APPENDIX A Integral Bilinear Forms and Dynkin Diagrams

(AI) Definition. A lattice (M, ( , )) is a pair consisting of a finitely generated free abelian group M together with a bilinear form

( , ): M x M -+ 7l. which is either:

(i) symmetric, i.e., (x, y) = (y, x) for all x, y EM; (ii) skew-symmetric, i.e. (x, y) = -(y, x) for all x, y E M.

For simplicity, we refer to M as a lattice when the bilinear form ( , ) is clear. The main example which we have in mind is the Milnor lattice Lx = (iin(F), < , » of an n-dimensional IHS X: f = 0 as defined in (3.3.6).

(A2) Definition. A symmetric lattice M is even if (x, x) == 0 (mod 2) for any xEM. A symmetric lattic M is odd if it is not even.

Note that the Milnor lattice Lx of an even dimensional IHS X is an even lattice by (3.3.7).

(A3) Definition. The lattice M is nondegenerate if it satisfies the following two equivalent conditions:

(i) Rad(M):= {x E M; (x, y) = 0 for all y EM} = 0; (ii) the natural group homomorphism

iM : M -+ M' := Hom(M, 7l.), x 1--+ (x, .) is injective.

The lattice M is unimodular if iM is an isomorphism. When the lattice M is non degenerate, we call the finite group

D(M) = coker(iM ) 220 Appendix A. Integral Bilinear Forms and Dynkin Diagrams the discriminant group of M and denote its order ID(M)I by det(M}. The quo• tient lattice M = M/Rad M is called the reduced lattice associated to M.

(A4) Exercise. (i) Show that

D(M) = D( -M), D(Ml Ef) M 2 ) = D(Ml) ® D(M2 ). Here - M is the lattice (M, - ( , » obtained by changing to signs of all the products (x, y) and Ml Ef) M2 is the direct sum lattice, i.e.,

(Xl + X2' Yl + Y2) = (Xl' yd + (x2, Y2) for all Xl' Yl E Ml and X2' Y2 E M 2· (ii) Show that det(Lx} = IA(I)I, for a nondegenerate Milnor lattice Lx, where A denotes the characteristic polynomial of the monodromy operator of X. Hint. Recall the proof of (3.4.7).

(A5) Lemma. If N c M is a sub lattice in the nondegenerate lattice M such that rk M = rk M, then the quotient group M/N is finite and its order satisfies the relation IM/NI2 det M = det N.

Proof Use the structure of the subgroup N as described in [La], p. 393. 0

(A6) Definition. Let M and N be two lattices. A group homomorphism qJ: M ~ N is called a lattice morphism if (x, y) = (qJ(x), cp(y)) for all x, Y E M. A lattice morphism qJ is called an embedding (resp. an isomorphism) if qJ is a group monomorphism (resp. isomorphism).

(A 7) Theorem (Structure of Skew-Symmetric Lattices). Any skew-symmetric lattice M is isomorphic to a direct sum of "elementary" skew-symmetric lattices

(Z2, ( )d,) 61'" EB (Z2, ( }dk ) Ef) (zm, ( )0) where Z2 = Ze l + Ze2 and (e l , e2 )d = d; for some integers d; > 0 and (x, y)o = 0 for all x, y E zm. Moreove'r the positive integers d; are uniquely determined if we ask in addition that dl ld 2 •• ·Idk •

For a proof, see [La], p. 3S0. In particular, M is nondegenerate if and only if m = 0 and then det M = d i ... dt. And M is unimodular if and only if m = 0 and d 1 = ... = dt = 1. We use the notation 12k for this unimodular skew-symmetric lattice of rank 2k.

(AS) Exercise. Let X: f = 0 be a reduced plane curve singularity with Milnor number Il and number of irreducible branches r. Then the Milnor lattice Lx is isomorphic to the direct sum Appendix A. Integral Bilinear Forms and Dynkin Diagrams 221

Hint. Use the Wang exact sequence (3.1.18) associated to the Milnor fibration for X.

When M is a symmetric lattice we can obtain a real bilinear form tensoring by IKt All the usual terminology for the latter applies to M and hence we can speak about the signature sign M = (m_, mo, m+). For instance, M is nega• tive definite when mo = m+ = 0, and M is indefinite when m_ > 0 and m+ > O. The difference m+ - m_ is called the index of the lattice. The classification of the indefinite unimodular lattices is due to Milnor [M2] and we do not recall it here since it is more complicated than (A 7). Moreover, the Milnor lattices Lx for the most familiar singularities X are neither indefinite nor unimodular, so the best way to introduce them to the reader is just by listing them. As remarked in (3.3.23), to each IHS X there are two naturally associated Milnor lattices, a symmetric one Lx and a skew• symmetric one V;. In what follows, we list the Dynkin diagrams for several important classes of singularities, namely for all simple and unimodular sin• gularities in Arnold's lists [AGVl]. These Dynkin diagrams determine at once the symmetric lattices L"x and they determine also the skew-symmetric lattices Lx via Gabrielov's result (3.3.22'). Recall that each vertex in a Dynkin diagram D corresponds to a vanishing cycle 11, with = -2. Two distinct vertices, corresponding to vanishing cycles l1i and I1j , respectively, are joined by kedges (resp. k dotted edges) if their intersection number (l1 i, I1j ) is k (resp. -k). Hence

j • •

j • •

Note also that the vertices in a Dynkin diagram are numbered (corresponding to the order of the vanishing cycles 111 ... , 11/l in the associated distinguished basis 11), unless any order is good (i.e., any order of l1/s corresponds to a distinguished basis). All the singularities discussed in what follows are stably equivalent to sur• face singularities in (:3. The reader has already encountered the notations for (and the equations of) these singularities in Chapter 2, §4, so here we describe only their Dynkin diagrams and some of their properties. 222 Appendix A. Integral Bilinear Forms and Dynkin Diagrams

(A9) Dynkin Diagrams for the Simple Singularities.

Type Dykin diagram

Ak,k~l --... - (kpoints)

Dk,k~4 -- ... ---< (kpoints) • • I • • • • • I • • • • • • I • •

We let All: denote the (symmetric) Milnor lattice of the singularity A" and use the same convention for all the other singularities discussed. A first striking fact about the Dynkin diagrams ofthe simple singularities is that they coincide with their resolution graphs, see (2.4.3). This remark gives us the first part ofthe following result (for a complete proof we refer to [Df4J).

(AIO) Proposition.

(i) The Milnor lattices A", Dk , E6 , E7 and Es are negative definite. (ii) Any IHS X whose symmetric Milnor lattice is negative definite is a simple singularity. (iii) det All: = k + 1, det D" = 4, det E, = 9 - I fori = 6,7,8.

(All) Dynkin Diagrams for the Simple-Elliptic Singularities £6, £7' and Ea. First we associate to each of these simple-elliptic singularities £ a triple of positive integers (p, q, r) as follows

£6 1--+ (3, 3, 3), £7 1--+ (2,4,4), £8 1--+ (2, 3, 6). A look at the equations for these singularities given in (2.4.9) will explain to the reader where these triples come from! See also (A13). Next to each triple Appendix A. Integral Bilinear Forms and Dynkin Diagrams 223

(p, q, r) we associate the following diagram:

p-l q - 1

T(p,q,r) - ... >--..... -

with a numbering such that the vertices denoted by i and i + 1 get consecutive indices.

(AI2) Proposition.

(i) The correspondence E 1--+ (p, q, r) 1--+ T(p, q, r) associates to each simple• elliptic singularity E,,(k = 6, 7, 8) a Dynkin diagram. (ii) The lattice E" is negative semidefinite, its radical has rank 2 and the cor• responding reduced Milnor lattice E" is exactly the Milnor lattice E". (iii) Any IHS X whose symmetric Milnor lattice is negative semidefinite is a simple-elliptic singularity.

For a proof, see, for instance, [Df4]. For a proof of the following more complicated results we refer to Ebeling [EI] and [E2].

(A 13) Proposition. (i) A Dynkin diagram for the cusp singularity T",q,. is given by the diagram T(p, q, r) described above. (ii) sign(T",q,.) = (p + q + r - 3, 1, 1). (iii) det(J;"q,.) = pqr(l - lip - llq - llr).

This is not a surprise, since we can regard the simple-elliptic singularities E" as special cases of the T",q,.-singularities for 1 1 1 -+-+-=1. p q r

(AI4) Proposition. (i) A Dynkin diagram for the triangle singularity Dp,q,. is given by the follow• ing diagram: 224 Appendix A. Integral Bilinear Forms and Dynkin Diagrams

p' - 1 q' - 1 r T(p,q,r) _ ... ~-..... -

with a numbering such that the vertices denoted by i and i + I get consecu• tive indices and (p', q', r') are the Gabrielov numbers given in (2.4.7). (ii) sign(Dp,q,r) = (p' + q' + r' - 2,0, 2). (iii) det(Dp,q,r) = p' q'r'(l - lip' - l/q' - l/r').

Recall that Dp.q,r denotes in fact two singularities: one weighted homoge• neous and the other one semiweighted homogeneous. By our discussion in (3.1.19) it follows that the two singularities have the same topological in• variants, in particular, the same Dynkin diagrams.

(A15) Exercise. Check that the Dolgacev numbers (p, q, r) and the Gabrielov numbers (p', q', r') associated to a given triangle singularity Dp,q,r in (2.4.7) satisfy the following relation

pqr(l- ~ - ~ -~) = p'q'r'(l- ~ - ~ - ~). p q r p' q' r' This is part of a "strange duality" which is explained in [EW].

(A16) Exercise. Show that the skew-symmetric Milnor lattice L"; associated to the triangle singularity

Sl1: X4 + y2z + XZ2 = 0 is isomorphic to the lattic 110 ® (Z, ( )0)' Hint. Use (A14) and (3.3.22').

Recall now the notion of an embedding of lattices cp: M -+ N from (A6). The lattice N is then called a supralattice of the lattice M. The embedding cp is called primitive if coker cp has no torsion. Assume from now on that M is a nondegenerate lattice. We would like to have a control over the set of all the possible torsion groups arising from various embeddings of a given lattice M. In other words, we have to consider the set

(AI7) T(M) = {Tors(NIM); N a supralattice of M} where Tors(G) denotes the torsion part of a finitely generated abelian group G. Appendix A. Integral Bilinear Forms and Dynkin Diagrams 225

(A18) Exercise. Show that T(M) = {Tors(N/ M); N is a supralattice of M and rk N = rk M}.

A first restriction on the finite groups F in T(M) is given by the next

(A19) Lemma. FE T(M)=> IFI2 dividesdet M. In particular, T(M) = {O} when M is a unimodular lattice.

Proof. Use (AS). o

To get a finer description of the groups in T(M) we proceed as follows. The bilinear form ( , ) on M has a natural extension to a bilinear form on M' = Hom(M, Z) with values in Q which can be defined in one of the follow• ing two equivalent ways. Identify M with the image of the monomorphism iM : M -+ M ', X f-+ (x, .) and let u, v E M'. (i) Take an integer k such that ku = x is in M and define 1 (u, v) = k vex).

(ii) Take two integers k, I such that ku = x and Iv = yare in M and define 1 (u, v) = ki (x, y).

This new bilinear form induces by passing to the quotient a bilinear form (A20) called the bilinear discriminant form.

(A21) Lemma. The bilinear discriminant form bM is nondegneerate in the sense that for all u E D(M) => v = O.

Proof. Let k be the smallest positive integer such that ku belongs to M. There is an element v E M' such that v(ku) = 1 and hence (u, v) = 1jk. Since bM(u, v) = 0, it follows that k = 1, i.e., u E M and hence u = 0 in D(M). 0

(A22) Definition. A subgroup H c D(M) is called isotropic if bM I H x H = O.

(A23) Proposition. There is a bijection between the set T(M) and the set of isotropic subgroups in D(M). 226 Appendix A. Integral Bilinear Forms and Dynkin Diagrams

Proof. Let F = N/M be a quotient in T(M). Then we have the obvious inclusions McNcM' since any element n E N induces an element in M' by m 1--+ (n, m) E 71.. The subgroup F = N/M c M'/M = D(M) is clearly isotropic. This argument can be reversed and hence gives rise to a one-to-one correspondence between the set T(M) and the set of isotropic subgroups in D(M). D

(A24) Exercise. (i) Show that bMlffiM2 = bM, EEl bM2 . (ii) Let Md = (1'2, ( )d) be the "elementary" skew-symmetric lattice consid• ered in (A 7). Then show that and

A A At a1 b2 -a2 b1 bM .(a1 , az), (b 1 , Dz»= d E iQI/l'.

Show that T(Md ) = {O} if and only ifthe lattice Md is unimodular.

In the symmetric case the results are much more interesting. Assume from now on that M is an even nondegenerate symmetric lattice. Then its dis• criminant bilinear form bM is determined by its quadratic form qM' defined as follows: (A25) qM(X + M) = (x, x) + 21'. Moreover, a subgroup F c D(M) is isotropicif and only if qM IF = O.

Using the description of the Milnor lattices Ak , Db Ek given in (A9), we can prove the following. (A26) (i) D(Ad = l'/(k + 1)1', q(l) = -k(k + 1)-1.

(ii) T(Ak) = {71./e71.; e21 k + 1 and k(k + l)e-Z E 21'}. In particular, T(Ak) = {O} for k = 1, ... ,6. (A27) (i) For keven, D(Dk) = (71./21')Z and the generators u1, Uz of D(Dk) can be chosen such that q(u1) = 1 and q(u z) = - k/4. (ii) Fork odd, D(Dd = 1'/471. and q(1) = - k/4.

In particular, T(Dd = {O, 71./271.} for k == 0 (mod 8) and T(Dk ) = {O} other• wise. (A28) (i) D(E6) = 71./371. and q(l) = 2/3. (ii) D(E?) = 71./271. and q(1) = 1/2. (iii) D(Es) = O.

In particular, T(Ek ) = {O} for k = 6,7,8. Other explicit computations of qua• dratic forms for Milnor lattices of singularities can be found in [EW]. Appendix A. Integral Bilinear Forms and Dynkin Diagrams 227

Two important results on lattices depending on the notions introduced above are the following ones, due to Nikulin [NkJ, see also [Dg2J.

(A29) Theorem (Uniqueness of a Primitive Embedding). Let i: M --. N be a primitive embedding of an even nondegenerate lattice M of signature (m_, m+) into an even nondegenerate lattice N of signature (n_, n+). Then this embedding is unique up to an automorphism of N provided the following conditions are satisfied: (i) n_ z m_, n+ Z m+; (ii) rk(N) - rk(M) z I(D(M» + 2, where I(F) denotes the minimal number of generators of a finite abelian group F.

(A30) Theorem (Uniqueness of a Milnor Lattice). The symmetric Milnor lattice L = V(X) of an IHS X is determined uniquely by its signature (L, 10' 1+) and the discriminant form q: D(L) --. 0./271. associated to its reduced lattice L.

This result (A30) can be used to determine the Milnor lattices for large classes of singularities (but not their Dynkin diagrams since it does not pro• vide us with distinguished bases in L). Consider the class of weighted homo• geneous surface singularities in (:3. Then the signature of the Milnor lattice L = V(X) of such a singularity (X, 0) can be determined using results of Steenbrink, [S2J, see Appendix C. To determine the remaining ingredient, namely, the discriminant form one can proceed as follows [Dg2], [LW]. We use integer coefficients for (co)homology and the subscript t stands for the "torsion part" of a finitely generated abelian group. If IX: G --. H is a group homomorphism between two such groups, then the restricted homomor• phism IXt : Gt --. Ht is well defined. Let M be a compact closed oriented (2n - I)-manifold and recall the defi• nition of the linking form of M

lk: Hn- 1 (M)t x Hn- 1 (M)t --. 0./71..

Given two classes v1 , V 2 E Hn- 1 (M)/, we can represent them by two disjoint cycles V1 and V2 (use transversality and the fact that dim V1 + dim V2 < dim M). Some integral multiple kV1 of V1 bounds an n-chain C1 in M and we set 1 lk(vl' v2 ) = kC1' V2 (mod 71.)

(compare to Definition (3.3.12». Suppose now that M bounds an oriented compact 2n-manifold N such that

Hn- 1 (N)t = O. The composite 228 Appendix A. Integral Bilinear Forms and Dynkin Diagrams is the adjoint of the intersection product ( , ) on H.(N), recall (2.3.6). It is obvious that H.(N), c Rad( , ) and hence R.(N) := H.(N)/Rad( , ) is a nondegenerate lattice. Let (DN' bN ) be the corresponding discriminant bilinear form.

(A31) Proposition. There is a natural isomorphism

(D N , bN ) ~ (Hn-I(M)" -lk).

Proof Let WI' Wz be two cycles in H.(N, M) and choose representatives WI' Wz in (N, M) for them which are transversal and aWl n awz = 0. Under these conditions, the intersection number WI· Wz is a well-defined integer, but not an invariant of the classes WI' Wz as remarked in [AGV2], p. 12. Let "\, Wz be the elements in DN corresponding to the cycles WI and W2 respectively. Let Vi = aWi be the associated cycles in Hn-I(M). We consider only classes WI' Wz such that the classes VI' Vz are torsion elements. Then there exists an integer k such that kVI = 0, i.e., kVl = aCI for some n-cycle CI in M. But then kWI - CI is an absolute n-cycle in N and hence we have by definition 1 bN(W1 , wz) = k(kWI - Cl> Wz)·

Indeed, note that k WI - CI - k WI in H.(N, aN) and we use here the intersec• tion number of an absolute cycle with a relative cycle, exactly like in [AGV2], p. 11. Again by definition we have t t lk(v l , vz) = k(CI , awz) = k(CI , Wz ). Hence

bN(W1 , Wz) + Ik(v1 , vz) = (WI' W2 ) E 7L which ends the proof. D

(A32) Example. Let M = K be the link of an n-dimensional IHS X and let

N = F be the corresponding Milnor fiber. The the condition H n- 1 (F), = 0 follows from (3.2.1). The discriminant bilinear form (DN , bN ) is in this case exactly the discriminant bilinear form (D(L), br) associated with the reduced Milnor lattice L of the singularity X. Proposition (A3t) says that the linking form of the link K is essentially the same as the discriminant bilinear form bI.

(A33) Example. Let M = L(X, 0) be the link of a normal surface singularity (X, 0) and N the total space of a resolution of the singularity (X,O). Then HI (M), = 0 by (2.3.1). Proposition (A31) says in this case that the linking form of the link M can be computed from the intersection matrix of the resolution (i.e., from the corresponding dual resolution graph). Appendix A. Integral Bilinear Forms and Dynkin Diagrams 229

Combining (A32) and (A33) we deduce that the discriminant bilinear form (and hence also the quadratic form) of the reduced Milnor lattice L of an isolated surface singularity in 1[3 can be computed from its dual resolution graph. In particular, using the result by Orlik-Wagreich (2.4.21), the formula for the signature in Steen brink [S2], and Nikulin's result (A30), we can deter• mine the Milnor lattices of all the weighted homogeneous surface singularities in 1[3. A last lattice-theoretic fact that we need in our book is the following.

(A34) Lemma. Let N be a unimodular lattice and MeN be a nondegenrate sublattice. Consider the orthogonal complement of M in N M.l = {x E N; (x, y) = ofor all y EM}. Then M.l is a nondegenerate lattice such that det M.l = det M ·ITors(N/M)I-2.

Proof. Let M = {x E N; kx E M for some integer k}. Then the embedding MeN is primitive and Tors(N/M) = M/M. Moreover, we have by (A5)

det(M) = det(M)·IM/MI- 2 • Using this, we can clearly assume that the embedding MeN is primitive. Let iM: M -+ M' and iN: N":' N' be the usual embeddings. Then one has det M·det M.l = det(M + Ml) = IN/(M + Ml)12 and N N' M' M + Ml ~ iN(M + Ml) ~ iM(M) ~ D(M). Hence det Ml = det(M) as claimed in (A34). o APPENDIX B Weighted Projective Varieties

Weighted projective varieties are increasingly important in all branches of algebraic geometry and even in mathematical physics, see, for instance, COg!], [S2], and [Ro]. In this appendix we survey some of their basic prop• erties, with special attention to their topological properties. The proofs for the statements below which are not included here can be found in [Ogl], if the reader is not explicitly referred to a different source. Let wo, WI' ..• , Wn be a set of strictly positive integers and consider the associated C* action on the affine space Cn +1, namely, (Bl)

as in (3.1.1 0). The integers Wi are called weights and we denote by w = (wo, ... , wn) the set ofall these weights. The weighted projective space oftype w, denoted by pew) can be defined either geometrically, as the quotient

(B2) pew) = cn +1 \ {O}/C* or algebraically, as (B3) pew) = Proj C[xo,' '" Xn] where the grading for the polynomial ring comes from the relations deg(x,) = wt(x;) = Wi' For the general definition of the scheme Proj S, for any graded ring S, we refer to [Hn], p. 76. We can regard the weighted projective space pew) as a quotient space in two other useful ways. Let S denote the unit sphere in cn+1 and note that the subgroup SI c C* has a naturally induced action on the sphere S. It is also clear that

(B4) pew) = SIS 1• Let G(m) be the multiplicative group of all the mth roots of unity. Consider the product group (B5) If we let G(w) act on the projective space pn coordinate wise, then it is clear Appendix B. Weighted Projective Varieties 231 that

(B6) Ifl>(W) = Ifl>njG(w).

Moreover, the quotient map n: Ifl>" -+ Ifl>(w) is given explicitly by the following formula (B7) Being a quotient variety of Ifl>" under a finite group action, it follows that Ifl>(w) is a complete normal variety, having only quotient singularities, see (2.3.14). We can obviously assume that the weights w satisfy the following condition

(BS) g.c.d.(wo, ... , wn) = 1 which means that the action (B1) is effective. It is often possible to assume a stronger condition on the weights w, namely

(B9) for all i = 0, ... , n. This stronger restriction is useful in many cases. One example of such a situa• tion is the following description of the singular locus Ifl>(W)'ing of a weighted projective space, see [DDv].

(BlO) Proposition. Assume that the weights w satisfy the condition (B9). Then

x E Ifl>(w)'ing<*g.c.d. {Wj; Xj -1= O} > 1.

(Bl1) Exercise. (i) Identify the type of the Hirzebruch-lung singularities in 1fl>(2, 3, 5)'lng' (ii) Why is the statement (BlO) is obviously false when the weights w do not satisfy (B9)?

A first basic topological property of the weighted projective spaces is the following.

(B12) Proposition.

Proof. Use the definition (B4) for the weighted projective space Ifl>(w), the fact thatn1 (S) = °and [Ar]. 0 The projection map S -+ SIS! = pew) is the analog of the Hopf map, but it is no longer an Sl-bundle map. Hence there is no associated Gysin sequence with Z-coefficients. However, since all the isotropy groups of this Sl-action on S are finite, there is a Smith-Gysin sequence with 0- or C-coefficients, see [Bd], p. 162. It follows that Proposi• tion (5.1.6) has the following analog in the weighted case: 232 Appendix B. Weighted Projective Varieties

(B13) Proposition. The rational cohomology algebra H*(!P(w); 0) for n = dim !P(w) 2 1 is a truncated polynomial algebra

o [IX]/(IX n +1) generated by an element IX of degree 2.

(B14) Remark. See [Kws] for a method to compute the integral cohomology algebra H*(!P(w); Z) in terms of the weights w.

Now let /; E C [xo, ... , xn] be weighted homogeneous polynomials of degree dj with respect to the weights w, for i = 1, ... , c. Consider the affine variety in e+1 (a weighted cone) (B15) CV: f1 = ... = !c = 0 and the subvariety V in !P(w) defined by these equations, i.e., (B16) V = CV\{O}/C* = Proj C[xo, ... , Xn]/(fl' ... ,fe). When dim V = n - c, we refer to V as being a weighted complete intersection. When c = 1, we call Va weighted hypersurface.

(BI7) Definition. A weighted complete intersection V is called quasismooth if the associated weighted affine cone CV has an isolated singularity at the origin.

We would like to emphasize that a quasismooth weighted complete inter• section V is not smooth as an algebraic variety. A discussion of the singular locus V.ing of such a variety V, in particular sufficient conditions for the relation to hold, can be found in [D3]. What is true, is that all the singularities of a quasismooth variety are quotient singularities. Such varieties are sometimes called V-manifolds, see for instance, [S2], [S6]. One of the main topological properties of the V-manifolds is the following.

(B18) Proposition. Any V-manifold M is a O-homology manifold of dimension 2 dim M.

Proof. Ifm = dim M, we have to show that

H:(M; 0) = Hti(D2m; 0), where x E M is an arbitrary point and D2m is the closed 2m-dimensional disc. The problem being a local one on M, we can assume that the gerrn(M, x) is a quotient singularity (em/G,O) as in (2.3.14). Moreover we can assume that G c U(m), i.e., the unit sphere s2m-l in cm is G-invariant. Then the link K of the singularity (Cm/G, 0) can be identified with the quotient S2m-1/G, a gener• alized lens space. D Appendix B. Weighted Projective Varieties 233

At the Q-cohomology group level, we have the equalities

H~(M; Q) = Ho(CmjG; Q) = W-1(S2m-ljG; Q) ~ W-1(S2m-l; Q)G = W-1(S2m-l; Q) = Ho(D2m; Q). The equality ex comes from a general result in transformation groups, see [Bd], p. 120. Moreover, the fact that the induced action of G on the cohomology H' (s2m-l; Q) is trivial follows from the fact that the group U(m) is connected (any element hE U(m) can be continuously deformed into the identity).

(BI9) Corollary. The rational cohomology of a weighted projective space P(w) or of a quasismooth weighted complete intersection V c P(w) satisfies Poincare duality.

Let Ky = S n CV be the link associated to a weighted complete intersec• tion. In analogy to (B4) we have (B20)

(B21) Corollary. A weighted complete intersection V is connected for dim V ~ I and simply-connected for dim V ~ 2.

Proof. Exactly as in (BI2), using the relation rc 1 (K y) = 0 from (3.2.12). D

(B22) Lefschetz Theorem over Q. If V is a weighted complete intersection in P(w), then the morphism induced by the inclusion is an isomorphism for k < dim V and a monomorphism fork = dim V

Proof. The proof given for (5.2.6) can be applied to this situation, since: (i) As shown in (BI8), the weighted projective space P(w) is a Q-homology manifold, hence all the duality theorems (Poincare, Alexander, Lefschetz) can be used with Q-coefficients. (ii) The complement in P(w) of a weighted hypersurface is still an affine vari• ety. This follows from the description (B6) and the fact that the quotient of an affine variety under a finite group is affine, see for instance [KK], p. 314, where the corresponding result for Stein spaces is proved.

Using the ramified covering rc: pn --+ lP(w) from (B7) we can associate to each weighted variety V c lP(w) the subvariety V = rc-1(V) in pn. Since V = VjG(w), it follows from [Bd], p. 120, that we have (B23) H*(V; Q) = H*(V; Q)G(W). 234 Appendix B. Weighted Projective Varieties

However this formula is not very useful in practice, since the dimension of the singular locus of V can be high, even if we start with a quasismooth complete intersection V. 0

(B24) Exercise. Consider the weights w = (3, 2, 3, 2) and the weighted hyper• surface V: x 3 + xy3 + Z3 + zt3 = 0 in pew). Show that V is quasismooth, but dim 'V.ing = 1, i.e., the surface V is not even normal.

We can use (B23) (the version with C-coefficients) to prove the following.

(B25) Lemma/Definition (A = IR, C).

(i) The morphism/: Hk(p(W); A) -+ Hk(V; A) induced by the inclusionj: V-+ pew) is a monomorphism for all k :::;; 2 dim V. (ii) H~(V; A) = coker / is called the primitive cohomology of the variety V.

(iii) m(V; A) = H 2n - k - 1 (U; A), where U = P(w)\ V.

When V is a weighted hypersurface in pew) associated to a weighted homo• geneous polynomial f of degree N with respect to the weights w, we have the following additional facts. Let F: f - 1 = 0 be the corresponding affine Milnor fiber and let h: F -+ F be the monodromy homeomorphism as in (3.1.19). Let H*(F)o be the fixed part cohomology in H*(F; q under the monodromy operator. Since U = F/

(B26) H*(U) = H*(F)o.

(B27) Exercise. (i) Use the previous relation to obtain an analog of Theorem (5.2.11) in the weighted hypersurface case. (ii) Let d be a divisor of N = deg f, d > 1. Consider the weighted hyper- surface V d: fd(xo, ... , Xn, t) = f(xo, ... , xn) + t d = 0 in the weighted projec- tive space P(wo, ... , wn, N d -1). This hypersurface is the d-fold covering of pew) ramifed along the hypersurface V, see [DDv]. Let U d be the complement of the weighted hypersurface V d• Show that dim HS(U d ) is equal to the number of eigenvalues (counted with multiplicities) of the monodromy operator h* on H S - 1 (F; C) which are dth roots of unity and different from 1. Hint. Use the Thom-Sebastiani formula.

Let us consider now in more detail quasismooth weighed complete inter• sections. Their topology is completely determined by their weights w =

(wo, ... , wn}andmultidegreed = (d 1 , ••• , del·

(B28) Proposition. Let V, V' be two quasismooth weighted complete intersec• tion in pew) having the same multidegree d. Then the varieties V and V' are homeomorphic. Appendix B. Weighted Projective Varieties 235

For a proof of this result, based on an S1-equivariant Ehresmann Fibration theorem we refer to [Dl]. Let V be a quasismooth complete inter• section in pew) and let Kv be the associated link. Then Kv is a smooth mani• fold with an induced S1-action, recall (B20). For a point x E K v , the isotropy group S; is the multiplicative group ofw(x)-roots of unity, where

(B29) w(x) = g.c.d.{wi ; Xi #- O} (compare with (BlO».

(B30) Definition. The weighted complete intersection V is called strongly smooth if it is quasismooth and if all the isotropy groups S; for x E K v are the same.

In such a case the projection Kv -+ V is an S1 = Sl/S~-bundle map. In particular V gets in this way the structure of a smooth manifold.

(B31) Example. Consider a set of weights w = (wo, ... , wn ) such that g.c.d.(wi, wJ = 1 for all i #- j. Let N be a common multiple of these weights Wi and consider the weighted hypersurface

V: f(x) = x~/wo + ... + x:/Wn = O. Then the hypersurface V is strongly smooth.

The main properties of this important class of weighted complete intersec• tions are contained in the following result, see [01].

(B32) Proposition. Let V be a weighted complete intersection of weights wand multidegree d which is strongly smooth. (i) Any other quasismooth weighted complete intersection with the same weights wand multidegree d is also strongly smooth and diffeomorphic to v: (ii) The integral cohomology algebra H*(V) is torsion free.

Now let V be a quasismooth weighted complete intersection in pew) of multidegree d. Using (819), (B22), and (B13), it follows that (B33) for i #- m, where m = dim V. In view of (B28) it is natural to try to compute in terms of w and d the following topological invariants for V: (i) the middle Betti number bm(V) or, alternatively, the Euler characteristic x(V) of V; (ii) (when m = dim V is even) the index r( V) of the cup product

Hm(v; 0) x Hm(v; 0) -+ H2m(v; 0) = 0. It turns out that the simplest way to compute these topological invariants is by computing some analytical invariants, namely the mixed Hodge numbers hp,q(V), see Appendix C. 236 Appendix B. Weighted Projective Varieties

The general case is treated in [H2] and [S6]. Here we present the result only for the hypersurface case, see [S2], [Dg1]. Assume that the hypersurface V is defined by a weighted homogeneous polynomialf of type (wo, ... , wn ; N). Let W = Wo + ... + Wn be the sum of all these weights. Note that the Milnor algebra M(f) = lC[xo, ... , xn] (aflaxo, ... , af/axn )

(see [D4], p. 111) is a graded algebra, using deg Xi = wt Xi = Wi for i = 0, ... , n. Let M(f). denote the homogeneous component of degree s in M(f) with respect to this grading.

(B34) Theorem (Steenbrink). The mixed Hodge numbers of the primitive cohomology group H(j-l(V) of a quasismooth weighted hypersurface V in [pew) are given by

(B35) Examples. (i) dim V = 1. Then V is a smooth connected curve (there are no quotient singularities in dimension 1) of genus

g(V) = hg·l(V) = dim M(f)N-w' To have a numerical example, consider the curve V: x~ + x~ + x~ = 0. Here w = (15,10,6), N = 30. Since N - W = -1 < 0, it follows that g(V) = 0, i.e., V ~ IP I, the projective line. (ii) dim V = 2. Then V is a normal surface and its index r(V) is given by the formula reV) = 1 + 2 dim M(f)N-w - M(fhN-w' Here we have used the general formula for reV) in terms of mixed Hodge numbers hp.q(V), see Appendix C as well as the equality dim M(f)(i+1lN-w = dim M(f)(n-ilN-w holding for any i, which is a well-known property of this Milnor algebra, see [D4], p. 113. To have again a numerical example, let C: g(x, y, z) = °be a smooth sex tic curve in [p2. Let V be the double covering of [p2 ramified along the curve C. Then an equation for V is V: f(x, y, z, t) = g(x, y, z) + t2 = ° in the weighted projective space 1P(1, 1, 1, 3), i.e., Wo = WI = W2 = I, W3 = 3, N = 6. A simple computation (if you need take g(x, y, z) = x 6 + y6 + Z6!) shows that reV) = -16. This should be compared to (5.3.33) since the double covering V is also a classical model for K3 surfaces. Appendix B. Weighted Projective Varieties 237

(iii) Closures of Affine Milnor Fibers. Let F:f(xl, ... ,xn)-l =0 be the affine Milnor fiber of the weighted homogeneous polynomial f of type (WI"'" Wn; N), having an isolated singularity at the origin. Assume that n is odd and let (/L, Jio, Ji+) be the signature ofthe Milnor lattice Hn - l (F).

Consider the compactification IC" c lP(w), where w = (1, WI' ••• , wn ). The closure F corresponding to this compactification is the weighted hypersurface given by J(xo, ... , xn) = xg - f(x l , ... , x n) = O. The Milnor algebras M(f) and M(J) are related by the following obvious relation M(J) = M(f) ® C [xo]/(xg-l ). Using this it follows that

h~n-i-I(F) = dim M(f)U+I)N-W-1 = L dim M(f)iN-w+s, s=I.N-I where W = WI + ... + wn • Using [S2] (see also (C26» it turns out that r(F) = 1 + Ji+ - Ji-. This formula can be regarded as an analog of(5.3.24).

Beyond the case of quasismooth complete intersections, one knows very little about the topology of weighted varieties. Assume, for instance, that we are again in the hypersurface case and dim CY.ing = 1 (this is the analog of a hypersurfaces with isolated singularities in the usual projective space lPn, the case considered in Chapter 5, §4). In the weighted case, even the formulas for the Euler characteristics X(V) and X(F) are not completely clear. However, there are conjectural formulas which are known to hold in many special cases, see [D5]. APPENDIX C Mixed Hodge Structures

In this appendix we survey some of the basic definitions and properties of mixed Hodge structures. A similar introduction to mixed Hodge structures can be found in Durfee [DfS], while for details and complete proofs we refer to the original papers by Deligne [Del] and to the forthcoming book by Steenbrink [S6]. For a smooth manifold X the famous de Rham theorem says that we have an isomorphism (Cl) i.e., any real singular cohomology class [c] for X can be represented by a closed differential form w on X, see, for instance, [GH], p. 44. However, the differential form w is by no means uniquely determined by the class [cJ. Assume from now on that X is an oriented compact smooth manifold which has a Riemannian metric g. Then we can define the classical *-operator (C2) where n = dim X, and ES(X) denotes the vector space of all differential s-forms on X, see [We], p. 158. Using this operator, we define

where d: ES(X) -+ p+l(X) is the exterior differentiation offorms. Let

(C3) .1. = dJ + Jd be the corresponding Laplace operator. A differential form w E ES(X) is called harmonic if it satisfies one of the following equivalent conditions: (i) .1.w = 0; (ii) dw = Jw = O. Let HS(X) be the vector space of all harmonic forms in ES(X). Appendix C. Mixed Hodge Structures 239

(C4) Theorem (Hodge). H*(X; IR) = H*(X), i.e., each cohomology class [c] E HS(X; IR) has a unique harmonic representative WE W(X).

For a proof of this famous result we refer to [We]. Assume from now on that X is a compact complex manifold (which also implies a natural orientation on X). Let ES(X; C) = ES(X) ® IC be the vector space of all the differentiable s-forms on X with IC-values. Recall that there is a natural decomposition (C5) Em(x; C) = EB Ep,q(X, C), p+q=m the so-called decomposition into (p, q)-types. Note that the Laplace operator A extends to this new space Em(x; C) and so does the definition of harmonic forms. Moreover, we clearly have where the left-hand side denotes the space of all harmonic complex m-forms on X. Let h be a Hermitian metric on X. Then the real part g = Re(h) of this metric is a Riemannian metric on X, while its imaginary part 0 = Im(h) is a 2-form of type (1, 1), i.e., 0 E E1.1(X; C).

(C6) Definition. The Hermitian metric h is called Kahler if dO = O. A complex manifold X is called Kahler if X has a Hermitian metric h which is Kahler.

(C7) Example. (i) The Fubini-Study metric on P" is Kahler, see for details [We], p. 190. (ii) If X is a Kahler manifold and Y c X is a closed complex submanifold, then Y is again Kahler. In particular, all the smooth projective varieties are Kahler.

Assume from now on that X is a compact Kahler manifold.

(C8) Proposition. Let W = Lp+q=m wp•q be the decomposition of an m-form on X according to (p, q)-types. Then the form W is harmonic if and only if all the components Wp,q are harmonic.

In other words, we have a decomposition (C9) Hm(x; C) = EB Hp,q(X), p+q=m where Hp,q(X) denotes the IC-vector space of all harmonic forms of type (p, q) on X. Note that the vector space Em(x, C) has a natural conjugation coming 240 Appendix C. Mixed Hodge Structures from the complex conjugation on C and denoted by OJ ~ ro. Moreover, we obviously have (ClO) Let us define the Hodge numbers of the manifold X by the formula (Cll) Note that (ClO) implies hP4(X) = hqP(X). The decomposition (C9) for the cohomology of X has some direct topological consequences, see, for instance, [We], pp. 198-208, or [GH], pp. 117-126.

(C12) Corollary. Let X be a compact Kahler manifold. Then:

(i) the odd Betti numbers b2k+1 (X) are even; (ii) the even Betti numbers b2k(X) for 0 ~ k ~ dim X are nonzero; (iii) assume that n = dim X is even; then the index .(X) of the manifold X can be computed in terms of the Hodge numbers of X via the following formula: .(X) = L (-1)Php,q(x). p=q(mod2)1 ~p,qsn

To go further, it is convenient to define abstractly the structure we have obtained in (C9), (ClO),

(C13) Definition. A (pure) Hodge structure of weight m is a pair (H, F), where H is a finite dimensionallR-vector space and F is a decreasing filtration on He = H ® C (called the Hodge filtration) such that:

(i) F is a finite filtration, i.e., there exist s, t E 7L with F"He = He and FtHe = 0; (ii) He = FPHe EEl (Fm pH He) for all p E 7L, where the conjugation on He is induced from the complex conjugation on C.

If we define Hp,q = FPHe n (FqHd for any pair (p, q) with p + q = m, then we have the following relations: «(X) He = E!3lp+q=m Hp,q; ({3) Hp,q = Hq,p; i.e., the abstract analog of (C9) and (C10) holds true. Conversely, starting with a finite direct sum decomposition of He satisfying the properties (IX) and ({3), we can define the Hodge filtration by the formula (C14) FPHe = EB H",m-s. s;;,p It is easy to check that the properties (i) and (ii) in Definition (C13) are then fulfilled. When there is no danger of confusion, we denote the Hodge structure (H, F) simply by H. Appendix C. Mixed Hodge Structures 241

(CIS) Definition. Let H and H' be pure Hodge structures of the same weight m. An IR-linear map

(C16) Remarks. (i) When

(CI7) Example. (i) If X is a compact Kahler manifold it follows from (C9), (CIO), and our discussion in Definition (C13), that the space Hm(x; IR) has a pure Hodge structure of weight m. (ii) Let f: X -+ X' be a complex analytic map between the compact Kahler manifolds X and X'. Then the induced morphism

f*: Hm(X'; IR) -+ Hm(x; IR) is a morphism of Hodge structures. This property comes from the obvious fact that f*: Em(X'; q -+ Em(x; q preserves the decompositions into (p, q)• types.

We can now define the central concept of this appendix.

(CI8) Definition (Deligne [Del]). A mixed Hodge structure (MHS) is a triple (H, W, F) where: (i) H is a finite dimensionallR-vector space; (ii) W is a finite increasing filtration on H called the weight filtration; (iii) F is a finite decreasing filtration on He called the Hodge filtration, such that (Gr!" H, F) is a Hodge structure of weight k for all k.

More explicitly, we look at the graded pieces Gr!" H = w"H/w,,-t H with respect to the weight filtration and ask that the filtration induced by F on (Gr!" H)c (denoted again by F) is a Hodge filtration as in (C13). This induced Hodge filtration is explicitly given by P(Gr!" H)c = PHe n WkHe + w,,-l HC/w,,-l Hc·

When (H, W, F) is a MHS (usually denoted simply by H) we can define the associated mixed Hodge numbers hp,q(H) by the formula (CI9) 242 Appendix C. Mixed Hodge Structures

(C20) Exercise. Let (H, W, F) be a MHS and let k E 7l be any integer. Show that the following data (H(k), W,F) where H(k) = H, WmH(k) = Wm+2kH, PH(k)c = P+kHc for all m, p E 7l again define a MHS. Show that hp.q(H(k» = hP+k,q+k(H).

(C2l) Definition. (i) Let H and H' be two MHS and let

(ii) An IR-linear map

(C22) Proposition (Deligne [Del]). If

The following simple example shows that MHS do arise quite naturally on the cohomology of algebraic varieties.

(C23) Example. Let Xl and X2 be two smooth projective varieties such that Xl n X2 is again smooth and let X = Xl U X2 be their union (e.g., take Xl and X2 to be smooth curves in 1Jb2). The Mayer-Vietoris sequence in cohomology with IR-coefficients (which are not written explicitly in order to simplify the notation) associated with the closed covering X = X I U X 2 looks like

1 -+Hk-I(Xd EB H k- (X2 ) ~ Hk-I(Xl n X 2 )! Hk(X)!.. Hk(Xd EB H k(X2 )-+.

The morphisms 01: and P here are essentially induced by inclusions and as a result they are morphisms of Hodge structures. Using (Cl6(i» it follows that:

(i) coker 01: is a Hodge structure of weight k - 1; (ii) im Pis a Hodge structure of weight k. The exact sequence

o -+ coker 01: -+ Hk(X) -+ im P-+ 0 shows that the quotients Gr,!Y Hk(X) have a natural Hodge structure of weight s if we set

The fundamental fact about MHS is that the cohomology of any algebraic variety has in a canonical way such a structure. More precisely, we have the following result due to Deligne [Del]. Appendix C. Mixed Hodge Structures 243

(C24) Theorem. There is a functorial MHS on H*(X; IR) for any algebraic variety X such that the following properties hold ftJr all m ~ 0: (i) the weight filtration Won Hm(x; IR) satisfies

0= W- 1 C Wo c··· C W2m = Hm(x; IR);

for m ~ n = dim X, we also have W2n = ... = W2m; (ii) the Hodge filtration F on Hm(x; q satisfies Hm(x; q = F O :::J ••• :::J Fm+1 = O. For m ~ n = dim X, we also have Fn+1 = 0; (iii) if X is a smooth variety, then Wm- 1 = 0 (i.e., all weights on Hm(x; IR) are ~ m) and Wm = j* Hm(x) for any compactiflcations

j:X~x;

(iv) If X is a projective variety, then Wm = Hm(x) (i.e., all weights on Hm(x; IR)

are :s; m) and Wm- 1 = ker p* for any proper map

p: X ~ X with X smoooth.

(C25) Remarks. (i) The word "functorial" above means that any morphism f: X ~ X' of algebraic varieties induces a MHS morphism foranym.

(ii) Using (iii) and (iv) in (C24), it follows that Hm(x; IR) has a pure Hodge structure when X is smooth and projective. This structure coincides precisely with the Hodge structure coming from (C9). (iii) The statement (C24(iii» holds for the larger class of varieties X which are Q-homology manifolds. Using (BI8) this covers the class of V-manifolds, in particular, the weighted projective spaces pew) and the quasismooth weighted complete intersections. Since these latter varieties are also projec• tive, it follow as in (ii) that their cohomology even has a pure Hodge structure. Using (CI6(i», it follows that the primitive cohomology Hi)(X; IR) has a natu• ral MHS too. The corresponding mixed Hodge numbers hp.q(Hi)(X; IR) are simply denoted by hg,q(X). This notation has already been used in (B24). Note also that for a projective Q-homology manifold X all the results in (CI2) hold true.

(C26) Example (Steenbrink [S2]). Let f be a weighted homogeneous polyno• mial of type (wo, ... , wn ; N) having an isolated singularity at the origin of en+1. Let F:f - 1 = 0 be the corresponding affine Milnor fiber. From (C24) it follows that the middle cohomology group H"(F; IR) has a MHS, such that all the weights are ~ n. More precisely, we have

o = w,,-l C w" C w,,+1 = H"(F; IR), and the corresponding mixed Hodge numbers are given by the following 244 Appendix C. Mixed Hodge Structures formulas: hP"'P(F) = L dim M(f)pN-w+i, i=!,N-] where M(f) is the graded Milnor algebra C[xo,'" ,x.] as in (B34) and w = Wo + ... + w., see [S2]. Assume now that n is even and let (jL, /10' /1+) be the signature of the Milnor lattice. Then we have

/10 = L hP,.+l-p (F), p /1- = L hP,'-P(F), peven

/1+ = L hP,'-P(F), podd

Note that the monodromy homeomorphism h: F ..... F is in fact an algebraic morphism by (3,1.19) and hence h*: H"(F; IR) ..... H'(F; IR) is a MHS morphism.

(C27) Remark. There are some other classes of cohomology groups of objects familiar in Algebraic Geometry (or even in Analytic Geometry) which carry natural MHS, see, for instance, [AGV], [Del], [Df7], [Na], [Sl], [S3]. We mention here explicitly the following situations:

(i) The cohomology of the Milnor fiber of a hypersurface singularity (X, 0), It is important to point out that the monodromy operator h* is no longer a MHS morphism in general as it is in the special case (C26), However, the semisimple part h: of the monodromy operator h* is a MHS morphism and using it we can define the spectrum of the singularity (X, 0), This spectrum has very interesting semicontinuity properties, see [S4] and [AGV2], the latter for many interesting applications also. (ii) Relative cohomology groups H*(X, Y; IR) where (X, Y) is a pair of alge• braic varieties. These MHS are functorial with respect to algebraic maps of pairs. A special case of this situation is the cohomology with supports H:(X; IR) = H*(X,X\Y; IR), where Y is a closed subvariety in X. (iii) Local cohomology of analytic spaces H:(X; IR) where X is a complex analytic space and x E X is a point. It is clear that these cohomology groups depend only on the singularity (X, x). The corresponding MHS are functorial with respect to analytic map germs f: (X, x) ..... (X', x'). Appendix C. Mixed Hodge Structures 245

Assume now that (X, x) is an isolated singularity and let K = L(X, x) be the corresponding link. The isomorphisms H;(X) = Hm(x, X\{x}) = Hm-1(X\{x}) = Hm-l(K) show that it is natural to consider also MHS on the cohomology of links of isolated singularities. These MHS have some useful properties described in the following result.

(C28) Proposition. Let K be the link of an n-dimensional isolated singularity (X, x). Then the natural MHS on Hm(K) has all the weights> m (resp. ::; m) for m ~ n (resp. for m < n).

(C29) Example (Links of Normal Surface Singularities). Let (X, x) be a normal surface singularity and let (X, D) -+ (X, x) be a very good resolution, as in Chapter 2, §3. Let K be the link corresponding to (X, x). Then the MHS on H2(K; ~) has all the weights >2 according to (C28). It can be shown, as in [Df7], that the weight filtration here looks like

0= W2 C W3 C W4 = H2(K; ~). Moreover, we have dim W3 = 2h 1 ,2 = 2 L g(D;), where D; are the irreducible components of the exceptional divisor D, and dim(W4 /W3) = h2 ,2 is equal to the number of cycles in the dual graph of the resolution. Hence these two numbers are independent of the resolution (X, D) chosen for the singularity (X, x), see also (2.3.2). To have a concrete example, consider a Tp,q,r-singularity 111 X: xyz + x p + yq + z' = 0 with - + - + - < 1. p q r Using the description of the very good resolution given in (2.4.5), it follows that the MHS on H2(K; IR) has the following mixed Hodge numbers:

(C30) Remark. Most of the morphisms occurring in the usual exact sequences are MHS morphisms in an obvious way. Here are two such examples: (i) In the exact sequence

! H';'(X) -+ Hm(x) -+ Hm(x\ Y) ! H,;,+1 (X) -+

all the morphisms (including <5) are MHS morphisms. (ii) Consider the Gysin sequence from (2.2.14)

-+ Hm(M) ~ Hm(M\D)!. Hm-l(D)! Hm+l(M) -+, 246 Appendix C. Mixed Hodge Structures

where M is a smooth algebraic variety and D is a smooth hypersurface. Then the residue morphism R is a MHS morphism of type (-1, -1), while the connecting homomorphism b (which is, in fact, Poincare dual to the morphism H*(M) -+ H*(D) induced by the inclusion D eM) is a MHS morphism oftype (1, 1). In case we prefer to work only with MHS morphisms, we can write the above exact sequence in the following form (recall (C20)):

Hm(M) ~ Hffl(M\D)!. H'n-l(D)( -1) ~ H m+ 1 (M) -+.

(C31) Remark. The MHS on the cohomology of algebraic varieties behave in a nice way with respect to Poincare, Alexander, or Lefschetz duality. For this we refer to Fujiki [Fj] and mention here only the following result. Let V be a weighted hypersurface in a weighted projective space lP(w), with dim lP(w) = n. Then we have the following relation between the mixed Hodge numbers for U = lP(w)\ V and the mixed Hodge numbers for the primitive cohomology of V (recall also (B25(iii» hp.q(Hm(u» = hn-v,n-q(Hgn-m-l(v)).

In the final part of this appendix we discuss some nice topological applica• tions of the existence of MHS on the cohomology of algebraic varieties and links. The first result is so crystal clear that we just state it and refer for a proof to Durfee [Df8].

(C32) Proposition. Let X and Y be smooth projective varieties and suppose that

Y = Y1 U ... U Y", are disjoint decompositions of X and Y into quasiprojective varieties Xi' lj. Assume, moreover, that the variety Xi is algebraically isomorphic to the variety YJor all i = 1, ... , n. Then for all pairs (p, q).

I n particular, the Betti numbers bk(X) and bk( Y) coincide for all k.

(C33) Remark. It might be interesting to compare (C32) with our example (5.4.29).

Now let (X, 0) be an ICIS at the origin of CN with dim (X, 0) = n + 1. Let f: (X, 0) -+ (C, 0) be an analytic function germ such that (Xo, 0) = U-1 (0), 0) is again an ICIS. Let K = K(X, 0) and Ko = K(Xo, 0) be the corresponding links. The morphism

r: Hm(K; IR) -+ Hm(Ko; IR) induced by the inclusionj: Ko -+ K is obviously trivial for m "# n. Indeed, in this range, at least one of the groups Hffl(K; IR) or Hffl(Ko; IR) is trivial by Appendix C. Mixed Hodge Structures 247

(3.2.12). On the other hand, the following statement is not at all obvious from a purely topological point of view.

(C34) Proposition. The morphismr is trivial.

Proof. Using (C28) it follows that the MHS on H"(Ko) has weights :s;; n, while the MHS on H"(K) has weights > n. Since r is a MHS morphism it should preserve the weight filtrations. But this is possible only whenr = o. 0

(C35) Remark. R. Randell has introduced in [RaJ the concept of ad-negligible plane curve singularity as follows. Let C: g = 0 be an isolated plane curve singularity and let d ~ 2 be an integer. Consider the surface singularity (the so-called "d-suspension of(C, 0)") defined by

S: g(x, y) + t d = 0 in C 3 •

Let Kc and Ks be the corresponding links and letj: Kc -+ Ks be the inclu• sion (x, y) 1----+ (x, y, 0). The singularity (C, 0) is called d-negligible if the morphism is surjective.

(C36) Corollary. If the singularity (C, 0) is d-negligible, then Hl (Ks; IR) = o.

Proof. If the above morphism j* is surjective, it follows by duality that the morphism j*: Hl(Ks; IR) -+ Hl(K6 IR) is injective. According to (C34) this can happen only if Hl(Ks; IR) = o. 0 References

[AC1] A'Campo, N.: Sur la monodromie des singularites isolces d'hypersurfaces complexes, Invent. Math.:ZO (1973),147-169. [AC2] A'Campo, N.: Le nombre de Lefschetz d'une monodromie, Indag. Math. 35(1973),113-118. [AC3] A'Campo, N.: La fonction zeta d'une monodromie, Comm. Math. Helv. 50 (1975),233-248. [AM] Akbulut, S., McCarthy, J.D.: Casson's Invariant for Oriented Homology 3-Spheres, an Exposition, Mathematical Notes 36, Princeton University Press, Princeton, 1990. [Ar] Armstrong, M.A.: Calculating the fundamental group of an orbit space, Proc. Amer. Math. Soc. 84 (1982), 267-271. [AGV1] Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differ• and entiable Maps, Vols. 1 and 2, Monographs in Mathematics 82 and 83, [AGV2] Birkhauser, Boston,1985 and 1988. [AB] Artal-Bartolo, E.: Sur Ie premier nombre de Betti de la fibre de Milnor du cone sur une courbe projective plane et son rapport avec la position des points singuliers, preprint 1990, University of Wisconsin, Madison. [ABG] Atiyah, M.F., Bott, R., Garding, L.: Lacunas for hyperbolic differential operators with constant coefficients II, Acta Math. 131 (1973),145-206. [BS] Banica, c., Stiinasila, 0.: Methodes Algebriques dans la Theorie Globale des Espaces Complexes, Gauthier-Villars, Paris, 1977. [BPV] Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces, Erg. der Math., 3. Folge, Band 4, Springer-Verlag, Berlin, 1984. [BD] Barthel, G., Dimca, A.: On complex projective hypersurfaces which are homology P"'s (preprint 1989). [BaK] Barthel, G., Kaup, L.: Topologie des Surfaces Complexes Compactes Singulieres, Scm. Math. Sup. 80, Les Presses de I'Universitc de Montreal, 1982. [BO] Bauer, S., Okonek, c.: The algebraic geometry of representation spaces associated to Seifert fibered homology 3-spheres, Math. Ann. 286 (1990), 45-76. [Bi] Birman, J.S.: Braids, links, and Mapping Class Groups, Annals of Mathe• matical Studies 82, Princeton, 1974. [BT] Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82. Sprin)1;er-Verlag. New York. 1982. 250 References

[Bd] Bredon, G.E.: Introduction to Compact Transformation Groups, Academic Press, New York, 1972. [BSl] Brian~on, J., Speder, J.P.: La trivialite topologique n'implique pas les con• dition de Whitney, C. R. Acad. Sci. Paris, Ser. A 280 (1975),365-367. [BS2] Brian~on, J., Speder, J.P.: Les conditions de Whitney impliquent J.l* con• stant, Ann. Inst. Fourier (Grenoble), 26 (1976),153-163. [Bl] Brieskorn, E.: Beispiele zur DitTerentialtopologie von Singularitiiten, In• vent. Math. 2 (1966),1-14. [B2] Brieskorn, E.: Singular Elements of Semi-Simple Algebraic Groups, Actes Congres Int. Math. Nice 1970, Vol. 2, 279-284. [B3] Brieskorn, E.: Die Monodromie der isolierten Singuiaritiiten von Hyperfliichen, M anuscripta Math. 2 (1970), 103-161. [B4] Brieskorn, E.: Sur les groupes de tresses (d'apres V.1. Arnold), Sem. Bourbaki 1971/72, Exp. 401, Lecture Notes in Mathematics 317, Springer• Verlag, Berlin, 1973. [B5] Brieskorn, E.: The unfolding of exceptional singularities, Nova Acta Leopoldina (N F 52), 240 (1981),65-93. [BK] Brieskorn, E., Knorrer, H.: Plane Algebraic Curves, Birkhiiuser, Boston, 1986. [Btl] Broughton, S.A.: On the topology of polynomial hypersurfaces, Proc. Symp. Pure Math. 40, Part I (Arcata Singularities Conference), American Mathematical Society, 1983, pp. 167-178. [Bt2] Broughton, S.A.: Milnor numbers and the topology of polynomial hyper• surfaces, Invent. Math. 92 (1988), 217-241. [Bc] Bruce, J.W.: Stratifying jet spaces-some special cases, Math. Proc. Cam• bridge Phi/os. Soc. 85 (1979) 439-444. [BG] Bruce, J.W., Giblin, P.J.: A stratification of the space of plane quartic curves, Proc. London Math. Soc. 42 (1981),270-298. [BW] Bruce, J.W., Wall, C.T.C.: On the classification of cubic surfaces, J. London Math. Soc. 19 (1979),245-256. [BV] Burghelea, D., Verona, A.: Local homological properties of analytic sets, M anuscripta Math. 7 (1972), 55-66. [ChI] Cheniot, D.: Le theoreme de Van Kampen sur Ie groupe fondamental du complementaire d'une courbe algebrique plane, Sem. F. Norguet 1971/72, Lecture Notes in Mathematics 409, Springer-Verlag, Berlin, 1974. [Ch2] Cheniot, D.: Vne demonstration du theoreme de Zariski sur les sections hyperplanes d'une hypersurface projective et du theoreme du Van Kampen sur Ie group fondamental du compiementaire d'une courbe pro• jective plane, Compositio Math. 27 (1973), 141-158. [Cd] Choudary, A.D.R.: On the resultant hypersurface, Pacific J. Math. 142 (1990),259-263. [Cl] Clemens, C.H.: Double solids, Adv. in Math. 47 (1983),107-230. [CM] Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups, 2nd edn., Springer-Verlag, Berlin, 1965. [CF] Crowell, R., Fox, R.: Knot Theory, Ginn, Boston, 1963. [Del] Deligne, P.: Theorie de Hodge, II and III, Publ. Math. IHES 40 (1971), 5-58 and 44 (1974), 5-77. [De2] Deligne, P.: Le groupe fondamental du complement d'une courbe plane n'ayant que des points double ordinaires est abelian, Slim. Bourbaki, 1979/ References 251

80, Lecture Notes in Mathematics 842, Springer-Verlag, Berlin 1981, pp. 1-10. [DD] Deligne, P., Dimca, A.: Filtrations de Hodge et par I'ordre du pole pour les hypersurfaces singulieres, Ann. Sci. Ecole Norm. Sup. 23 (1990), 645- 656. [tDP] tom Dieck, T., Petrie, T.: Contractible affine surfaces of Kodaira dimen• sion one, Japan J. Math. 16 (1990), 147-169. [Dt] Dimca, A.: Monodromy and Betti numbers of weighted complete intersec• tions, Topology 24 (1985),369-374. [D2] Dimca, A.: On the homology and cohomology of complete intersections with isolated singularities, Compositio Math. 58 (1986), 321-339. [D3] Dimca, A.: Singularities and coverings of weighted complete intersections, J. Reine Angew. Math. 366 (1986),184-193. [D4] Dimca, A.: Topics on Real and Complex Singularities, Vieweg Advanced Lectures in Mathematics, Vieweg, Braunschweig/Wiesbaden 1987. [D5] Dimca, A.: On the Milnor fibrations of weighted homogeneous polynomi• als, Compositio Math. 76 (1990),19-47. [D6] Dimca, A.: Betti numbers of hypersurfaces and defects of linear systems, Duke Math. J. 60 (1990), 285-298. [D7] Dimca, A.: On the connectivity of affine hypersurfaces, Topology 29 (1990), 511-514. [D8] Dimca, A.: Differential forms and hypersurface singularities, in: Singu• larity Theory and its Applications, Warwick 1989, Proceedings, Lecture Notes in Mathematics 1462, Springer-Verlag, Berlin, 1991, pp. 122-153. [D9] Dimca, A.: On the de Rham cohomology of a hypersurface complement, Amer. J. Math. 113(1991),763-771. [DDv] Dimca, A., Dimiev, S.: On analytic coverings of weighted projective spaces, Bull. London Math. Soc. 17 (1985), 234-238. [Dol] Dold, A.: Partitions of unity in the theory of fibrations, Ann. of Math. 78 (1963),223-255. [D02] Dold, A.: Lectures on Algebraic Topology, 2nd ed., Springer-Verlag, New York, 1980. [Dgl] Dolgachev, I.: Weighted projective varieties, in: Group Actions and Vector Fields, Proceedings 1981, Lecture Notes in Mathematics 956, Springer• Verlag, Berlin, 1982, pp. 34-71. [Dg2] Dolgachev, I.: Integral quadratic forms: applications to algebraic geome• try (after V. Nikulin). Sem. Bourbaki 1982/83, Exp. 611, Astl?risque 105- 106, Soc. Math. de France, 1983. [DgLi] Dolgachev, I., Libgober, A.: On the fundamental group ofthe complement to a discriminant variety, in: Algebraic Geometry, Chicago 1980, Lecture Notes in Mathematics 862, Springer-Verlag, New York, 1981, pp. 1-25. [DS] van Doorn, M.G.M., Steenbrink, J.H.M.: A supplement to the mono• dromy theorem, Abh. Math. Sem. Univ. Hamburg 59(1989), 225-233. [Dfl] Durfee, A.H.: Foliations of odd-dimensional spheres, Ann. of Math. 96 (1972),407-411. [Df2] Durfee, A.H.: The characteristic polynomial of the monodromy, Pacific J. Math. 59 (1975), 21-26. [Df3] Durfee, A.H.: The signature of smoothings of complex surface singularities, Math. Ann. 232 (1978),85-98. 252 References

[Df4] Durfee, A.H.: Fifteen characterizations of rational double points and simple critical points, Enseign. Math. 25 (1979), 131-163. [Df5] Durfee, A.H.: A naive guide to mixed Hodge theory, Proc. Symp. Pure Math. 40, Part I (Arcata Singularities Conferences), American Mathemati• cal Society, 1983, pp. 313-320. [Df6] Durfee, A.H.: Neighborhoods of algebraic sets. Trans. Amer. Math. Soc. 276 (1983),517-530. [Df7] Durfee, A.H.: Mixed Hodge structures on punctured neighborhoods, Duke Math. J. 50 (1983),1017-1040. [DfS] Durfee, A.H.: Algebraic varieties which are disjoint union of subvarieties, in: Geometry and Topology: Manifolds, Varieties, and Knots, Marcel Dekker, New York, 1987, pp. 99-102. [DfK] Durfee, A.H., Kauffman, L.: Periodicity of branched cyclic covers, Math. Ann. 218(1975),157-174. [El] Ebeling, W.: Quadratische Formen und Monodromiegruppen von Singu• laritiiten, Math. Ann. 255 (1981), 463-498. [E2J Ebeling, W.: Milnor lattices and geometric bases of some special singu• larities, in: Noeuds, Tresses et Singularites, Monographie No. 31 de I'Enseignement Math., Universite de Geneve, 1983, pp. 129-146; Enseign. Math. 29 (1983), 263-280. [E3J Ebeling, W.: The Monodromy Groups of Isolated Singularities of Complete Intersections, Lecture Notes in Mathematics 1293, Springer-Verlag, Berlin, 1987. [E4J Ebeling, W.: An example of two homeomorphic, nondiffeomorphic com• plete intersection surfaces, Invent. Math 99 (1990), 651-654. [EOJ Ebeling, W., Okonek, c.: Donaldson invariants, monodromy groups and singularities, Internat. J. Math. 1 (1990),233-250. [EW] Ebeling, W., Wall, C.T.C.: Kodaira singularities and an extension of Arnold's strange duality, Compositio Math. 56 (1985) 3-77. [EhJ Ehresmann, c.: Sur les espaces fibres differentiables, c. R. Acad. Sci. Paris 224(1947),1611-1612. [EmJ Ephraim, R.: C1-preservation of multiplicity, Duke Math. J. 43 (1976), 797-803. [EsJ Esnault, H.: Fibre de Milnor d'un cone sur une courbe plane singuliere, Invent. Math. 68 (1982) 477-496. [FSJ Fintushel, R., Stern, R.: Instanton homology of Seifert fibered homology 3-spheres, Proc. London Math. Soc. 61 (1990), 109-137. [Fr] Fried, D.: Monodromy and dynamical systems, Topology 25 (1986), 443- 453. [Fj] Fujiki, A.: Duality of mixed Hodge structures of algebraic varieties, Publ. Res. Inst. Math. Sci. Kyoto Univ.16 (1980) 635-667. [FJ Fulton, W.: On the fundamental group ofthe complement of a node curve, Ann. of Math. 111 (1980),407-409. [GKJ Gabrielov, A.M., Kushnirenko, A.G.: Description of deformations with constant Milnor number for homogeneous functions, Functional Anal. App/. 9 (1975),329-331. [GW] van Geemen, B., Werner, J.: Nodal quintics in jp4 in: Arithmetic of Com• plex Manifolds, Proceedings, 1988, Lecture Notes in Mathematics 1399, Springer-Verlag, Berlin, 1988, pp. 48-59. References 253

[GWPL] Gibson, CG., Wirthmiiller, K., du Plessis, A.A., Looijenga, EJ.N.: Topo• logical Stability of Smooth Mappings, Lecture Notes in Mathematics 552, Springer-Verlag, Berlin, 1976. [GG] Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities, Graduate Text in Mathematics 14, Springer-Verlag, Berlin, 1973. [GM] Goresky, M., MacPherson, R.: Stratified Morse Theory, Ergebnisse der Math., 3. Folge, Band 14, Springer-Verlag, Berlin, 1988. [Gn] Greenberg, M.: Lectures on Algebraic Topology, W.A. Benjamin, New York,1966. [Gl] Greuel, G.-M.: Der Gauss-Manin-Zusammenhang isolierter Singu• laritiiten von vollstiindigen Duchschnitten, Math. Ann. 214 (1975), 235- 266. [G] Griffiths, Ph.: On the periods of certain rational integrals, I, II, Ann. of Math. 90 (1969), 460-541. [GH] Griffiths, Ph., Harris, J.: Principles of Algebraic Geometry, Wiley, New York,1978. [Gr] Grothendieck, A.: On the de Rham cohomology of algebraic varieties, Publ. Math. IRES 29 (1966),351-358. [HL] Hit Huy Vui, Le Dung Trang: Sur la topologie des polynomes complexes, Acta Math. Vietnam 9 (1984),21-32. [HI] Hamm, H.: Lokale topologische Eigenschaften komplexer Riiume, Math. Ann. 191 (1971),235-252.

[H2] Hamm, H.: Genus Xy of quasihomogeneous complete intersections, Func• tional Anal. Appl. 11 (1978), 78-79. [H3] Hamm, H.: Zum Homotopietyp Steinscher Raume, J. Reine Angew. Math. 338 (1983),121-135. [H4] Hamm, H.: Lefschetz theorems for singular varieties, Proc. Symp. Pure Math. 40, Part I (Arcata Singularities Conference), American Mathemati• cal Society, 1983, pp. 547-557. [HLel] Hamm, H., Le Dung Trang: Un theofl!me de Zariski du type de Lefschetz, Ann. Sci. Ecole Norm. Sup. 6 (1973),317-366. [HLe2] Hamm, H., Le Dung Trang: Rectified homotopical depth and Grothendieck conjectures, in: The Grothendieck Festschrift I I, Progress in Mathematics 87, Birkhiiuser, Boston, 1990, pp. 311-351. [Ha] Hardt, R.: Topological properties of sub analytic sets, Trans. Amer. Math. Soc. 211 (1975),57-70. [Hs] Harris, J.: On the Severi problem, Invent. Math. 84 (1986), 445-461. [Hn] Hartshorne, R.: Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, Berlin, 1977. [Hil] Hironaka, H.: Subanalytic sets, in: Number Theory, Algebraic Geometry and Commutative Algebra, volume in honor of A. Akizuki, Kinokunyo, Tokyo, 1973, pp. 453-493. [Hi2] Hironaka, H.: Triangulations of algebraic sets, Proc. Symp. Pure Math. 29 (Arcata Algebraic Geometry Conference), American Mathematical Society, 1974,pp.165-185. [Hr] Hirsch, M.W.: Differential Topology, Graduate Texts in Mathematics 33, Springer-Verlag, Berlin, 1976. [Hzl] Hirzebruch, F.: Topological Methods in Algebraic Geometry, 3rd ed., Springer-Verlag, Berlin, 1966. 254 References

[Hz2] Hirzebruch, F.: Singularities and exotic spheres, Sem. Bourbaki 1966/67, Exp.314. [Hz3] Hirzebruch, F.: Some examples of threefolds with trivial canonical bundle, in: "Collected Papers II", Springer-Verlag, New York, 1987, pp. 757- 770. [Hu] Husemoller, D.: Fibre Bundles, McGraw-Hill, New York, 1966. [Ii] Iitaka, S.: Algebraic Geometry, Graduate Texts in Mathematics 76, Springer-Verlag, New York, 1982. [vK] van Kampen, E.R.: On the fundamental group of an algebraic curve, Amer. J. Math. 55 (1933),255-260. [Ks] Karchyauskas, K.K.: A generalized Lefschetz theorem, Functional Anal. Appl. 11 (1977), 312-313. [Kp] Karpishpan, Y.: Pole order filtration on the cohomology of algebraic links, Compositio Math. 78(1991), 213-226. [Ka] Kato, M.: Topology of k-regular spaces and algebraic sets, in: Manifolds, Tokyo 1973, Proceedings, University of Tokyo Press, Tokyo, 1975, pp. 153- 159. [KM] Kato, M., Matsumoto, Y.: On the connectivity of the Milnor fiber of a holomorphic function at a critical point, in: Manifolds, Tokyo 1973, Pro• ceedings,pp.131-136, University of Tokyo Press, Tokyo, 1975. [KK] Kaup, B., Kaup, L.: Holomorphic Functions of Several Variables, de Gruyter Studies in Mathematics 3,1983. [Kws] Kawasaki, T.: Cohomology of twisted projective spaces and lens com• plexes, Math. Ann. 206 (1973),243-248. [KeMi] Kervaire, M., Milnor, J.: Groups of homotopy spheres I, Ann. of Math. 77 (1963),504-537. [Kw] Kirwan, F.: An introduction to intersection homology theory, Pitman Re• search Notes in Mathematics, Longman, London, 1988. [Kl] Kleiman, S.L.: The enumerative theory of singularities, in: Real and Com• plex Singularities (Oslo 1976), Sijthoff and Noordhoff, Amsterdam 1977, pp. 297 - 396. [Ku] Kushnirenko, A.G.: Polyedres de Newton et nombres de Milnor, Invent. Math. 32 (1976), 1-3l. [Ll] Lamotke, K.: Die Homologie isolierter Singularitiiten, Math. Z. 143 (1975),27-44. [U] Lamotke, K.: The topology of complex projective varieties after S. Lefschetz, Topology 20 (1981),15-51. [L3] Lamotke, K.: Regular Solids and Isolated Singularities, Vieweg Advanced Lectures in Mathematics, Vieweg, Braunschweig/Wiesbaden 1986. [La] Lang, S.: Algebra, Addison-Wesley, Boston, 1965. [Lft] Laufer, H.: Normal Two-Dimensional singularities, Annals of Mathemati• cal Studies 71, Princeton University Press, Princeton, 1971. [Lw] Lawson, H.B.: Codimension-one foliations of spheres, Ann. of Math. 94 (1971),494-503. [LeI] Le Dung Trang: Sur les noeuds algebriques, Compositio Math. 25 (1972), 281-321. [Le2] Le Dung Trang: Topologie des singularites des hypersurfaces complexes, Asterisque 7/8 (Singularites a Cargese) 1973, pp. 171-182. [Le3] Le Dung Trang: Some remarks on relative monodromy, in: Real and Com- References 255

plex Singularities (Oslo 1976), Sijthoff and Noordhoff, Amsterdam, 1977, pp.397-403. [UR] U: Dung Trang, Ramanujam, c.P.: The invariance of Milnor number implies the invariance of the topological type, Amer. J. Math. 98 (1976), 67-78. [LeT] Le Dung Trang, Teissier, B.: Cycles evanescents et conditions de Whitney II, Proc. Symp. Pure Math. 40, Part 2 (Arcata Singularities Conference), American Mathematical Society 1983, pp. 65-103. [Lf2] Lefschetz, S.: Analysis Situs et la Geometrie Algebrique, Gauthier-Villars, Paris, 1924. [Lil] Libgober, A.: Some properties of the signature of complete intersections, Proc. Amer. Math. Soc. 79 (1980), 373-375. [Li2] Libgober A.: Alexander polynomial of plane algebraic curves and cyclic multiple planes, Duke Math. J. 49 (1982),833-851. [Li3] Libgober, A.: Alexander invariants of plane algebraic curves, Proc. Symp. Pure Math. 40, Part 2 (Arcata Singularities Conference). American Mathe• matical Society, 1983, pp. 135-144. [Li4] Libgober, A.: Homotopy groups of the complements to singular hyper• surfaces, Bull. Amer. Math. Soc. 13 (1985), 49-5\. [Li5] Libgober, A.: Invariants of plane algebraic curves via representations of the braid groups, Invent. Math. 95 (1989), 25-30. [Li6] Libgober, A.: Fundamental groups of the complements to plane singular curves, Proc. Symp. Pure Math. 46, Part 2 (Bowdoin Algebraic Geometry Conference), American Mathematical Society 1987, pp. 29-45. [LiWl] Libgober, A., Wood, J.W.: On the topological structure of even-dimen• sional complete intersections, Trans. Amer. Math. Soc. 267 (1981), 637- 660. [LiW2] Libgober, A., Wood, J.W.: Differentiable structures on complete intersec• tions I, Topology 21 (1982),469-482. [LiW3] Libgober, A., Wood, IW.: Uniqueness ofthe complex structure on Kahler manifolds of certain homotopy types, J. Differential Geom. 32 (1990),139- 154. [LV] Loeser, F., Vaquie, M.: Le polynome d'Alexander d'une courbe plane pro• jective, Topology 29 (1990),163-173. [Lo] Lojasiewicz S.: Ensembles semianalytiques, preprint IHES, 1972. [Lg] Looijenga, E.J.N.: Isolated Singular Points on Complete Intersections, Lon• don Mathematical Society Lecture Note Series 77, Cambridge University Press, Cambridge, 1984. [LgWa] Looijenga, E.IN., Wahl, J.: Quadratic functions and smoothing surface singularities, Topology 25 (1986),261-291. [My] Massey, D.B.: The U varieties, J, Invent. Math. 99 (1990),357-376. [Ma] Mather, J.: Notes on Topological Stability, Harvard University, Cam• bridge, 1970. [Ms] Matsumura, H.: Commutative Algebra, W.A. Benjamin, New York, 1970. [Mc] McClearly, J.: User's Guide to Spectral Sequences, Publish or Perish, 1985. [Ml] Milnor, J.: Construction of universal bundles II, Ann. of Math. 63 (1956), 430-436. [M2] Milnor, J.: On simply-connected 4-manifolds, Int. Symp. Alg. Topology, Mexico City 1958, pp. 122-128. 256 References

[M3] Milnor, J.: Morse Theory, Annals of Mathematical Studies 51, Princeton University Press, Princeton, 1963. [M4] Milnor, J.: Topology from the Differential Viewpoint, The University Press of Virginia, Charlottesville, 1965. [M5] Milnor, J.: Singular Points of Complex Hypersurfaces, Annals of Mathe• matical Studies 61, Princeton University Press, Princeton, 1968. [M6] Milnor, J.: On the 3-dimensional Brieskorn manifolds M(p, q, r), in: Knots, Groups, and 3-Manifolds-papers dedicated to the memory of R.H. Fox, Annals of Mathematical Studies 84, Princeton University Press, Princeton, 1975, pp. 175-225. [MH] Milnor, J., Husemoller, D.: Symmetric Bilinear Forms, Springer-Verlag, Berlin, 1973. [MO] Milnor, J., Orlik. P.: Isolated singularities defined by weighted homoge• neous polynomials, Topology 9 (1970),385-393. [Mii] Muller, G.: Reduktive Automorphismengruppen analytischer C-Algebren, J. Reine Angew. Math. 364 (1986),26-34. [Mul] Mumford, D.: The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math. JHES 9, (1961), 5-22. [Mu2] Mumford, D.: Algebraic Geometry J, Complex Projective Varieties, Grundlehren der math. Wiss. 221, Springer-Verlag, Berlin, 1976. [Ng] Nagata, M.: Polynomial rings and affine spaces, Reg. Con! Series in M ath• ematics 37, American Mathematical Society, 1978. [Na] Navarro Aznar, V.: Sur la theorie de Hodge-Deligne, Invent. Math. 90 (1987),11-76. [Nl] Nemethi, A.: Theorie de Lefschetz pour les varietes algebriques affines. C. R. Acad. Sci. Paris, 303, Serie T, No. 12 (1986),567-570. [N2] Nemethi, A.: On the fundamental group of the complement of certain singular plane curves, Math. Proc. Cambridge Phi/os. Soc. 102 (1987), 453- 457. [N3] Nemethi, A.: Lefschetz theory for complex affine varieties, Rev. Roumaine Math. Pures Appl. 33 (1988),233-250. [NZ] Nemethi, A., Zaharia, A.: On the bifurcation set of a polynomial function and Newton boundary, Publ. Res. Inst. Math. Sci. Kyoto Univ. 26 (1990), 681-689. [NW] Neumann, W., Wahl, J.: Casson invariant of links of singularities, Com• ment. Math. H elv. 65 (1990), 58-78. [Nk] Nikulin, V.V.: Integral symmetric bilinear forms and some of their appli• cations,lzv. Akad. Nauk SSR 43(1979),111-177. [01] Oka, M.: On the homotopy types of hypersurfaces defined by weighted homogeneous polynomials, Topology 12 (1973),19-32. [02] Oka, M.: On the cohomology structure of projective varieties, in: Mani• folds, Tokyo 1973, Proceedings, University of Tokyo Press, Tokyo, 1975, pp.137-143. [03] Ok a, M.: The monodromy of a curve with ordinary double points, Invent. Math. 27 (1974),157-164. [04] Oka, M.: Some plane curves whose complements have non-abelian funda• mental groups, Math. Ann. 218 (1975), 55-65. [05] Oka, M.: On the fundamental group of the complement of certain plane curves, J. Math. Soc. Japan 30 (1978),579-597. References 257

[06] Oka, M.: Symmetric plane curves with nodes and cusps, preprint 1991. [Om] Olum, P.: Nonabelian cohomology and van Kampen's theorem, Ann. of Math. 68 (1958),658-668. [OR] Orlik, P., Randell, R.: The monodromy of weighted homogeneous singu• larities, Invent. Math. 39 (1977),199-211. [OS] Orlik, P., Solomon, L.: Singularities I, Hypersurfaces with an isolated sin• gularity, Adv. in Math. 27 (1978),256-272. [OW] Orlik, P., Wagreich, Ph.: Equivariant resolution of singularities with C*• action, in: Proceedings of the Second Conference on Compact Transforma• tion Groups II, Lecture Notes in Mathematics 299, Springer-Verlag, Berlin, 1972, pp. 270-290. CPa] Parusinski, A.: A generalization of the Milnor number, Math. Ann. 281 (1988),247-254. [Ph] Pham, F.: Formules de Picard-Lefschetz generalisees et ramification des integrales, Bull. Soc. Math. France 93 (1965),333-367. [P] Prill, D.: Local classification of quotients of complex manifolds by discon• tinuous groups, Duke Math. J. 31 (1964), 375-386. [Rm] Ramanujam, C.P.: A topological characterization ofthe affine plane as an algebraic variety, Ann. of Math. 94 (1971),69-88. [Ra] Randell, R.: On the fundamental group of the complement of a singular plane curve, Quart. J. Math. Oxford 31 (1980), 71-79. [Rd] Reid, M.: Young person's guide to canonical singularities, Proc. Amer. Math. Soc. Summer Institute Bowdoin 1985, Proc. Sympos. Pure Math. 46 (1987), Part I, pp. 345-414. [Ro] Roan, S.-S.: On Calabi-Yau orbifolds in weighted projective spaces, Inter• nat. J. Math. 1 (1990), 211-232. [R] Rolfsen, D.: Knots and Links, Mathematical Lecture Series 7, Publish or Perish, 1976. [Ru] Russell, P.: Simple birational extensions of two-dimensional affine ratio• nal domains, Compositio Math. 33 (1976),197-208. [Sa] Sakamoto, K.: Milnor fiberings and their characteristic maps, in: Mani• folds, Tokyo 1973, Proceedings, University of Tokyo Press, Tokyo, 1975, pp. 145-150. [Sy] Sathaye, A.: On linear planes, Proc. Amer. Math. Soc. 56(1976),1-7. [Sch] Schoen, c.: Algebraic cycles on certain desingularized nodal hypersurfaces, Math. Ann. 270 (1985), 17-27. CST] Sebastiani, M., Thorn, R.: Un result at sur la monodromie, Invent. Math. 13 (1971),90-96.

[Se] Serre, I.-P.: Arbres, amalgames, SL2 , Asterisque 46, Soc. Math. France, 1977. [Sil] Siersma, D.: Classification and deformation of singularities, Ph.D. Thesis, Amsterdam, 1974. [Si2] Siersma, D.: Isolated line singularities, Proc. Symp. Pure Math. 40, Part 2 (Arcata Singularities Conference), American Mathematical Society, 1983, pp. 485-496. [Si3] Siersma, D.: Quasihomogeneous singularities with transversal type AI' Contemporary Mathematics 90, American Mathematical Society, 1989, pp.261-294. 258 References

[Si4] Siersma, D.: Variation mappings on singularities with a I-dimensional critical locus, Topology 30 (1991), 445-469. [Sm] Smale, S.: On the structure of manifolds, Amer. J. Math. 84 (1962), 387- 399. [Sp] Spanier, E.H.: Algebraic Topology, McGraw-Hill, New York, 1966. [S I] Steenbrink, J.H.M.: Mixed Hodge structures on the vanishing cohomology, in: Real and Complex Singularities (Oslo 1976), SijthofT and NoordhofT, Amsterdam, 1977, pp. 525-563. [S2] Steenbrink, J.H.M.: Intersection form for quasi-homogeneous singulari• ties, Compositio Math. 34 (1977),211-223. [S3J Steen brink, J.H.M.: Mixed Hodge structures associated with isolated sin• gularities. Proc. Symp. Pure Math. 40, Part II (Arcata Singularities Confer• ence), American Mathematical Society, 1983, pp. 513-536. [S4] Steenbrink, J.H.M.: Semicontinuity of the singularity spectrum, Invent. Math. 79(1985),557-565. [S5] Steenbrink, J.H.M.: talk at ICM'90 (Kyoto). [S6] Steenbrink, J.H.M.: Mixed Hodge Structures and Singularities (in prepara• tion). [Sn] Stein, K.: Analytische Zerlegungen Komplexer Riiume, Math. Ann. 132 (1956),63-93. [Sv] Stevens, J.: Periodicity of branched cyclic covers of manifolds with open book decomposition, Math. Ann. 273 (1986),227-239. [Sr] van Straten, D.: On the Betti numbers of the Milnor fiber of a certain class of hypersurface singularities, Lecture Notes in Mathematics 1273, Springer-Verlag, Berlin, 1987, pp. 203-220. [Sz] Szczepanski, S.: Criteria for topological equivalence and the Le• Ramanujan theorem for three complex variables, Duke Math. J. 58 (1989), 513-530. [Tl] Teissier, B.: Cycles evanescents, sections planes et conditions de Whitney, Asterisque 718 (Singularites a Cargese) 1973, pp. 285-362. [T2] Teissier, B.: Introduction to equisingularity problems, Proc. Symp. Pure Math. 29 (Arcata Algebraic Geometry Coriference), American Mathemati• cal Society, 1974, pp. 593-632. [Td] Teodosiu, G.: A class of analytic coverings ramified over u3 = v2 , J. Lon• don Math. Soc. (2) 38 (1988), 231-242. [Tr] Trotman, D.: Comparing regularity condit' ;)ns on stratifications, Proc. Symp. Pure Math. 40, Part 2 (Arcata Singularities Co"!ference), American Mathematical Society, 1983, pp. 575-586. [V] Verdier, J.-L.: Stratifications de Whitney et theoreme de Bertini-Sard, Invent. Math. 36(1976), 295-312. [Wa] Wagreich, Ph.: The structure of quasihomogeneous singularities, Proc. Symp. Pure Math. 40, Part 2 (Arcata Singularities Conference), American Mathematical Society, 1983, pp. 593-611. [WI] Wall, C.T.c.: Regular stratifications, in: Dynamical Systems-Warwick 1974, Lecture Notes in Mathematics, 468, Springer-Verlag, Berlin, 1974, pp.332-344. [W2] Wall, c.T.c.: Classification of unimodal isolated singularities of complete intersections, Proc. Symp. Pure Math. 40 Part 2 (Arcata Singularities Con• ference), American Mathematical Society, 1983, pp. 625-640. References 259

[We] Wells, R.O.: Differential Analysis on Complex Manifolds, Graduate Texts in Mathematics 65, Springer-Verlag, Berlin, 1980. [Wn] Werner, J.: Kleine Auflosungen spezieller dreidimensionaler Varietaten, Bonner Math. Schriften 186 (1987). [Whl] Whitney, H.: Local properties of analytic varieties, in: Differential and Combinatoric Topology, Princeton University Press, Princeton, 1965, pp. 205-244. [Wh2] Whitney, H.: Tangents to an analytic variety, Ann. of Math. 81 (1965), 496-549. [Wdl] Wood, J.: Some criteria for finite and infinite monodromy of plane alge• braic curves, Invent. Math. 26 (1974), 179-185. [Wd2] Wood, J.: A connected sum decomposition for complete intersections, Proc. Symp. Pure Math. 32, Part 2, American Mathematical Society, 1978, pp.191-193. [XY] Xu, Y.-I., Yau, S.S.-T.: Durfee conjecture and coordinate free characteriza• tion of homogeneous singularities, preprint 1990, University of Illinois at Chicago. [Ym] Yamamoto, M.: Classification of isolated algebraic singularities by their Alexander polynomials, Topology 23 (1984), 277-287. [Za] Zaidenberg, M.G.: An analytic cancellation theorem and exotic algebraic structures on C', n ;?: 3, Max-Planck-Institut, Preprint, 1991. [ZI] Zariski, 0.: On the problem of existence of algebraic functions of two variables possessing a given branch curve, Amer. J. Math. 51 (1929). [Z2] Zariski, 0.: Algebraic Surfaces, Springer-Verlag, Berlin, 1935, 2nd supplemented Edn. 1971. [Z3] Zariski, 0.: On the Poincare group of rational plane curves, Amer. J. Math. 58(1936), 1-14. [Z4] Zariski, 0.: A theorem on the Poincare group of an algebraic hypersurface, Ann. of Math. 38 (1937),131-141. [Z5] Zariski, 0.: Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481-491 Index

Ak-singularities 13,59-60,95,203,222, cubic curve 194,216 226 cubic surfaces 165-166 adjunction formula 141 curve singularity 41-49, 100, 117-120, affine hypersurface 19-22,28, 103, 125,220 174-176,244 cusp singularities 13, 60-62,207,223, affine variety 26,28, 182,233 245 Alexander ideal 38 CW-complex 27,139 Alexander invariant 35-36 cylindric structure 26, 75, 157 Alexander matrix 38, 92 Alexander polynomial ofa hypersurface 106-110,206-216 Dk-binary dihedral group 111 of a knot 38-40, 44 Dk-singularities 13, 59-60, 222, 226 local 206-207 defect of a linear system 207-208 degeneration 121, 126, 128, 138, 214-215 Borromean rings 41 de Rham-Koszul complex 190 braid groups 113-115 differential forms 177-180 BrianIVon-Speder example 12 discriminant Brieskorn-Pham singularities 94-99, form 225-226 105-106,211 group 220, 226 hypersurface 81, 132 distinguished basis 83 Casson invariant 98-99 Dolgachev numbers 63, 224 Cayley-Bacharach theorem 212 dual hypersurface 14,131 Chern classes 151-152 Durfee conjecture 159 compact polyhedron 27, 75 du Val singularities 13,59-60,222, complete intersection 226 locally 25 Dynkin diagram 60,221-224 projective 142-176 singularity 24-25, 76,80-81,90 conic structure 23, 163 Ek-singularities 13,59-60,222,226 connectivity 76, 79 Ek-singularities 13, 63-64, 203, contraction 222-223 of a resolution graph 56, 60, 62 Ehresmann's Fibration Theorem 15 with a vector field 179-180 Eilenberg-MacLane space 32, 115, covering 146 cyclic 34-35,90,234 n-equivalence 25 universal 28, 112 equivalent germs 2 262 Index

essential singularity 205, 207 form 37,52,86, 89 Euler vector field 179, 181 matrix 52, 83 exceptional singularities 13, 62-63, number 45,89, 100 223-224

join 27,87-88,135-137 filtration lfodge 185-188,202,217,240 polar 184-188,202,217 K3 surface 160, 236 weight 241,245 Kahler manifold 239 Fitting ideal 38, 44 Knot 29 foliations of a sphere 98 algebraic 42 free product of groups 56, 111, 134 fibered 39 fundamental group 27,105, 116 (p,q) torus 31,33-35,43 of a hypersurface complement 26, trefoil 31,40 80,101-138 tri vial 30, 33 of a knot complement 32-40 of a link 53-58 local 2,103-104 lattice 219 even/odd 154-155,219 morphism of 220 Gabrielov numbers 63, 224 nondegenerate 219 graph 94, 113 reduced 220 of a resolution 50,60,218,229,245 skew-symmetric 220 Grothendieck residue 49 unimodular 219 group Lefschetz number 75, 108 isotropic (subgroup) 225 Lefschetz theorem 25 isotropy 197 lens space 59 of a knot 32 line configuration 212-213 reflection 57 link 23,29,66,245 small 57-58 algebraic 42 Gysin sequence 46, 77, 198 fibered 39 at infinity 28, 175 trivial 30,41 lfeisenberg group 64 linking form 227-228 lfirzebruch Index Theorem 153 linking number 33, 45 lfirzebruch -lung singularities 58,231 lfodge numbers 236,240-241,246 lfodge structures 240-247 Milnor homology manifold 164,232, 243 algebra 193,236,244 lfopf bundle 32, 139, 142 fiber 68-75,82,133,163,237 hyperbolic singularities 13,60-62,207, lattice 83,89-90,157-158,161,167, 223, 245 170-172,220,227-229 (hyper)plane section 15,25-26,133- Milnor number 10, 78, 162-163 135, 137 global 21 hypersurface /l-constant 10, 12 nodal 17-19,208-210,218 /l*-constant 12 quadratic 194 /l-determinacy 125 smooth projective 15, 109, 152 /l-equivalent 10 /l*-invariant 11 -12 monodromy index 98,221,237,240 homeomorphism 40, 73-74 inflectional tangent 120,127,186 operator 40,44, 73-75, 86 intersection relation 119-120, 127 cohomology 217 morsification 82 Index 263 normal bundle 141, 152, 155 sphere normal crossing singularity 79, 105 exotic 97 -98 normal singularity 50, 55, 104 homology 53, 60, 93-97 stabilization of singularities 89 Stein variety 26,69, 182 parallelizable manifold 75,97 stratification 3 Picard-Lefschetz transformation 85 Samuel 4 pinch point 4, 11, 24 topologically locally trivial 8 Plucker formula 14 Whitney (regular) 4,6-7, 17, 135 Poincare icosahedral sphere 60 stratum 3 Poincare-Leray residue 46-47, surface singularity 49-67, 245 192-193,200 Poincare series 65 pole (order of the) 184 Tp .•. r -singularities 13, 60-62, 207, 223, Pontrjagin classes 153 245 presentation 38, Ill, 127 Thorn's First Isotopy Lemma 16 matrix 38-40 Thom-Sebastiani construction 27, primitive (co)homology 146-147,234 86-90, 135 primitive embedding 90, 224, 227 Tjurina number 81 projective cone 169-170 transversal singularity 197 projective space 139-142,230-232 triangle singularities 13, 62-63, 223- (co)homology 109, 145, 147-148, 224 167 "tubular" neighborhood 148-151 quartic curves 126, 129-133 unimodular matrix/lattice 37,53,164 quasismooth 232 quotient singularities 57-60,231-233 vanishing cycle 83 rational double points 13,59-60,222, Veronese embedding 15 226 Veronese variety 15 resolution of singularities 50, 52, 59- 63,66-67,218,228,243,245 resultant hypersurface 80 Wang sequence 61, 73 weight 64 weighted homogeneous Seifert projective space 105,230-237 invariants 65 projective variety 230-237 matrix 36-38,85-86 singularity 64-67, 71-74, 79, 87, surface 34 202-204,236,243-244 semi weighted homogeneous Whitney regularity 3, 12 singularity 12, 74, 125,204 Whitney umbrella 4, 11, 24 signature 221,244 simple-elliptic singularities 13, 63-64, 203,222-223 Zariski conjecture 2 Smith-Gysin sequence 231 Zariski sextic curves 18, 134-138, Smith theory 173-174 210-211 Universitext ( conlinued)

Nikulin/Shafarevich: Geometries and Groups 0ksendal: Stochastic Differential Equations Rees: Notes on Geometry Reisel: Elementary Theory of Metric Spaces Rey: Introduction to Robust and Quasi-Robust Statistical Methods Rickart: Natural Function Algebras Rotman: Galois Theory Rybakowskl: The Homotopy Index and Partial Differential Equations Samelson: Notes on Lie Algebras Smith: Power Series From a Computational Point of View Smorynskl: Logical Number Theory I: An Introduction Smorynskl: Self-Reference and Modal Logic Stillwell: Geometry of Surfaces Stmock: An Introduction to the Theory of Large Deviations Sunder: An Invitation to von Neumann Algebras Tondeur: Foliations on Riemannian Manifolds Ve.'hulst: Nonlinear Differential Equations and Dynamical Systems Zaanen: Continuity, Integration and Fourier Theory