Chern currents of coherent sheaves

. Given a ﬁnite locally free resolution of a coherent analytic sheaf F , equipped with Hermitian metrics and connections, we construct an explicit current, obtained as the limit of certain smooth Chern forms of F , that represents the Chern class of F and has support on the support of F . If the connections are (1 , 0) -connections and F has pure dimension, then the ﬁrst nontrivial component of this Chern current coincides with (a constant times) the fundamental cycle of F . The proof of this goes through a generalized Poincaré–Lelong formula, previously obtained by the authors, and a result that relates the Chern current to the residue current associated with the locally free resolution.


Introduction
Let X be a complex manifold and let F be a coherent analytic sheaf on X of positive codimension, i.e. such that the support supp F has positive codimension.Assume that F has a locally free resolution of the form (1.1) where E k are holomorphic vector bundles on X, and O(E k ) denote the corresponding locally free sheaves.
Then the (total) Chern class of F equals where c(E k ) is the (total) Chern class of E k , see Section 3.1.In this paper we construct explicit representatives of the nontrivial part of c(F ) with support on supp F .Let us briefly describe our construction; the representatives of c(F ) will be currents obtained as limits of certain Chern forms.Assume that (E, ϕ) is a locally free resolution of F of the form (1.1).Moreover assume that each vector bundle E k is equipped with a Hermitian metric and a connection D k (that is not necessarily the Chern connection of the metric).Let σ k be the minimal inverse of ϕ k , see Section 2.3, let χ : R ≥0 → R ≥0 be a smooth cut-off function such that χ(t) ≡ 0 for t 1 and χ(t) ≡ 1 for t 1, let s be a (generically non-vanishing) holomorphic section of a vector bundle such that {s = 0} ⊇ supp F , let χ = χ(|s| 2 / ), and let D k be the connection Our construction of Chern currents is inspired by the paper [BB72] by Baum and Bott, where singular holomorphic foliations are studied by expressing characteristic classes associated to a foliation as certain cohomological residues, more precisely as push-forwards of cohomology classes living in the singular set of the foliation.A key point in the proofs in [BB72] are the concepts of connections compatible with and fitted to a complex of vector bundles.One may check that their constructions of fitted connections (with some minor adaptations) correspond to connections of the form (1.3).For the results in [BB72], it was sufficient to consider Chern forms associated to connections (1.3) for small enough, but fixed, while in the present paper, we study the limit of such forms when → 0.
Example 1.3.Let us compute c Res (E, D) when F is the structure sheaf O Z of a divisor Z ⊂ X, defined by a holomorphic section s of a holomorphic line bundle L over X, and (E, ϕ) is the locally free resolution Assume that L is equipped with a connection D L ; equip E 1 = L * with the induced dual connection D L * , and E 0 with the trivial connection.The minimal inverse of s is 1/s and Ds = D L s, so D 1 = χ (D L s/s) + D L * , and D 0 is the trivial connection.The curvature form of D 1 equals Θ 1 = dχ ∧ (D L s/s) − (1 − χ )Θ L , where Θ L is the curvature form of D L (which equals minus the curvature form of D L * ).By an appropriate formulation of the Poincaré-Lelong formula, (1.7) lim where [Z] is the current of integration along (the cycle of) Z.Note that Thus, the first Chern current coincides with [Z], which can be seen as a canonical representative of c 1 (O Z ) with support on supp O Z .By further calculations of terms of all degrees, one can show that Our next result is an explicit description of the first nontrivial Chern current c Res p (E, D) in the case when F has pure codimension p, i.e. supp F has pure dimension dim X − p, that generalizes (1.8).Recall that the (fundamental) cycle of F is the cycle (1.9) [F ] = (considered as a current of integration), where Z i are the irreducible components of supp F , and m i is the geometric multiplicity of Z i in F , see e.g.[Ful98, Chapter 1.5].
Theorem 1.4.Assume that F is a coherent analytic sheaf of pure codimension p > 0 that admits a locally free resolution (E, ϕ) of the form (1.1).Moreover, assume that each E k is equipped with a Hermitian metric and a In case F has codimension p, but not necessarily pure codimension p, then Theorem 1.4 still holds if we replace the first equation by (1.12) c Res p (E, D) = (−1) p−1 (p − 1)![F ] p , where [F ] p denotes the part of [F ] of codimension p, i.e. in (1.9), one only sums over the components Z i of codimension p.
In particular, c Res p (E, D) is independent of the choice of Hermitian metrics and (1, 0)-connections on (E, ϕ).Moreover, it follows that on cohomology level where now the right hand side should be interpreted as a de Rham class.When F is the pushforward of a vector bundle from a subvariety, that (1.13) holds is a well-known consequence of the Grothendieck-Riemann-Roch theorem, cf.[Ful98, Examples 15.2.16 and 15.1.2].The proof of Theorem 1.4 relies on a generalization of the Poincaré-Lelong formula.Given a complex (1.1) equipped with Hermitian metrics, Andersson and the second author defined in [AW07] an associated so-called residue current R E = R = R k with support on supp F , where R k is a Hom(E 0 , E k )-valued (0, k)-current for k = 0, . . ., N , see Section 2.3.The construction involves the minimal inverses σ k of ϕ k .If (E, ϕ) is the complex (1.6), then R E coincides with the residue current ∂(1/s) = lim →0 ∂χ (1/s); more generally if (E, ϕ) is the Koszul complex of a complete intersection, then R E coincides with the classical Coleff-Herrera residue current, [CH78].Using residue currents, we can write the Poincaré-Lelong formula (1.7) as indeed, the left hand side in (1.7) equals lim ∂χ ∧ (D L s)/s.Given a Hom(E , E )-valued current α, let tr α denote the trace of α.In [LW18,LW21] we proved the following generalization of the Poincaré-Lelong formula: Assume that R E is the residue current associated with a finite locally free resolution (E, ϕ) of a coherent analytic sheaf F of pure codimension p. Moreover assume that D is a connection on End E induced by arbitrary (1, 0)connections on E k .Then If F has codimension p, but not necessarily pure codimenion, (1.14) still holds if we replace [F ] by [F ] p , cf.
[LW18, Theorem 1.5].In view of this, Theorem 1.4, as well as (1.12), are direct consequences of the following explicit description of (the components of low degree of) c Res (E, D) in terms of R E .
Theorem 1.5.Assume that F is a coherent analytic sheaf of codimension p > 0 that admits a locally free resolution (E, ϕ) of the form (1.1).Moreover, assume that each E k is equipped with a Hermitian metric and a (1, 0)-connection D k .Let R be the associated residue current and D the connection on End E induced by the D k .Then Moreover (1.10) and (1.11) hold.
In fact, we formulate and prove our results in a slightly more general setting.We consider the Chern class c(E) of a generically exact complex of vector bundles (E, ϕ) that is not necessarily a locally free resolution of a coherent sheaf.Theorem 5.1 below asserts that c(E, D ) as well as products of such currents have well-defined limits when → 0 and represent the corresponding (products of) Chern classes.In particular, Theorem 1.1 follows.In Theorem 6.1, if (E, ϕ) is exact outside a variety of codimension p, we give an explicit description of c Res p (E, D) := lim →0 c p (E, D ) terms of residue currents that generalizes Theorem 1.5.From this and a more general version of the Poincaré-Lelong formula (1.14) it follows that if the cohomology groups are of pure codimension p, then c Res p (E, where [E] is the cycle of (E, ϕ), see Corollary 6.7 and (2.19).
Our results could alternatively be formulated in term of the Chern character ch(E) of E. From Theorem 1.1, for > 0, we obtain a current ch Res (E, D) that represents the th graded piece ch (E) of the Chern character, see Section 6. Theorems 1.4 and 1.5 are then equivalent to ch Res (E, D) = 0 for < p, and We refer to the currents in Theorem 1.1 as Chern currents, in analogy with the usual Chern forms representing Chern classes.In works of Bismut, Gillet, and Soulé [BGS90a,BGS90b] appears the similarly named concept of Bott-Chern currents, that are certain explicit dd c -potentials in a transgresssion formula in a Grothendieck-Riemann-Roch theorem, and not directly related to our currents.
There are some similarities between our results and results by Harvey and Lawson.In [HL93] they study characteristic classes of morphisms ϕ : E 0 → E 1 of vector bundles, and only in very special situations there is overlap between their results and ours.We remark that the connection (1.3) that plays a crucial role in our work essentially appears and is important in [HL93], see, in particular, [HL93, Section I.4].
Chern classes of coherent sheaves, without the assumption of the existence of a global locally free resolution, were studied in the thesis of Green, [Gre80], as well as in various recent papers, including [Gri10,Hos20a,Hos20b,Qia16,BSW21,Wu20].Several of these papers also concern classes in finer cohomology theories than de Rham cohomology, as for example (rational or complex) Bott-Chern or Deligne cohomology.
In the present paper, our focus has been to find explicit representatives of Chern classes of a coherent sheaf with support on the support of the sheaf, a type of result which as far as we can tell, none of the above mentioned works seems to consider.By incorporating the construction of residue currents associated with a twisted resolution from [JL21], it might be possible to extend our results to arbitrary coherent sheaves, without any assumptions about the existence of a global locally free resolution.We plan to explore this in future work.The currents we study provide representatives of the Chern classes in de Rham cohomology.Our methods unfortunately do not seem to yield representatives in the finer cohomology theories mentioned above, since for example Chern classes in complex Bott-Chern cohomology as in [Qia16,BSW21], are naturally obtained from Chern forms of the Chern connection of a hermitian metric, while our construction, building on the techniques in [BB72], involve Chern forms of connections that are not Chern connections of a hermitian metric.
The paper is organized as follows.In Section 2 we give some necessary background on (residue) currents.In Section 3 we describe Chern forms and Chern characters, and in Section 4 we discuss compatible connections.The proofs of (the generalized versions of) Theorems 1.1 and 1.5 occupy Sections 5 and 6, respectively.Finally in Section 7 we compute c Res (E, D) for an explicit choice of a locally free resolution (E, ϕ) of a coherent sheaf F .In particular, we compute c Res (E, D) for > codim F in this case.

Currents associated with complexes of vector bundles
We say that a function χ : R ≥0 → R ≥0 is a smooth approximand of the characteristic function χ if χ is smooth and χ(t) ≡ 0 for t 1 and χ(t (2.1)

Pseudomeromorphic currents
Let f be a (generically nonvanishing) holomorphic function on a (connected) complex manifold X. Herrera and Lieberman [HL71], proved that the principal value exists for test forms ξ and defines a current [1/f ].It follows that ∂[1/f ] is a current with support on the zero set Z(f ) of f ; such a current is called a residue current.Assume that χ ∼ χ [1,∞) and that F is a (generically nonvanishing) section of a Hermitian vector bundle such that Z(f f , see e.g.[AW18].In particular, the limits are independent of χ and F.
In the literature there are various generalizations of residue currents and principal value currents.In particular, Coleff and Herrera [CH78] introduced products like In order to obtain a coherent approach to questions about residue and principal value currents was introduced in [AW10] the sheaf P M X of pseudomeromorphic currents on X, consisting of direct images under holomorphic mappings of products of test forms and currents like (2.3).See e.g.[AW18, Section 2.1] for a precise definition; in particular it follows from the definition that P M is closed under push-forwards of modifications.Also, we refer to [AW18] for the results mentioned in this subsection.The sheaf P M X is closed under ∂ and under multiplication by smooth forms.Pseudomeromorphic currents have a geometric nature, similar to closed positive (or normal) currents.For example, the dimension principle states that if the pseudomeromorphic current µ has bidegree ( * , p) and support on a variety of codimension strictly larger than p, then µ vanishes.
The sheaf P M X admits natural restrictions to constructible subsets.In particular, if W is a subvariety of the open subset U ⊆ X, and F is a section of a vector bundle such that {F = 0} = W , then the restriction to U \ W of a pseudomeromorphic current µ on U is the pseudomeromorphic current where χ ∼ χ [1,∞) as above.This definition is independent of the choice of F and χ.
A pseudomeromorphic current µ on X is said to have the standard extension property (SEP) if 1 U \W µ = µ| U for any subvariety W ⊆ U of positive codimension, where U ⊆ X is any open subset.By definition, it follows that if µ has the SEP and F 0 is any holomorphic section of a vector bundle, then

Superstructure and connections on a complex of vector bundles
Let (E, ϕ) be a complex of vector bundles over X.As in [AW07], see also [LW18], we will consider the complex (E, ϕ) to be equipped with a so-called superstructure, i.e. a Z 2 -grading, which splits E := ⊕E k into odd and even parts E + and E − , where E + = ⊕E 2k and E − = ⊕E 2k+1 .Also End E gets a superstructure by letting the even part be the endomorphisms preserving the degree, and the odd part the endomorphisms switching degrees.This superstructure affects how form-and current-valued endomorphisms act.Assume that α = ω ⊗ γ is a section of E • (End E), where γ is a holomorphic section of Hom(E , E k ), and ω is a smooth form of degree m.Then we let deg f α = m and deg e α = k − denote the form and endomorphism degrees, respectively, of α.The total degree is where η is a scalar form, and ξ is a section of E, both homogeneous in degree, then the action of α on β is defined by If furthermore, α = ω ⊗ γ , where γ is a holomorphic section of End E, and ω is a smooth form, both homogeneous in degree, then we define For an (m × n)-matrix A and an (n × m)-matrix B, we have that tr(AB) = tr(BA), while for the morphisms α and α above, we get such an equality with a sign due to the superstructure, (2.7) Note that ∂ extends in a way that respects the superstructure to act on End E-valued morphisms.In particular, (2.8) ∂(αα ) = ∂αα + (−1) deg α α ∂α .
We will consider the situation when (E, ϕ) is equipped with a connection D = D E = (D 0 , . . ., D N ), where D k is a connection on E k .Then there is an induced connection ⊕D k on E, that we also denote by D E .This in turn induces a connection D End on End E that takes the superstructure into account, defined by (2.9) To simplify notation, we will sometimes drop the subscript End and simply denote this connection by D. If Θ k denotes the curvature form of D k , and α : E k → E , then, by (2.9), (2.11) The above formulas hold also when α and α are current-valued instead of form-valued, as long as the involved products of currents are well-defined.We let D k and D k denote the (1, 0)-and (0, 1)-parts of D k , respectively, and we let where (•) (1,1) denotes the component of bidegree (1, 1).
Since (E, ϕ) is a complex and ϕ k has odd degree, it follows from (2.10) that (2.13)

Residue currents associated to a complex
Let us briefly recall the construction in [AW07].Assume that we have a generically exact complex (E, ϕ) of vector bundles over a complex manifold X of the form (2.5), and assume that each E k is equipped with some Hermitian metric.If Z k is the analytic set where ϕ k has lower rank than its generic rank, then outside of Let Z be the set where (E, ϕ) is not pointwise exact.It follows from the definition of σ k that (2.15) and furthermore applying (2.10) to this equality, we get that (2.17) Lemma 2.1.Let X, (E, ϕ), Z, and σ k be as above.Assume that for each j = 1, . . ., m, s j is an entry of σ k , ∂σ k , or ∂σ k for some k in some local trivialization, and let which proves the last part of the lemma.Here, in the second equality, we have used that the two limits exist and are both equal to ∂µ by (2.4).
In particular where R k = R 0 k .Given a complex (E, ϕ) of vector bundles of the form (2.5), following [LW21], we define the cycle where H k is the homology sheaf of (E, ϕ) at level k.Note that if (E, ϕ) is a locally free resolution of a coherent analytic sheaf F , then In [LW21] we prove the following generalization of (1.14).

Chern classes and forms
Assume that E is a holomorphic vector bundle of rank r equipped with a connection D. Then recall that the (total where Θ is the curvature matrix of D in a local trivialization; in particular, c (E, D) is a form of degree 2 .The de Rham cohomology class of c(E, D) is the (total) Chern class c(E) = c (E) of the vector bundle E.
If (E, ϕ) is a complex of vector bundles of the form (2.5) that is not necessarily a locally free resolution of a coherent analytic sheaf, in line with the Chern theory of virtual bundles as in e.g.[BB72, Section 4] or [Suw98, Section II.8.C], we let Moreover, if (E, ϕ) is equipped with a connection D = (D k ), cf.Section 2.2, we let and we let c (E, D) = c(E, D) be the component of degree 2 .Consider now a coherent analytic sheaf F with a locally free resolution (1.1).We define the Chern class of F by (1.2), i.e. c(F ) = c(E), and if (E, ϕ) is equipped with a connection D, then this class may be represented by (3.1).This definition of Chern classes of coherent sheaves may be motivated in terms of K-theory.However, it is typically considered only on manifolds with the so-called resolution property.Recall that a complex manifold X is said to have the resolution property if any coherent analytic sheaf F on X has a finite locally free resolution (1.1).For such manifolds, the definition (1.2) is the unique extension of the definition of Chern classes from locally free sheaves to coherent analytic sheaves that satisfies the following Whitney formula: In this paper, we define Chern classes of coherent sheaves by (1.2) also on manifolds which do not have the resolution property, but then necessarily only for coherent sheaves with a locally free resolution (1.1).Note that if we are on a manifold for which the resolution property does not hold, it is not immediate that the de Rham cohomology class of (1.2) is well-defined, i.e. independent of the resolution.However, that it is well-defined follows from a construction of Chern classes of arbitrary coherent analytic sheaves on arbitrary complex manifolds by Green, [Gre80], see also [TT86], since in case one has a global locally free resolution of finite length, the definition in [Gre80] coincides with the one in (1.2).

The Chern character (form) of a vector bundle
Assume that E is a holomorphic vector bundle of rank r.Then formally we can write where α i are the so-called Chern roots of E, see e.g.[Ful98, Remark 3.2.3].In particular, this means that c (E) = e (α 1 , . . ., α r ), where e is the th elementary symmetric polynomial The Chern character of E may formally be defined as the symmetric polynomial ch(E) = r i=1 e α i in the Chern roots, see e.g.[Ful98, Example 3.2.3].In particular, the th graded piece is where p is the th power sum polynomial Since any symmetric polynomial in x i may be expressed as a unique polynomial in e j (x), there are polynomials Q (t 1 , . . ., t ), ≥ 1, such that p (x) = Q (e 1 (x), . . ., e (x)); these are sometimes called Hirzebruch-Newton polynomials.If t j is given weight j, then Q (t 1 , . . ., t ) is a weighted homogenous polynomial of degree .Written out explicitly, Definition (3.2) should be read as If E is equipped with a connection D, one can analogously define Chern character forms The polynomials Q may be computed recursively through Newton's identities, In particular, it follows that the Q are independent of r.Moreover, ch (E, D) is of the form where Q is a weighted homogeneous polynomial of degree , and conversely, where Q is a weighted homogeneous polynomial of degree .
Example 3.1.We obtain from (3.5) that p 1 = e 1 and p 2 = e 2 1 −2e 2 .Thus, We have the following (formal) relationship between e (x) and p (x), and thus Q (e 1 , . . ., e ): (3.9) ln This follows e.g. by integrating [Mac95, Chapter I, Equation (2.10')] with respect to t.Since e 1 , . . ., e r are algebraically independent, (3.9) holds if we replace the e by a in any commutative ring.In particular, if we apply (3.9) to e = c (E, D) and take the components of degree 2 (the coefficents of t ) we get here () denotes the part of form degree 2 .

The Chern character of a complex of vector bundles
Let (E, ϕ) be a complex of vector bundles of the form (2.5).Then the Chern character can be defined as cf., e.g., [EH16, Chapter 14.2.1] and [Kar08, Chapter V.3].
If (E, ϕ) is equipped with a connection D = (D k ), for ≥ 1 we define a Chern character form ch (E, D) through (3.3).Then ch (E, D) inherits properties from the vector bundle case.In particular (3.6) and (3.7) hold.Also (3.10) holds and, using that In particular, ch (E, D) represents ch (E).
Let Θ k denote the curvature matrix of D k (in some local trivialization) and define 1 (3.12) In view of (3.4) and (3.11), for ≥ 1, Assume that D = (D k ) is a (1, 0)-connection.Let () (q,r) denote the part of bidegree (q, r) of a form.Since the curvature matrices Θ k (in local trivializations) consist of forms of bidegree (2, 0) and (1, 1), it follows that 1 To be consistent with (3.2) we should have a factor (i/2π) in the definition of p (E, D), cf.(3.13).However, the normalization (3.12) is more convenient to work with.
and by (3.13) that

Connections compatible with a complex
Assume that (E, ϕ) is a complex of vector bundles of the form (4.1) Moreover assume that each E k is equipped with a connection D k .Then, following [BB72], we say that the connection for k = 0, . . ., N .In terms of the induced connection D = D End on End E, this can succinctly be written as Dϕ k = 0. Note that in contrast to above, (4.1) starts at level −1.The typical situation we consider is when we start with a complex (2.5) that is pointwise exact outside an analytic variety Z and then restrict to X \ Z; then E −1 = 0. Remark 4.1.By [BB72, Lemma 4.17], given a complex (E, ϕ) of vector bundles of the form (4.1) one can always extend a given connection D −1 on E −1 to a connection D = (D k ) that is compatible with (E, ϕ) where it is pointwise exact.In fact, Lemma 4.4 below gives an explicit formula for such a connection, see Remark 4.5.Remark 4.2.In [BB72], the condition of being compatible is stated without the minus sign in (4.2); our condition on D is actually the same, but we need to introduce the minus sign since we use the conventions of the superstructure.Indeed, if ξ is a section of E k of form-degree 0, then D k−1 • ϕ k ξ is defined in the same way with or without the superstructure, while the action of ϕ k on D k ξ changes sign depending on whether the superstructure is used or not since D k ξ has form-degree 1, cf. (2.6).
Compatible connections satisfy the following Whitney formula, [BB72, Lemma 4.22], cf.Section 3.1.Lemma 4.3.Assume that (E, ϕ) is an exact complex of vector bundles of the form (4.1) that is equipped with a connection D = (D k ) that is compatible with (E, ϕ).Then

The connection D
We will consider a specific situation and choice of compatible connection.As in previous sections, let (E, ϕ) be a complex of vector bundles of the form (2.5) that is pointwise exact outside the analytic set Z.Moreover, let χ be a smooth approximand of χ [1,∞) , let F be a (generically nonvanishing) section of a vector bundle such that Z ⊆ {F = 0}, and let χ = χ(|F| 2 / ).Then χ ≡ 0 in a neighborhood of Z. Consider now a fixed choice of connection D = (D k ) on (E, ϕ), and for > 0, define a new connection D = ( D k ) on (E, ϕ) through Note that if D is a (1, 0)-connection, then so is D .
Remark 4.5.Assume that (E, ϕ) is a pointwise exact complex of vector bundles equipped with some connection D = (D k ).Then, as in the proof above, it follows that the connection D defined by Assume that θ k is a connection matrix for D k in a local trivialization, i.e.

The Chern current c Res (E, D)
In this section we prove that the limits as → 0 of products of Chern forms c (E, D ), where D is the connection from the previous section, give the desired currents in (1.5).More generally, we prove the following generalization of Theorem 1.1.
Theorem 5.1.Assume that (E, ϕ) is a complex of Hermitian vector bundles of the form (2.5) that is pointwise exact outside a subvariety Z of positive codimension.Moreover assume that D = (D k ) is a connection on (E, ϕ) and let D be the connection defined by (4.3).Then, for 1 , . . ., m ∈ N >0 , where the right side is defined by (3.1), is a well-defined closed pseudomeromorphic current that is independent of the choice of χ , has support on Z, and represents Theorem 1.1 is an immediate consequence of Theorem 5.1.
We first prove that lim →0 M exists and is a pseudomeromorphic current.This is a local statement and we may therefore work in a local trivialization where D k is determined by the connection matrix θ k .By (4.4), Θ k is a (form-valued) matrix of the form where α k = Θ k is smooth and β k , β k and β k are polynomials in σ k , Dϕ k , θ k and exterior derivatives of such factors.In particular α k , β k , β k , and β k are independent of .
Since M is a polynomial in the entries of Θ 0 , . . ., Θ N , see Section 3.1, we can write where A, B j , and B j are independent of , A is smooth, and B j and B j are polynomials in entries of σ k , Dϕ k , θ k and exterior derivatives of such factors.Let χ = χ(|F| 2 / ), where χ = χ j ∼ χ [1,∞) , cf.Section 2.
Then by Lemma 2.1, the limits of as → 0 exist and are pseudomeromorphic currents that are independent of χ .It follows that the limit (5.1) exists and is a pseudomeromorphic current that is independent of χ .By Lemma 4.4, D is compatible with (E, ϕ) where χ ≡ 1 and therefore, by Lemma 4.3, c(E, D ) = 0 there.It follows that M has support where χ 1.Note that the σ k are smooth outside of Z.By Lemma 2.1, the limit (5.1) is independent of the choice of χ .In particular, we may assume that the section F defining χ = χ(|F| 2 / ) is locally defined such that {F = 0} = Z.It then follows that the limit (5.1) has support on Z.That (5.1) represents c 1 (E) ∧ • • • ∧ c m (E) follows by Poincaré duality, since the forms on the right hand side of (5.1) represent this class for all > 0. Also (5.1) is closed since the forms on the right hand side are for all > 0.

An explicit description of Chern currents of low degrees
In this section we study the Chern current c Res (E, D) of a complex (E, ϕ) that is equipped with a (1, 0)connection D. Our main result is the following generalization of Theorem 1.5 that is an explicit description of c Res p (E, D) in terms of the residue current R associated with (E, ϕ).
Theorem 6.1.Assume that (E, ϕ) is a complex of Hermitian vector bundles of the form (2.5) that is pointwise exact outside a subvariety Z of codimension p, and let R be the corresponding residue current.Moreover, assume that D = (D k ) is a (1, 0)-connection on (E, ϕ) and let c Res (E, D) be the corresponding Chern current.Then In fact, Theorem 6.1 follows from the following formulation in terms of the Chern character (forms).For (E, ϕ) and D as in the theorem and for 1 , . . ., m ≥ 1 we let (6.4) where D is the connection defined by (4.3).By Theorem 5.1 this is a well-defined current with support on Z that represents ch 1 (E) Theorem 6.2.Assume that (E, ϕ), D, R, and p are as in Theorem 6.1.For ≥ 1, let ch Res (E, D) be the corresponding Chern character current (6.4).Then Moreover (6.6) ch Res (E, D) = 0 for < p and (6.7) ) is a polynomial of weighted degree we get that (6.8) , where s ≥ 2 and λ 1 + • • • + λ s = .Thus (6.8) vanishes by (6.7) for ≤ p.Now (6.1) and (6.2) follow by (3.7), (6.5), and (6.6).Also, the left hand side of (6.3) is a sum of terms of the form (6.9) and thus it vanishes.Remark 6.3.Taking Theorem 6.1 for granted, by similar arguments as in the proof above, using (3.6), we get Theorem 6.2.Thus Theorems 6.1 and 6.2 are equivalent.
Recall from Section 2.3 that if (E, ϕ) is a locally free resolution of a sheaf F of codimension p, then R k = 0 for > 0. Thus the only nonvanishing residue current in (6.1) is R p = R 0 p , and hence Theorem 1.5 follows.It may be noted that our proof of Theorem 6.1 does not become simpler in the situation of Theorem 1.5.
To organize the proof of Theorem 6.2 we will introduce a certain class O Z, , of forms depending on > 0 that in the limit are pseudomeromorphic currents with support on Z that vanish if ≤ codim Z.Throughout this section, let (E, ϕ) be fixed as the complex from Theorems 6.1 and 6.2 and let σ k be the minimal inverse of ϕ k as in Section 2.3.Let E q, denote smooth forms of bidegree ( * , q) that can be written as polynomials in χ , ∂χ , entries of σ k , ∂σ k or ∂σ k in some local trivialization for k = 1, . . ., N , and smooth forms independent of .Here χ = χ(|F| 2 / ), where χ is a smooth approximand of χ [1,∞) and F is a generically non-vanishing section of a holomorphic vector bundle such that Z = {F = 0}.We say that ψ ∈ E := ⊕E q, is in O Z, , if ψ is a sum of terms of the form a ∧ b , where a is a smooth form that is independent of , and b is in E q, , where q < , and vanishes where χ ≡ 1.In particular, if ψ ∈ E q, vanishes where χ ≡ 1, then ψ ∈ O Z, , for any > q.Note that (6.10) Lemma 6.4.Assume that Z has codimension p, and let ψ be a form in O Z, , with ≤ p. Then lim →0 ψ = 0.
Proof.Consider a term a ∧ b of ψ as above.Then b ∈ E q, , where q < ≤ p.By Lemma 2.1, the limit b := lim →0 b exists and is a pseudomeromorphic current of bidegree ( * , q).Since b ≡ 0 where χ ≡ 1, b has support on Z and thus b = 0 by the dimension principle, see Section 2.1.Since a is smooth and independent of , it follows that lim →0 (a ) , let χ and D be defined as in Section 4, and let c(E, D ) = c (E, D ) be the corresponding Chern form defined by (3.1).Since the limits in Theorem 5.1 are independent of the choice of χ , and the results in this section are local statements, we may assume locally that the section F in the definition of χ = χ(|F| 2 / ) is such that {F = 0} = Z.Lemma 6.5.For ≥ 1 and > 0, we have Proof.We may work in a local trivialization; let Θ k be the curvature matrix of D k .By Remark 5.2, since the D k are (1, 0)-connections, ch (E, D ) consists of components of bidegree ( + q, − q) with q ≥ 0. From the proof of Theorem 5.1 it follows that c(E, D ) is in E and vanishes where χ ≡ 1, and consequently, the same holds for ch(E, D ).It follows that ch ( +q, −q) (E, D ) ∈ O Z, , for q > 0, so (6.12) Since D is a (1, 0)-connection, by (3.15), (6.13) ch ( , where p ( , ) is given by (3.14).To prove the lemma it thus suffices to show that (6.14) To prove (6.14), first note in view of (4.4) that since D is a (1, 0)-connection, where Θ k is the curvature matrix of D k .We make the following decomposition: and expand the product.Note that γ k ∈ O Z,1, , and thus by (6.10) all terms with a factor γ k are in O Z, , .Next, note that since ( ∂χ ) 2 = 0, α 2 k = 0, and since α k and β k have total degree 4 and endomorphism degree 2, all terms containing one α k and the remaining − 1 factors being β k are all equal to tr(α k ∧ β −1 k ) by (2.7).To conclude, (6.17) (2.1), and in the middle expression all terms having a factor (Θ k ) (1,1) also contain a factor ∂χ , and thus are in O Z, , .Moreover, by (2.8) and (2.12), and hence since all terms containing a factor (Θ k ) (1,1) or (Θ k−1 ) (1,1) also contain a factor ∂χ .Note that α 0 = 0 since ϕ 0 = 0.It thus follows from (6.17) and (6.18) that Thus, to prove (6.14) it suffices to show that .Since c(j,j) vanishes for any j ≥ 1, p = 0. Note that , where Θ k is the curvature matrix corresponding to D k .Thus for > 0. This concludes the proof of (6.14).
Lemma 6.6.For ≥ 1 and > 0, we have If > N , the sum on the right hand side should be interpreted as 0.
Proof.For = 1 the sums differ only by a shift in the indices, so we may assume ≥ 2. For fixed k ∈ Z and m, r, s ≥ 0, let If m = 0, then the factor ∂σ k+m • • • ∂σ k+1 should be interpreted as 1.Moreover since (E, ϕ) starts at level 0 and ends at level N , we interpret ϕ j and σ j as 0 if j > N or j < 1, and consequently we interpret ρ r,s k,m as 0 if k + m ≥ N or k ≤ 0.
We claim that (6.21) ∂χ ∧ tr Let us take (6.21) for granted and let ρ t k,m = t r=0 ρ r,t−r k,m .Then by inductively applying (6.21) to r = 0, . . ., t with s = t − r, we get , .Thus, since ρ r,s 0,m and ρ r,s N −m,m vanish, (6.23) Assume that 2 ≤ ≤ N .By inductively applying (6.23) to m = 0, . . ., − 2 with t = − 2 − m, we get which after a shift in indices is exactly (6.20).If > N and we perform the same induction, after N − 1 steps we end up with which by (6.22) equals ρ ; thus (6.20) holds also in this case.It remains to prove (6.21).To do this let us replace the first factor Dϕ k ∂σ k in the left hand side of (6.21) by the right hand side of (2.17); we then get three terms.The term corresponding to the second term ∂σ k−1 Dϕ k−1 in (2.17) is precisely the first term in the right hand side of (6.21).Next, by (2.13) and (2.16), Applying this repeatedly we get Since Dϕ k+1 Next, by Theorem 5.1 and Remark 5.2, ch Res (E, D) is a pseudomeromorphic current with support on Z and with components of bidegree ( + q, − q) where q ≥ 0. Therefore it vanishes by the dimension principle when < p, see Section 2.1.This proves (6.6).
Assume that (E, ϕ) is a complex of Hermitian vector bundles of the form (2.5) such that H k (E) has pure codimension p or vanishes for k = 0, . . ., N , and let Z = ∪ supp H k (E).Then (E, ϕ) is pointwise exact outside Z, which has codimension p. Now, by combining Theorem 2.2 and Theorem 6.1, we obtain the following generalization of Theorem 1.4.Corollary 6.7.Assume that (E, ϕ) is a complex of Hermitian vector bundles of the form (2.5) such that H k (E) has pure codimension p or vanishes for k = 0, . . ., N .Moreover, assume that D . Moreover (1.10) and (1.11) hold.
Here [E] is the cycle of (E, ϕ) defined by (2.19).In particular, it follows from Corollary 6.7 that c Res p (E, D) is independent of the choice of Hermitian metric and (1, 0)-connection D on (E, ϕ).

An example
We will compute (products of) Chern currents c Res (E, D) for an explicit choice of (E, ϕ) and D. Let J ⊆ O P 2 [t,x,y] be defined by J = J (y k , x y m ), where m < k, and let O Z := O P 2 /J .Then Z has pure dimension 1, since Z red = {y = 0}, which is irreducible.However, note that J has an embedded prime J (x, y) of dimension 0. Now F = O Z has a locally free resolution of the form where in the trivialization in the coordinate chart Let us start by computing the Chern class of F .Let ω denote c 1 (O(1)).Then here the sum ends in degree 2, since we are in dimension 2. Thus, by (1.2), In particular,

Chern currents
Assume that each E k in (7.1) is equipped with the metric induced by the standard metric on O(1) → P 2 , in turn induced by the standard metric on C 3 , and let D k the corresponding Chern connection.