl]. We pay special attention to the relation between Vand the dual to it connection V*. For Gauss-Manin connections, deter-mined by the homological and cohomological fibrations, this leads to theperiod matrix of a singularity / being a solution of system of lineardifferential equations - the Picard-Fuchs equation of the singularity.

In §3 we introduce the De Rham cohomology sheaf J^^R(X/S) =J%fn(f*Q'x,s), which is a natural extension of the sheaf j^? to the wholedisk. This sheaf establishes the connection between topology and algebraand analysis, and reflects the analytic nature of the singularity / . §4contains Brieskorn's calculation of the Gauss-Manin connection of thesheaf ^ ( " 2 ) = J&£R(X/S) in terms of differential forms. This calculationnaturally leads to the sheaf ^ ( ~ 1 } = /*QJ f s /d( f *£2£^.) which contains^ ( ~ 2 ) and is also an extension of the s h e a f s . We introduce the notion ofthe meromorphic Gauss-Manin connection on the sheaf ^M =3¥(~2>) <S> &s[t~1]. We also explain how to give a more conceptual descrip-tion of the calculation of the Gauss-Manin connection on J^(~2) as aconnecting homomorphism in an exact cohomology sequence. In §5 themain personage of this survey appears. This is the Brieskorn lattice ^ ( 0 ) ,the third natural extension of the sheaf 3$ to the whole disk S. The lattice

is defined in terms of (n + l)-forms. The identification of M{0) andx^ on *S" is realized by means of the Leray derivative (the Poincare

residue), co i-> co/df = Res [co/(f - t)]. All three lattices ^ ( " ° ,i = 0, 1,2, are the terms of an increasing filtration on ,J&. The correlationbetween them is contained in the diagram shown in (1.5.3.4). In §6 fromMalgrange's theorem, which claims that the periods of an «-form co havethe limit l im^o J K O ^ = ^' w e °btain a result of Sebastiani about theabsence of torsion in the sheaves J$f(~l\ and hence we obtain that thesesheaves are locally free ^ -modules of rank /a.

In §7 we recall the classical definition of a regular singular point of asystem of linear differential equations and Sauvage's theorem on the






Introduction xvii

regularity of a simple singular point. We give the definition of a connectionwith regular singularity and of the residue of such a connection wrt asaturated lattice. In §8 we prove a fundamental theorem on the regularity ofthe Gauss-Manin connection. We give two proofs of this important fact.Firstly, we prove that periods of holomorphic forms give solutions of thePicard-Fuchs equation. Then from Malgrange's theorem it follows thathorizontal sections of the Gauss-Manin connection have a moderategrowth. Secondly, using a resolution of the singularity and the sheaves ofdifferential forms with logarithmic poles, we construct a saturated lattice inthe meromorphic Gauss-Manin connection. In §9 from the regularity ofthe Gauss-Manin connection we deduce the monodromy theorem claimingthat all eigenvalues of the monodromy are roots of unity. We give abeautiful proof of this due to Brieskorn, based on the positive solution ofthe seventh Hilbert problem. The part of the monodromy theorem concern-ing the size of Jordan blocks is proved in (II.3.5.9). We obtain corollariesof the monodromy theorem about the decomposition into series of theperiods of integrals of differential forms.

In §10 we consider non-isolated hypersurface singularities. Starting fromthe construction of the Gauss-Manin connection as a connecting homo-morphism in an exact sequence of complexes and in the spirit of [Sr], weobtain a natural generalization of Brieskorn lattices 3@^ to the case ofnon-isolated singularities. We give Van Straten's criterion on the absenceof torsion in the sheaves 3^^ and its application to the non-isolatedsingularities with a one-dimensional critical set.

In Chapter II we consider the limit MHS appearing on the vanishingcohomology Hn{Xu C) of an isolated hypersurface singularity. In §1 werecall very briefly the necessary basic definitions from the theory of MHSdeveloped by Deligne. In §2 we introduce the limit MHS constructed bySchmid [Sm] for a variation (H, JF') of pure Hodge structures F\ on avector space //, parametrized by points of the punctured disk *S". The limitHodge filtration F' appears on the zero fibre 2SItSS of the canonicalextension 3! of the sheaf 3& = H®<9S' to the point 0. The weightfiltration Wt is the weight filtration W(N) of the nilpotent operatorN — — (l/2jri)log Tu. In §3 we introduce the limit MHS on tfye vanishingcohomology H = Hn(XO0, C) (X^ is the canonical fibre of the Milnorfibration) according to Steenbrink. Steenbrink used an embedding of theMilnor fibration X —> S to a projective family Y —• S. He introduced thelimit MHS on Hn(Yoc, C) using a resolution of singularities of the zerofibre and the complex of relative differential forms with logarithmic poles.We give Steenbrink's construction only very schematically, without going






xviii Introduction

into technical details. The limit MHS on Hn(XOO9 C) constructed bySteenbrink can be considered as a quotient of the MHS on Hn(YOO9 C).From here it follows the symmetry of Hodge numbers of the MHS onHn(XO09 C), and also the monodromy theorem on the size of Jordanblocks.

In §4 we consider the Hodge theory of a non-singular hypersurface Y in anon-singular manifold X developed by Griffiths and extended by Deligneto the case of divisors with normal crossings. This theory enables us tocalculate the cohomology of Yand of the complement X\Y by means ofdifferential forms on X with poles on Y9 and also relates the Hodgefiltration F' to the pole order filtration P\ When Y has singularities, thefiltrations F' and P' do not, generally, coincide [Kal, D-Di]. We apply thistheory for the description of Hn(Xt), where the hypersurface Xt c X isthe fibre of the Milnor fibration. The relative variant of this theory enablesus to obtain a natural extension 3@x of the sheaf 3$ to S. Then 3$x isdescribed as the cohomology sheaf 3$x = 3¥n+l(K\ d) of the bicomplex(K\ di, d2), the terms of which are differential forms with poles, and thedegrees are defined by the order of a differential form and the order of itspole. In the bicomplex K" the Poincare complex and the Koszul complexare intertwined, and its differentials are exterior differentiation di = d ofdifferential forms and exterior multiplication by the form df. The Gauss-Manin connection and the Hodge filtration F' on 3%x are defined in termsof this bicomplex. In fact 3&x is a D-module, or a differential system,which appears in the papers of Pham [Phi, Ph2], Scherk and Steenbrink[Sc-S]. The study of this is extended in the papers of M. Saito [Sal-SalO]in the frames of the theory of D-modules and of the 'monstrous' theory ofmixed Hodge modules developed by him. This latter theory raises the levelof abstractness by at least one more order, and moves further from the soil,in which all this began to grow. The 'new' homological algebra, derivedcathegories, perverse sheaves, the Riemann-Hilbert correspondence, etc.are essentially used in it. Under Saito's influence the theory becomes moreand more abstract and technical. One of the aims of this book was not tofollow this line but to evolve the theory using the 'traditional' (i.e. habitualto algebraic geometricians) language of sheaves, connections, spectralsequences etc. and to avoid using the language of the theory of/^-modules,the theory of mixed Hodge modules etc. This is possible because of thefollowing circumstance. As is shown in §5 the operator dt is invertibL3%x, 5%x contains the Brieskorn lattice J^?(0) = Fn3@x as well ascanonical lattice S, i ^ ( 0 ) C 32 C 3VX- That is the pair JBf<°) c % <tains all the information which is of interest to us. The localization oi






Introduction xix

differential system 3%x leads us to the classical frames of the mero-morphic connection ^M = (33x\t)> and as is explained in the supplementto §7, the pair 3%^ c SZ happily stands localization and removes to^ # , 3&^ C 2? C *J&. All this enables us to work in the classical framesof the meromophic connection ^M.

In §6 we study the structure of the meromorphic connection ^M, itsdecomposition into the sum ^M = 0 a Qe of root subspaces of the operatortdt, the K-filtration on ^S. We determine the isomorphism xp:H^ S$ jtSZ = 0-i<aôCa between the canonical fibre of the fibration Hand the zero fibre of the canonical lattice 3$ — V>~x^&, and it enables usto introduce the MHS on Z/in terms of the lattice 5%.

In §7 we introduce the MHS on H = Hn(Xoo, C), according to Varch-enko [V3] and Scherk and Steenbrink [Sc-S]. At first we define the limitHodge filtration F' according to Scherk and Steenbrink, who transformedthe approach of Varchenko and gave the definition of F' in terms of thedifferential system 3%x and the embedding 3%{G) c J^7, and then we defineF' according to Varchenko. Such an order of exposition is connected withthe fact that, in our opinion, the construction of Scherk and Steenbrink hasgenetically a more natural motivation. In the beginning we observe thesequence of steps leading to the construction of F" by Scherk andSteenbrink.

In §8 we study the main discrete invariant of a hypersurface singularity- its spectrum Sp(/), and a more detailed invariant Spp(/) the set ofspectral pairs. To give Spp(/) is equivalent to giving Hodge numbers H^q.The spectrum Sp(/) is a set of pi rational numbers a\, . . . , a^, whereoij — —(l/2jti)Xj are logarithms of the eigenvalues lj of the monodromyT. It codes the relation between the semisimple part of the monodromy andthe limit Hodge filtration F\ We study the properties of spectrum. Weshow that Sp(f) C (— 1, n) and that the spectrum is symmetric wrt thecentre of this interval (n — l) /2. We develop the techniques for thecalculation of the spectrum. We explain: how to find the spectrum of a(semi)quasihomogeneous singularity; how to find the spectrum in terms ofthe Newton filtration defined by the Newton boundary; and in particular,how to find the negative part of spectrum, the degree of which,Y^-Kaôâ = Pg, is equal to the geometric genus of the singularity [S3];how to find the spectrum of the join of isolated singularities, and inparticular, how the spectrum changes under adding squares of newvariables. This technique enables us to find spectra of all the simple, uni-and bi-modal singularities, which we gather together in a table. Finally, westudy variations of MHS of families of hypersurface singularities. We deal






xx Introduction

with this more widely in Chapter III, but in §8 we consider the behavior ofdiscrete invariants under deformations. We give the results of Varchenkoand Steenbrink on the semicontinuity of spectrum and on its stability under//-const deformations. At the end of §8, following Steenbrink, we define thespectrum of non-isolated singularities and give a theorem on the relation ofthe spectrum of a singularity with a one-dimensional critical set and thespectra of isolated singularities of its Iomdin series.

In §1 of Chapter III we begin to study //-const deformations fy(x) ofisolated singularities parametrized by the points ye Y of a non-singularvariety. We explain how to glue the objects, associated earlier with an'individual' singularity, to a family parametrized by Y. In particular, weobtain the family of Milnor fibrations X(y), the family of cohomologicalfibrations H = (J H(y), the family of meromorphic connections^Jg, — ®^C^, etc. In §2 we obtain the formula for differentiating wrtparameter y geometric sections s[(o](t, y), defined by a holomorphic(n + l)-form co = g(x, y)dx. The same formula is used for differentiatingthe root components s(a), (5) of the geometric sections st^Ky) =Y^p>-\s(w, P)(y)* I n the case of upper diagonal deformations of a quasi-homogeneous singularity f{x) from this formula we obtain a formula fordecomposition of s(co, /3)(y) into Taylor series in degrees of y In §3 firstwe define the period map <I>: Y —>H defined by the embedding ofBrieskorn lattices. For all Brieskorn lattices J$(°\y) we have inclusionsV> ~l D J%?{0)(y) 3 Vn~l. The period map takes a point y to the subspaceJ^f(0)O)mod Vn~x in the finite-dimensional vector space V>~l/Vn~\ i.e.to the point in the Grassman manifold II. Then we define the period mapO defined by the MHS on the vanishing cohomology. This period maptakes a point y to the Hodge filtration F(y) of the limit MHS, i.e. to thepoint in the flag manifold. We give, following Hertling, explicit calcula-tions of the period maps of universal families of unimodular singularities.In §4 we calculate the tangent map of the period map and prove thehorizontality theorem and the infinitesimal Torelli theorem for the periodmap O. We compare the period maps <E> and O of the miniversal //-constdeformation of a quasihomogeneous singularity. From this comparison itfollows that the Torelli theorem for the period map O in general is false.Finally, in §5 we consider the 'global' Torelli problem. Varchenko intro-duced the notion of the Picard-Fuchs singularity PFS(f) of a singularity /in terms of framed Picard-Fuchs equations. Hertling reformulated this interms of embeddings of Brieskorn lattices. We interpret it as a point in thequotient U/G\ of the period space by the group G\ C GL{Hj) ofautomorphisms commuting with the monodromy. Then the Torelli problem






Introduction xxi

for the family fy,y€ Y, concerns the injectivity of the map 4>i:Y / ~ —• IT/Gi, where ' — ' i s the equivalence relation induced by the in-equivalence of singularities. We give Hertling's results [Hel, He2].

The reader is expected to have the knowledge and training usual inalgebraic and analytic geometry. This includes knowledge of sheaf theoryand the technique of spectral sequences.

The list of references reflects the development of singularity theory. Itincludes: firstly, the original papers of E. Brieskorn, B. Malgrange, J.H.M.Steenbrink, J. Scherk, A.N. Varchenko, F. Fham, M. Saito, Ya. Karpishpanand C. Hertling, directly connected with the considered topic; secondly,some general papers on MHS and periods of integrals (P. Griffiths, P.Deligne, W. Schmid); thirdly, some books and surveys [Mi, AGV, Phi, Dl,Di3, B-Z]. Some other references are also included.

Now about numbering and cross-references in this book. Each of threechapters is divided into sections, and each section is divided into subsec-tions. In a subsection all claims, remarks and displayed formulae arenumerated successively in a uniform way by three numbers, the first ofwhich is the number of the section and the second the number of thesubsection. For example, the tag (1.3.2) means claim (or formula) 2 insubsection 3 of section 1. To refer to a claim etc. we use three numberswithin a given chapter, and four numbers, the first of which is a romannumeral, to refer to other chapters. Thus (II.7.5.2) refers the reader to claim2 of subsection 5 of section 7 of chapter II.

The preparation of this book was partially supported by Grant. No. 95-01-01575 from the Russian Foundation for Fundamental Research andGrant No. 4373 from the INTAS.






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