ONE PARAMETER REGULARIZATIONS OF PRODUCTS OF RESIDUE CURRENTS. Dedicated to the memory of Mikael Passare

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1 ON PARAMTR RGULARIZATIONS OF PRODUCTS OF RSIDU CURRNTS MATS ANDRSSON & HÅKAN SAMULSSON KALM & LIZABTH WULCAN & ALAIN YGR Abstract We show that Coleff-Herrera type products of residue currents can be defined by analytic continuation of natural functions depending on one complex variable Dedicated to the memory of Mikael Passare 1 Introduction Let f be a holomorphic function defined on a domain in C n It is proved in [15] using Hironaka s desingularization theorem that if ϕ is a test form then ϕ/f lim ɛ 0 + f 2 >ɛ exists and defines the action of a current, denoted 1/f The -image, (1/f), is the residue current of f and it has the useful property that it is annihilated by a holomorphic function g if and only if g is in the ideal generated by f If f 1,, f q are holomorphic functions then the Coleff-Herrera product of the currents (1/f ) is defined as follows For a test form ϕ of bidegree (n, n q) consider the residue integral I ϕ f (ɛ) = ϕ, f 1 f q T (ɛ) where T (ɛ) = q 1 { f 2 = ɛ } It is proved in [12] that the limit of ɛ I ϕ f (ɛ) exists if ɛ = (ɛ 1,, ɛ q ) 0 along a path in R q + such that ɛ /ɛ k +1 0 for all k N and = 1,, q 1; such a path is said to be admissible Moreover, the limit defines the action of a current, the Coleff-Herrera product 1 (11) f 1 ϕ := lim I ϕ q f 1 ɛ 0 f (ɛ), where lim means the limit along an admissible path as above Following Passare [19], let χ be a smooth approximation of the characteristic function 1 [1, ) and consider the smooth form χ( f q 2 /ɛ q ) (12) χ( f 1 2 /ɛ 1 ) f q f 1 It follows from [16, Theorem 2] or the proof of [19, Proposition 2] that the limit in the sense of currents of (12) as ɛ 0 along an admissible path equals the Coleff-Herrera product, and moreover, that one gets the same result if one first lets ɛ 1 0, then Date: April 28, Mathematics Subect Classification 32A26, 32A27, 32B15, 32C30 First three authors partly supported by the Swedish Research Council

2 2 M ANDRSSON & H SAMULSSON KALM & WULCAN & A YGR lets ɛ 2 0 and so on The Coleff-Herrera product is thus indeed the result of an iterative procedure In general there are no obvious commutation properties, eg, (1/zw) (1/z) = 0 whereas (1/z) (1/zw) = (1/z 2 ) (1/w), where the last product is simply a tensor product However, if f = (f 1,, f q ) defines a complete intersection, ie, codim {f = 0} = q, then the Coleff-Herrera product depends in an anticommutative way of the ordering of the tuple f; in fact by [11] the smooth form (12) then converges unconditionally Moreover, also in the complete intersection case, a holomorphic function annihilates the Coleff-Herrera product if and only if it is in the ideal f 1,, f q ; this last property is called the duality property and it was proved independently by Dickenstein-Sessa, [13], and Passare, [18] In this paper we consider another approach to Coleff-Herrera type products; it is based on analytic continuation and has been studied in, eg, [6, 7, 10, 20, 27] For λ C with Re λ 0, let fq Γ ϕ f (λ 2λq 1,, λ q ) = f 1 2λ 1 ϕ, f 1 f q where ϕ is a test form It is standard to see that λ 1 Γ ϕ f (λ 1,, λ q ) has an analytic continuation to a neighborhood of 0 and that Γ ϕ f (0, λ 2,, λ q ) equals f q 2λq f q f 2 2λ2 f 2 1 f 1 ϕ From [5, Proposition 21] it follows that λ 2 Γ ϕ f (0, λ 2,, λ q ) is analytic at 0, that λ 3 Γ ϕ f (0, 0, λ 3,, λ q ) is too, and so on Once one knows that the Coleff-Herrera product is obtained by letting ɛ 0 successively in (12) it is not that hard to see that 1 f q 1 f 1 ϕ = Γ ϕ f (λ 1,, λ q ) λ1 =0 λq=0, where the expression on the right hand side means that we first let λ 1 0, then let λ 2 0 etc; see, eg, [16, Theorem 2] However, from an algebraic point of view, cf [8, Theorem 32], it is often desirable to have a current given as the value at 0 of a single one-variable analytic function; this is the motivation for this paper From Theorem 12 below it follows that if µ 1 > > µ q > 0 are integers, then λ Γ ϕ f (λµ 1,, λ µq ), a priori defined for Re λ 0, has an analytic continuation to a neighborhood of [0, ) C and that the value at λ = 0 equals the Coleff- Herrera product (11) Notice that this way of letting (λ 1,, λ q ) 0 is analogous to limits along admissible paths in the sense that λ goes to zero much faster than λ +1, = 1,, q 1 We remark that if f defines a complete intersection then it is showed in [23] that Γ ϕ f (λ) is analytic in a neighborhood of the half-space {Re λ 0, = 1,, q} Let us now consider a more general setting Let f be a section of a Hermitian vector bundle of rank m over a reduced complex space X of pure dimension n In [22] and [1] were introduced currents U and R, generalizing the currents 1/f and (1/f), respectively These currents are based on Bochner-Martinelli type expressions To be precise, let f = f 1 e f m e m, where {e k } k is a local holomorphic frame for with dual frame {e k } k, and let s = s 1 e s me m be the section of the dual

3 ON PARAMTR RGULARIZATIONS OF PRODUCTS OF RSIDU CURRNTS 3 bundle with pointwise minimal norm such that f s = f 2 For λ C, Re λ 0, we let (13) U λ := m f 2λ s ( s) k 1, f 2k where (0, 1)-forms anticommute with the e k It turns out, [1], [22], that λ U λ, considered as a current-valued map, has an analytic continuation to a neighborhood of 0 The value at λ = 0 is a current U on X that takes values in Λ ; U is the standard extension of k s ( s) k 1 / f 2k across {f = 0} If has rank 1, then U = (1/f)e for any choice of metric Let (14) R λ := 1 f 2λ + m f 2λ s ( s) k 1 f 2k Letting f := δ f, where δ f denotes interior multiplication with f, one can check that R λ = 1 f U λ, see [1] for details It follows that λ R λ has an analytic continuation to a neighborhood of 0 and the value at λ = 0 is the current R; it is straightforward to check that R has support on {f = 0} If has rank 1 then R = (1/f) e and more generally, if f defines a complete intersection then R = (1/f m ) (1/f 1 ) e 1 e m for any choice of metric, see [1] and [22] The value at λ = 0 of the term 1 f 2λ of Rλ is the restriction 1 {f=0} to the zero set of f, see [5] In itself it is zero unless f vanishes identically on some components of X in which case it simply is 1 there However, when forming products of R s the role of 1 {f=0} is much more significant, cf [3] and xample 14 Remark 11 For future reference we notice that if π : X X is a modification such that the pullback of the ideal sheaf defined by f is principal, then one can write π f = f 0 f, where f 0 is a section of the line bundle L X corresponding to the exceptional divisor and f is a non-vanishing section of L 1 π quipping L with some Hermitian metric, for instance by setting f 0 L := π f π, we can thus write π f = f 0 L f L 1 π Locally on X we can identify f 0 and f by a holomorphic function and a non-vanishing holomorphic tuple, respectively, still denoted f 0 and f Hence, locally on X we have π f = f 0 u for some smooth positive function u Let f be a section of a Hermitian vector bundle of rank m, let U and R be the associated currents, and let U,λ and R,λ be the corresponding λ-regularizations Following, eg, [3] and [16] we define products of the currents R recursively as follows Having defined R k 1 R 1, consider the current-valued function λ R k,λ R k 1 R 1, a priori defined for Re λ 0 It turns out, see, eg, [5] or [16], that it can be analytically continued to a neighborhood of 0, and we define R k R 1 to be the value at λ = 0 Theorem 12 Let µ 1 > > µ q be positive integers Then the current-valued function λ R q,λµq R 1,λµ1, a priori defined for Re λ 0, has an analytic continuation to a neighborhood of the half-axis [0, ) C and the value at λ = 0 is R q R 1

4 4 M ANDRSSON & H SAMULSSON KALM & WULCAN & A YGR To connect with Coleff-Herrera type products, let χ be the characteristic function 1 [1, ) or a smooth regularization thereof and let m R,ɛ := 1 χ( f 2 /ɛ ) + χ( f 2 /ɛ ) s ( s ) k 1 f 2k If ϕ is a test form on X, then the limit of (15) R q,ɛq R 1,ɛ 1 ϕ X as ɛ 0 along an admissible path exists and equals the action of R q R 1 on ϕ, see [16] Let us mention a version of Theorem 12 with connection to intersection theory Let f be a section of and let M λ := 1 f 2λ + f 2λ log f 2 2πi k 1 (dd c log f 2 ) k 1, where dd c = /2πi It is showed in [3] that λ M λ has an analytic continuation to a neighborhood of 0 and that the value at λ = 0 is a positive closed current, which we denote by M One can give a meaning to the product (dd c log f 2 )k for arbitrary k that extends the classical one for k codim {f = 0}, and from [3] it follows that M = 1 Z + k 1 1 Z (dd c log f 2 ) k, where 1 Z is the restriction to the zero set Z of f The current M is closely connected to R For instance, if X is smooth and D is the Chern connection on then it follows from [2] that M k = R k (Df/2πi) k /k!, where the subscript k means the component of bidegree (, k) Let f 1,, f q be sections of Hermitian vector bundles and let M 1,, M q be the associated currents One can define products of the M recursively as for the R and we have the following analogue of Theorem 12 Theorem 13 Let µ 1 > > µ q be positive integers Then the current-valued function λ M q,λµq M 1,λµ1, a priori defined for Re λ 0, has an analytic continuation to a neighborhood of the half-axis [0, ) C and the value at λ = 0 is M q M 1 xample 14 (xample 56 in [3]) Let J x O X,x be an ideal and let h 1,, h n J x be a generic Vogel sequence of J x ; see, eg, [3] for the definition By the Stückrad- Vogel procedure, [24], adapted to the local situation, [17], [25], one gets an associated Vogel cycle V h ; the multiplicities of the components of various dimensions of V h are the Segre numbers, [14], used in excess intersection theory By Theorem 13 we have that n ( λ 1 hk 2λµk + h k 2λµk log h k 2 /2πi ) is analytic at 0 and by [3] the value there is the Lelong current associated with V h ; see [3] for more details

5 ON PARAMTR RGULARIZATIONS OF PRODUCTS OF RSIDU CURRNTS 5 Remark 15 Assume that codim {f = 0} = m m q Then M = (dd c log f 2 ) m = [f = 0], where [f = 0] is the Lelong current of the fundamental cycle of f, and more generally, M q M 1 = [f q = 0] [f 1 = 0], ie, the current representing the proper intersection of the cycles [f = 0] In this case the current-valued function (λ 1,, λ q ) R q,λq R 1,λ 1 has an analytic continuation to a neighborhood of the origin in C q, [16], and the value at λ = 0 is the R-current associated to f, [26] Moreover, by [16], (15) depends Hölder continuously on ɛ [0, ) q if χ is smooth The smoothness of χ is necessary in view of the example in [21, Section 1] 2 Proof of Theorems 12 and 13 We will actually prove a slightly more general result than Theorem 12; we will allow mixed products of U and R k Let P denote either U or R and let P,λ be the corresponding λ-regularization, (13) or (14) One defines products of the P recursively as above Theorem 12 Let µ 1 > > µ q be positive integers Then the current-valued function λ P q,λµq P 1,λµ1, a priori defined for Re λ 0, has an analytic continuation to a neighborhood of the half-axis [0, ) C and the value at 0 is P q P 1 Let π : X X be a smooth modification of X such that {π f = 0}, = 1,, q, and {π f = 0} are normal crossings divisors Then locally in X we can write π f = f 0f, where f 0 is a monomial in local coordinates and f is a non-vanishing holomorphic tuple It follows that s = f 0s, where s is a smooth section A straightforward computation shows that π R,λ = 1 f 0 2λ u 2λ + m ( f 0 2λ u 2λ ) (f 0 )k ϑ k, π U,λ = m f 0 2λ u 2λ (f 0 )k ϑ k, where u is a smooth non-vanishing function and ϑ k = s ( s )k 1 /u 2k is a smooth form, cf, Remark11 In the same way, π M f,λ = 1 f 0 2λ u 2λ + ( f 0 2λ u 2λ ) log( f 0 2 u 2 ) ω k, k 1 where ω k is smooth, cf [3, Section 4] Taking the identity log( f 0 2 u 2 ) = df 0 /f u /u into account, Theorems 12 and 13 are consequences of the following quite technical lemma

6 6 M ANDRSSON & H SAMULSSON KALM & WULCAN & A YGR Lemma 21 Let u 1,, u r be smooth non-vanishing functions defined in some neighborhood of the origin in C n, with coordinates (x 1,, x n ) For λ = (λ 1,, λ r ) C r, Re λ 0, α 1,, α r N n, and k 1,, k r N, let Γ(λ) := u rx αr 2λr u p+1 x α p+1 2λ p+1 up x αp 2λp u 1 x α 1 2λ 1 x krαr x k 1α 1 ; here x k lα l = x k lα l,1 1 x k lα l,n n if α l = (α l,1,, α l,n ) If σ is a permutation of {1,, r}, write Γ σ (λ 1,, λ r ) := Γ(λ σ(1),, λ σ(r) ) Let µ 1,, µ r be positive integers Then Γ σ (κ µ 1,, κ µr ) has an analytic continuation to a connected neighborhood of the half-axis [0, ) in C, and if µ 1 > > µ r, then (21) Γ σ (κ µ 1,, κ µr ) κ=0 = Γ σ (λ 1,, λ r ) λ1 =0 λr=0 The reason for the permutation σ is that we have mixed products of U s and R s in Theorem 12 Proof To begin with let us assume that all u = 1 A straightforward computation shows that r =1 Γ(λ) = λ 1 λ xα 2λ d x i1 d x ip p A I =: λ 1 λ p Γ I, x i1 x ip x r =1 k α I where the sum is over all increasing multi-indices I = {i 1,, i p } {1,, n} and A I is the determinant of the matrix (α l,i ) 1 l p,1 p Pick a non-vanishing summand Γ I ; without loss of generality, assume that I = {1,, p} and A I = 1 With the notation b k (λ) := r l=1 λ lα l,k for 1 k n, n Γ I = x k 2b k(λ) x r =1 k α d x 1 d x p x 1 x p = 1 b 1 (λ) b p (λ) p x k 2b k(λ) n k=p+1 x k 2b k(λ) x r =1 k α Now the current-valued function p Γ x =1 2b (λ) n =p+1 I : (λ 1,, λ r ) x 2b (λ) x k α has an analytic continuation to a neighborhood of the origin in C r ; in fact, it is a tensor product of one-variable currents In particular, ΓI (κ µ 1,, κ µr ) κ=0 = Γ I (λ) λ1 =0 λr=0 Let λ 1 λ p γ(λ) = b 1 (λ) b p (λ) and γ σ = γ(λ σ(1),, λ σ(r) ) We claim that if µ 1 > > µ r, then γ σ (λ) λ1 =0 λr=0= γ σ (κ µ 1,, κ µr ) κ=0, where it is a part of the claim that both sides make sense Let us prove the claim Since A I = 1, reordering the factors b 1,, b p and multiplying γ(λ) by a non-zero constant, we may assume that α kk = 1, k = 1,, p, so that λ 1 λ p γ(λ) = λ 1 + α 21 λ α r1 λ r α p1 λ λ p + + α rp λ r I

7 ON PARAMTR RGULARIZATIONS OF PRODUCTS OF RSIDU CURRNTS 7 For < r set τ := λ /λ +1 and γ σ (τ 1,, τ r 1 ) := γ σ (λ); notice that γ σ is 0- homogeneous, so that γ σ is well-defined Then λ = τ τ r 1 λ r, and therefore γ σ consists of p factors of the form τ k τ r 1 (22) α k1 τ 1 τ r τ k τ r α k,r 1 τ r 1 + α kr Observe that (22) is holomorphic in τ in some neighborhood of the origin Indeed, if α kr 0, then (22) is clearly holomorphic, whereas if α kr = 0 we can factor out τ r 1 from the denominator and numerator In the latter case (22) is clearly holomorphic if α k,r 1 0 etc; since α kk = 1 this procedure eventually stops Hence, γ σ (τ) is holomorphic in a neighborhood of 0 It follows that γ σ (κ µ 1,, κ µr ) = γ σ (κ µ 1 µ 2,, κ µ r 1 µ r ) is holomorphic in a neighborhood of 0 and since the denominator of γ σ (κ µ 1,, κ µr ) is a polynomial in κ with nonnegative coefficients it is in fact holomorphic in a neighborhood of [0, ) Moreover, γ σ (λ 1,, λ r ) is holomorphic in = { λ 1 /λ 2 < ɛ,, λ r 1 /λ r < ɛ} Let us now fix λ 2 0,, λ r 0 in Then γ σ (λ) is holomorphic in λ 1 in a neighborhood of the origin Next, for λ 3 0,, λ r 0 fixed in, γ σ (λ) λ1 =0 is holomorphic in λ 2 in a neighborhood of the origin, etc It follows that γ σ (λ) λ1 λr=0 = γ σ (τ) τ=0 = γ σ (κ µ 1,, κ µr ) κ=0, which proves the claim Thus (21) follows in the case u = 1, = 1,, r Now, consider the general case Replace each u 2λ in Γ(λ) by u 2ω, where ω C Then Γ is a sum of terms of the following representative form: r p p r (23) u 2ω u 2ω u 2ω =p +1 xα 2λ p x =1 α 2λ x krαr x k 1α 1 =p+1 =1 p +1 Fixing all λ and ω except for λ σ(1) and ω σ(1), (23) becomes an analytic (currentvalued) function g(λ σ(1), ω σ(1) ) in a neighborhood of 0 C 2 Thus, the value at 0 of g(λ σ(1), λ σ(1) ) is the same as first letting ω σ(1) = 0 (which corresponds to setting u σ(1) = 1) and then letting λ σ(1) = 0 in g(λ σ(1), ω σ(1) ) Continuing analogously for (λ σ(2), ω σ(2) ) and so on, it follows that the right hand side of (21) is independent of the u To see that the left hand side of (21) is independent of u, replace each λ in (23) by κ µ σ() and denote the resulting expression by g(κ, ω 1,, ω r ) Then g is clearly analytic in the ω and by the first part of the proof it is also analytic in a neighborhood of [0, ) C κ Hence, g is analytic in a neighborhood of 0 C r+1 The left hand side of (21) is obtained by evaluating κ g(κ, κ µ σ(1),, κ µ σ(r)) at κ = 0; this is thus the same as evaluating g(κ, 0) (which corresponds to setting all u = 1) at κ = 0 Hence also the left hand side of (21) is independent of the u and the lemma follows References [1] M Andersson: Residue currents and ideals of holomorphic functions, Bull Sci math 128 (2004), [2] M Andersson: Residues of holomorphic sections and Lelong currents, Ark Mat 43 (2005), [3] M Andersson, H Samuelsson, Wulcan, A Yger: Local intersection numbers and a generalized King formula, arxiv:

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