On the spectrum of curve singularities

Transcription

1 On the spectrum of curve singularities by András Némethi The Ohio State University Columbus, OH 43210, USA. Dedicated to Egbert Brieskorn on the occasion of his 60th birthday 1. Introduction There is the following crucial problem in the singularity theory: how can we recognize that a singular germ is hypersurface, or complete intersection singularity. Or: is there any criterion which distinguishes the isolated hypersurface singularities among the isolated singularities? These kind of questions arise in many areas of the algebraic geometry and singularity theory, we mention here only one. A. Durfee [3] conjectured that the Milnor fiber of an isolated complete intersection singularity of dimension two is negative. Although J. Wahl [23] found a smoothing of a (non-hypersurface) singularity with positive signature, there is still a strong belief that the conjecture is true for hypersurfaces. But, if the conjecture is true, then what is special in hypersurfaces, which makes their signature negative? In this paper we try to understand this question for hypersurfaces of type f(x, y) + z N. In [12, 10], we give a series of properties of the resolution graphs of germs f : (X, x) (C, 0), defined on a normal surface singularity (X, x), which distinguish the plane curve singularities. Here, we present some properties ({P(N)} 2 N ) of the spectrum, which codifies a part of the information of the mixed Hodge structure on the vanishing cohomology of the germs. The property P(N), for plane curve singularities is equivalent to the negativity of the equivariant signature σ 1 (f + z N ) of the germ f(x, y) + z N. We prove that these properties are valid for plane curve singularities (in particular, we prove Durfee s conjecture for hypersurfaces of type f(x, y) + z N ) (cf. 5.2), and we give germs f : (X, x) (C, 0) (with (X, x) non-smooth) which do not satisfy any of the properties P(N). In the paper we emphasize the arithmetical flavour of the spectrum and its connections with number theory, namely with generalized Dedekind sums. 2. The properties P(N) (2 N ). 1

2 We start this section with some definitions (their motivation will come later, cf. 2.7) Definitions. Let N Q be the free abelian monoid generated by Q: its elements are finite sums of the form r Q n r (r), where n r N = {0, 1,...}. We will use the notation S N for 1 k<n(k/n) (N 2). If S = r (0,1] n r (r), we define the symmetric element with respect r = 1 by S = n 1 (1)+ r (0,1)( nr (r) +n r (2 r) ). The degree of S = n r (r) is given by µ(s) = n r N, the signature of S by σ(s) = r n r sign sin πr = r Z( 1) [r] n r (where [ ] denotes the integer part), and m(s) = r n r ( 1) k m ( [r, 1 + r] [k, k + 1] ), k Z where, for an interval I R, m(i) denotes its length. The multiplicative structure of N Q is generated by (r) (s) = (r + s). Then it is easy to see that m(s) = lim N σ(s S N ). N 2.2. Definition. a) Fix an element S = r (0,1] n r (r) and an integer N 2. We say that S satisfies the property P(N) if σ( S S N ) < 0. b) An element S = r (0,1] n r (r) satisfies the property P( ) if m( S) < Remark. If S = r (0,1) n r (r), then σ( S S N ) = 2σ(S S N ) and similarly m( S) = 2m(S). Therefore, P(N) (resp. P( )) is equivalent to σ(s S N ) < 0 (resp. m(s) < 0). We are really interested in elements S provided by the spectrum of an isolated curve singularity Example. Let (X, x) be a normal surface singularity, with the property that its link K X is rational homology sphere (i.e. H 1 (K X, Q) = 0). (This condition is not really necessary, but it will simplify our presentation. For the general situation, see for example, [14].) Let f : (X, x) (C, 0) be a germ of an analytic function which defines a one dimensional isolated singularity. We consider an embedded resolution ϕ : (Y, D) (X, f 1 (0)) of (f 1 (0), x) (X, x) (here D = ϕ 1 (f 1 (0))). Let E = ϕ 1 (x) be the exceptional divisor and E = w W E w be its decomposition in irreducible divisors. All these 2

3 irreducible divisors are rational. Let a A S a be the irreducible decomposition of the strict trasform S of f 1 (0). Then D = E S. Let G f be the resolution graph of f, i.e. its vertices V = W A consist of the nonarrowhead vertices W (corresponding to the irreducible exceptional divisors), and arrowhead vertices A (correponding to the strict transform divisors of D). We will assume that no irreducible exceptional divisor has an self intersection and W =. If two irreducible divisors corresponding to v 1, v 2 V have an intersection point then (v 1, v 2 ) (= (v 2, v 1 )) is an edge of G f. The set of edges is denoted by E. For any w W, we denote by V w the set of vertices v V adjacent to w. The graph G f is decorated by the self intersection (or Euler-) numbers e w := E w E w for any w W. For any v V let m v be the multiplicity of f ϕ along the irreducible divisor corresponding to v. In particular, for any a A one has m a = 1. The multiplicities satisfy the following relations. For any w W one has: (2.5) e w m w + v V w m v = 0. These relations determine the multiplicities {m w } w W in terms of the autointersection numbers {e w } w. For an edge e = (v 1, v 2 ) define m e = g.c.d.(m v1, m v2 ). The vanishing cohomology H 1 (F, C) of the Milnor fiber F of f has a natural mixed Hodge structure. The associated spectrum Sp(f) is in the interval ( 1, 1), and it is symmetric with respect to r = 0 (for details, see for example, [21, 20, 14]). We define S(f) = (r + 1), where the sum is over the spectral elements r of Sp(f) which are in the interval ( 1, 0]. Obviously S(f) is exactly Sp(f) shifted one unit to the right. In particular µ( S(f)) = dim H 1 (F, C). The element S(f) can be computed from the graph G f as follows; 2.6. Proposition. [20, 14] Define Rw s := v V w {sm v /m w }, where { } denotes the fractional part. Then: S(f) = (#A 1)(1) + e E ( s ) + 0<s<m e m e w W ( 1 + Rw)( s s ). 0<s<m w m w 2.7. Remark. The relation between the above example and the properties P(N) is the following. If g : (C 3, 0) (C, 0) is an isolated singularity, then its equivariant signature σ λ for λ 1 can be computed from its spectrum 3

4 Sp(g) only (for the exact formula, see e.g. [9, 11]). In general σ 1 cannot be computed from Sp(g) alone (cf [11]). If g(x, y, z) = f(x, y) + z N, then Sp(g), shifted by one unit, is exactly S(f) S N (cf. [18, 22]); and σ 1 (f + z N ) = σ( S(f) S N ). The equivariant signature σ 1 (f + z N ) is zero if the monodromy h 1 (g) corresponding to the eigenvalue one has no Jordan blocks of size two. In particular, (cf. [6]) if f is irreducible, then σ 1 (g) = 0. In general σ 1 (g) = h 22 1 (g) 0 (cf [11]), hence σ( S(f) S N ) σ(f +z N ). Now, the conjecture of Durfee [3], mentioned in the introduction, says that the signature of any hypersurface singularity g : (C 3, 0) (C, 0) is negative. Property P(N) (2 N < ) for S(f) corresponds exactly to this conjecture. The function N σ( S(f) S N ) is a sum of a periodic and a linear function (say P (N) and c N); (cf. 4.2). Hence P( ) is equivalent to c < 0. We expect that the above results (valid in the hypersurface situation) have the corresponding analogues in the general case, (i.e. when f is not plane curve singularity). On the other hand, the properties P(N) make sense for arbitrary germ f : (X, 0) (C, 0). The goal of the present paper is to show that the properties P(N) distinguish the plane curve singularities among the general germs f : (X, 0) (C, 0). We end this section with the following remark: for any fixed 2 N 1 < N 2 <, the properties P(N 1 ), P(N 2 ) and P( ) are independent Example. Consider a sufficiently small 1 >> r > 0. a) S = 2(1/2 r) + (1 r) satisfies P( ) and P(3), but not P(2); b) S = (r) + 2(1/2 + r) satisfies P(2), but not P(3) and P( ). 3. Positive results. The case of plane curve singularities. In this section we study the set S(f), where f : (C 2, 0) (C, 0) is an isolated singularity. I. Combinatorial arithmetical approach Since S(f) is constant under the µ constant deformations, S(f) depends only on the topological type of the embedding (f 1 (0), 0) (C 2, 0) (this fact can be read also from 2.6). Moreover, in the irreducible case, the topological invariants of this embedding behave rather additively with respect to the splice decomposition of the of f 1 (0) (or of the graph G f ) (see, for example, 4

5 [4]). Hence, one can expect that for irreducible f the invariant S(f) have a nice representation in terms of the Newton pairs of f. In the Brieskorn case f(x, y) = x p +y q, ((p, q) = 1) one has S(f) = (l/p+k/q), where the sum is over 0 < l < p, 0 < k < q, l/p+k/q < 1. It is convenient to use the notation S(p, q) for this element of N Q. One can expect that for general irreducible f, S(f) can be obtained by iterating the Brieskorn case. The following theorem provides such a formula (cf. 2.2 in [20], and an unpublished paper of M. Saito) Theorem. Assume that f : (C 2, 0) (C, 0) is irreducible with Newton pairs (p i, q i ) r i=1. Define the integers {a i } r i=1 by a 1 = q 1 and a i+1 = q i+1 + p i+1 p i a i if i 1. (These numbers are the decorations of the splice diagram of f, cf. [4].) Then: S(f) = r i=1 S i, where S i = ( k/a i + l/p i + t p i+1 p i+2 p r ), where the second sum is over 0 < k < a i, 0 < l < p i, k/a i + l/p i < 1 and 0 t p i+1 p r 1 (if i = r, then S r = S(a r, p r )). Proof. If #V w > 2, then w is called rupture point. Consider the relation given by (2.6). If #V w = 1, then (by 2.5) R s w = 0, if #V w = 2, then R s w = 1 iff m w sm v (v V w ). By an elementary computation one can show that S(f) is concentrated only in the rupture points {w 1,..., w r } of G f, namely: (3.2) S(f) = r i=1 0<s<mw i mw i sd i ( 1 + {sm v /m wi } ) ( s ), v V wi m wi where d i := g.c.d. ( ) (m v ) v Vwi, m wi (cf. [20] and [19]). Now, m wi = a i p i p i+1 p r (for details, see e.g. [4]). Consider the integers b i and c i such that b i p i + c i a i = 1. Then the multiplicities {m v } (v V wi ), modulo m wi, are: ( b i p i p i+1 p r, c i a i p i+1 p r, p i+1 p r ) (cf. [15, 19]). Hence: d i = p i+1 p r. Therefore Rw s i = { b i s/a i } + { c i s/p i } + {s/a i p i }. Now, write s = kp i + la i + ta i p i, where 0 k < a i, 0 l < p i, 0 t < p i+1 p r. Then m wi sd i iff (k, l) = (0, 0), and Rw s i = { k/a i } + { l/p i } + {k/a i + l/p i }, hence the result follows. 5

6 By (3.1) the Milnor number µ(f) of f is 2µ(S(f)) = 2 i µ(s i ), where 2µ(S i ) = (a i 1)(p i 1)p i+1 p r. The next theorem gives a similar additivity formula for the signature σ(f + z N ) of f(x, y) + z N. Define ((x)) by {x} 1/2 if x Z, and ((x)) = 0 otherwise Theorem. Let f be as in (3.1), and D i := g.c.d.(n, p i+1 p r ) for 1 i < r, and D r = 1. Then r σ(s(f) S N ) = D i σ(s(a i, p i ) S N/Di ). i=1 Proof. First we consider two lemmas Lemma. Let α (0, 1), and N 2 an integer. Then: σ((α) S N ) = 2((Nα)) 2N((α)). The proof is easy and it is left to the reader (write i/n α < (i + 1)/N) Lemma. P 1 t=0 Proof. Use the Fourier sine expansion: ((N x + t P )) = (N, P ) (( N (N, P ) x)). ((x)) = 1 π k=1 1 sin 2πkx, k (or an elementary, longer computation). Now (3.4) and (3.5) applied for x = k/a i + l/p i gives: σ(s i S N ) = D i σ(s(a i, p i ) S N/Di ), which ends the proof. The discussion (2.7) gives: 3.6. Corollary. Let f be as in (3.1), and D i as in (3.3). Then: r σ(f + z N ) = D i σ(x a i + y p i + z N/D i ). i=1 Now, for the Brieskorn singularities it is well known that σ(x a + y p + z N ) < 0, and actually σ(x a + y p + z N ) < (a 1)(p 1)(N 1)/3, therefore lim N σ(x a + y p + z N )/N < 0 as well (cf. 4.3). This gives: 6

7 3.7. Corollary. For any irreducible plane curve singularity f, the element S(f) satisfies the properties P(N), (2 N ). Unfortunately, the above arithmetical results (i.e. 3.1 and 3.6) have no analogs in the reducible case (at least known by the author). In the next subsection, we will prove the analog of (3.7) for reducible plane curves. II. Geometrical approach We start with the following theorem, which supports the belief that Durfee s conjecture ([3]) is true for hypersurface singularities Theorem. Assume that the germ f : (C 2, 0) (C, 0) defines an isolated singularity and it has r irreducible components. Then the signature of g(x, y, z) = f(x, y) + z N satisfies: σ(f + z N ) (N 1)(r 1) 0. Proof. We start with the following remark. Assume that g : (C 3, 0) (C, 0) defines an isolated singularity with Milnor lattice L(g). If g t is a deformation of g such that g 0 = g and g t, for t 0 small, has k singular points with Milnor lattices L 1,..., L k, then there is a lattice embedding k i=1l i L(g). If c is the codimension of the embedding, then: ( ) σ(g) c + i σ(l i ). Now consider a deformation {f t } t of f such that f 0 = f and f t, for t 0 small, has exactly δ(f) nodes, where µ(f) = 2δ(f) r +1 (cf. [7]). The deformation g t = f t +z N gives an embedding of δ(f) copies of L(x 2 +y 2 +z N ) in L(f +z N ) with codimension c = (N 1)(µ δ), hence ( ) gives the wanted inequality. Since σ( S(f) S N ) σ(f + z N ) (cf. 2.7), (3.7) and (3.8) gives: 3.9. Corollary. For any plane curve singularity f, the element S(f) satisfies the properties P(N), (2 N ). 4. The arithmetical approach revisited. Dedekind sums. We will consider again a germ f : (X, x) (C, 0) as in (2.4). For simplicity, we will assume that the graph G f has only one rupture point w. Since in (2.6) the contribution of the non rupture points is cancelled with the contribution of the edges, one has: (4.1) S(f) = (#A 1)(1) + 0<s<m w ( v V w {sm v /m w })( s m w ).

8 Using (3.4) and 0<s<b((as/b)) = 0 (apply 3.5 for x = 0), one has: σ( S(f) S N ) = (#A 1)(N 1)+4 0<s<m w (( sm v )) ( (( Ns )) N(( s )) ). v V w m w m w m w We recall that the generalized Dedekind sum (cf. [17, 24]) is defined by: s(b, c; a) = 0<s<a (( sb )) ((sc)) (where a > 0). a a Notice that s(b, 1; a) = s(1, b; a) is exactly the classical Dedekind sum Corollary. Let (X, x) and f be as above. Then: σ( S(f) S N ) = (#A 1)(N 1)+4 ( s(mv, N; m w ) N s(m v, 1; m w ) ), and: v V w m( S(f)) = (#A 1) 4 v V w s(m v, 1; m w ). In particular, the properties P(N) are related to the arithmetical properties of the Dedekind sums (associated with the multiplicity structure of G f ) Example. Consider again f : (C 2, 0) (C, 0), f(x, y) = x p + y a, (a, p) = 1, a > 1, p > 1. Fix b and c such that bp + ca = 1. Then m w = ap and {m v } = { bp, ca, 1} (cf. the proof of 3.1). We recall the following properties of the Dedekind sums: b s(b, c; a) = (a, b, c) s( (a, b), c (a, b, c) ; s(b, c; a) = s(kb, kc; a) if (k, a) = 1. a ), and : (a, b) Using these identities one has: m( S(f))/4 = s(p, 1; a)+s(a, 1; p) s(1, 1; ap). Finally, the famous Dedekind s Reciprocity Law says that: s(b, 1; a) + s(a, 1; b) = a2 + b 2 + (a, b) 2. 12ab Therefore: m( S(f)) = (a 2 1)(p 2 1)/(3ap) < 0, in particular: σ(x a + y p + z N ) = N (a2 1)(p 2 1) 3ap v V w s(m v, N; ap).

9 Therefore our Corollary (4.2) gives, in this very particular case, (via Brieskorn formula of the signature [2]), the number of lattice points in the tetrahedron (0, 0, 0), (0, 0, a), (0, p, 0), (N, 0, 0) in terms of Dedekind sums. This problem is well known in number theory, it was solved by Mordell [8] (in the case when a, p, N are relatively prime numbers), and recently by Pommersheim [16] (in the general case). We invite the reader to complete our example for the case (a, p) 1 (in order to provide a new proof of Pommersheim s formula, cf. [12]). 5. Germs without property P(N) Fix an integer k 1. Consider the graph G(k): (2) (3) (2k) (2k + 1) (2k) (3) (2) (1) (1) (2) k 1 (1) One can verify that the intersection matrix of the graph is negative definite (here it is useful the method given in [4], page 154). Then by a theorem of Grauert [5], there exists a germ f k : (X k, x) (C, 0) as in (2.4), such that G fk = G(k). Then, (4.1) gives: S(f) = 2(1) + therefore: m( S(f k )) = s k 1 s k s ( 2k + 1 ), (1 2s 2k + 1 ) = 2 + 2k2 2k + 1. This shows that there is no upper bound for m( S(f)), in general. By (4.2): σ (k) N := σ( S(f k ) S N ) = 2 + m( S(f k )) N + P (k) (N), where P (k) (N) is an odd periodic function in N, of period 2k + 1. If k = 2, then σ (2) N = 2 2N/5 + P (2) (N), where for N = 0, ; ±1; ±2 one has: P (2) (N) = 0; 8/5; ±4/5. In particular, σ (2) N 0 iff N = 2, 3, 4, 5, 7, 9. 9

10 If k = 3, then σ (3) N = 2 + 4N/7 + P (3) (N), where for N = 0; ±1; ±2; ±3 one has: P (3) (N) = 0; 18/7; ±6/7; ±2/7. In particular, σ (3) N > 0 for any N 2. The case N = 2 also can be computed explicitly. By the Dedekind s Reciprocity Law one has: s(1, 1; 2k + 1) = s(2, 2; 2k + 1) = k(2k 1) k(k 2), s(2, 1; 2k + 1) = 6(2k + 1) 6(2k + 1). Therefore, (4.2) gives: σ (k) 2 = 2 + 2k 0. Now, assume that k 4 and N 3. Then by the inequality: s(b, c; a) (cf. [12, 13]), and by (4.2): (a 1)(a 2) ; (a > 0), 12a σ (k) N 2 + N( 2 + 2k2 2k(2k 1) ) 2k + 1 2k + 1 > 0. Therefore, for k 3, S(f k ) does not satisfy the property P(N) for any 2 N, and for k = 2, P(N) is satisfied only if N {2, 3, 4, 5, 7, 9} Final Remarks. a). Actually, for any c R, we can define similar properties as above: we say that S = r (0,1] n r (r) satisfies P(N, c) if σ( S S N ) < c µ( S S N ). The case c = 1/3 is optimal: if c < 1/3, then for sufficiently large d, P(d, c) is not true for S(x d + y d ). By another conjecture of Durfee ([3]), P(N, 1/3) is true for any plane curve singularity f. b). For a plane curve singularity f, the negativity of the signature of f + z N was firstly proved by Ashikaga [1] in a very long paper with a rather sophisticated method. Actually, he proved that the order of σ(f + z N ) is less than N m(f) 2 /3, where m(f) is the multiplicity of f. In [12, 13], by a more conceptual approach, we reprove these facts, together with some stronger results. For example, in [12] we prove that P(N, 1/3) is true for any N and any irreducible plane curve singularity f. In [13] we consider the case when f is not irreducible, and we prove that P(N, 1/3) is true with the following restrictions: we assume that if N is smaller than a bound B(f) then λ = 1 is not an eigenvalue of the monodromy of f + z N. 10

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