Universal abelian covers of rational surface singularities and multi-index filtrations
aa r X i v : . [ m a t h . AG ] J un Universal abelian covers of rational surfacesingularities and multi-index filtrations ∗ A. Campillo F. Delgado † S.M. Gusein-Zade ‡ In [1] and [2], there were computed the Poincar´e series of some (multi-index)filtrations on the ring of germs of functions on a rational surface singularity.These Poincar´e series were written as the integer parts of certain fractionalpower series, an interpretation of whom was not given. Here we show that, upto a simple change of variables, these fractional power series are specializationsof the equivariant Poincar´e series for filtrations on the ring Ø e S , of germs offunctions on the universal abelian cover ( e S ,
0) of the surface ( S , S ,
0) be an isolated complex rational surface singularity and let π :( X, D ) → ( S ,
0) be a resolution of it (not necessarily the minimal one). Here X is a smooth complex surface, the exceptional divisor D = π − (0) is a normalcrossing divisor on X , all components E σ ( σ ∈ Γ) of the exceptional divisor D are isomorphic to the complex projective line CP and the dual graph of theresolution is a tree.Let Ø S , be the ring of germs of analytic functions on ( S , σ ∈ Γ, i.e.for a component E σ of the exceptional divisor, and for f ∈ Ø S , , let v σ ( f ) be theorder of zero of the lifting f ◦ π of the function f to the space X of the resolutionalong the component E σ . Let us choose several components E , . . . , E s of theexceptional divisor D ( { , . . . , s } ⊂ Γ). The valuations v , . . . , v s define amulti-index filtration { J ( v ) } on the ring Ø S , : for v = ( v , . . . , v s ) ∈ Z s ≥ , J ( v ) = { f ∈ Ø S , : v ( f ) ≥ v } (here v ( f ) = ( v ( f ) , . . . , v s ( f )) ∈ Z s ≥ , v ′ ≥ v ∗ Math. Subject Class. 14H20, 14J17, 32S25. Keywords: universal abelian covers, ratio-nal surface singularities, Poincar´e series. † First two authors were partially supported by the grant MTM2004-00958. Address:University of Valladolid, Dept. of Algebra, Geometry and Topology, 47011 Valladolid, Spain.E-mail: [email protected], [email protected] ‡ Partially supported by the grants RFBR-007-00593, INTAS-05-7805, NWO-RFBR047.011.2004.026, and RFBR-JSPS 06-01-91063. Address: Moscow State University, Facultyof Mathematics and Mechanics, Moscow, GSP-2, 119992, Russia. E-mail: [email protected]
1f and only if v ′ i ≥ v i for all i = 1 , . . . , s ). In [1], there was computed thePoincar´e series P ( t , . . . , t s ) of this filtration (the definition of the Poincar´eseries of a multi-index filtration can be found e.g. in [1, 2, 3]). Let ( E σ ◦ E δ )be the intersection matrix of the components of the exceptional divisor. For σ = δ , the intersection number E σ ◦ E δ is equal to 1 if the components E σ and E δ intersect (at one point) and is equal to zero if they don’t intersect; theself-intersection number E σ ◦ E σ of each component E σ is a negative integer.Let d = det( − ( E σ ◦ E δ )) and let ( m σδ ) = − ( E σ ◦ E δ ) − . All entries m σδ arepositive and m σδ ∈ (1 /d ) Z . For σ ∈ Γ, let m σ := ( m σ , . . . , m σs ) ∈ Q s ≥ .Let • E σ be the “smooth part” of the component E σ in the exceptionaldivisor D , i.e., E σ minus intersection points with all other components of theexceptional divisor D .For a fractional power series S ( t , . . . , t s ) ∈ Z [[ t /d , . . . , t /ds ]], let Int S ( t , . . . , t s ) be its “integer part”, i.e., the sum of all monomials from S ( t , . . . , t s ) with integer exponents. In [1] it was shown that P ( t , . . . , t s ) = Int Y σ ∈ Γ (1 − t m σ ) − χ ( • E σ ) , (1)where t m := t m · . . . · t m s s , χ ( X ) is the Euler characteristic of the space X .A similar formula was obtained in [2] for the Poincar´e series of the multi-index filtration on the ring Ø S , defined by orders of a function germ on irre-ducible components of a curve ( C, ⊂ ( S , Q ( t ) = Y σ ∈ Γ (1 − t m σ ) − χ ( • E σ ) (2)(and a similar one in [2]) participated as a formal expression convenient towrite the formula (1) for the Poincar´e series P ( t , . . . , t s ). There was no in-terpretation of it.In [3], there was defined an equivariant Poincar´e series for an “equivariant”filtration on the ring Ø V, of germs of functions on a germ ( V,
0) of a complexanalytic variety with an action of a finite group G . This Poincar´e series wascomputed for a divisorial filtration on the ring Ø C , and for the filtrationdefined by branches of a G -invariant plane curve singularity ( C, ⊂ ( C , C was equipped with a G -action.Let p : ( e S , → ( S ,
0) be the universal abelian cover of the surface sin-gularity ( S , H ( S \ { } ) be the first homology group of the (nonsingular) surface S \ { } . The order of the group G is equal to the determinant d of the minus2ntersection matrix − ( E σ ◦ E δ ) and moreover G is the cokernel Z Γ / Im I of themap I : Z Γ → Z Γ defined by this matrix.The group G acts on the germ ( e S ,
0) and the restriction p | e S\{ } of themap p to the complement of the origin is a (usual, nonramified) covering e S \ { } → S \ { } with the structure group G . One can lift the map p to a(ramified) covering p : ( e X, e D ) → ( X, D ) where e X is a normal surface (generallyspeaking not smooth) and e X \ e D ∼ = e S \ { } :( e X, e D ) e π −→ ( e S , ↓ p p ↓ ( X, D ) π −→ ( S , e X as the normalization of the fibre product X × S e S of thevarieties X and e S over S ).Let g σ , σ ∈ Γ be the element of the group G represented by the loop in X \ D going around the component E σ in the positive direction. The group G is generated by the elements g σ for all σ ∈ Γ. For a point x ∈ • E σ and fora point e x from the preimage p − ( x ) of it, locally, in a neighbourhood of thepoint e x , the map p : e D → D is an isomorphism and the map p : e X → X is aramified (over D ) covering, the order d σ of which coincides with the order ofthe generator g σ of the group G . Lemma 1
The order d σ of the element g σ ∈ G is the minimal natural k suchthat km δσ is an integer for all δ ∈ Γ . Proof . This follows immediately from the fact that Z Γ / Im I ∼ = Im m/ Z Γ where m : Z Γ → Q Γ is the map given by the matrix ( m σδ ) (i.e. minus theinverse of the map I ). (cid:3) Let R ( G ) be the ring of (virtual) representations of the group G . For σ ∈ Γ, let α σ be the one-dimensional representation G → C ∗ = GL (1 , C )of the group G defined by α σ ( g δ ) = exp( − π √− m σδ ) (here the minus signreflects the fact that the action of an element g ∈ G on the ring Ø e S , is definedby ( g · f )( x ) = f ( g − ( x ))).Let us choose any component e E i of the preimage p − ( E i ) of the compo-nent E i and let e v i be the corresponding divisorial valuation on the ring Ø e S , .On the space S α Ø α S , of all G -equivariant functions on ( e S ,
0) ( α runs over allnonequivalent 1-dimensional representations of the group G ) the valuation e v i does not depend on the choice of the component e E i .In [3], there was defined the equivariant Poincar´e series of the multi-indexfiltration defined by the divisorial valuations e v , . . . , e v s .3 heorem 1 The equivariant Poincar´e series P G ( t , . . . , t s ) of the s -index fil-tration defined by the set of divisorial valuations { e v , . . . , e v s } is given by theformula: P G ( t , . . . , t s ) = Y σ ∈ Γ (1 − α σ t d m σ ) − χ ( • E σ ) = Y σ ∈ Γ (1 − α σ t d m σ · . . . · t d s m sσ s ) − χ ( • E σ ) . (3)For a power series S ( t , . . . , t s ) = P v ∈ Z s ≥ s v t v ∈ R ( G )[[ t , . . . , t s ]] ( R ( G ) isthe ring of representations of the group G ), let its reduction red S ( t , . . . , t s )be the series P v ∈ Z s ≥ (dim s v ) t v ∈ Z [[ t , . . . , t s ]]. Corollary . One has red P G ( t , . . . , t s ) = Q ( t d , . . . , t d s s ), where Q ( t ) is thefractional power series defined by (2). Proof of Theorem 1
For short we shall say that an effective divisor on • D = S • E σ (or on • e D = p − ( • D )) is Cartier if it is the intersection with • D (or with • e D ) of the strict transform of a Cartier divisor on ( S ,
0) (or on ( e S , P G ( t ) is equal to the integralwith respect to the Euler characteristic of the monomial α t e v over the spaceof G -invariant effective Cartier divisors on • e D . Here α ∈ R ( G ) and e v ∈ Z s ≥ are functions (in fact semigroup homomorphisms) on the space of G -invariantCartier divisors on • e D : a G -invariant Cartier divisor defines the orders of zeroof the corresponding ( G -equivariant) function along the components • E i andalso the corresponding 1-dimensional representation of the group G .Thus to compute the equivariant Poincar´e series P G ( t ) one has to describethe space of G -invariant effective Cartier divisors on • e D and the correspondingfunctions v and α on it. Lemma 2
Any G -invariant effective divisor on • e D is a Cartier divisor. Proof . It is sufficient to show this for the divisor P e x ∈ p − ( x ) e x for a point x ∈ • E σ ,i.e. for the G -orbit of a point from • e E σ . The isotropy group G e x of a point e x ∈ p − ( x ) is the cyclic subgroup of the group G of order d σ generated by theelement g σ (this element acts trivially on p − ( E σ )).4et us take the germ at the point x of a smooth curve L σ on ( X, D ) transver-sal to the exceptional divisor D . By the Artin criterion (see, e.g., [7], Lemmaon page 156), the divisor d · L σ is the strict transform of a Cartier divisoron ( S ,
0) (in fact already d σ L σ is one with this property), i.e. there exists afunction f σ : S → C such that the strict transform of the divisor { f σ = 0 } is d · L σ . Let f σ = f σ ◦ π be the lifting of the function f σ to the space X of theresolution and let e f σ = f σ ◦ π ◦ p be the lifting of the function f σ to the space e X of the modification of the universal abelian cover ( e S ,
0) ( e f σ is a G -invariantfunction on e X ). Let us describe the divisor { e f σ = 0 } . Let e L σ, e x ⊂ e X be thegerm at the point e x ∈ p − ( x ) of the preimage under the map p of the curve L σ ⊂ X .The order of zero of the function e f σ along e L σ, e x is equal to d . The order ofzero of the function f σ along the component • E δ is equal to d · m σδ . The ramifi-cation order of the map p over the component • E δ is equal to d δ . Therefore theorder of zero of the function e f σ = f σ ◦ p along the preimage of the component • E δ is equal to d · d δ · m σδ . This (integer) number is divisible by d (since d δ m σδ is an integer: see Lemma 1). Therefore the zero divisor of the function e f σ isdivisible by d , i.e. the order of zero of this function along each component ofits zero set is divisible by d . This means that a root d q e f σ of degree d of thefunction e f σ (i.e. a branch of this root) is well defined up to multiplication bya root of degree d of 1 G -equivariant complex analytic function on e X and thusit is the lifting of a G -equivariant function on ( e S ,
0) (see e.g. [4, page ?]). (cid:3)
Corollary . Each G -invariant divisor on the universal abelian cover ( e S ,
0) ofthe rational surface singularity ( S ,
0) is a Cartier one.Lemma 2 means that the space of G -invariant effective Cartier divisors on • e D is in one to one correspondence with the space of all effective divisors on • e D .As it follows from the proof of Lemma 2, the order of zero of the G -equivariantfunction e f σ (corresponding to one point x ∈ • E σ ) along the component e E i isequal to d i m σi . One has to find the (one-dimensional) representation α σ withrespect to which the function e f σ is G -equivariant. Lemma 3 α σ ( g δ ) = exp( − π √− m σδ ) . Proof . The element g δ of the group G acts trivially on the preimage p − ( • E δ )of the component • E δ of the exceptional divisor and acts by multiplication by5xp( 2 πd δ √−
1) on the normal line to it. The order of zero of the function e f σ along the preimage p − ( • E δ ) is equal to m σδ d δ . Therefore g δ · f σ f σ = exp( − π √− m σδ d δ d δ ) = exp( − π √− m σδ ) . (cid:3) Now Theorem follows from the usual arguments used e.g. in [1, 3]. Thespace of effective divisors on • D = S • E σ is the direct product of the spaces ofeffective divisors on the components • E σ . Each of the latter ones is the disjointunion of symmetric powers S k • E σ of the component • E σ . Therefore P G ( t , . . . , t s ) = Y σ ∈ Γ ∞ X k =0 χ ( S k • E σ ) · α kσ t kd m σ ! , (this follows from the fact that v and α are semigroup homomorphisms). Thewell-known formula ∞ X k =0 χ ( S k X ) t k = (1 − t ) − χ ( X ) implies the equation (3). (cid:3) A similar result holds for the filtration on the ring Ø e S , defined by ordersof a function germ on branches of a G -invariant curve ( e C, ⊂ ( e S , e C = r S i =1 e C i where e C i are irreducible G -invariant components of the curve e C (generally speaking each curve e C i consists of several irreducible compoentspermuted by the group G ). Each curve e C i is the preimage under the map p of an irreducible curve C i on ( S , e C = r S i =1 e C i defines an r -indexfiltration on the space S α Ø α e S , of G -equivariant functions on the surface ( e S , S α Ø α e C, of G -equivariant functions on the cuvre ( e C, ϕ i : ( C , → ( e S ,
0) be a parametrization (uniformization) of an irreduciblecomponent of the curve e C i . For a G -equivariant function germ f , let e w i ( f )be the order of zero of the function f ◦ ϕ i at the origin: f ◦ ϕ i ( τ ) = aτ e w i ( f ) +terms of higher degree, a = 0. The valuations e w , . . . , e w r define a multi-indexfiltration in the usual way.Let π : ( X, D ) → ( S ,
0) be a resolution of the surface singularity ( S , C, ⊂ ( S , = r S i =1 C i . Let C i be the strict trasnform of the curve C i in X . Let E , . . . , E s be all the components of the exceptional divisor D of the resolution. Let ◦ E i bethe “smooth part” of the component E i in the total transform π − ( C ) of thecurve C , i.e. E i minus intersection points with all other components of the totaltransform π − ( C ). Let m i = ( m i , . . . , m is ) ∈ Q s ≥ , d = ( d , . . . , d s ) ∈ Z s ≥ ,and a 1-dimensional representation α of the group G ( i = 1 , . . . , s ) be definedas above. The same arguments as in the proof of Theorem 1 imply the followingstatement. Theorem 2