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Épijournal de Géométrie Algébrique
epiga.episciences.org
Volume 6 (2022), Article Nr. 14
Chern currents of coherent sheaves
Richard Lärkäng and Elizabeth Wulcan
Abstract.
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.
Keywords. Chern classes; coherent sheaves; residue currents
2020 Mathematics Subject Classification. 32A27; 14C17; 32C30; 14F06; 53C05
Received by the Editors on November 3, 2021, and in final form on January 19, 2022.
Accepted on March 16, 2022.
Richard Lärkäng
Department of Mathematics, Chalmers University of Technology and the University of Gothenburg, S-412 96 Gothenburg, Sweden
e-mail: larkang@chalmers.se
Elizabeth Wulcan
Department of Mathematics, Chalmers University of Technology and the University of Gothenburg, S-412 96 Gothenburg, Sweden
e-mail: wulcan@chalmers.se
The first author was partially supported by the Swedish Research Council (2017-04908) and the second author was partially
supported by the Swedish Research Council (2017-03905) and the Göran Gustafsson Foundation for Research in Natural Sciences
and Medicine.
© by the author(s) This work is licensed under http://creativecommons.org/licenses/by-sa/4.0/
arXiv:2105.14843v5 [math.CV] 29 Jul 2022
2 R. Lärkäng and E. Wulcan2 R. Lärkäng and E. Wulcan
Contents
1.Introduction................................ 2
2. Currents associated with complexes of vector bundles . . . . . . . . . . . . . . . 6
3. Chern forms and Chern characters. . . . . . . . . . . . . . . . . . . . . . . 10
4. Connections compatible with a complex . . . . . . . . . . . . . . . . . . . . 13
5. The Chern current cRes(E,D)......................... 14
6. An explicit description of Chern currents of low degrees . . . . . . . . . . . . . . 15
7.Anexample................................ 21
References.................................. 25
1. 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
suppF
has positive codimension. Assume that
F
has a locally free resolution of the
form
(1.1) 0→ O(EN)ϕN
−−−→ · ·· ϕ1
−−→ O(E0)→ F → 0,
where
Ek
are holomorphic vector bundles on
X
, and
O(Ek)
denote the corresponding locally free sheaves.
Then the (total) Chern class of Fequals
(1.2) c(F) =
N
Y
k=0
c(Ek)(−1)k,
where
c(Ek)
is the (total) Chern class of
Ek
, see Section 3.1. In this paper we construct explicit representatives
of the nontrivial part of c(F)with support on suppF.
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
Ek
is equipped with a Hermitian metric and a connection
Dk
(that is not necessarily
the Chern connection of the metric). Let
σk
be the minimal inverse of
ϕk
, see Section 2.3, let
χ:R≥0→R≥0
be a smooth cut-offfunction such that
χ(t)≡0
for
t1
and
χ(t)≡1
for
t1
, let
s
be a (generically
non-vanishing) holomorphic section of a vector bundle such that
{s= 0} ⊇ suppF
, let
χ=χ(|s|2/)
, and
let b
D
kbe the connection
(1.3) b
D
k=−χσkDϕk+Dk;
here
D
is a connection on
EndE
, where
E=LEk
, induced by the
Dk
, see Section 2.2. Then clearly the
Chern form
(1.4) c(E, b
D) :=
N
Y
k=0
c(Ek,b
D
k)(−1)k
Chern currents of coherent sheaves 3Chern currents of coherent sheaves 3
is a representative for
c(F)
. Throughout this paper we consider Chern classes as de Rham cohomology
classes of smooth forms or currents. Our first main result asserts that the limit of this form is a current with
the desired properties.
Theorem 1.1.
Assume that
F
is a coherent analytic sheaf of positive codimension that admits a locally free
resolution of the form
(1.1)
. Moreover, assume that each
Ek
is equipped with a Hermitian metric and a connection
Dk
. Let
c(E, b
D)
be the Chern form of
F
defined by
(1.4)
, and let
c`(E, b
D)
denote the component of degree
2`
. Let
`1,...,`m∈N>0. Then
(1.5) cRes
`1(E, D)∧ · ·· ∧ cRes
`m(E, D) := lim
→0c`1(E, b
D)∧ · ·· ∧ c`m(E, b
D)
is a well-defined closed current, independent of
χ
, that represents
c`1(F)∧ ··· ∧ c`m(F)
and has support on
suppF.
The Chern currents
(1.5)
are pseudomeromorphic in the sense of [
AW10
], see Theorem 5.1, which means
that they have a geometric nature similar to closed positive (or normal) currents, see Section 2.1. We let
cRes(E,D) = 1 + cRes
1(E, D) + cRes
2(E, D) + ··· .
The first nontrivial component of
cRes(E,D)
is (the current of integration along) a cycle, see Theorem 1.4
below. We do not know whether Chern currents of higher degree are of order 0in general.
Remark 1.2.
If all the connections
Dk
are
(1,0)
-connections, i.e. the
(0,1)
-part of each
Dk
equals
¯
∂
, then so
are the connections
b
D
k
. However, even if the
Dk
are Chern connections, the
b
D
k
are not Chern connections
in general. Thus, it might be the case that the involved forms and currents in
(1.5)
contain terms of bidegree
(`+m,` −m)with m > 0(but only when ` > codim Fby Theorem 1.4 below).
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
cRes(E,D)
when
F
is the structure sheaf
OZ
of a divisor
Z⊂X
, defined by a
holomorphic section sof a holomorphic line bundle Lover X, and (E,ϕ)is the locally free resolution
(1.6) 0→ O(L∗)s
→ O → OZ→0.
Assume that
L
is equipped with a connection
DL
; equip
E1=L∗
with the induced dual connection
DL∗
, and
E0
with the trivial connection. The minimal inverse of
s
is
1/s
and
Ds =DLs
, so
b
D
1=χ(DLs/s) + DL∗
, and
b
D
0
is the trivial connection. The curvature form of
b
D
1
equals
b
Θ
1=dχ∧(DLs/s)−(1 −χ)ΘL
, where
ΘL
is the curvature form of
DL
(which equals minus the curvature form of
DL∗
). By an appropriate formulation
of the Poincaré-Lelong formula,
(1.7) lim
→0dχ∧DLs
s= 2πi[Z],
where [Z]is the current of integration along (the cycle of) Z. Note that
c(E, b
D) = c(E1,b
D
1)−1= 1 −c1(E1,b
D
1) + c1(E1,b
D
1)2− · ·· .
Thus, since ΘLis smooth,
(1.8) cRes
1(E, D) = lim
→0−c1(E1,b
D
1) = −lim
→0
i
2πb
Θ
1= [Z].
4 R. Lärkäng and E. Wulcan4 R. Lärkäng and E. Wulcan
Thus, the first Chern current coincides with
[Z]
, which can be seen as a canonical representative of
c1(OZ)
with support on suppOZ.
By further calculations of terms of all degrees, one can show that
cRes(E,D) = 1 + [Z](1 + c1(L, D) + ···+cn−1
1(L,D)).
Our next result is an explicit description of the first nontrivial Chern current
cRes
p(E, D)
in the case when
F
has pure codimension
p
,i.e.
suppF
has pure dimension
dimX−p
, that generalizes
(1.8)
. Recall that the
(fundamental) cycle of Fis the cycle
(1.9) [F] = X
i
mi[Zi]
(considered as a current of integration), where
Zi
are the irreducible components of
suppF
, and
mi
is the
geometric multiplicity of Ziin 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
Ek
is equipped with a Hermitian metric and a
(1,0)-connection Dk. Then
cRes
p(E, D)=(−1)p−1(p−1)![F].
Moreover
(1.10) cRes
`(E, D)=0 for 0< ` < p
and
(1.11) cRes
`1(E, D)∧ · ·· ∧ cRes
`m(E, D)=0 for m≥2and 0< `1+···+`m≤p.
Here the Chern currents on the left hand sides are defined by (1.5).
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) cRes
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
Zi
of
codimension p.
In particular,
cRes
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
(1.13) cp(F)=(−1)p−1(p−1)![F],
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
RE=R=PRk
with support on
suppF
, where
Rk
is a
Hom(E0,Ek)
-valued
(0,k)
-current for
k= 0,...,N
, see Section 2.3. The construction involves the minimal inverses
σk
of
ϕk
.
If
(E, ϕ)
is the complex
(1.6)
, then
RE
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
RE
coincides with the classical
Coleff-Herrera residue current, [
CH78
]. Using residue currents, we can write the Poincaré-Lelong formula
(1.7) as
¯
∂1
s∧DLs= 2πi[Z];
Chern currents of coherent sheaves 5Chern currents of coherent sheaves 5
indeed, the left hand side in
(1.7)
equals
lim ¯
∂χ∧(DLs)/s
. Given a
Hom(E`,E`)
-valued current
α
, let
trα
denote the trace of
α
. In [
LW18
,
LW21
] we proved the following generalization of the Poincaré-Lelong
formula:
Assume that
RE
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
EndE
induced by arbitrary
(1,0)
-
connections on Ek. Then
(1.14) 1
(2πi)pp!tr(Dϕ1···DϕpRp)=[F].
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) cRes (E, D)in terms of RE.
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
Ek
is equipped with a Hermitian metric and a
(1,0)
-connection
Dk
. Let
R
be the associated residue current and
D
the connection on
EndE
induced by the
Dk
.
Then
cRes
p(E, D) = (−1)p−1
(2πi)pptr(Dϕ1···DϕpRp).
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, b
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
cRes
p(E, D) := lim→0cp(E, b
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
cRes
p(E, D)=(−1)p−1(p−1)![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
chRes
`(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
chRes
p(E, D) = 1
(2πi)pp!tr(Dϕ1···DϕpRp)=[F],
chRes
`(E, D)=0 for ` < p, and
chRes
`1(E, D)∧ · ·· ∧ chRes
`m(E, D)=0 for m≥2and `1+···+`m≤p,
see Theorem 6.2 and Remark 6.3.
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
ddc
-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:E0→E1
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
6 R. Lärkäng and E. Wulcan6 R. Lärkäng and E. Wulcan
[
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 2we give some necessary background on (residue) currents.
In Section 3we describe Chern forms and Chern characters, and in Section 4we discuss compatible
connections. The proofs of (the generalized versions of) Theorems 1.1 and 1.5 occupy Sections 5and 6,
respectively. Finally in Section 7we compute
cRes(E,D)
for an explicit choice of a locally free resolution
(E, ϕ)of a coherent sheaf F. In particular, we compute cRes
`(E, D)for ` > codim Fin this case.
Acknowledgements
This paper is very much inspired by an ongoing joint project with Lucas Kaufmann, which aims to
understand Baum-Bott residues in terms of (residue) currents. We are greatly indebted to him for many
valuable discussions on this topic. We would also like to thank Dennis Eriksson for many important
discussions and helpful comments on a previous version of this paper.
2. 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[1,∞)
of the
interval [1,∞)and write
χ∼χ[1,∞)
if
χ
is smooth and
χ(t)≡0
for
t1
and
χ(t)≡1
for
t1
. Note that if
χ∼χ[1,∞)
and
ˆ
χ=χ`
, then
ˆ
χ∼χ[1,∞)and
(2.1) dˆ
χ=`χ`−1dχ.
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
lim
→0Z|f|2>
ξ
f
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= 0}. Then
(2.2) [1/f ] = lim
→0
χ(|F|2/)
fand ¯
∂[1/f ] = lim
→0
¯
∂χ(|F|2/)
f,
Chern currents of coherent sheaves 7Chern currents of coherent sheaves 7
see e.g. [AW18]. In particular, the limits are independent of χand F.
In the literature there are various generalizations of residue currents and principal value currents. In
particular, Coleffand Herrera [CH78] introduced products like
(2.3) [1/f1]···[1/fr]¯
∂[1/fr+1]∧ · ·· ∧ ¯
∂[1/fm].
In order to obtain a coherent approach to questions about residue and principal value currents was introduced
in [
AW10
] the sheaf
P MX
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 MX
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 MX
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 \ Wof a pseudomeromorphic current µon Uis the pseudomeromorphic current
1U \Wµ:= lim
→0χ(|F|2/)µ|U,
where χ∼χ[1,∞)as above. This definition is independent of the choice of Fand χ.
A pseudomeromorphic current
µ
on
X
is said to have the standard extension property (SEP) if
1U \Wµ=µ|U
for any subvariety
W⊆ U
of positive codimension, where
U ⊆ X
is any open subset. By definition, it follows
that if µhas the SEP and F.0is any holomorphic section of a vector bundle, then
(2.4) lim
→0χ(|F|2/)µ=µ.
2.2. Superstructure and connections on a complex of vector bundles
Let (E, ϕ)be a complex
(2.5) 0−→ EN
ϕN
−−−→ EN−1
ϕN−1
−−−−→ · · · ϕ2
−−→ E1
ϕ1
−−→ E0−→ 0,
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
Z2
-grading, which splits
E:= ⊕Ek
into odd and even parts
E+
and
E−
, where
E+=⊕E2k
and
E−=⊕E2k+1
. Also
EndE
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•(EndE)
, where
γ
is a holomorphic section of
Hom(E`,Ek)
, and
ω
is a smooth form of degree
m
. Then we let
degfα=m
and
degeα=k−`
denote the form and endomorphism degrees, respectively, of
α
. The total degree is
degα= degfα+ degeα
. If
β
is a form-valued section of
E
,i.e.
β=η⊗ξ
, 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
(2.6) α(β) := (−1)(degeα)(degfβ)ω∧η⊗γ(ξ).
If furthermore,
α0=ω0⊗γ0
, where
γ0
is a holomorphic section of
EndE
, and
ω0
is a smooth form, both
homogeneous in degree, then we define
αα0:= (−1)(degeα)(degfα0)ω∧ω0⊗γ◦γ0.
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 α0above, we get such an equality with a sign due to the superstructure,
(2.7) tr(αα0)=(−1)(degα)(degα0)−(degeα)(degeα0)tr(α0α),
see [LW18, Equation (2.14)].
8 R. Lärkäng and E. Wulcan8 R. Lärkäng and E. Wulcan
Note that
¯
∂
extends in a way that respects the superstructure to act on
EndE
-valued morphisms. In
particular,
(2.8) ¯
∂(αα0) = ¯
∂αα0+ (−1)degαα¯
∂α0.
We will consider the situation when
(E, ϕ)
is equipped with a connection
D=DE= (D0,...,DN)
, where
Dk
is a connection on
Ek
. Then there is an induced connection
⊕Dk
on
E
, that we also denote by
DE
. This
in turn induces a connection DEnd on EndEthat takes the superstructure into account, defined by
(2.9) DEndα:= DE◦α−(−1)deg αα◦DE,
if αis a EndE-valued form. It satisfies the following Leibniz rule, [LW18, Equation (2.4)], cf. (2.8)
(2.10) DEnd(αα0) = DEndαα0+ (−1)deg ααDEndα0.
To simplify notation, we will sometimes drop the subscript
End
and simply denote this connection by
D
. If
Θkdenotes the curvature form of Dk, and α:Ek→E`, then, by (2.9),
(2.11) DDα =Θ`α+ (−1)degα+degα+1 αΘk=Θ`α−αΘk.
The above formulas hold also when
α
and
α0
are current-valued instead of form-valued, as long as the
involved products of currents are well-defined.
We let
D0
k
and
D00
k
denote the
(1,0)
- and
(0,1)
-parts of
Dk
, respectively, and we let
D0= (D0
k)
and
D00 = (D00
k)
denote the corresponding
(1,0)
- and
(0,1)
-parts of
DE= (Dk)
. We say that
DE
is a
(1,0)
-
connection if each
Dk
is a
(1,0)
-connection, i.e.
D00
k=¯
∂
. We will use the following consequence of
(2.11)
:
assume that
DE
is a
(1,0)
-connection, and
α:Ek→E`
is a holomorphic (or more generally a
¯
∂
-closed
form-valued) morphism. Then
(2.12) ¯
∂Dα = (Θ`)(1,1)α−α(Θk)(1,1),
where (·)(1,1) denotes the component of bidegree (1,1).
Since (E, ϕ)is a complex and ϕkhas odd degree, it follows from (2.10) that
(2.13) ϕk−1Dϕk=Dϕk−1ϕk.
2.3. 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
Ek
is equipped with some
Hermitian metric. If
Zk
is the analytic set where
ϕk
has lower rank than its generic rank, then outside of
Zk
the minimal (or Moore-Penrose) inverse
σk:Ek−1→Ek
of
ϕk
is determined by the following properties:
ϕkσkϕk=ϕk
,
imσk⊥imϕk+1
, and
σk+1σk= 0
. One can verify that
σk
is smooth outside of
Zk
. Since
σkσk−1= 0 and σkhas odd degree, by (2.8),
(2.14) σk¯
∂σk−1=¯
∂σkσk−1.
Let Zbe the set where (E,ϕ)is not pointwise exact. It follows from the definition of σkthat
(2.15) ϕkσk+σk−1ϕk−1= IdEk−1
outside Z, or more generally outside Zk∪Zk−1. Applying (2.8) to (2.15), we obtain that outside Z
(2.16) ϕk¯
∂σk=¯
∂σk−1ϕk−1
and furthermore applying (2.10) to this equality, we get that
(2.17) Dϕk¯
∂σk=D¯
∂σk−1ϕk−1+¯
∂σk−1Dϕk−1+ϕkD¯
∂σk.
Chern currents of coherent sheaves 9Chern currents of coherent sheaves 9
Lemma 2.1.
Let
X
,
(E, ϕ)
,
Z
, and
σk
be as above. Assume that for each
j= 1,...,m
,
sj
is an entry of
σk
,
∂σk
,
or
¯
∂σk
for some
k
in some local trivialization, and let
s=s1···sm
. Assume that
χ∼χ[1,∞)
and that
F
is a
(generically nonvanishing) holomorphic section of a vector bundle over Xsuch that Z⊂ {F= 0}. Then the limits
lim
→0χ(|F|2/)sand lim
→0
¯
∂χ(|F|2/)∧s
exist and define pseudomeromorphic currents on
X
that are independent of the choices of
χ
and
F
; the support of
the second current is contained in Z. Furthermore,
lim
→0∂χ(|F|2/)∧s= 0.
Proof.
By Hironaka’s theorem there is a holomorphic modification
π:˜
X→X
, such that for each
k
,
π∗σk
is locally of the form
(1/γk)˜
σk
, where
γk
is holomorphic with
Z(γk)⊂˜
Z:= π−1Z
, and
˜
σk
is smooth, see
[
AW07
, Section 2]. Now, where
χ(|π∗F|2/).0
,
¯
∂π∗σk= (1/γk)¯
∂˜
σk
and
∂π∗σk=∂(1/γk)˜
σk+ (1/γk)∂˜
σk
.
Since each holomorphic derivative
∂/∂zi(1/γk)
is a meromorphic function with poles contained in
˜
Z
it
follows that
π∗sj
equals (a sum of terms of the form)
(1/gj)˜
sj
, where
gj
is holomorphic with
Z(gj)⊂˜
Z
, and
˜
sj
is smooth. Thus
π∗s
equals (a sum of terms of the form)
(1/g)˜
s
, where
g
is holomorphic with
Z(g)⊂˜
Z
,
and ˜
sis smooth. In view of (2.2),
lim
→0χ(|π∗F|2/)π∗sand lim
→0
¯
∂χ(|π∗F|2/)∧π∗s
are well-defined pseudomeromorphic currents on
˜
X
independent of
χ
and
F
; the second current has support
on
˜
Z
. Since
P M
is closed under push-forwards of modifications, cf. Section 2.1, this proves the first part of
the lemma.
As proved above, the limit
µ:= lim
→0χ(|F|2/)s
exists. This current is in fact a so-called almost semi-meromorphic current, cf. [
AW18
, Section 4], and in
particular, it has the SEP. By [AW18, Theorem 3.7], ∂µ also has the SEP. Thus,
lim
→0∂χ(|F|2/)∧µ= lim
→0∂(χ(|F|2/)∧µ)−lim
→0χ(|F|2/)∂µ =∂µ −∂µ = 0,
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
(2.18) R`
k:= lim
→0
¯
∂χ(|F|2/)∧σk¯
∂σk−1··· ¯
∂σ`+1
is a
Hom(E`,Ek)
-valued pseudomeromorphic current of bidegree
(0,k −`)
with support on
Z
; in fact, it
follows from the proof that the support is contained in
Z`+1 ∪ · ·· ∪ Zk
. If
`=k−1
, then the right hand side
of
(2.18)
should be interpreted as
lim→0¯
∂χ(|F|2/)∧σk
. The residue current
RE=R:= PR`
k
associated
with
(E, ϕ)
was introduced in [
AW07
], cf. the introduction. Assume that
(E, ϕ)
is a locally free resolution of
a coherent analytic sheaf
F
. Then
R`
k
vanishes for
` > 0
by [
AW07
, Theorem 3.1]. In this case
R=PRk
,
where Rk=R0
k.
Given a complex (E,ϕ)of vector bundles of the form (2.5), following [LW21], we define the cycle
(2.19) [E] =
N
X
k=0
(−1)k[Hk(E)],
where
Hk
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 [E]=[F]. In [LW21] we prove the following generalization of (1.14).
10 R. Lärkäng and E. Wulcan10 R. Lärkäng and E. Wulcan
Theorem 2.2.
Let
(E, ϕ)
be a complex of Hermitian vector bundles of the form
(2.5)
such that
Hk(E)
has pure
codimension p > 0or vanishes, for k= 0,...,N, and let Dbe an arbitrary (1,0)-connection on (E, ϕ). Then,
1
(2πi)pp!
N−p
X
k=0
(−1)ktr(Dϕk+1 ···Dϕk+pRk
k+p)=[E].
3. Chern forms and Chern characters
3.1. 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) Chern form c(E,D) = 1 + c1(E,D) + ···+cr(E,D)is defined by
r
X
`=0
c`(E, D)t`= detI+i
2πΘt,
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) = Pc`(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
c(E) =
N
Y
k=0
c(Ek)(−1)k.
Moreover, if (E, ϕ)is equipped with a connection D= (Dk),cf. Section 2.2, we let
(3.1) c(E, D) =
N
Y
k=0
c(Ek,Dk)(−1)k
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
afinite 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: if
0→ F 0→ F → F 00 →0
is a short exact sequence of sheaves, then
c(F) = c(F00 )c(F00 )
,
cf. [BS58, Théorème 2] or [EH16, Chapter 14.2].
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).
Chern currents of coherent sheaves 11Chern currents of coherent sheaves 11
3.2. The Chern character (form) of a vector bundle
Assume that Eis a holomorphic vector bundle of rank r. Then formally we can write
1 + c1(E)t+···+cr(E)tr=
r
Y
i=1
(1 + αit),
where
αi
are the so-called Chern roots of
E
, see e.g. [
Ful98
, Remark 3.2.3]. In particular, this means that
c`(E) = e`(α1,...,αr), where e`is the `th elementary symmetric polynomial
e`(x) = e`(x1,...,xr) = X
1≤i1<···<i`≤r
xi1···xi`.
The Chern character of
E
may formally be defined as the symmetric polynomial
ch(E) = Pr
i=1 eαi
in the
Chern roots, see e.g. [Ful98, Example 3.2.3]. In particular, the `th graded piece is
(3.2) ch`(E) = 1
`!p`(α1,...,αr),
where p`is the `th power sum polynomial
p`(x) = p`(x1,...,xr) =
r
X
i=1
x`
i.
Since any symmetric polynomial in
xi
may be expressed as a unique polynomial in
ej(x)
, there are polynomi-
als
Q`(t1,...,t`)
,
`≥1
, such that
p`(x) = Q`(e1(x),...,e`(x))
; these are sometimes called Hirzebruch–Newton
polynomials. If
tj
is given weight
j
, then
Q`(t1,...,t`)
is a weighted homogenous polynomial of degree
`
.
Written out explicitly, Definition (3.2) should be read as
ch`(E) = 1
`!Q`c1(E),...,c`(E).
If Eis equipped with a connection D, one can analogously define Chern character forms
(3.3) ch`(E, D) = 1
`!Q`c1(E, D),...,c`(E,D)
and
ch(E, D) = Pch`(E, D)
representing the Chern character. If
Θ
is the curvature corresponding to
D
(in
a local trivialization), then
(3.4) ch`(E, D) = 1
`!tri
2πΘ`
,
cf. e.g. [Tu17, §B.4-6].
The polynomials Q`may be computed recursively through Newton’s identities,
(3.5) p`(x)=(−1)`−1`e`(x) +
`−1
X
i=1
(−1)`−i−1e`−i(x)pi(x), ` ≥1.
In particular, it follows that the Q`are independent of r. Moreover, ch`(E,D)is of the form
(3.6) ch`(E, D) = (−1)`−1
(`−1)! c`(E, D) + e
Q`c1(E, D),...,c`−1(E,D),
where e
Q`is a weighted homogeneous polynomial of degree `, and conversely,
(3.7) c`(E, D)=(−1)`−1(`−1)! ch`(E,D) + b
Q`ch1(E, D),...,ch`−1(E, D),
where b
Q`is a weighted homogeneous polynomial of degree `.
12 R. Lärkäng and E. Wulcan12 R. Lärkäng and E. Wulcan
Example 3.1.
We obtain from
(3.5)
that
p1=e1and p2=e2
1−2e2
. Thus,
Q1(t1) = t1
and
Q2(t1,t2) = t2
1−2t2
,
so
(3.8) ch1(E, D) = c1(E, D)and ch2(E, D ) = 1
2(c1(E, D)2−2c2(E, D)).
We have the following (formal) relationship between e`(x)and p`(x), and thus Q`(e1,...,e`):
(3.9) ln
X
`≥0
e`(x)t`
=X
`≥1
(−1)`−1
`p`(x)t`=X
`≥1
(−1)`−1
`Q`(e1,...,e`)t`.
This follows e.g. by integrating [
Mac95
, Chapter I, Equation (2.10’)] with respect to
t
. Since
e1,...,er
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
(3.10) lnc(E, D)`= (−1)`−1(`−1)!ch`(E, D);
here ()`denotes the part of form degree 2`.
3.3. 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
ch(E) =
N
X
k=0
(−1)kch(Ek),
cf.,e.g., [EH16, Chapter 14.2.1] and [Kar08, Chapter V.3].
If
(E, ϕ)
is equipped with a connection
D= (Dk)
, 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
lnc(E, D)= ln
N
Y
k=0
c(Ek,Dk)(−1)k
=
N
X
k=0
(−1)klnc(Ek,Dk),
we get that
(3.11) ch`(E, D) =
N
X
k=0
(−1)kch`(Ek,Dk).
In particular, ch`(E, D)represents ch`(E).
Let Θkdenote the curvature matrix of Dk(in some local trivialization) and define1
(3.12) p`(E, D) =
N
X
k=0
(−1)ktrΘ`
k.
In view of (3.4) and (3.11), for `≥1,
(3.13) ch`(E, D) = i`
(2π)``!p`(E, D).
Assume that
D= (Dk)
is a
(1,0)
-connection. Let
()(q,r)
denote the part of bidegree
(q, r)
of a form. Since
the curvature matrices
Θk
(in local trivializations) consist of forms of bidegree
(2,0)
and
(1,1)
, it follows that
(3.14) p(`,`)(E,D) =
N
X
k=0
(−1)ktr(Θk)`
(1,1)
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.
Chern currents of coherent sheaves 13Chern currents of coherent sheaves 13
and by (3.13) that
(3.15) ch(`,`)(E,D) = i`
(2π)``!p(`,`)(E,D).
4. Connections compatible with a complex
Assume that (E, ϕ)is a complex of vector bundles of the form
(4.1) 0→EN
ϕN
−−−→ · ·· ϕ1
−−→ E0
ϕ0
−−→ E−1→0.
Moreover assume that each
Ek
is equipped with a connection
Dk
. Then, following [
BB72
], we say that the
connection D= (D−1,...,DN)on (E,ϕ)is compatible with (E,ϕ)if
(4.2) Dk−1◦ϕk=−ϕk◦Dk
for
k= 0,...,N
. In terms of the induced connection
D=DEnd
on
EndE
, this can succinctly be written as
Dϕk= 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= (Dk)
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
Ek
of form-degree
0
, then
Dk−1◦ϕkξ
is defined in the
same way with or without the superstructure, while the action of
ϕk
on
Dkξ
changes sign depending on
whether the superstructure is used or not since Dkξhas 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= (Dk)that is compatible with (E, ϕ). Then
c(E−1,D−1) =
N
Y
k=0
c(Ek,Dk)(−1)k.
4.1. The connection b
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= (Dk)
on
(E, ϕ)
, and for
> 0
, define a new connection
b
D= (b
D
k)
on
(E, ϕ)
through
(4.3) b
D
k=−χσkDϕk+Dk.
Note that if Dis a (1,0)-connection, then so is b
D.
Lemma 4.4. The connection b
Dis compatible with (E, ϕ)where χ≡1.
14 R. Lärkäng and E. Wulcan14 R. Lärkäng and E. Wulcan
Proof. Using (2.9), (2.13) and (2.15) we obtain that
b
Dϕk=b
D
k−1◦ϕk+ϕk◦b
D
k
=χ(−σk−1Dϕk−1ϕk−ϕkσkDϕk)+Dk−1◦ϕk+ϕk◦Dk
=−χ(σk−1ϕk−1+ϕkσk)Dϕk+Dk−1◦ϕk+ϕk◦Dk
= (1 −χ)Dϕk.
In particular, b
Dis compatible with the complex where χ≡1.
Remark 4.5.
Assume that
(E, ϕ)
is a pointwise exact complex of vector bundles equipped with some
connection D= (Dk). Then, as in the proof above, it follows that the connection e
Ddefined by
e
Dk=−σkDϕk+Dk
is compatible with (E, ϕ). Moreover, e
D−1=D−1,cf. Remark 4.1.
Assume that
θk
is a connection matrix for
Dk
in a local trivialization, i.e.
Dkα=dα +θk∧α
. Then the
connection matrix for b
D
kis
ˆ
θ
k=−χσkDϕk+θk
and thus the curvature matrix of b
D
kequals
(4.4) b
Θ
k=dˆ
θ
k+ ( ˆ
θ
k)2=−d(χσkDϕk) + χ2
σkDϕkσkDϕk−χ(θkσkDϕk+σkDϕkθk) + Θk,
where Θkis the curvature matrix of Dk.
5. The Chern current cRes(E,D)
In this section we prove that the limits as
→0
of products of Chern forms
c`(E, b
D)
, where
b
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= (Dk)
is a connection on
(E, ϕ)
and let b
Dbe the connection defined by (4.3). Then, for `1,...,`m∈N>0,
(5.1) cRes
`1(E, D)∧ · ·· ∧ cRes
`m(E, D) = lim
→0c`1(E, b
D)∧ · ·· ∧ c`m(E, b
D),
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 c`1(E)∧ · ·· ∧ c`m(E).
Theorem 1.1 is an immediate consequence of Theorem 5.1.
Proof. Let
(5.2) M=c`1(E, b
D)∧ · ·· ∧ c`m(E, b
D).
We first prove that
lim→0M
exists and is a pseudomeromorphic current. This is a local statement and we
may therefore work in a local trivialization where
Dk
is determined by the connection matrix
θk
. By
(4.4)
,
b
Θ
kis a (form-valued) matrix of the form
b
Θ
k=αk+χβ0
k+χ2
β00
k+dχ∧β000
k,
where
αk=Θk
is smooth and
β0
k
,
β00
k
and
β000
k
are polynomials in
σk
,
Dϕk
,
θk
and exterior derivatives of
such factors. In particular αk,β0
k,β00
k, and β000
kare independent of .
Chern currents of coherent sheaves 15Chern currents of coherent sheaves 15
Since Mis a polynomial in the entries of b
Θ
0,...,b
Θ
N, see Section 3.1, we can write
M=A+X
j≥1
χj
B0
j+X
j≥1
χj−1
dχ∧B00
j,
where
A
,
B0
j
, and
B00
j
are independent of
,
A
is smooth, and
B0
j
and
B00
j
are polynomials in entries of
σk
,
Dϕk
,
θk
and exterior derivatives of such factors. Let
ˆ
χ=ˆ
χ(|F|2/)
, where
ˆ
χ=χj∼χ[1,∞)
,cf. Section 2.
Then by Lemma 2.1, the limits of
χj
B0
j=ˆ
χB0
jand χj−1
dχ∧B00
j=dˆ
χ∧B00
j/j
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,
b
D
is compatible with
(E, ϕ)
where
χ≡1
and therefore, by Lemma 4.3,
c(E, b
D)=0
there. It follows that
M
has support where
χ.1
. Note that the
σk
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.
Remark 5.2.
Assume that
D= (Dk)
in Theorem 5.1 is a
(1,0)
-connection. Then
b
Θ
k
only has components
of bidegree
(2,0)
and
(1,1)
,cf.
(4.4)
. It follows that
(5.2)
and consequently
(5.1)
consist of components of
bidegree (`+q,` −q)with q≥0, where `=`1+···+`m.
6. An explicit description of Chern currents of low degrees
In this section we study the Chern current
cRes(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 cRes
p(E, D)in terms of the residue current Rassociated 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= (Dk)is a (1,0)-connection on (E, ϕ)and let cRes(E,D)be the corresponding Chern current. Then
(6.1) cRes
p(E, D) = (−1)p−1
(2πi)pp
N−p
X
k=0
(−1)ktr(Dϕk+1 ···Dϕk+pRk
k+p).
Moreover
(6.2) cRes
`(E, D)=0 for 0< ` < p
and
(6.3) cRes
`1(E, D)∧ · ·· ∧ cRes
`m(E, D)=0 for m≥2and 0< `1+···+`m≤p.
In fact, Theorem 6.1 follows from the following formulation in terms of the Chern character (forms). For
(E, ϕ)and Das in the theorem and for `1,...,`m≥1we let
(6.4) chRes
`1(E, D)∧ · ·· ∧ chRes
`m(E, D) := lim
→0ch`1(E, b
D)∧ · ·· ∧ ch`m(E, b
D),
where
b
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)∧ · ·· ∧ ch`m(E).
16 R. Lärkäng and E. Wulcan16 R. Lärkäng and E. Wulcan
Theorem 6.2.
Assume that
(E, ϕ)
,
D
,
R
, and
p
are as in Theorem 6.1. For
`≥1
, let
chRes
`(E, D)
be the
corresponding Chern character current (6.4). Then
(6.5) chRes
p(E, D) = 1
(2πi)pp!
N−p
X
k=0
(−1)ktr(Dϕk+1 ···Dϕk+pRk
k+p).
Moreover
(6.6) chRes
`(E, D)=0 for ` < p
and
(6.7) chRes
`1(E, D)∧ · ·· ∧ chRes
`m(E, D)=0 for m≥2and `1+···+`m≤p.
Proof of Theorem 6.1.Since b
Q`in (3.7) is a polynomial of weighted degree `we get that
(6.8) b
Q`chRes
1(E, D),...,chRes
`−1(E, D)
is a sum of terms
(6.9) chRes
λ1(E, D)∧ · ·· ∧ chRes
λs(E, D),where s≥2and λ1+···+λs=`.
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
Rp=R0
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
OZ,`,
of forms depending on
> 0
that in the limit are pseudomeromorphic currents with support on
Z
that vanish if
`≤codimZ
.
Throughout this section, let
(E, ϕ)
be fixed as the complex from Theorems 6.1 and 6.2 and let
σk
be the
minimal inverse of
ϕk
as in Section 2.3. Let
Eq,
denote smooth forms of bidegree
(∗,q)
that can be written
as polynomials in
χ
,
¯
∂χ
, entries of
σk
,
∂σk
or
¯
∂σk
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:= ⊕Eq,
is in
OZ,`,
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
Eq,
, where
q < `
, and vanishes where
χ≡1
. In particular, if
ψ∈ Eq,
vanishes where χ≡1, then ψ∈OZ,`, for any ` > q. Note that
(6.10) Eq, ∧OZ,`, ⊆OZ,`+q,.
Lemma 6.4.
Assume that
Z
has codimension
p
, and let
ψ
be a form in
OZ,`,
with
`≤p
. Then
lim→0ψ= 0.
Proof.
Consider a term
a∧b
of
ψ
as above. Then
b∈ Eq,
, where
q < ` ≤p
. By Lemma 2.1, the limit
b:= lim→0b
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∧b) = a∧b= 0.
Throughout this section, let
D= (Dk)
be a
(1,0)
-connection on
(E, ϕ)
, let
χ∼χ[1,∞)
, let
χ
and
b
D
be
defined as in Section 4, and let
c(E, b
D) = Pc`(E, b
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.
Chern currents of coherent sheaves 17Chern currents of coherent sheaves 17
Lemma 6.5. For `≥1and > 0, we have
(6.11) ch`(E, b
D) = 1
(2πi)``!
¯
∂χ`
∧
N
X
k=1
(−1)ktrσkDϕk(¯
∂σkDϕk)`−1+OZ,`, .
Proof.
We may work in a local trivialization; let
b
Θ
k
be the curvature matrix of
b
D
k
. By Remark 5.2, since the
Dk
are
(1,0)
-connections,
ch`(E, b
D)
consists of components of bidegree
(`+q, ` −q)
with
q≥0
. From the
proof of Theorem 5.1 it follows that
c(E, b
D)
is in
E
and vanishes where
χ≡1
, and consequently, the same
holds for ch(E, b
D). It follows that ch(`+q,`−q)(E, b
D)∈OZ,`, for q > 0, so
(6.12) ch`(E, b
D) = ch(`,`)(E, b
D) + OZ,`,.
Since b
Dis a (1,0)-connection, by (3.15),
(6.13) ch(`,`)(E, b
D) = i`
(2π)``!p(`,`)(E, b
D),
where p(`,`)is given by (3.14). To prove the lemma it thus suffices to show that
(6.14) p(`,`)(E, b
D)=(−1)`¯
∂χ`
∧
N
X
k=1
(−1)ktrσkDϕk(¯
∂σkDϕk)`−1+OZ,`, .
To prove (6.14), first note in view of (4.4) that since Dis a (1,0)-connection,
(6.15) (b
Θ
k)(1,1) =−¯
∂(χσkDϕk) + (Θk)(1,1),
where Θkis the curvature matrix of Dk. We make the following decomposition:
−¯
∂(χσkDϕk) + (Θk)(1,1) =−¯
∂χ∧σkDϕk−χ¯
∂(σkDϕk)−(Θk)(1,1)+ (1 −χ)(Θk)(1,1)
=: αk+βk+γk.
(6.16)
Let us consider
tr(b
Θ
k)`
(1,1) = tr(αk+βk+γk)`
and expand the product. Note that
γk∈OZ,1,
, and thus by
(6.10)
all terms with a factor
γk
are in
OZ,`,
.
Next, note that since
(¯
∂χ)2= 0
,
α2
k= 0
, and since
αk
and
βk
have total degree
4
and endomorphism
degree
2
, all terms containing one
αk
and the remaining
`−1
factors being
βk
are all equal to
tr(αk∧β`−1
k)
by (2.7). To conclude,
(6.17) tr(b
Θ
k)`
(1,1) =`tr(αk∧β`−1
k) + tr β`
k+OZ,`,.
We have that
`tr(αk∧β`−1
k)=(−1)`¯
∂χ`
∧trσkDϕk¯
∂(σkDϕk)−(Θk)(1,1)`−1
= (−1)`¯
∂χ`
∧trσkDϕk¯
∂(σkDϕk)`−1+OZ,`, ,
since
`χ`−1
¯
∂χ=¯
∂χ`
,cf.
(2.1)
, and in the middle expression all terms having a factor
(Θk)(1,1)
also contain
a factor ¯
∂χ`
, and thus are in OZ,`, . Moreover, by (2.8) and (2.12),
¯
∂(σkDϕk) = ¯
∂σkDϕk−σk¯
∂(Dϕk) = ¯
∂σkDϕk−σk(Θk−1)(1,1)ϕk+σkϕk(Θk)(1,1),
and hence
(6.18) `tr(αk∧β`−1
k)=(−1)`¯
∂χ`
∧trσkDϕk¯
∂σkDϕk`−1+OZ,`, ,
18 R. Lärkäng and E. Wulcan18 R. Lärkäng and E. Wulcan
since all terms containing a factor
(Θk)(1,1)
or
(Θk−1)(1,1)
also contain a factor
¯
∂χ`
. Note that
α0= 0
since
ϕ0= 0. It thus follows from (6.17) and (6.18) that
p(`,`)(E, b
D) =
N
X
k=0
(−1)ktr(b
Θ
k)`
(1,1)
= (−1)`¯
∂χ`
∧
N
X
k=1
(−1)ktrσkDϕk¯
∂σkDϕk`−1+
N
X
k=0
(−1)ktrβ`
k+OZ,`,.
Thus, to prove
(6.14)
it suffices to show that
PN
k=0(−1)ktr β`
k
vanishes for
`≥1
. Outside
Z
, let
e
D
be the
connection on (E, ϕ)defined by
(6.19) e
Dk:= −σkDϕk+Dk
and let
˜
c=c(E, e
D)
be the corresponding Chern form defined by
(3.1)
. If follows from Lemma 4.4 that
e
D
is
compatible with
(E, ϕ)
and thus by Lemma 4.3,
˜
cj
vanishes for
j≥1
. For
`≥1
, let
˜
p`:= p(`,`)(E, e
D)
, where
p(`,`)
is given by
(3.14)
. By
(3.15)
and
(3.6)
,
˜
p`
is a polynomial in
˜
c(1,1),..., ˜
c(`,`)
. Since
˜
c(j,j )
vanishes for any
j≥1,˜
p`= 0. Note that βk=χ(e
Θk)(1,1), where e
Θkis the curvature matrix corresponding to e
Dk. Thus
N
X
k=0
(−1)ktrβ`
k=χ`
N
X
k=0
(−1)ktr(e
Θk)`
(1,1) =χ`
˜
p`= 0
for > 0. This concludes the proof of (6.14).
Lemma 6.6. For `≥1and > 0, we have
(6.20) ¯
∂χ∧
N
X
k=1
(−1)ktrσkDϕk(¯
∂σkDϕk)`−1=
(−1)`
N−`
X
k=0
(−1)k¯
∂χ∧trσk+`¯
∂σk+`−1··· ¯
∂σk+1Dϕk+1 ···Dϕk+`+OZ,`, +¯
∂OZ,`,.
If ` > N , the sum on the right hand side should be interpreted as 0.
Here ¯
∂OZ,`, means forms of the form ¯
∂ψ, where ψ∈OZ,`,.
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
ρr,s
k,m =¯
∂χ∧trσk+m+1 ¯
∂σk+m··· ¯
∂σk+1(¯
∂σkDϕk)rD¯
∂σk(Dϕk¯
∂σk)sDϕk···Dϕk+mϕk+m+1.
If
m= 0
, then the factor
¯
∂σk+m··· ¯
∂σk+1
should be interpreted as
1
. Moreover since
(E, ϕ)
starts at level
0
and ends at level
N
, we interpret
ϕj
and
σj
as
0
if
j > N
or
j < 1
, and consequently we interpret
ρr,s
k,m
as
0
if k+m≥Nor k≤0.
We claim that
(6.21) ¯
∂χ∧trσk+m¯
∂σk+m−1··· ¯
∂σk(¯
∂σk−1Dϕk−1)r(Dϕk¯
∂σk)s+1Dϕk···Dϕk+m=
¯
∂χ∧trσk+m¯
∂σk+m−1··· ¯
∂σk(¯
∂σk−1Dϕk−1)r+1(Dϕk¯
∂σk)sDϕk···Dϕk+m
+ρr,s
k−1,m +ρr,s
k,m +OZ,r+s+m+2, +¯
∂OZ,r +s+m+2,.
Chern currents of coherent sheaves 19Chern currents of coherent sheaves 19
Let us take
(6.21)
for granted and let
ρt
k,m =Pt
r=0 ρr,t−r
k,m
. Then by inductively applying
(6.21)
to
r= 0,...,t
with s=t−r, we get
(6.22) ¯
∂χ∧trσk+m¯
∂σk+m−1··· ¯
∂σk(Dϕk¯
∂σk)t+1Dϕk···Dϕk+m=
¯
∂χ∧trσk+m¯
∂σk+m−1··· ¯
∂σk−1(Dϕk−1¯
∂σk−1)tDϕk−1···Dϕk+m
+ρt
k−1,m +ρt
k,m +OZ,t+m+2, +¯
∂OZ,t+m+2,.
It follows that, for fixed mand t,
N−m
X
k=1
(−1)k¯
∂χ∧trσk+m¯
∂σk+m−1··· ¯
∂σk(Dϕk¯
∂σk)t+1Dϕk···Dϕk+m=
N−m
X
k=1
(−1)k¯
∂χ∧trσk+m¯
∂σk+m−1··· ¯
∂σk−1(Dϕk−1¯
∂σk−1)tDϕk−1···Dϕk+m
−ρt
0,m + (−1)N−mρt
N−m,m +OZ,t+m+2, +¯
∂OZ,t+m+2,.
Thus, since ρr,s
0,m and ρr,s
N−m,m vanish,
(6.23)
N−m
X
k=1
(−1)k¯
∂χ∧trσk+m¯
∂σk+m−1··· ¯
∂σk(Dϕk¯
∂σk)t+1Dϕk···Dϕk+m=
−
N−m−1
X
k=1
(−1)k¯
∂χ∧trσk+m+1 ¯
∂σk+m··· ¯
∂σk(Dϕk¯
∂σk)tDϕk···Dϕk+m+1
+OZ,t+m+2, +¯
∂OZ,t+m+2,.
Assume that 2≤`≤N. By inductively applying (6.23) to m= 0,...,`−2with t=`−2−m, we get
N
X
k=1
(−1)k¯
∂χ∧trσk(Dϕk¯
∂σk)`−1Dϕk
=−
N−1
X
k=1
(−1)k¯
∂χ∧trσk+1 ¯
∂σk(Dϕk¯
∂σk)`−2DϕkDϕk+1+OZ,`, +¯
∂OZ,`,
=···
= (−1)`−1
N−`+1
X
k=1
(−1)k¯
∂χ∧trσk+`−1¯
∂σk+`−2··· ¯
∂σkDϕk···Dϕk+`−1+OZ,`, +¯
∂OZ,`,,
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
(−1)N¯
∂χ∧trσN¯
∂σN−1··· ¯
∂σ1(Dϕ1¯
∂σ1)`−NDϕ1···DϕN+OZ,`, +¯
∂OZ,`,,
which by (6.22) equals ρ`−N−1
0,N−1+ρ`−N−1
1,N−1+OZ,`, =OZ ,`,; thus (6.20) holds also in this case.
It remains to prove
(6.21)
. To do this let us replace the first factor
Dϕk¯
∂σk
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
¯
∂σk−1Dϕk−1in (2.17) is precisely the first term in the right hand side of (6.21). Next, by (2.13) and (2.16),
ϕk−1Dϕk¯
∂σk=Dϕk−1ϕk¯
∂σk=Dϕk−1¯
∂σk−1ϕk−1.
Applying this repeatedly we get
ϕk−1(Dϕk¯
∂σk)s= (Dϕk−1¯
∂σk−1)sϕk−1.
20 R. Lärkäng and E. Wulcan20 R. Lärkäng and E. Wulcan
Using this and (2.13) (to “move” the D), we get
σk+m¯
∂σk+m−1··· ¯
∂σk(¯
∂σk−1Dϕk−1)rD¯
∂σk−1ϕk−1(Dϕk¯
∂σk)sDϕk···Dϕk+m=
σk+m¯
∂σk+m−1··· ¯
∂σk(¯
∂σk−1Dϕk−1)rD¯
∂σk−1(Dϕk−1¯
∂σk−1)sDϕk−1···Dϕk+m−1ϕk+m.
It follows that the term corresponding to the first term in (2.17) equals ρr,s
k−1,m.
Finally we consider the term corresponding to the last term in
(2.17)
. As above, using
(2.13)
and
(2.16)
, we
get that
(¯
∂σk−1Dϕk−1)rϕk=ϕk(¯
∂σkDϕk)r
and thus, using this and (2.14) (to “move” the ¯
∂),
(6.24) σk+m¯
∂σk+m−1··· ¯
∂σk(¯
∂σk−1Dϕk−1)rϕkD¯
∂σk(Dϕk¯
∂σk)sDϕk···Dϕk+m=
¯
∂σk+m··· ¯
∂σk+1σkϕk(¯
∂σkDϕk)rD¯
∂σk(Dϕk¯
∂σk)sDϕk···Dϕk+m.
In view of (2.15) we can replace the factor σkϕkby IdEk−ϕk+1σk+1:
(6.25) ¯
∂σk+m··· ¯
∂σk+1σkϕk(¯
∂σkDϕk)rD¯
∂σk(Dϕk¯
∂σk)sDϕk···Dϕk+m=
¯
∂σk+m··· ¯
∂σk+1(¯
∂σkDϕk)rD¯
∂σk(Dϕk¯
∂σk)sDϕk···Dϕk+m+
(−¯
∂σk+m··· ¯
∂σk+1ϕk+1σk+1(¯
∂σkDϕk)rD¯
∂σk(Dϕk¯
∂σk)sDϕk···Dϕk+m) =: ξ+δ.
By repeatedly using (2.16) we get that
¯
∂σk+m··· ¯
∂σk+1ϕk+1σk+1 =¯
∂σk+m··· ¯
∂σk+2ϕk+2 ¯
∂σk+2σk+1 =··· =ϕk+m+1 ¯
∂σk+m+1 ··· ¯
∂σk+2σk+1.
It follows, using (2.14), that δin (6.25) equals
δ=−ϕk+m+1σk+m+1 ¯
∂σk+m··· ¯
∂σk+1(¯
∂σkDϕk)rD¯
∂σk(Dϕk¯
∂σk)sDϕk···Dϕk+m=: −ϕk+m+1β.
Note that degβ= 4κ+ 2 and degeβ= 2κ+ 1, where κ=m+r+s+ 1, and since
degϕk+m+1 = degeϕk+m+1 = 1,
we get, in view of (2.7), that tr(ϕk+m+1β) = −tr(βϕk+m+1)and it follows that this terms equals ρr,s
k,m.
It remains to consider ξin (6.25). Let
η:= σk+m¯
∂σk+m−1··· ¯
∂σk+1(¯
∂σkDϕk)rD¯
∂σk(Dϕk¯
∂σk)sDϕk···Dϕk+m
Then, by
(2.8)
,
¯
∂η =ξ+ξ0
, where
ξ0
consists of a sum of terms with a factor
¯
∂Dϕj
or
¯
∂D ¯
∂σj
. Let
q=m+r+s+2
. Then, note that
¯
∂χ∧η
is in
OZ,q,
, and thus
¯
∂χ∧¯
∂η =−¯
∂(¯
∂χ∧η)∈¯
∂OZ,q,
. Moreover,
by
(2.12)
, each term in
ξ0
has a factor that is a smooth
(1,1)
-form. Therefore
¯
∂χ∧ξ0∈OZ,q,
, and hence
tr ¯
∂χ∧ξ=−tr( ¯
∂χ∧ξ0) + tr( ¯
∂χ∧¯
∂η)∈OZ ,q, +¯
∂OZ,q, . This concludes the proof of (6.21).
Proof of Theorem 6.2.
We first prove
(6.5)
. Since
Z
has codimension
p
and
χp∼χ[1,∞)
, by Lemmas 6.4,6.5,
and 6.6, and by (2.18), we have
chRes
p(E, D) = lim
→0chp(E, b
D)
=1
(2πi)pp!lim
→0
¯
∂χp
∧
N
X
k=1
(−1)ktrσkDϕk(¯
∂σkDϕk)p−1
=(−1)p
(2πi)pp!
N−p
X
k=0
(−1)klim
→0
¯
∂χp
∧trσk+p¯
∂σk+p−1··· ¯
∂σk+1Dϕk+1 ···Dϕk+p
=(−1)p
(2πi)pp!
N−p
X
k=0
(−1)ktrRk
k+pDϕk+1 ···Dϕk+p.
Chern currents of coherent sheaves 21Chern currents of coherent sheaves 21
Since
Dϕk+1 ···Dϕk+p
and
Rk
k+p
both have total degree
2p
and endomorphism degree
p
, it follows from
(2.7) that
trRk
k+pDϕk+1 ···Dϕk+p= (−1)ptr Dϕk+1 ···Dϕk+pRk
k+p,
and thus (6.5) follows.
Next, by Theorem 5.1 and Remark 5.2,
chRes
`(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).
It remains to prove (6.7). From the beginning of the proof of Lemma 6.5 and (6.10) it follows that
(6.26) ch`1(E, b
D)∧ · ·· ∧ ch`m(E, b
D) = Cp(`1,`1)(E, b
D)∧ · ·· ∧ p(`m,`m)(E, b
D) + OZ,`1+···+`m, ,
for some appropriate constant C. By (6.14), the fact that (¯
∂χ)2= 0, and (6.10), it follows that
p(`1,`1)(E, b
D)∧ · ·· ∧ p(`m,`m)(E, b
D)∈OZ,p,
if m≥2and `1+···+`m≤p. Thus the limit of (6.26) vanishes in this case, which proves (6.7).
Assume that
(E, ϕ)
is a complex of Hermitian vector bundles of the form
(2.5)
such that
Hk(E)
has pure
codimension
p
or vanishes for
k= 0,...,N
, and let
Z=∪suppHk(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
Hk(E)
has pure codimension
p
or vanishes for
k= 0,...,N
. Moreover, assume that
D= (Dk)
is a
(1,0)
-connection on
(E, ϕ). Then
cRes
p(E, D)=(−1)p−1(p−1)![E].
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
cRes
p(E, D)
is independent of the choice of Hermitian metric and (1,0)-connection Don (E, ϕ).
By equipping a complex of vector bundles
(E, ϕ)
with Hermitian metrics and
(1,0)
-connections and
taking cohomology we get the following generalization of (1.13).
Corollary 6.8.
Assume that
(E, ϕ)
is a complex of vector bundles of the form
(2.5)
such that
Hk(E)
has pure
codimension por vanishes for k= 0,...,N. Then
h(−1)p−1(p−1)![E]i=cp(E).
7. An example
We will compute (products of) Chern currents
cRes(E,D)
for an explicit choice of
(E, ϕ)
and
D
. Let
J ⊆ OP2
[t,x,y]
be defined by
J=J(yk,x`ym)
, where
m<k
, and let
OZ:= OP2/J
. Then
Z
has pure
dimension
1
, since
Zred ={y= 0}
, which is irreducible. However, note that
J
has an embedded prime
J(x, y)of dimension 0. Now F=OZhas a locally free resolution of the form
(7.1) 0→ O(−k−`)ϕ2
−−→ O(−k)⊕ O(−`−m)ϕ1
−−→ OP2→ F → 0,
where in the trivialization in the coordinate chart C2=C2
(x,y)
(7.2) ϕ2="−x`
yk−m#and ϕ1=hykx`ymi.
22 R. Lärkäng and E. Wulcan22 R. Lärkäng and E. Wulcan
Let us start by computing the Chern class of F. Let ωdenote c1(O(1)). Then
c(E0) = c(OP2)=1
c(E1) = cO(−k)⊕ O(−`−m)= 1 −(k+`+m)ω+k(`+m)ω2
c(E2) = cO(−k−`)= 1 −(k+`)ω,
see e.g. [Ful98, Chapter 3]. Moreover,
(7.3) c(E1)−1= 1 −c1(E1) + c1(E1)2−c2(E1);
here the sum ends in degree 2, since we are in dimension 2. Thus, by (1.2),
c(F) = c(E0)c(E1)−1c(E2)=1−c1(E1) + c1(E2) + c1(E1)2−c2(E1)−c1(E1)c1(E2)
= 1 + mω +m2+`(m−k)ω2.
In particular,
(7.4) c1(F) = mω and c2(F) = m2+`(m−k)ω2.
7.1. Chern currents
Assume that each
Ek
in
(7.1)
is equipped with the metric induced by the standard metric on
O(1) →P2
, in
turn induced by the standard metric on
C3
, and let
Dk
the corresponding Chern connection. Let
D= (Dk)
,
let χ∼χ[1,∞), and let χand b
Dbe as in Section 4. By Theorem 1.4,
cRes
1(E, D)=[F]=[Z] = m[y= 0],
which clearly is a representative of
c1(F)
, see
(7.4)
, with support on
suppF=Zred ={y= 0}
. We want to
compute
cRes
2(E, D)
and
(cRes
1)2(E, D)
. Note that these currents are not covered by Theorem 1.5, since
p= 1
in this case.
Let
ˆ
p`=p(`,`)(E, b
D)
, where
p(`,`)
is given by
(3.14)
. For degree reasons,
ch2(E, b
D) = ch(2,2)(E, b
D)
and
ch2
1(E, b
D) = ch2
(1,1)(E, b
D),cf. Remark 5.2. It follows in view of (3.8) and (3.15) that
(7.5) c2(E, b
D) = i
2π21
2(ˆ
p2
1−ˆ
p2)and c2
1(E, b
D) = i
2π2
ˆ
p2
1.
Thus, to compute cRes
2(E, D)and (cRes
1)2(E, D), it suffices to calculate the limits of ˆ
p2
1and ˆ
p2as →0.
Note first that only two of the standard coordinate charts of
P2
intersect
Z
. In
C2
(t,y)
, we have that
ϕ1=ymhyk−m1i
, so
σ1= (1/ym)σ0
1
, where
σ0
1
is smooth. By using
(4.4)
one can check that the limits of
ˆ
p2
1
and
ˆ
p2
put no mass at
{t=y= 0}
. Thus it is enough to compute the limits in the coordinate chart
C2
(x,y)
where
ϕj
are given by
(7.2)
. Note that
ϕ1=ymϕ0
1
, where
ϕ0
1= [yk−mx`]
has rank
1
outside of the origin.
Then σ1= (1/ym)σ0
1, where σ0
1is smooth outside the origin, and
(7.6) σ0
1ϕ0
1|{y=0}="0 0
0 1 #
when
x,0
and
ϕ0
1σ0
1= 1
outside the origin. Also note that
σ2
is smooth outside the origin, since
ϕ2
has
constant rank there. Let
O0
Z,`,
be defined as in the beginning of Section 6but with
σ1
replaced by
σ0
1
, and
let
O=O0
Z,2,
. Then
ψ∈O
is smooth outside the origin and, by arguments as in the proof of Lemma 6.4,
lim→0ψ= 0.
Next, let ˆ
ω= (2π/i)ω, where ωnow denotes the Fubini-Study form. Then
(7.7) Θ1="−k0
0−(`+m)#ˆ
ω, Θ2=−(k+`)ˆ
ω.
Chern currents of coherent sheaves 23Chern currents of coherent sheaves 23
In particular,
Θk
is of bidegree
(1,1)
. Let
e
D= ( e
Dk)
be the connection on
P2\Z
defined by
(6.19)
and let
e
Θkbe the corresponding curvature forms. Then a computation, cf. (6.15) and (6.16), yields
(7.8) (b
Θ
k)(1,1) =−¯
∂χ∧σkDϕk+χ(e
Θk)(1,1) + (1 −χ)Θk.
Let us start by computing
ˆ
p2
1
. Recall from the proof of Lemma 6.5 that
˜
pj=−tr(e
Θ1)j
(1,1) + tr(e
Θ2)j
(1,1)
vanishes where
χ.0
for
j= 1,2
. Moreover, note in view of
(7.7)
that
−trΘ1+ tr Θ2=mˆ
ω
. It follows that
ˆ
p1=−tr(b
Θ
1)(1,1) + tr(b
Θ
2)(1,1) =¯
∂χ∧tr(σ1Dϕ1)−tr(σ2Dϕ2)+ (1 −χ)mˆ
ω.
Note that (1 −χ)¯
∂χ= (1/2) ¯
∂˜
χ, where ˜
χ= 2(χ−χ2/2) ∼χ[1,∞). Using this and that (¯
∂χ)2= 0, we get
ˆ
p2
1= 2mˆ
ω∧(1 −χ)¯
∂χ∧tr(σ1Dϕ1)−tr(σ2Dϕ2)+ (1 −χ)2m2ˆ
ω2
=mˆ
ω∧¯
∂˜
χ∧tr(σ1Dϕ1) + O.
Note that
(7.9) σ1Dϕ1=−Dym
ymσ0
1ϕ0
1+θ1σ0
1ϕ0
1+σ0
1Dϕ0
1.
Therefore, in view of the Poincaré-Lelong formula, cf.
(1.7)
and
(7.6)
, since
σ0
1ϕ0
1
is smooth outside the origin,
(7.10) ¯
∂˜
χ∧σ1Dϕ1=−¯
∂˜
χ∧Dym
ymσ0
1ϕ0
1+O−−−−→
→0
2π
im[y= 0]"0 0
0 1 #
outside the origin. Since the limit is a pseudomeromorphic
(1,1)
-current,
(7.10)
holds everywhere by the
dimension principle. It follows that
(7.11) lim
→0ˆ
p2
1= lim
→0mˆ
ω∧¯
∂˜
χ∧tr(σ1Dϕ1) = 2π
im2ˆ
ω∧[y= 0].
Let us next consider ˆ
p2=−tr(b
Θ1)2
(1,1) + tr(b
Θ2)2
(1,1). A computation using (2.7), cf. (7.8), yields
tr(b
Θ
k)2
(1,1) = tr−¯
∂χ∧σkDϕk−χ¯
∂(σkDϕk) + Θk2
(7.12)
=¯
∂χ2
∧trσkDϕk¯
∂(σkDϕk)−2¯
∂χ∧tr(σkDϕkΘk) + trχ(e
Θk)(1,1) + (1 −χ)Θk2.
Again using that ˜
pj=−tr(e
Θ1)j
(1,1) + tr(e
Θ2)j
(1,1) vanishes where χ.0, we get
(7.13) −χ(e
Θ1)(1,1) + (1 −χ)Θ12+χ(e
Θ2)(1,1) + (1 −χ)Θ22=O→0.
Note that ¯
∂χ∧σ2Dϕ2Θ2is in O. Therefore, in view of (7.7) and (7.10),
(7.14) 2¯
∂χ∧tr(σ1Dϕ1Θ1)−2¯
∂χ∧tr(σ2Dϕ2Θ2) =
−2¯
∂χ∧Dym
ymtr(σ0
1ϕ0
1Θ1) + O−−−−→
→0−2π
i2m(`+m)ˆ
ω∧[y= 0].
Let us next consider the contribution from the first term
(7.15) ¯
∂χ2
∧trσkDϕk¯
∂(σkDϕk)=¯
∂χ2
∧trσkDϕk¯
∂σkDϕk−¯
∂χ2
∧trσkDϕkσk¯
∂(Dϕk)
in
(7.12)
. We start by considering the contribution from the first term in
(7.15)
. By arguments as in the proof
of Lemma 6.6, we get that
(7.16) −¯
∂χ2
∧trσ1Dϕ1¯
∂σ1Dϕ1+¯
∂χ2
∧trσ2Dϕ2¯
∂σ2Dϕ2=
¯
∂χ2
∧trσ2¯
∂σ1Dϕ1Dϕ2−¯
∂χ2
∧trD¯
∂σ1Dϕ1+¯
∂χ2
∧trD¯
∂σ2Dϕ2.
24 R. Lärkäng and E. Wulcan24 R. Lärkäng and E. Wulcan
Taking the limit of the first term in the right hand side of (7.16), we get
lim
→0
¯
∂χ2
∧tr(σ2¯
∂σ1Dϕ1Dϕ2) = lim
→0
¯
∂χ2
∧tr(Dϕ1Dϕ2σ2¯
∂σ1)
= tr(Dϕ1Dϕ2R0
2)
=2π
i2
`(2k−m)[0].
(7.17)
Here, the first equality follows from
(2.7)
, the second equality from
(2.18)
, and the third equality is computed
in [
LW18
, Example 5.2]. Next, by
(2.7)
,
tr(D¯
∂σ1Dϕ1) = −tr(Dϕ1D¯
∂σ1)
. Using
(2.10)
,
(2.11)
and the fact that
ϕ0
1σ0
1= 1, so ϕ0
1¯
∂σ0
1= 0, we have
¯
∂χ2
∧Dϕ1D¯
∂σ1=¯
∂χ2
∧D(Dϕ1¯
∂σ1) = ¯
∂χ2
∧D(Dϕ0
1¯
∂σ0
1) = ¯
∂χ2
∧Dϕ0
1D¯
∂σ0
1.
If we let
f
be the section of
O(k−m)⊕ O(`)
defined by
f=hyk−mx`i
, then
ϕ2,ϕ0
1
are the morphisms
in the Koszul complex defined by (contraction with)
f
. If we let
σ
be the minimal inverse of
f
, when
f
is
viewed as a section of
Hom(O(−(k−m)) ⊕ O(−`),O)
, then
σ2
and
σ0
1
are given by multiplication with
σ
.
One may verify that
Dϕ2
and
Dϕ0
1
are given by contraction with
Df
, and that
D¯
∂σ2
and
D¯
∂σ0
1
are given
by multiplication with D¯
∂σ. A calculation then yields that
trDϕ0
1D¯
∂σ0
1=−trD¯
∂σ2Dϕ2,
so by (7.16),
(7.18) −¯
∂χ2
∧trσ1Dϕ1¯
∂σ1Dϕ1+¯
∂χ2
∧trσ2Dϕ2¯
∂σ2Dϕ2=¯
∂χ2
∧trσ2¯
∂σ1Dϕ1Dϕ2.
Thus, in view of (7.18) and (7.17),
(7.19) −¯
∂χ2
∧trσ1Dϕ1¯
∂σ1Dϕ1+¯
∂χ2
∧trσ2Dϕ2¯
∂σ2Dϕ2−−−−→
→02π
i2
`(2k−m)[0].
Next, let us consider the contribution from the second term in (7.15). As above, using (2.12), cf. (7.9),
¯
∂χ2
∧σ1Dϕ1σ1¯
∂(Dϕ1)=−¯
∂χ2
∧σ1Dϕ1σ1ϕ1Θ1=¯
∂χ2
∧Dym
ymσ0
1ϕ0
1Θ1+O.
Note that ¯
∂χ2
∧σ2Dϕ2σ2¯
∂(Dϕ2)is in O. Thus, by (7.10) and (7.7),
(7.20) ¯
∂χ2
∧trσ1Dϕ1σ1¯
∂(Dϕ1)−¯
∂χ2
∧trσ2Dϕ2σ2¯
∂(Dϕ2)−−−−→
→0
2π
im(`+m)ˆ
ω∧[y= 0],
cf. (7.14).
From (7.15), (7.19), and (7.20), we conclude that
(7.21) −¯
∂χ2
∧trσ1Dϕ1¯
∂(σ1Dϕ1)+¯
∂χ2
∧trσ2Dϕ2¯
∂(σ2Dϕ2)−−−−→
→0
2π
i2
`(2k−m)[0] + 2π
im(`+m)ˆ
ω∧[y= 0].
Next, from (7.12), (7.13), (7.14), and (7.21), we conclude that
(7.22) ˆ
p2=−tr(b
Θ1)2
(1,1) + tr(b
Θ2)2
(1,1) −−−−→
→0−(2π/i)m(`+m)ˆ
ω∧[y= 0] + (2π/i)2`(2k−m)[0].
Finally from (7.5), (7.11), and (7.22) we conclude that
cRes
2(E, D) = i
2π21
2lim
→0(ˆ
p2
1−ˆ
p2) = 1
2m(2m+`)ω∧[y= 0] −`(2k−m)[0]
and that
(cRes
1)2(E, D) = i
2π2
lim
→0ˆ
p2
1=m2ω[y= 0].
Chern currents of coherent sheaves 25Chern currents of coherent sheaves 25
Taking cohomology, since [[0]] = [[y= 0] ∧ω]=[ω2], we get
hcRes
2(E, D)i=m2+`(m−k)[ω2] = c2(F)and h(cRes
1)2(E, D)i=m2[ω2] = c1(F)2,
see (7.4).
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