HIGHER ARITHMETIC K-THEORY Introduction The aim of this paper is to provide a new definition ...

76 Pages·2012·0.47 MB·English

HIGHER ARITHMETIC K-THEORY
YUICHIRO TAKEDA
Introduction
The aim of this paper is to provide a new de(cid:12)nition of higher K-theory in Arakelov
geometry and to show that it enjoys the same formal properties as the higher algebraic
K-theory of schemes.
LetX beaproperarithmeticvariety,thatis,letX bearegularschemewhichis(cid:13)at,proper
and of (cid:12)nite type over Z, the ring of integers. In the research on arithmetic Chern characters
of hermitian vector bundles on X, Gillet and Soul(cid:19)e de(cid:12)ned the arithmetic K -group K (X)
0 0
of X [10]. It can be viewed as an analogue in Arakelov geometry of the K -group of vector
0
bundles on a scheme. b
After the advent of K (X), its higher extension was discussed in some papers, such as
0
[5, 6, 15]. In these papers one common thing was suggested that higher arithmetic K-theory
should be obtained as thbe homotopy group of the homotopy (cid:12)ber of the Beilinson’s regulator
map. That is to say, it was predicted that there would exist a group KM (X) for each n 0
n
(cid:21)
satisfying the long exact sequence
Kn+1(X) (cid:26) HD2p(cid:0)n(cid:0)1(X;R(p)) KMn(X) Kn(X) ;
(cid:1)(cid:1)(cid:1) ! ! (cid:8)p ! ! ! (cid:1)(cid:1)(cid:1)
where Hn(X;R(p)) is the real Deligne cohomology and (cid:26) is the Beilinson’s regulator map.
D
To get the homotopy (cid:12)ber, a simplicial description of the regulator map is necessary. And
it was given by Burgos and Wang in [5]. For a compact complex manifold M, they de(cid:12)ned
an exact cube of hermitian vector bundles on M and associated with it a di(cid:11)erential form
called a higher Bott-Chern form. This leads us to a homomorphism of complexes
ch : ZS (M) D (M;p)[2p+1]
(cid:3)
(cid:3) !
from the complex ZS (M) associated with the S-construction of the category of hermitian
vector bundles on M(cid:3)to the complebx D (M;p) computing the real Deligne cohomology of
(cid:3)
M, which is de(cid:12)nedbin [3]. It is the main theorem of [5] that the following map coincides
with the higher Chern character map with values in Deligne cohomology:
(cid:26) : Kn(M) (cid:25)n+1(S(M)) Hurewicz Hn+1(S(M)) H(ch) HD2p(cid:0)n(M;R(p)):
’ (cid:0)! (cid:0)!
Applying the theory of higher Bott-Chern forms to an arithmetic variety, we can obtain a
b b
simplicial description of the regulator map.
In this paper, we will give another de(cid:12)nition of higher arithmetic K-theory of a proper
arithmetic variety by means of higher Bott-Chern forms, which di(cid:11)ers from the one coming
1991 Mathematics Subject Classi(cid:12)cation. Primary 14G40; Secondary 11G35, 19E08.
Key words and phrases. K-theory, Arakelov geometry, higher Bott-Chern form.
1
2
from the homotpy (cid:12)ber of the regulator map. One of the remarkable features of our arith-
metic K-theory is that it is given as an extension of the usual K-theory by the cokernel of
the regulator map.
Beforeexplainingourmethod,letusrecallthede(cid:12)nitionofK (X). Foraproperarithmetic
0
variety X, let Ap;p(X) be the space of real (p;p)-forms ! on X(C) such that F(cid:3) ! = ( 1)p!
(cid:0)
for the complex conjugation F : X(C) X(C) and let A(bX) = Ap;p(X)=1Im@ +Im@.
1 ! (cid:8)p
Then K (X) is de(cid:12)ned as a factor group of the free abelian group generated by pairs (E;!)
0 e
of a hermitian vector bundle E on X and ! A(X). The relation on pairs is given by each
2
short ebxact sequence E : 0 E E E 0 as follows:
0 00
! ! ! !
e
(E ;! )+(E ;! ) = (E;! +! +ch(E));
0 0 00 00 0 00
where ch(E) is the Bott-Chern secondary characteristic class of E and ! ;! A(X).
0 00
e 2
We can interpret the above de(cid:12)nition of K (X) in terms of loops and homotopies on
0
S(X) ,ethe topological realization of the S-construction of the category of hermeitian vector
j j
bundles on X. Let us consider a pair (l;!), wbhere l is a pointed simplicial loop on S(X)
j j
abnd ! A(X). Two pairs (l;!) and (l ;! ) are de(cid:12)ned to be homotopy equivalent if there
0 0
2
is a cellular homotopy H : S1 I= I S(X) from l to l such that the Bott-Cbhern
0
(cid:2) f(cid:3)g(cid:2) ! j j
secondaryecharacteristic class ch(H) of H, which is de(cid:12)ned in a natural way, is equal to the
di(cid:11)erence ! !. Let (cid:25) ( S(X) ;ch) denote thebset of all equivalence classes of such pairs.
0 1
(cid:0) j j
Then it carries the structure ofean abelian group and the map
b b Ke (X) (cid:25) ( S(X) ;ch)
0 1
! j j
de(cid:12)ned by (E;!) (l ; !), where l is the simplicial loop on S(X) determined by E, is
7! E (cid:0) b E b b e j j
shown to be bijective.
In this paper, we will generalize the above construction of (cid:25) tob higher homotopy groups
1
to de(cid:12)ne higher arithmetic K-groups. Strictly speaking, we will de(cid:12)ne the n-th arithmetic
K-group K (X) as the set of all homotopy equivalence classes of pairs (f;!), where f :
n b
Sn+1 S(X) is a pointed cellular map and ! is a real di(cid:11)erential form on X(C) modulo
! j j
exact formbs. We will employ the theory of higher Bott-Chern forms in de(cid:12)ning a homotopy
equivalencbe relation on these pairs.
Let us describe the content of the paper in more detail.
In 1we introduce some materials used inthe present paper, such as S-construction, cubes
x
and higher Bott-Chern forms. Furthermore, by renormalizing the higher Bott-Chern forms,
we obtain a homomorphism of complexes
chGS : ZS (X) A (X)[1];
(cid:3) ! (cid:3)
where A (X) is a complex of vector spaces of real di(cid:11)erential forms on X(C) satisfying a
(cid:3) b
certainHodgetheoreticcondition. In 2weproposethenotionofmodi(cid:12)edhomotopygroups,
x
which is a higher generalization of the above (cid:25) . In 3 we de(cid:12)ne higher arithmetic K-groups
1
x
K (X) as the homotopy groups of S(X) modi(cid:12)ed by the homomorphism chGS. We show
(cid:3) j j
the following exact sequence concerning K (Xb):
n
b b
K (X) A (X) K (X) K (X) 0;
n+1 ! n+1 b ! n ! n !
e b
3
where An+1(X) = An+1(X)=ImdA. When n = 0, this exact sequence has already been
obtained in [10]. Moreover, we de(cid:12)ne the Chern form map
e chGS : K (X) A (X)
n n ! n
in the same way as ch : K (X) A(X) in [10]. The group KM (X), which is characterized
0 n
! b
as the long exact sequence as mentioned before, is realized as the kernel of chGS. When
n
we (cid:12)x an F -invariant Kba(cid:127)hler metric h on X(C), we de(cid:12)ne Arakelov K-group K (X) of
X n
1
X = (X;h ) as the subgroup of K (X) consisting of all elements x K (X) such that
X n n
2
chGS(x) is harmonic with respect to h . The group K (X) (cid:12)ts into the exact sequence
n X n
Kn+1(X) (cid:26) HD2p(cid:0)bn(cid:0)1(X;R(p)) Kn(X) Kn(X) 0;b
! (cid:8)p ! ! !
where (cid:26) is the Beilinson’s regulator map.
The next two sections concern product structure on K (X). In 4 we prove a product
(cid:3) x
formula for higher Bott-Chern forms. It provides an alternative proof of the fact that the
regulator map respects the products. In 5, we de(cid:12)ne abproduct in higher arithmetic K-
x
theory. We show that this product is graded commutative up to 2-torsion. A striking
property of the product is the lack of the associativity. In other words, for x;y;z K (X),
2 (cid:3)
(x y) z is not equal to x (y z) in general. We compute this di(cid:11)erence explicitly.
(cid:2) (cid:2) (cid:2) (cid:2)
Moreover, we show that there is an associative product in the Arakelov K-theory. b
In 6, we de(cid:12)ne a direct image morphism in higher arithmetic K-theory. To do this we
x
employ a higher analytic torsion form of an exact metrized cube de(cid:12)ned by Roessler [14].
Moreover we establish the projection formula.
Contents
1. Preliminaries 4
2. Modi(cid:12)ed homotopy groups 13
3. De(cid:12)nition of arithmetic K-groups 18
4. A product formula for higher Bott-Chern forms 25
5. Product 33
6. Direct images 60
Appendix A. Some identities satis(cid:12)ed by binomial coe(cid:14)cients 73
References 76
4
1. Preliminaries
1.1. Conventions on complexes. Let us (cid:12)rst settle some conventions on complexes. For
a complex A = (An;d ) of an abelian category A and n Z, the n-th translation A[n] is
(cid:3) A (cid:3)
2
de(cid:12)ned as A[n]k = An+k and d = ( 1)nd . For a morphism of complexes u : A B ,
A[n] A (cid:3) (cid:3)
(cid:0) !
the mapping cone Cone(u) is de(cid:12)ned by
Cone(u)k = Ak+1 Bk
(cid:8)
and the di(cid:11)erential d : Ak+1 Bk Ak+2 Bk+1 is de(cid:12)ned by
(cid:8) ! (cid:8)
d(a;b) = ( d (a);u(a)+d (b)):
A B
(cid:0)
A homological complex is a family An n Z of objects of A with morphisms
@ : A A such that @2 = 0. For af hogm2ological complex (A ;@ ), we can de(cid:12)ne
aAcompnlex!A nb(cid:0)y1An = A anAd d = @ . The n-th translation of a nhomAological complex
(cid:3) n A A
A is de(cid:12)ned by A[n] = A(cid:0) and @ = ( 1)n@ .
k k n A[n] A
(cid:3) (cid:0) (cid:0)
1.2. S-construction. InthissubsectionwerecallS-constructiondevelopedbyWaldhausen
[16]. Let [n] be the (cid:12)nite ordered set 0;1; ;n and Ar[n] the category of arrows of [n].
For a small exact category A, let S Afbe th(cid:1)e(cid:1)s(cid:1)et ogf functors
n
E : Ar[n] A; i j E
i;j
! (cid:20) 7!
satisfying the following conditions:
(1) E = 0 for any 0 i n.
i;i
(2) For any i j k,(cid:20)E (cid:20) E E is a short exact sequence of A.
i;j i;k j;k
(cid:20) (cid:20) ! !
For example, S A = 0 , S A is the set of objects of A and S A is the set of short exact
0 1 2
sequences of A. The ffungctor
SA : [n] S A
n
7!
becomes a simplicial set with the base point 0 S A.
0
The set S A can be identi(cid:12)ed with the set of2sequences of injections
n
E (cid:26) (cid:26) E
0;1 0;n
(cid:1)(cid:1)(cid:1)
with quotients E E =E for each i < j. By using this identi(cid:12)cation, we can describe
i;j 0;j 0;i
the boundary maps’and the degeneracy maps of SA as follows:
E (cid:26) (cid:26) E ; k = 0;
@ (E (cid:26) (cid:26) E ) = 1;2 (cid:1)(cid:1)(cid:1) 1;n
k 0;1 (cid:1)(cid:1)(cid:1) 0;n E (cid:26) (cid:26) E (cid:26) E (cid:26) (cid:26) E ; k 1
( 0;1 0;k 1 0;k+1 0;n
(cid:1)(cid:1)(cid:1) (cid:0) (cid:1)(cid:1)(cid:1) (cid:21)
and
0 (cid:26) E (cid:26) (cid:26) E ; k = 0;
s (E (cid:26) (cid:26) E ) = 0;1 (cid:1)(cid:1)(cid:1) 0;n
k 0;1 (cid:1)(cid:1)(cid:1) 0;n (E (cid:26) (cid:26) E (cid:26)id E (cid:26) (cid:26) E ; k 1:
0;1 0;k 0;k 0;n
(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:21)
Theorem 1.1. [16, 1.9] There is a homotopy equivalence map between the topological
x
realizations of simplicial sets
SA BQA ;
j j ’ j j
5
where BQA is the classifying space of the Quillen’s Q-construction of A [13]. Therefore for
n 0, it follows that
(cid:21)
(cid:25) ( SA ;0) K (A):
n+1 n
j j ’
1.3. Exact n-cubes. Let us recall the notion of an exact n-cube. For more details, see
[4, 5]. Let < 1;0;1 > be the ordered set consisting of three elements and < 1;0;1 >n
its n-th power(cid:0). For a small exact category A, a functor :< 1;0;1 >n A i(cid:0)s called an
n-cube of A. Let denote the image of an object (F(cid:11) ; (cid:0);(cid:11) ) of <! 1;0;1 >n. For
integers i and j sFa(cid:11)t1is;(cid:1)f(cid:1)(cid:1)y;(cid:11)inng 1 i n and 1 j 1, a1n(cid:1)(cid:1)(n(cid:1) n1)-cube(cid:0)@j is de(cid:12)ned
(cid:20) (cid:20) (cid:0) (cid:20) (cid:20) (cid:0) iF
by (@j ) = . It is called a face of . For an object (cid:11) of
< 1;i0F;1(cid:11)1>;(cid:1)(cid:1)n(cid:1);(cid:11)1n(cid:0)a1nd anF(cid:11)in1;t(cid:1)(cid:1)e(cid:1)g;(cid:11)ei(cid:0)r1i;j;s(cid:11)ai;t(cid:1)(cid:1)i(cid:1)s;(cid:11)fyn(cid:0)in1g 1 i n, a 1-cube @(cid:11)F called an edge of is
(cid:0) (cid:0) (cid:20) (cid:20) icF F
de(cid:12)ned by
:
F(cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)i(cid:0)1;(cid:0)1;(cid:11)i;(cid:1)(cid:1)(cid:1);(cid:11)n(cid:0)1 ! F(cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)i(cid:0)1;0;(cid:11)i;(cid:1)(cid:1)(cid:1);(cid:11)n(cid:0)1 ! F(cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)i(cid:0)1;1;(cid:11)i;(cid:1)(cid:1)(cid:1);(cid:11)n(cid:0)1
An n-cube is said to be exact if all edges of are short exact sequences. Let C A denote
n
F F
the set of all exact n-cubes of A. If is an exact n-cube, then any face @j is also exact.
F iF
Hence @j induces a map
i
@j : C A C A:
i n ! n(cid:0)1
Let be an exact n-cube of A. For an integer i satisfying 1 i n+1, let s1 be an
F (cid:20) (cid:20) iF
exact (n+1)-cube de(cid:12)ned as follows:
0; (cid:11) = 1;
(s1 ) = i
iF (cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)n+1 (F(cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)i(cid:0)1;(cid:11)i+1;(cid:1)(cid:1)(cid:1);(cid:11)n+1; (cid:11)i 6= 1
and the morphism (s1 ) (s1 ) is the identity of
.iFIn (cid:11)a1d;(cid:1)d(cid:1)(cid:1)i;(cid:11)tii(cid:0)o1n;(cid:0), 1le;(cid:11)ti+s1;(cid:1)(cid:1)1(cid:1);(cid:11)n+b1e!an exiFact(cid:11)1(;n(cid:1)(cid:1)(cid:1)+;(cid:11)i(cid:0)11);-0c;(cid:11)uib+1e;(cid:1)(cid:1)d(cid:1);e(cid:11)(cid:12)n+n1ed as follows:
F(cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)i(cid:0)1;(cid:11)i+1;(cid:1)(cid:1)(cid:1);(cid:11)n+1 i(cid:0) F
0; (cid:11) = 1;
(s 1 ) = i (cid:0)
(cid:0)i F (cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)n+1 (F(cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)i(cid:0)1;(cid:11)i+1;(cid:1)(cid:1)(cid:1);(cid:11)n+1; (cid:11)i 6= (cid:0)1
and the morphism (s 1 ) (s 1 ) is the identity
(cid:0)i F (cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)i(cid:0)1;0;(cid:11)i+1;(cid:1)(cid:1)(cid:1);(cid:11)n+1 ! (cid:0)i F (cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)i(cid:0)1;1;(cid:11)i+1;(cid:1)(cid:1)(cid:1);(cid:11)n+1
of . Then the map
F(cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)i(cid:0)1;(cid:11)i+1;(cid:1)(cid:1)(cid:1);(cid:11)n+1
sj : C A C A
i n ! n+1
is also de(cid:12)ned. An exact cube written as sj is said to be degenerate.
Let ZC A be the free abelian group geneirFated by C A. Let D ZC A be the subgroup
n n n n
(cid:26)
generated by all degenerate exact n-cubes. Let ZC A = ZC A=D and
n n n
n 1
@ = ( 1)i+j+1@je: ZC A ZC A:
(cid:0) i n ! n(cid:0)1
i=1j= 1
XX(cid:0)
e e
Then ZC A = (ZC A;@) becomes a homological complex.
n
We can(cid:3)construct an exact (n 1)-cube Cub(E) associated with an element E S A in
n
an indeuctive waey. In the case o(cid:0)f n = 1, let us de(cid:12)ne Cub(E) = E for E S A2. When
1
2
6
an exact (m 1)-cube is associated with any element of S A for m < n, an (n 1)-cube
m
Cub(E) assoc(cid:0)iated with E S A is de(cid:12)ned as (cid:0)
n
2
@ 1Cub(E) = s1 s1(E );
1(cid:0) n(cid:0)2(cid:1)(cid:1)(cid:1) 1 0;1
@0Cub(E) = Cub(@ E);
1 1
@1Cub(E) = Cub(@ E):
1 0
Then Cub : S A C A induces a homomorphism of complexes
n n 1
! (cid:0)
Cub : ZS A[1] ZC A:
(cid:3) ! (cid:3)
Remark: The di(cid:11)erential of ZC A de(cid:12)ned aboveeis the minus of the one de(cid:12)ned in [5].
But since the di(cid:11)erential ofZS A[1](cid:3)is alsothe minus ofthe one ofZS A, the homomorphism
(cid:3) (cid:3)
Cub is compatible with the di(cid:11)eerentials of complexes.
1.4. Deligne cohomology. Let M be a complex algebraic manifold of dimension n, that
is, let M be the analytic space consisting of all C-valued points of a smooth algebraic variety
of dimension n over C. Then higher Chern character map from higher algebraic K-theory
to a reasonable cohomology theory
ch : K (M) H2j i(M;(cid:0)(j))
i i (cid:0)
! (cid:8)j
has been developed in [8]. In the case of i = 0, it is given by the Chern characters of vector
bundles. But when i > 0 and M is compact, ch with values in the singular cohomology is
i
known to be trivial.
In order to obtain a nontrivial higher Chern character map, we should consider Deligne
cohomology. Fromnow onwe assume thatM is compact. Let (cid:10) bethe complex ofanalytic
(cid:3)M
sheaves of holomorphic di(cid:11)erential forms and let Fp(cid:10) be the de Rham (cid:12)ltration of (cid:10) ,
(cid:3)M (cid:3)M
that is,
Fp(cid:10) = (0 (cid:10)p (cid:10)p+1 (cid:10)n ):
(cid:3)M ! M ! M ! (cid:1)(cid:1)(cid:1) ! M
The real Deligne complex R(p)D on M is a complex of sheaves de(cid:12)ned as follows:
Cone(R(p)(cid:8)Fp(cid:10)(cid:3)M (cid:0)"(cid:0)!(cid:19) (cid:10)(cid:3)M)[(cid:0)1];
where " and (cid:19) are natural inclusions. It is obvious that R(p)D is quasi-isomorphic to the
complex
0 ! R(p) ! OM ! (cid:10)1M ! (cid:1)(cid:1)(cid:1) ! (cid:10)pM(cid:0)1 ! 0;
where R(p) is of degree zero. The real Deligne cohomology Hi (M;R(p)) is de(cid:12)ned as the
D
hypercohomology of R(p), that is,
HDi (M;R(p)) = Hi(M;R(p)D):
In fact, Deligne cohomology can be de(cid:12)ned when M is neither smooth nor compact [7].
Moreover, it can be a target of a nontrivial higher Chern character map, that is, there is a
homomorphism
chi : Ki(M) HD2p(cid:0)i(M;R(p));
! (cid:8)p
which is far from trivial.
7
Let Ep(M) be the space of real smooth di(cid:11)erential forms of degree p on M and Ep(M) =
R
Ep(M) C. Let Ep;q(M) be the space of complex di(cid:11)erential forms of type (p;q) on M. We
R
(cid:10)R
set
En 1(M)(p 1) Ep0;q0(M); n < 2p;
R(cid:0)
(cid:0) \p0+q(cid:8)0=n 1
(cid:0)
Dn(M;p) = 8 p0<p;q0<p
>>><E2Rp(M)(p)\Ep;p(M)\Kerd; n = 2p;
0; n > 2p
and de(cid:12)ne a di(cid:11)erential dD :>>>Dn(M;p) Dn+1(M;p) by
: !
(cid:25)(d!); n < 2p 1;
(cid:0) (cid:0)
dD(!) = 8 2@@!; n = 2p 1;
(cid:0) (cid:0)
><0; n > 2p 1;
(cid:0)
where (cid:25) : En(M) Dn(M;p) is the canonical projection.
>
! :
Theorem 1.2. [3, Thm.2.6] Let M be a compact complex algebraic manifold. Then
(D(cid:3)(M;p);dD) becomes a complex of R-vector spaces and there is a canonical isomorphism
Hn(D(cid:3)(M;p);dD) HDn(M;R(p))
’
if n 2p.
(cid:20)
1.5. Higher Bott-Chern forms. Inthissubsection werecallthehigherBott-Chernforms
developed by Burgos and Wang. For more details, the reader should consult the original
paper [5] or a survey [4]. A hermitian vector bundle E = (E;h) on a complex algebraic
manifold M is an algebraic vector bundle E on M with a smooth hermitian metric h. On
a hermitian vector bundle E, there is a unique connection that is compatible with both
rE
the metric and the complex structure. Let K denote the curvature form of . The Chern
E rE
form of E is de(cid:12)ned as
ch (E) = Tr(exp( K )) D2p(M;p):
0 (cid:0) E 2 (cid:8)p
Hereafter we assume that M is compact. An exact metrized n-cube on M is an exact
n-cube made of hermitian vector bundles on M. Let = E be an exact metrized n-cube
(cid:11)
F f g
on M. For an n-tuple (cid:11) = ((cid:11) ; ;(cid:11) ) with 1 (cid:11) 1 and an integer i satisfying (cid:11) = 1,
1 n k i
(cid:1)(cid:1)(cid:1) (cid:0) (cid:20) (cid:20)
there is a surjection E E . is called an emi-n-cube if the metric
(cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)i(cid:0)1;0;(cid:11)i+1;(cid:1)(cid:1)(cid:1);(cid:11)n ! (cid:11) F
on any E with (cid:11) = 1 coincides with the metric induced from E by the
(cid:11) i (cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)i(cid:0)1;0;(cid:11)i+1;(cid:1)(cid:1)(cid:1);(cid:11)n
above surjection.
Let (x : y) be the homogeneous coordinate of P1. Let (1) be the tautological line bundle
O
on P1 with the Fubini-Study metric. For a hermitian vector bundle E on M, let E(1) be the
hermitian vector bundle on M P1 given by (cid:25) E (cid:25) (1). Let (cid:27) ;(cid:27) H0(P1; (1)) be
(cid:2) 1(cid:3) (cid:10) 2(cid:3)O x y 2 O
global sections of (1) determined by x and y respectively.
O
For an emi-1-cube E : E E E , a map : E E (1) E (1) on M
1 0 1 1 0 1
P1 is de(cid:12)ned by e ((cid:19)(e)(cid:0) !(cid:27) ;e !(cid:27) ), where (cid:19) is the(cid:0)inj!ection E (cid:8) (cid:0) E . The 1(cid:2)-
x y 1 0
transgression bundle7!tr (E) of(cid:10)Eis a h(cid:10)ermitian vector bundle on M P1 gi(cid:0)ven!by the cokernel
1
(cid:2)
of with the induced metric. It follows from the de(cid:12)nition that tr (E) E and
1 M x=0 0
j (cid:2)f g ’
8
tr (E) E E . Hence tr (E) can be viewed as a family of hermitian vector
1 M y=0 1 1 1
bundlejs (cid:2)onf Mgp’aram(cid:0)et(cid:8)rized by P1 connecting E with E E .
0 1 1
(cid:0) (cid:8)
The n-transgression bundle of an emi-n-cube is de(cid:12)ned by iterating the above process.
The 1-transgression tr ( ) of an emi-n-cube is an emi-(n 1)-cube on M P1 de(cid:12)ned as
1
F F (cid:0) (cid:2)
tr ( ) = tr (@(cid:11) ( ))
1 F (cid:11) 1 nc F
for (cid:11) < 1;0;1 >n 1. The n-transgression bundle of is de(cid:12)ned as
(cid:0)
2 (cid:0) F
ntimes
tr ( ) = tr tr tr ( );
n 1 1 1
F (cid:1)(cid:1)(cid:1) F
which is a hermitian vector bundle on M z(P1)}n|. {
Let z = x=y be the Euclidean coordina(cid:2)te of P1 and z the i-th Euclidean coordinate of
i
(P1)n. For an integer i satisfying 1 i n, a di(cid:11)erential form with logarithmic poles Si on
(P1)n is de(cid:12)ned as (cid:20) (cid:20) n
dz dz dz(cid:22) dz(cid:22)
Si = ( 1)(cid:27)log z 2 (cid:27)(2) (cid:27)(i) (cid:27)(i+1) (cid:27)(n):
n (cid:0) j (cid:27)(1)j z ^(cid:1)(cid:1)(cid:1)^ z ^ z(cid:22) ^(cid:1)(cid:1)(cid:1)^ z(cid:22)
(cid:27)X2Sn (cid:27)(2) (cid:27)(i) (cid:27)(i+1) (cid:27)(n)
Let us de(cid:12)ne the Bott-Chern form of an emi-n-cube as
F
1
ch ( ) = ch (tr ( )) T D2p n(M;p);
n 0 n n (cid:0)
F (2(cid:25)p(cid:0)1)n Z(P1)n F ^ 2 (cid:8)p
where
( 1)n n
T = (cid:0) ( 1)iSi:
n 2n! (cid:0) n
i=1
X
Proposition 1.3. Let be an emi-n-cube on M. Then
F
ch (sj ) = 0
n+1 iF
for 1 i n+1 and j = 1.
(cid:20) (cid:20) (cid:6)
Proof: Let us (cid:12)rst consider the case of n = 0. For a hermitian vector bundle E on M, it
follows that s 1E = 0 E id E . Hence the 1-transgression bundle tr(s 1E) is isometric
(cid:0)1 ! ! (cid:0)1
to E(1). If r : P1 (cid:16)P1 is an involu(cid:17)tion given by r (z) = z 1, then r (E(1)) = E(1). Hence
(cid:3) (cid:0) (cid:3)
!
we have
1
ch (s 1E) = ch (E(1))log z 2
1 (cid:0)1 2(cid:25)p 1 P1 0 j j
(cid:0) Z
1
= r ch (E(1))log z 2
(cid:3) 0
2(cid:25)p 1 P1 j j
(cid:0) Z
1 (cid:0) (cid:1)
= (cid:0) ch (E(1))log z 2
0
2(cid:25)p 1 P1 j j
(cid:0) Z
= ch (s 1E);
(cid:0) 1 (cid:0)1
therefore ch (s 1E) = 0. In the same way we can show ch (s1E) = 0.
1 (cid:0)1 1 1
9
Let us move on to the general case. For 1 i n+1, let r : (P1)n+1 (P1)n+1 denote
i
(cid:20) (cid:20) !
an involution given by
z ; i = j;
j
r z = 6
i(cid:3) j z 1; i = j:
( j(cid:0)
Then we have r (tr (sj )) = tr (sj ) and r T = T . Therefore we can show
i(cid:3) n+1 iF n+1 iF i(cid:3) n+1 (cid:0) n+1
ch (sj ) = 0 in the same way as above. (cid:3)
n+1 iF
Let us extend the de(cid:12)nition of the Bott-Chern form to an arbitrary exact metrized n-
cube. Let be an exact metrized n-cube, not necessarily emi. For an integer i satisfying
F
1 i n, (cid:21)1 is de(cid:12)ned as
(cid:20) (cid:20) iF
E ; (cid:11) = 1 or 0;
((cid:21)1 ) = (cid:11) i (cid:0)
iF (cid:11) E ; (cid:11) = 1;
( 0(cid:11) i
whereE isthesamevectorbundleasE withthemetricinducedfromE .
Let (cid:21)2 0(cid:11)be an exact metrized n-cube de(cid:11)(cid:12)ned as (cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)i(cid:0)1;0;(cid:11)i+1;(cid:1)(cid:1)(cid:1);(cid:11)n
iF
E ; (cid:11) = 1;
(cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)i(cid:0)1;1;(cid:11)i+1;(cid:1)(cid:1)(cid:1);(cid:11)n i (cid:0)
((cid:21)2 ) = E ; (cid:11) = 0;
iF (cid:11) 8 0(cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)i(cid:0)1;1;(cid:11)i+1;(cid:1)(cid:1)(cid:1);(cid:11)n i
><0; (cid:11) = 1
i
(cid:21)an1d the(cid:21)m2 or.pAhinsmemEi-(cid:11)n1-;(cid:1)c(cid:1)(cid:1)u;(cid:11)bi(cid:0)e1;(cid:21)1;(cid:11)i+i1s;>:(cid:1)(cid:1)d(cid:1);e(cid:11)(cid:12)nn!edEas0(cid:11)1;(cid:1)(cid:1)(cid:1);(cid:11)i(cid:0)1;1;(cid:11)i+1;(cid:1)(cid:1)(cid:1);(cid:11)n is the identity. We set (cid:21)iF =
iF (cid:8) iF F
(cid:21) (cid:21) (cid:21) ; n 1;
n n 1 1
(cid:21) = (cid:0) (cid:1)(cid:1)(cid:1) F (cid:21)
F ; n = 0:
(
F
De(cid:12)nition 1.4. The Bott-Chern form of an exact metrized n-cube is an element of
D2p n(M;p) de(cid:12)ned as F
(cid:0)
(cid:8)p
1
ch ( ) = ch (tr ((cid:21) )) T :
n 0 n n
F (2(cid:25)p(cid:0)1)n Z(P1)n F ^
Remark: For an emi-n-cube , the Bott-Chern form ch ( ) de(cid:12)ned in Def.1.4 is the
n
F F
same form as the one we have de(cid:12)ned before, because there is a decomposition
(cid:21) = (a degenerate emi-n-cube):
F F (cid:8)
ForacompactcomplexalgebraicmanifoldM,letP(M)bethecategoryofhermitianvector
bundles on M and S(M) the S-construction of P(M). We note that S(M) is homotopy
equivalent to the S-construction of vector bundles bon M by the map forgetting metrics. In
particular, it followsbthat b b
(cid:25) ( S(M) ) K (M):
n+1 n
j j ’
b
10
LetZC (M) = ZC P(M)andZCemi(M)thesubcomplex ofZC (M)generatedbyemi-cubes
on M. (cid:3)Then (cid:3)(cid:21) induces a h(cid:3)omomorphism of complexes (cid:3)
F 7! F
e b e b e b e b
(cid:21) : ZC (M) ZCemi(M):
(cid:3) ! (cid:3)
e b e b
Theorem 1.5. [5] If is an exact metrized n-cube on M, then we have
F
dDchn( ) = chn 1(@F):
F (cid:0)
Hence the higher Bott-Chern forms induce a homomorphism of complexes
ch : ZC (M) D (M;p)[2p]:
(cid:3)
(cid:3) ! (cid:8)p
Moreover, the following map e b
Kn(M) = (cid:25)n+1(S(M)) Hurewicz Hn+1(S(M)) Cub Hn(ZC (M)) ch HD2p(cid:0)n(M;p)
(cid:0)! (cid:0)! (cid:3) ! (cid:8)p
agrees with the higherbChern character withbvalues in the Deebligne cohomology.
This is the main theorem of [5]. Here let us prove dDchn( ) = chn 1(@ ) in a di(cid:11)erent
way from [5]. To do this we introduce another description ofFthe logari(cid:0)thmFic forms Si.
n
For integers (cid:11) ; ;(cid:11) with 1 (cid:11) n, a k-form with logarithmic poles ((cid:11) ; ;(cid:11) ) on
1 k i 1 k
(P1)n is given as (cid:1)(cid:1)(cid:1) (cid:20) (cid:20) (cid:1)(cid:1)(cid:1)
((cid:11) ; ;(cid:11) ) = dlog z 2 dlog z 2:
1 (cid:1)(cid:1)(cid:1) k j (cid:11)1j ^(cid:1)(cid:1)(cid:1)^ j (cid:11)kj
Let ((cid:11) ; ;(cid:11) )(i;k i) denote the (i;k i)-part of ((cid:11) ; ;(cid:11) ). Then we have
1 k (cid:0) 1 k
(cid:1)(cid:1)(cid:1) (cid:0) (cid:1)(cid:1)(cid:1)
n
Si = (i 1)!(n i)! ( 1)(cid:11)+1log z 2(1; ;(cid:11); ;n)(i 1;n i):
n (cid:0) (cid:0) (cid:0) j (cid:11)j (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:0) (cid:0)
(cid:11)=1
X
b
Lemma 1.6. It follows that
n
@Si = i!(n i)!(1 n)(i;n i) +(n i) ( 1)(cid:11)@@log z 2Si ;
n (cid:0) (cid:1)(cid:1)(cid:1) (cid:0) (cid:0) (cid:0) j (cid:11)j n 1;(cid:11)b
(cid:0)
(cid:11)=1
X
n
@Si = (i 1)!(n i+1)!(1 n)(i 1;n i+1) (i 1) ( 1)(cid:11)@@log z 2Si 1 ;
n (cid:0) (cid:0) (cid:1)(cid:1)(cid:1) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) j (cid:11)j n(cid:0)1;(cid:11)b
(cid:0)
(cid:11)=1
X
where Si is the logarithmic form on (P1)n with the same expression as Si for the
n 1;(cid:11)b n 1
coordinate(cid:0)(z ; ;z ;z ; ;z ). (cid:0)
1 (cid:11) 1 (cid:11)+1 n
(cid:1)(cid:1)(cid:1) (cid:0) (cid:1)(cid:1)(cid:1)
We will prove more general identities in Lem.4.3. Let us return to the proof of the identity
dDchn( ) = chn 1(@ ). Since (cid:21) : ZC (M) ZCemi(M) is a homomorphism of complexes,
F (cid:0) F (cid:3) ! (cid:3)
e b e b