LINEAR FREE DIVISORS AND THE GLOBAL LOGARITHMIC COMPARISON THEOREM MICHEL GRANGER, DAVID MOND, ALICIA NIETO REYES, AND MATHIAS SCHULZE Abstract. A complex hypersurface D in Cn is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for n at most 4. Analogous to Grothendieck’s comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for D if the complex of global logarithmic differential forms computes the complex cohomology of Cn \D. We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the Lie algebra of linear logarithmic vector fields is reductive. For n at most 4, we show that the GLCT holds for all LFDs. We show that LFDs arising naturally as discriminants in quiver representation spaces (of real Schur roots) fulfill the GLCT. As a by-product we obtain a simplified proof of a theorem of V. Kac on the number of irreducible components of such discriminants.
Contents 1. Introduction 2. Linear free divisors and subgroups of Gln (C) 3. Cohomology of the complement and Lie algebra cohomology 4. Linear free divisors in quiver representation spaces 5. Examples of linear free divisors 5.1. A non reductive GLCT example 5.2. Discriminants of quiver representations 5.3. Incomplete collections of maximal minors 6. Classification in small dimensions 6.1. Structure of logarithmic vector fields 6.2. The case s = n − 2 6.3. Classification up to dimension 4 6.4. Summary of the classification up to dimension 4 7. Strong Euler homogeneity and local quasihomogeneity 7.1. Example 5.3 again 7.2. Example 5.1 again References
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Date: July 30, 2007. 1991 Mathematics Subject Classification. 32S20, 14F40, 20G10, 17B66. DM is grateful to Ignacio de Gregorio for helpful conversations on the topics treated here. MS gratefully acknowledges financial support from EGIDE and the Humboldt Foundation. 1
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MICHEL GRANGER, DAVID MOND, ALICIA NIETO REYES, AND MATHIAS SCHULZE
1. Introduction K. Saito introduced the notion of free divisor in [Sai80]. A reduced divisor D = V (h) ⊂ Cn is free if the sheaf Der(− log D) of logarithmic vector fields is a locally free OCn -module. It is linear if furthermore there is a basis for Γ(Cn , Der(− log D)) as OCn -module consisting of vector fields all of whose coefficients, with respect to the standard basis ∂/∂x1 , . . . , ∂/∂xn of the space Der(Cn ) of holomorphic vector fields on Cn , are linear functions. With respect to the standard weighting of Der(Cn ), such vector fields have weight zero, and we will refer to them in this way. It turns out that linear free divisors are relatively abundant; the authors believe that in the current paper and in [BM06], recipes are given which allow the construction of more free divisors than have been described in the sum of all previous papers. Suppose D is a linear free divisor. Then because the determinant of the matrix of coefficients of a basis of Γ(Cn , Der(− log D)) is a reduced equation for D by [Sai80, Thm. 1.8.(ii)], it follows that D is homogeneous of degree n. And because Der(− log D) can have no members of negative weight, D cannot be isomorphic to the product of C with a lower dimensional divisor. Example 1.1. (1) The normal crossing divisor D = {x1 · · · xn = 0} ⊂ Cn is a linear free divisor; Der(− log D) has basis x1
∂ ∂ , . . . , xn . ∂x1 ∂xn
Up to isomorphism it is the only example among hyperplane arrangements, cf. [OT92, Ch. 4]. (2) In the space B2,3 of binary cubics, the discriminant D, which consists of binary cubics having a repeated root, is a linear free divisor. For f (U, V ) = xU 3 + yU 2 V + zU V 2 + wV 3 has a repeated root if and only if its Jacobian ideal does not contain any power of the maximal ideal (U, V ), and this in turn holds if and only if the four cubics ∂f ∂f ∂f ∂f U ,V ,U ,V ∂U ∂U ∂V ∂V are linearly dependent. Writing the coefficients of these four cubics as the columns of the 4 × 4 matrix 3x 0 y 0 2y 3x 2z y A := z 2y 3w 2z 0 z 0 3w we conclude that D has equation det A = 0. After division by 3 this determinant is −y 2 z 2 + 4wy 3 + 4xz 3 − 18wxyz + 27w2 x2 . In fact each of the columns of this matrix determines a vector field in Der(− log D); for the group Gl2 (C) acts linearly on B2,3 by composition on the right, and, up to a sign, the four columns here are the infinitesimal generators of this action corresponding to a basis of gl2 (C). Each is tangent to D, since the action preserves D. Further examples of irreducible linear free divisors can be found (though not under this name) in the paper [SK77] of Sato and Kimura. Besides our example,
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two, of ambient dimension 12 and 40, are described in [SK77, §5, Prop. 11, 15], and by repeated application of castling transformations, cf. [SK77, §2], it is possible to generate infinitely many more, of higher dimensions. In Section 5 of this paper we describe a number of further examples of linear free divisors, and in Section 6 we prove some results about linear bases for the module Γ(Cn , Der(− log D)), and go on to classify all linear free divisors in dimension n ≤ 4. Linear free divisors provide a new insight into a conjecture of H. Terao [Ter78, Conj. 3.1] relating the cohomology of the complement of certain divisors D to the cohomology of the complex Ω• (log D) of forms with logarithmic poles along D. For linear free divisors, the link between the complex Γ(Cn , Ω• (log D)) and H ∗ (Cn r D) can be understood as follows. First, when D is a linear free divisor, then it turns out that Cn r D is a single orbit of the identity component G◦D of the group GD := {A ∈ Gln (C) : A(D) = D}, so H ∗ (Cn r D; C) is isomorphic to the cohomology of G◦D ; this is explained in Section 2. Second, in this case H ∗ (Γ(Cn , Ω• (log D))) coincides with the Lie algebra cohomology, with complex coefficients, of the Lie algebra gD of G◦D . For compact connected Lie groups G, a well-known argument shows that the Lie algebra cohomology coincides with the topological cohomology of the group. For linear free divisors the group G◦D is never compact, but the isomorphism also holds good for the larger class of reductive groups, and for a significant class of linear free divisors, G◦D is indeed reductive. Theorem 1.2. Suppose that D is a linear free divisor and that G◦D is reductive. Then (1.1)
H ∗ (Γ(Cn , Ω• (log D))) ' H ∗ (Cn r D).
This is proved in Section 3. Among linear free divisors to which it applies are those arising as discriminants in representation spaces of quivers, as discussed in detail in [BM06] and briefly in Section 4 below. Terao’s conjecture remains open, though it has been answered in the affirmative for a very large class of arrangements in [WY97], using a technique developed in [CJNMM96]. For general free divisors, the additional hypothesis of local quasihomogeneity is enough to guarantee a local result from which the global isomorphism of (1.1) follows. A divisor D is locally quasihomogeneous if in some neighborhood of each point x ∈ D there is a system of local coordinates, centered at x, and a set of positive weights for these coordinates, with respect to which D has a weighted homogeneous defining equation. We shall see in Section 7 that all linear free divisors in dimension n ≤ 4 are actually locally quasihomogeneous. Theorem 1.3 ([CJNMM96]). Let D ⊂ Cn be a locally quasihomogeneous free divisor, let U = Cn r D, let j : U → Cn be inclusion, and let Ω• (log D) be the complex of sheaves of holomorphic differential forms with logarithmic poles along D. Then the de Rham morphism (1.2)
Ω•X (log D) → Rj∗ CU
is a quasi-isomorphism. Grothendieck’s Comparison Theorem [Gro66] asserts that a similar isomorphism holds for any divisor D, if instead of logarithmic poles we allow meromorphic poles of arbitrary order along D. Because of this similarity, we refer to the quasiisomorphism of (1.2) as the Logarithmic Comparison Theorem (LCT) and to the global isomorphism (1.1) as the Global Logarithmic Comparison Theorem (GLCT).
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Several authors have further investigated the range of validity of LCT, and established interesting links with the theory of D-modules, in particular in [CMNM02], [CJUE05], [GS06], [Tor04], and [Wal05]. Local quasihomogeneity was introduced in [CJNMM96] as a technical device to make possible an inductive proof of the isomorphism in 1.3. Subsequently it turned out to have a deeper connection with the theorem. In particular by [CMMNMCJ02], for plane curves the logarithmic comparison theorem holds if and only if all singularities are weighted homogeneous in suitable coordinates. The situation in higher dimensions remains unclear. There is as yet no counterexample to the conjecture that for free divisors the weaker condition of strong Euler homogeneity, in which at each point x ∈ D there is a germ of vector field χ ∈ mx DerCn ,x such that χ · h = h, is equivalent to LCT. In Section 7 we examine the examples described in Sections 5 and determine which are locally quasihomogeneous. It turns out that all linear free divisors we know are strongly Euler homogeneous. The optimistic reader could therefore conjecture that all linear free divisors are strongly Euler homogeneous and fulfill LCT and then also GLCT. We do not know any counter-example to these statements. In Subsection 7.1 we give examples of quivers Q and dimension vectors d for which the discriminant in Rep(Q, d) is a linear free divisor but is not locally quasihomogeneous. In such cases Theorem 1.3 therefore does not apply, but Theorem 1.2 does. In Subsection 7.2, we show that the group of a linear free divisor does not need to be reductive for LCT to hold. However we do not know whether reductiveness of the group implies LCT for linear free divisors. The property of being a linear free divisor is not local, and our proof of GLCT here is quite different from the proof of LCT in [CJNMM96]. The fact that linear free divisors in Cn arise as the complement of the open orbit of an n-dimensional algebraic subgroup of Gln (C), means that there is some overlap between the topic of this paper and of the paper [SK77], where Sato and Kimura classify irreducible prehomogeneous vector spaces, that is, triples (G, ρ, V ), where ρ is an irreducible representation of the algebraic group G on V , in which there is an open orbit. However, the hypothesis of irreducibility means that the overlap is slight. Any linear free divisor arising as the complement of the open orbit in an irreducible prehomogeneous vector space is necessarily irreducible by [SK77, §4, Prop. 12], whereas among our examples and in our low-dimensional classification (in Section 6) all the linear free divisors except one (Example 1.1(2)) are reducible. Even where G is reductive, the passage from irreducible to reducible representations in this context is by no means trivial, including as it does substantial parts of the theory of representations of quivers. 2. Linear free divisors and subgroups of Gln (C) Fix a coordinate system x = x1 , . . . , xn on Cn with corresponding partial derivatives ∂ = ∂1 , . . . , ∂n where ∂i = ∂/∂xi . With respect to the standard degree, deg xi = 1 = − deg ∂i . Let D ⊆ Cn be a divisor defined by a homogeneous polynomial f ∈ C[x1 , . . . , xn ] of degree d. Then LD := {xA∂ t | xA∂ t (f ) ∈ C · f } ⊆ Γ(Cn , Der(− log D))
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is the Lie algebra of degree 0 global logarithmic vector fields. Consider the subgroup GD = {A ∈ Gln (C) : A(D) = D} = {A ∈ Gln (C) : f ◦ A ∈ C · f } with identity component G◦D and Lie algebra gD . Lemma 2.1. G◦D is an algebraic subgroup of Gln (C) and gD = {A | xAt ∂ t ∈ LD }. P P Proof. Let f = |α|=d fα xα and define fαA by f ◦ A = |α|=d fαA xα . Then fαA , is a homogeneous polynomial of degree d in the entries aij of A. That f ◦ A ∈ C · f is equivalent to the vanishing of the 2 × 2-determinants fα fβA − fβ fαA for all multiindices α, β. Thus, these polynomial equations in the aij define GD in Gln (C). As GD is a subgroup by definition, it is an algebraic subgroup of Gln (C) by [Mil06, Thm. 4.8]. Then the same holds for G◦D . By [Mil06, §12], the Lie algebra of G◦D consists of all n × n-matrices A such that f ◦ (I + Aε) = a(ε) · f ∈ C[ε] · f 2
where C[ε] = C[t]/ht i 3 [t] = ε. Taylor expansion of this equation with respect to ε yields f + ∂(f ) · A · x · ε = (a(0) + a0 (0) · ε) · f and hence a(0) = 1 and, by transposing the ε-coefficient, xAt ∂ t ∈ LD . The argument can be reversed to prove the converse by setting a(ε) := 1 + (xAt ∂ t (f )/f ) · ε. Let us recall from the introduction what are our objects of interest. Definition 2.2. A divisor D ⊆ Cn is called linear free if LD contains a basis of Der(− log D). Lemma 2.3. The complement Cn r D of a linear free divisor is an orbit of G◦D . Proof. For x ∈ Cn , the orbit G◦D · x is a smooth constructible subset of Cn whose boundary is a union of strictly lower dimensional orbits, cf. [Hum75, Prop. 8.3]. The orbit map G◦D → G◦D · x sends In + Aε to x + xAt ε and induces a tangent map (2.1)
gD Tx (G◦D · x),
A 7→ xAt .
For x 6∈ D, Der(− log D)(x) and hence also LD (x) is n-dimensional. Then by Lemma 2.1 and (2.1) Tx G◦D · x and hence G◦D · x are n-dimensional. As this holds for all x 6∈ D, the boundary of G◦D · x must be D and then G◦D · x = Cn r D. Reversing our point of view we might try to find algebraic subgroups G ⊆ Gln (C) that define linear free divisors. This requires by definition that G is n-dimensional and connected and by Lemma 2.3 that there is an open orbit. The complement D is then a candidate for a free divisor. Indeed D is a divisor: comparing with (2.1), D is defined by the discriminant determinant ∆ = det A1 xt · · · An xt where A1 , . . . , An is a basis of the Lie algebra g of G. As the entries of the defining polynomial are linear, ∆ is a homogeneous polynomial of degree n. Thus, if ∆ is not reduced, D can not be linear free. We shall see examples where this happens in the next section. On the other hand, Saito’s criterion [Sai80, Lem. 1.9] shows the following.
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MICHEL GRANGER, DAVID MOND, ALICIA NIETO REYES, AND MATHIAS SCHULZE
Lemma 2.4. Let the n-dimensional algebraic group G act linearly on Cn with an open orbit. If ∆ is reduced then D is a linear free divisor. As a first step towards our main result, we now describe the cohomology of Cn r D in terms of G◦D . Proposition 2.5. Suppose that D ⊂ Cn is a linear free divisor and x ∈ Cn r D, and let G◦Dx be the isotropy subgroup of x under the action of G◦D on Cn . Then ◦
H ∗ (Cn r D; C) = H ∗ (G◦D ; C)GDx = H ∗ (G◦D ; C). Proof. By Lemma 2.3, Cn r D ∼ = G◦D /G◦Dx with finite G◦Dx and the first equality follows. The second equality holds because G◦D is path connected, which means that left translation by g ∈ G◦Dx is homotopic to the identity and thus induces the identity map on cohomology. Remark 2.6. The argument for the second equality also shows that if G◦D is a finite quotient of the connected Lie group G then H ∗ (Cn r D; C) ' H ∗ (G; C). We will use this below in calculating the cohomology of Cn r D. 3. Cohomology of the complement and Lie algebra cohomology Let g be a Lie algebra. The complex of Lie algebra cochains with coefficients in Vk Vk the complex representation V of g has kth term C HomC (g, V ) ∼ = HomC ( C g, V ), Vk Vk+1 and differential dL : Hom(g, V ) → Hom(g, V ) defined by (3.1) (dL ω)(v1 ∧ · · · ∧ vk+1 ) =
X
+
X
(−1)i+j ω([vi , vj ] ∧ v1 · · · ∧ vbi ∧ · · · ∧ vbj ∧ · · · ∧ vk+1 )
i 0 if ` ≥ i; in particular it is diagonal. In other words we have a triangular-type system of diagonal linear parts with positive terms on the i’th row and zeros on rows after the i’th. Proof. If si,j = 0 for any j ≥ i, and sk,i = 0 for any k ≤ i, we can take vi = ξi,i . If sk,i = 1, with k ≤ i, then we may apply Lemma 7.12 and a linear combination of the vector fields vi,i , vi,i+1 , . . . , vi,n does the trick. Finally if si,j = 1 for some j > i, we observe that ξi,i − ξj,j , is diagonal and has non zero positive eigenvalues in the positions {(i, i), . . . , (i, j − 1)} ∪ {(i, j + 1), . . . (i, n)}. Then we see that the vector field vi = vi,j + ξi,i − ξj,j + vi+1,j + · · · + vj,j + vj,j+1 + · · · + vj,n
LINEAR FREE DIVISORS
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does the trick since by adding vi,j we complete the row i by a positive eigenvalue at (i, j), and we cancel with the help of the appropriate vk,` all the negative eigenvalues with row indices k > i. Proposition 7.14. There is an Euler vector field v, v(f ) ∈ O · f vanishing at S, with linear part diagonal and having only strictly positive eigenvalues. Proof. We construct v, by decreasing induction on i, as a linear combination αn vn + · · · + α1 v1 with positive coefficients, with αi > 0 large enough following the choice of αn , . . . , αi+1 . By construction we have v(f ) = λf with λ ∈ O. This completes the proof of Proposition 7.11. References [AAA+ 97]
D. V. Anosov, S. Kh. Aranson, V. I. Arnold, I. U. Bronshtein, V. Z. Grines, and Yu. S. Il0 yashenko, Ordinary differential equations and smooth dynamical systems, Springer-Verlag, Berlin, 1997, Translated from the 1985 Russian original by E. R. Dawson and D. O’Shea, Third printing of the 1988 translation [Dynamical systems. I, Encyclopaedia Math. Sci., 1, Springer, Berlin, 1988; MR0970793 (89g:58060)]. MR MR1633529 (99h:58053) [Art68] M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277– 291. MR MR0232018 (38 #344) [BM06] Ragnar-Olaf Buchweitz and David Mond, Linear free divisors and quiver representations, Singularities and computer algebra, London Math. Soc. Lecture Note Ser., vol. 324, Cambridge Univ. Press, Cambridge, 2006, pp. 41–77. MR MR2228227 (2007d:16028) [CJNMM96] Francisco J. Castro-Jim´ enez, Luis Narv´ aez-Macarro, and David Mond, Cohomology of the complement of a free divisor, Trans. Amer. Math. Soc. 348 (1996), no. 8, 3037–3049. MR MR1363009 (96k:32072) [CJUE05] F. J. Castro-Jim´ enez and J. M. Ucha-Enr´ıquez, Logarithmic comparison theorem and some Euler homogeneous free divisors, Proc. Amer. Math. Soc. 133 (2005), no. 5, 1417–1422 (electronic). MR MR2111967 (2005h:32070) [CMMNMCJ02] Francisco J. Calder´ on Moreno, David Mond, Luis Narv´ aez Macarro, and Francisco J. Castro Jim´ enez, Logarithmic cohomology of the complement of a plane curve, Comment. Math. Helv. 77 (2002), no. 1, 24–38. MR MR1898392 (2003e:32047) [CMNM02] Francisco Calder´ on-Moreno and Luis Narv´ aez-Macarro, The module Df s for locally quasi-homogeneous free divisors, Compositio Math. 134 (2002), no. 1, 59–74. MR MR1931962 (2003i:14016) [Gab72] Peter Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71– 103; correction, ibid. 6 (1972), 309. MR MR0332887 (48 #11212) [Gro66] A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes ´ Etudes Sci. Publ. Math. (1966), no. 29, 95–103. MR MR0199194 (33 #7343) [GS06] Michel Granger and Mathias Schulze, On the formal structure of logarithmic vector fields, Compos. Math. 142 (2006), no. 3, 765–778. MR MR2231201 (2007e:32037) [HM90] Herwig Hauser and Gerd M¨ uller, The cancellation property for direct products of analytic space germs, Math. Ann. 286 (1990), no. 1-3, 209–223. MR MR1032931 (91a:32008) [Hum75] James E. Humphreys, Linear algebraic groups, Springer-Verlag, New York, 1975, Graduate Texts in Mathematics, No. 21. MR MR0396773 (53 #633) [Jac79] Nathan Jacobson, Lie algebras, Dover Publications Inc., New York, 1979, Republication of the 1962 original. MR MR559927 (80k:17001) [Kac82] V. G. Kac, Infinite root systems, representations of graphs and invariant theory. II, J. Algebra 78 (1982), no. 1, 141–162. MR MR677715 (85b:17003)
36
MICHEL GRANGER, DAVID MOND, ALICIA NIETO REYES, AND MATHIAS SCHULZE
[Mat69]
[Mil06] [NR05] [OT92]
[Sai71] [Sai80]
[Sch91] [Sch04] [Ser01]
[SK77]
[Ter78]
[Tor04]
[Wal05]
[WY97]
John N. Mather, Stability of C ∞ mappings. IV. Classification of stable germs ´ by R-algebras, Inst. Hautes Etudes Sci. Publ. Math. (1969), no. 37, 223–248. MR MR0275460 (43 #1215b) J.S. Milne, Algebraic groups and arithmetic groups, v1.01 ed., June 2006, http: //www.jmilne.org. Alicia Nieto Reyes, Master’s thesis, University of Warwick, Coventry, England, 2005. Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR MR1217488 (94e:52014) Kyoji Saito, Quasihomogene isolierte Singularit¨ aten von Hyperfl¨ achen, Invent. Math. 14 (1971), 123–142. MR MR0294699 (45 #3767) , Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 265–291. MR MR586450 (83h:32023) Aidan Schofield, Semi-invariants of quivers, J. London Math. Soc. (2) 43 (1991), no. 3, 385–395. MR MR1113382 (92g:16019) W. Schmid, Geometric methods in representation theory, arXiv.org math (2004), no. 0402306. Jean-Pierre Serre, Complex semisimple Lie algebras, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001, Translated from the French by G. A. Jones, Reprint of the 1987 edition. MR MR1808366 (2001h:17001) M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1–155. MR MR0430336 (55 #3341) Hiroaki Terao, Forms with logarithmic pole and the filtration by the order of the pole, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) (Tokyo), Kinokuniya Book Store, 1978, pp. 673– 685. MR MR578880 (82b:14004) Tristan Torrelli, On meromorphic functions defined by a differential system of order 1, Bull. Soc. Math. France 132 (2004), no. 4, 591–612. MR MR2131905 (2005m:32015) Uli Walther, Bernstein-Sato polynomial versus cohomology of the Milnor fiber for generic hyperplane arrangements, Compos. Math. 141 (2005), no. 1, 121– 145. MR MR2099772 (2005k:32030) Jonathan Wiens and Sergey Yuzvinsky, De Rham cohomology of logarithmic forms on arrangements of hyperplanes, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1653–1662. MR MR1407505 (97h:52013)
´matiques, Universite ´ d’Angers, 2 Bd. Lavoisier, 49045 M. Granger, Dep. de Mathe Angers, France E-mail address:
[email protected] D. Mond, Mathematics Institute, University of Warwick, Coventry CV47AL, England E-mail address:
[email protected] A. Nieto Reyes, Dep. de Matematicas, Estadistica y Computacion, Universidad de Cantabria, Spain E-mail address:
[email protected] M. Schulze, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, USA E-mail address:
[email protected]
