Cohomology of the complement of a free divisor

Cohomology of the Complement of a Free Divisor Author(s): Francisco J. Castro-Jimenez, Luis Narvaez-Macarro and David Mond Source: Transactions of the American Mathematical Society , Aug., 1996, Vol. 348, No. 8 (Aug., 1996), pp. 3037-3049 Published by: American Mathematical Society Stable URL: https://www.jstor.org/stable/2155121 JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Transactions of the American Mathematical Society This content downloaded from 52.18.63.169 on Wed, 09 Nov 2022 07:40:07 UTC All use subject to https://about.jstor.org/terms TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 348, Number 8, August 1996 COHOMOLOGY OF THE COMPLEMENT OF A FREE DIVISOR FRANCISCO J. CASTRO-JIMENEZ, LUIS NARVAEZ-MACARRO, AND DAVID MOND ABSTRACT. We prove that if D is a "strongly quasihomogeneous" free divisor in the Stein manifold X, and U is its complement, then the de Rham cohomology of U can be computed as the cohomology of the complex of meromorphic differential forms on X with logarithmic poles along D, with exterior derivative. The class of strongly quasihomogeneous free divisors, introduced here, includes free hyperplane arrangements and the discriminants of stable mappings in Mather's nice dimensions (and in particular the discriminants of Coxeter groups). 1. INTRODUCTION The purpose of this paper is to prove the following theorem: Theorem 1.1. Let D be a strongly quasihomogeneous free divisor in the complex manifold X, let U be the complement of D in X, and let j : U -- X be inclusion. Then the natural morphism from the complex Qx (log D) of differential forms with logarithmic poles along D to Rj* Cu is a quasi-isomorphism. The natural morphism referred to here is simply the de Rham morphism, which at presheaf level is just the map sending w e F(V,QP(logD)) to w: Hp(V-VnD; C) -+ C. This result should be contrasted with Grothendieck's Comparison Theorem ([4]): Theorem 1.2. If D is a divisor in the complex manifold X, then the de Rham morphism Qxc (* D) -- Ri* (C u) is a quasi-isomorphism. O Here Qx (*D) is the complex of merornorphic differential forms on X with meromorphic poles (of arbitrary order) along D; Qx (log D) is the subcomplex consisting of forms w such that both w and dw have at most a first-order pole along D. Logarithmic differential forms were first defined in full generality by Kyogi Saito in [12], where the notion of free divisor was also introduced. A hypersurface D C X is a free divisor if the Ox-module Der(log D) of (ambient) vector fields which are tangent to Received by the editors November 4, 1994. 1991 Mathematics Subject Classification. Primary 32S20, 32S25, 14F40; Secondary 52B30, 58C27. The first two authors were supported by DGICYT PB94-1435. (?)1996 American Mathematical Society 3037 This content downloaded from 52.18.63.169 on Wed, 09 Nov 2022 07:40:07 UTC All use subject to https://about.jstor.org/terms 3038 F. J. CASTRO-JIMEtNEZ, LUIS NARVAEZ-MACARRO, AND DAVID MOND D at each of its smooth points, is locally free. A smooth hypersurface is of course a free divisor, as is the union of the coordinate hyperplanes in Ct: in the latter case, Der(log D) is freely generated over 0 by the vector fields x10/Ox1,. ,xtOlxt. The determinant of the matrix of coefficients of the generators of Der(log D) in this example is actually an equation for D; this is always the case for a free divisor, and characterises them (this is "Saito's criterion"; see [12], 1.8). The notion of strong quasihomogeneity is a slight extension of that of quasihomogeneity. We shall use the term weighted homogeneous in the strict sense: a divisor D C Ct is weighted homogeneous if it has a defining equation h which is homogeneous with respect to positive weights wi. Definition 1.3. A divisor D in a t-dimensional complex manifold X is strongly quasihomogeneous if at each point q E D, there are local coordinates (U; x1, .. , xt) centred at q (i.e. with xi(q) = 0 for i = 1,... ,t) with respect to which D n U is weighted homogeneous. Similarly, the polynomial f(xi,... ,Xt) is strongly quasihomogeneous if f 1 (0) is strongly quasihomogeneous. For an example of a weighted homogeneous divisor which is not strongly quasihomogeneous, consider the surface D in 3-space given by f (x, y, z) = x5z + x3y3 + y5z = 0. Clearly D is weighted homogeneous, and thus its germ at each point q 4 0 is analytically trivial (i.e. a product) along the C*- orbit through q. By Lemma 2.3, if the germ (D, q) were isomorphic to a weighted homogeneous germ (D1, 0) then (D, q) would be isomorphic to a product (C, 0) x (D', 0) with D' weighted homogeneous. We show that this cannot be the case at any point q = (0, 0, zo) with zo =A 0. The singular locus of D is the z-axis, and so any product structure in the neighbourhood of a point on this axis must have the axis as a factor. However, if S is a smooth surface meeting the z-axis transversely at q, then the germ at q of fis is not in its own jacobian ideal (this can be easily checked by pulling back the equation f via an embedding (C2, 0) -+ (C3, q)). It follows that the germ of S n D at q is not weighted homogeneous with respect to any system of coordinates centred at q. Natural examples of strongly quasihomogeneous free divisors include the union of reflecting hyperplanes of a complex reflection group, and indeed any free hyperplane arrangement (see e.g [11], Chapter 4). Also if X and Y are complex manifolds with dim X > dim Y, and f: X -- Y is a stable mapping, then its discriminant D(f) is a free divisor; if moreover (dim X, dim Y) are nice dimensions in the sense of J. Mather, then D(f) is also strongly quasihomogeneous. This is discussed in more detail below. Theorem 1.1 has the following corollaries: Corollary 1.4. If X is a Stein manifold, D C X is a strongly quasihomogeneous free divisor, and U is its complement, then Hk (U; C) h k px, Q (log D))). (here we use the lower case hP to denote the cohomology of a complex). Proof. By a standard argument, HP(U; C) = HP(X, Rj*(Cu)), where H means hypercohomology. By Theorem 1.1, This content downloaded from 52.18.63.169 on Wed, 09 Nov 2022 07:40:07 UTC All use subject to https://about.jstor.org/terms COHOMOLOGY OF THE COMPLEMENT OF A FREE DIVISOR 3039 Finally, because X is a Stein space, we have HP(X, Q(log D))- hP(F(X, Q(log D))). OI In particular, Corollary 1.5. Let X and Y be complex manifolds of dimensions n and t, with n > t and (n, t) nice dimensions, and suppose in addition Y is Stein. Iff : X -+ Y is a locally stable map (i.e. each of its multi-germs is stable) with discriminant D, then for all k > 0, hk (F(Y, Qy (log D))) = Hk(y - D; C). Proof. i) D is a free divisor. Since this is a local question, it is enough to consider, for each y E D, the germ f : (x,f -(y) n r) --+(y,y) where E is the critical set of f . Freeness of (D, y) was proved first by Saito (when the germ of f is a deformation of a hypersurface singularity) and then by Looijenga in ([6], 6.13) in the case that the germ of f is a versal deformation of an isolated complete intersection singularity (ICIS). It is well-known (and easy to prove) that if n > t then f is stable if and only if f-1(0) is an ICIS and f is a versal deformation of f-1(0). ii) In the nice dimensions, every stable map-germ is quasihomogeneous, in approriate coordinates (see below). O Corollary 1.6. If D C Ct is the discriminant of a Coxeter group, then for all k > O, hk ((Ct, Qt(log D))) = Hk (Ct - D;C). Proof. It was shown by Brieskorn (cf. [1]) that D is the discriminant of the versal deformation of the isolated hypersurface singularity which bears the name of the group. Thus, as described in the preceding proof, D is a free divisor. All of the discriminants of the Coxeter groups are quasihomogeneous; moreover, at each point x 74 0, D is the product of the discriminant of a Coxeter group and a smooth factor (if irreducible at x) or the union of several such, meeting transversely. Strong quasihomogeneity follows. O It is conjectured by H. Terao and S. Yuzvinsky that for hyperplane arrangements in affine spaces, the algebraic version of Corollary 1.4 holds even without the hypothesis of freeness: Conjecture 1.7. Let A be an arrangement of affine hyperplanes in Ct. Let Qv (log A) be the complex of germs of rational differential forms on Ct with logarithmic poles along the arrangement, with exterior derivative d. Then hP(F(Ct, Qvj9 (log A))) = HP(Ct - A; C) for p > 0. They have proved the following partial affirmative answers: p = ,1,it, t= 1,2,3, A is in general position, and A is (central) generic. This content downloaded from 52.18.63.169 on Wed, 09 Nov 2022 07:40:07 UTC All use subject to https://about.jstor.org/terms 3040 F. J. CASTRO-JIMENEZ, LUIS NARVAEZ-MACARRO, AND DAVID MOND The nice dimensions were defined by Mather in [8] as those dimensions (n, t) such that for any two smooth manifolds N and T of dimensions n and t, stable mappings are dense in the space of all proper smooth maps N -+ T (with its Whitney topology). The nice dimensions can also be described in complex analytic terms as those dimensions (n, t) such that every map-germ f : (Cn, S) -) (Ct, 0) of finite Ae, codimension (i.e. isolated instability) has a 1-parameter deformation {fA}AE(c,o) such that fA is stable for A 74 0. Mather determined exactly which dimensions were nice in [9]. In fact niceness reduces to a purely local question on the existence of moduli for contact equivalence, and the computations in [9] involved listing stable map-germs in the nice dimensions. All of the normal forms in the list are weighted homogeneous. All stable germs in the nice dimensions are obtained from the list (up to biholomorphic coordinate change) by the use of one or both of the following steps: a. trivial unfolding: take fo from the list and define f : (Cn X Ce, 0) (CP x CI,O) by f(x,u) = (fo(x),u); b. (in the case of stable multigerms), by putting together trivial unfoldings of germs from the list, with different domains, in such a way that the isosingular loci of the different germs meet in general position in the target [7]. Moreover, the quasihomogeneity of all stable germs actually characterises the nice dimensions, a fact which surely deserves some explanation (which we cannot give). That D is strongly quasihomogeneous in the nice dimensions follows from what we have just said: for at each point, D is the discriminant of a stable map-germ. We do not know if the theorem is true for all free divisors. The authors are very grateful to Eduard Looijenga for suggesting the outline of the proof of Theorem 1.1, to Sergey Yuzvinsky and Hiroaki Terao for their information about the problem, and for helpful comments on an earlier version of the paper, to the University of Seville for its hospitality and financial support to the third named author in the period during which this work was begun, and to the Spanish Ministry of Education and British Council for a Joint Action-Accion Integrada grant, which made further joint work possible. 2. PROOFS The proof of the main result relies on an easy technical lemma due to Naruki: Lemma 2.1. Suppose that Oct,o (or C[xl,... , xt]) is filtered by giving each variable xi the weight wi, and let X = E wixiOlOxi be the associated Euler vector field (i.e. associated infinitesimal generator of the C*- action). If w is a homogeneous differential form, then Lx(w) = wt(w)w. (Here Lx(w) = dtx(w) + tx(dw) is the Lie derivative of w with respect to X.) Proof. Straightforward calculation. O If w is logarithmic then so is x (w). Thus, the Lie derivative can be used to provide a contracting homotopy on the subcomplex Q (log D)o of the stalk complex Q (logD)o consisting of all germs of forms of fixed weight w 74 0, showing it to be acyclic. It follows that Q (logD)o -4 Q (logD)o This content downloaded from 52.18.63.169 on Wed, 09 Nov 2022 07:40:07 UTC All use subject to https://about.jstor.org/terms COHOMOLOGY OF THE COMPLEMENT OF A FREE DIVISOR 3041 is a quasi-isomorphism. For if w E QP(log D) is closed, then so is each of its homogeneous parts. Writing w = Ej wj, where wj is homogeneous of weight j, we then have wj = 1Lx(wj) = d(Q1txwj) for j 74 0, so w = wo + du, where a = tx Z 1#0 7W1. There is no problem with convergence of the infinite sum, so that a really is a well-defined logarithmic form. Hence the classes of w and wo in hP(Q (log D)o) are equal. A similar argument shows that Q(*D)?- Q(*D)o, is a quasi-isomorphism, where Q (*D)o is the complex of germs at 0 of meromorphic forms with poles of arbitrary order along D. For future use, we record some further easy consequences of this argument. Lemma 2.2. Let X be a complex analytic manifold of dimension n and let D C X be a divisor. Let X := X x C, let 7r X -- X and q : X -- C be the projections, let D := D x C and let H:= X x {0}. Then the following hold: (i) The natural morphism ir-1Q (log D) -- Q!(log D) is a quasi-isomorphism. (ii) The natural morphism ir-1Q1C(*D) -- Qx (*D) is a quasi-isomorphism. (iii) The inclusion Qx (log D U H) >- Qx (log D) (*H) is a quasi-isomorphism. (iv) D c X is free if and only D c X is free. Moreover, in this case QL(log D) = Ir*Q> (log D) E Oy dt. (v) D c X is free if and only if D U H is free. Moreover, in this case Qx (log D U H) = 7r*Q1 (log D) @ O Proof. (i), (ii) and (iii): By giving t weight 1, and local coordinates on X weight 0, we ensure that the complex on the left hand side of each arrow is the weight 0 part of the complex on the right. (iv) This is an easy consequence of Saito's criterion. Suppose first that D is free. Since 0/Ot E Der(log D), we can choose a basis 4i, ..n. , 0/10t for Der(log D)(x.,t.) whose first n members have no 0/0t component. By Saito's criterion, the determinant of their matrix of coefficients (with respect to some coordinate system on X) is a local definiig equation h for D. The restrictions of the (i to a slice X x {to} are in Der(logD)x,, and the determinant of their matrix of coefficients is the restriction of h to X x {to}. It is thus a local equation for D. This proves that D is free. The converse is still easier. The second assertion does not in fact require the hypothesis of freeness, but its proof in the free case is an obvious consequence of the dual version of Saito's criterion: if g is a defining equation for a free divisor D at xo, then w1,... , wn generate Qx (log D)xo over Ox, x if and only if 1 w1 A... Awn x ~~~~~~~~~~9 is a generator of Qn It follows from this criterion (applied to D and to D) that the O0x-modules lr*Q> (log D) E? Oyx dt and QX (log D) are equal. (v) The proof here is similar to the proof of (iv). The only change is that at a point (x, 0) E D U H we take a basis of Der(log D U H) whose last element is tO/Ot instead of 0/0t. O The main theorem is proved below by induction on the dimension of the divisor: in outline, we use Proposition 2.4 (below) to show that we may assume the result This content downloaded from 52.18.63.169 on Wed, 09 Nov 2022 07:40:07 UTC All use subject to https://about.jstor.org/terms 3042 F. J. CASTRO-JIMEtNEZ, LUIS NARVAEZ-MACARRO, AND DAVID MOND of Theorem 1.1 true in a punctured neighbourhood of any point; the problem is then reduced to a calculation in local cohomology, where once again we make use of Lemma 1.1. First we need Lemma 2.3. Let D C Ct be a weighted homogeneous divisor, and suppose that there is a vector field , tangent to D with ,(0) 74 0. Then (D, 0) is isomorphic to a product (D' x C, (0, 0)), with D' weighted homogeneous. Proof. We preface the proof by noting that (as an easy argument shows) an isomorphism of germs of singular hypersurfaces always extends to an isomorphism of germs of ambient vector spaces; this fact is used implicitly in what follows, since the weighted homogeneity of a divisor D is actually a property of the pair (Ct, D). The proof of the lemma is very simple: clearly ( must be transverse to one of the coordinate hyperplanes in some neighbourhood of 0. Reordering the coordinates, we may suppose this to be Ct-i x {0}. The integral flow of ( then defines an isomorphism from D n (Ctix {0}) x V (where V is some neighbourhood of 0 in C) onto some neighbourhood of 0 in D. Evidently the divisor D n (Ctix {0}) is weighted homogeneous. O Now we have Proposition 2.4. Let X be a complex manifold of dimension t, let D be a strongly quasihomogeneous divisor in X, and let p E D. Then there is an open neighbourhood U of p such that for each q E U n D, q 74 p, the germ of pair (X, D, q) is isomorphic to a product (Ctix C, D' x C, (0,0)), where D' is a strongly quasihomogeneous divisor. Proof. By definition of strong quasihomogeneity, there are coordinates in some neighbourhood U of p, in which p is the origin and D has a weighted homogeneous defining equation. Let 0 74 q E D n U. Again by definition of strong quasihomogeneity, there is a neighbourhood V C U of q and an analytic isomorphism : (D nV, q) -+(Vl nD1,,0), where D1 is weighted homogeneous and Vi is some neighbourhood of 0 in Ct. Let X be the infinitesimal generator of the C*-action on U. Then X is tangent to D, and x(q) 74 0; hence ?*(X) is tangent to D1, and ?*(X)(0) 74 0. It follows from Lemma 2.3 that there is some neighbourhood V2 C Vi of 0 and an isomorphism : D, n V2 - (D'nW1) x W2 with b(0) = (0,0), where W1 and W2 are neighbourhoods of 0 in Ct-1 and C respectively, and D' is weighted homogeneous. It remains to show that D' is strongly quasihomogeneous. Let 0 74 q" E D', and let a = b-1 (q", 0). Clearly D1 is strongly quasihomogeneous, so it follows by the argument of the previous paragraph that (as germs) (Dl,, a) (D2 x C, (0, 0)) for some weighted homogeneous divisor D2. Hence we have (D' x C, (q", 0)) - (D2 x C, (0, 0)); the cancellation property for germs of reduced analytic spaces [3] now implies that (D', q") - (D2, 0), showing that D' is strongly quasihomogeneous. O I Before proving Theorem 1.1, we need one further preparatory result. This content downloaded from 52.18.63.169 on Wed, 09 Nov 2022 07:40:07 UTC All use subject to https://about.jstor.org/terms COHOMOLOGY OF THE COMPLEMENT OF A FREE DIVISOR 3043 Lemma 2.5. Let X be a complex manifold, D C X a divisor and j: U - + X the inclusion of the complement of D in X. The following properties are equivalent: (a) The natural morphism Qx (log D) -+ Rj* Cu is a quasi-isomorphism. (b) For each Stein open set V C X (or indeed each sufficiently small Stein open set), and for each integer p > 0, the natural morphism hP (F(V, Q (log D))) -) hP (F(V - D, Qu)) = HP (V - D; C) is an isomorphism. Proof. By the Poincare Lemma, property (a) is equivalent to the inclusion of complexes Q (log D) -)j iQu( Rj. (QU being a quasi-isomorphism. Suppose that (a) holds. Taking hypercohomology over an open set V C X, we have an isomorphism HP(V, Qx (log D)) - HP(V, Rj*Qu) = HP(V-D, Qu) for each p > 0. If V is Stein, then V - D is also, and as the Qx (log D) and Qi are coherent (see [12]), we have HP (V, Qxc(log D)) hP (IF(V, Qx(log D))), HP(V - D, Qu) - hP(F(V - D, Qu)), from which (b) follows. Now suppose that property (b) holds. Evidently the morphism of (a) is a quasiisomorphism outside D. So let x E D. We must show that the natural morphism hP(QxC(log D)x) --* ((j,*Qu)x) is an isomorphism for each p > 0. As taking the homology of a complex commutes with inductive limits, the first term is the inductive limit of hP(F(V, Qx (log D))) and the second is the inductive limit of hP(F(V - D, Qu)), where V runs through a fundamental system of neighbourhoods (which we may take to be Stein) of x, and thus (a) follows. C Proof of Theorem 1.1. We use induction on dimension. If t = 1, then D is a collection of isolated points and the result is obvious (especially bearing in mind Lemma 2.1). Now suppose t > 1, and let po E D. Let U be a neighbourhood of po as in the statement of Proposition 2.4. Then at any point q E D n U with q 74 po, D is locally a product C x D', where D' is a strongly quasihomogeneous free divisor of dimension t- 2. The induction hypothesis, applied to D', together with Lemma 2.2, implies that the result holds also for C x D', and hence Q (logD)q -* Rj* (Cu)q is a quasi-isomorphism. We now have to prove that the same holds at po. By Lemma 2.5 it is enough to show that for any open polycylinder V C U centred at po, the de Rham morphism defines an isomorphism hP(F(V, Q(logD))) --+ hP(F(V - V n D; Qu)) for p > O. To do this, we compare four spectral sequences. Take local coordinates x1,... , Xt centred at po, as in Definition 1.3. Let Vi = V n {xi 74 O}, and let Vi' = Vi -Vi n D. Then {l4}j=i,...,t and {V7}ij=1,...,t are Stein open covers of V - 0 and V - V n D This content downloaded from 52.18.63.169 on Wed, 09 Nov 2022 07:40:07 UTC All use subject to https://about.jstor.org/terms 3044 F. J. CASTRO-JIMENEZ, LUIS NARVAEZ-MACARRO, AND DAVID MOND respectively. Consider the double complexes q KP,q = @ r(n Vij ,QP (log D))v l<io< iq<t j=0 q kp,q= ] F(0 QP), 1<io< -<iq<t j=0 each of which comes equipped with Cech differential d and exterior derivative d. The restriction morphism p"l: K' - + kX commutes with both differentials; hence, it induces morphisms of the spectral sequences arising from the standard filtrations on the total complexes of these double complexes. Denoting the spectral sequences for K by 'E and "E, and those for k by 'E and "E, we have q 'Ep-q = hP(K.q) - @ hP(F (n VijI,Q(logD))), 1 <io < ... <iq <t j=0 q q tp,q= hp h(ir(n vij ,Q*)) (D HP( nvi/; c), l<io< - <iq<t j=0 l<io< <iq<t j=0 IElP-q = hq (KPX) = Hq(V - 0, QP (log D)), and = h (KP.) = H (V-V n D, QP). By the induction hypothesis, the morphism q 'ElPq _,' Elp'q is an isomorphism for all p, q. Hence, 'pPoq is an isomorphism also. On the other hand, the spectral sequences "E and "E both collapse at E2. For "F, this is clear: V - V n D is a Stein space and QP is coherent, so that ,,Ekp,q = 0 for q > 0. For "E, more is required. It is here that we make use of the quasihomogeneity of D. Because QP(logD) is a locally free 0-module, there is an isomorphism Op 0 dp- QP (log D), which carries over to an isomorphism O Ht-l(V_-,0)dP - Ht-l(V - 0, Q(logD)). By the standard facts about depth and local cohomology (cf. [5]), we have Hq(V - 0,0) = 0 for q = 0,t - 1, and Ht-l(V - 0,0) is the quotient of F(ft=1 Vi, 0) by the subspace Bt-1 of sections of the form Et=, fi Ivo, where fi E rF(njli Vj I0). Let Vo =ni=l Vi. Elements of F(V0, 0) can be represented by Laurent series ~~~1 . 2_ ail,...,tl i.,.XitEZ converging in V?, and Bt-1 consists of precisely those series in which all coefficients ail,...,it in which all ij are negative vanish. It follows that Ht-l(V - 0, 0) is isomorphic to the space of all Laurent series Eai4...itx,1 ..xt , converging on V?, in which ail...it = 0 if not all ij are negative. In fact all such series converge throughout nif=_(Ct- {xi = 0}), and so Ht-1(V - 0, 0) = Ht-1(Ct - 0, 0) and is independent of the open set V. Note that each term in Htl (V - 0,0) has filtration less than or equal to -Et wi. This fact will play a crucial role in the proof. This content downloaded from 52.18.63.169 on Wed, 09 Nov 2022 07:40:07 UTC All use subject to https://about.jstor.org/terms