Horizontal Forms of Chern Type on Complex Finsler Bundles
The aim of this paper is to construct horizontal Chern forms of a holomorphic vector bundle using complex Finsler structures. Also, some properties of these forms are studied.
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irk-123456789-1463502019-02-10T01:24:04Z Horizontal Forms of Chern Type on Complex Finsler Bundles Ida, C. The aim of this paper is to construct horizontal Chern forms of a holomorphic vector bundle using complex Finsler structures. Also, some properties of these forms are studied. 2010 Article Horizontal Forms of Chern Type on Complex Finsler Bundles / C. Ida // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 18 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53B40; 57R20 doi:10.3842/SIGMA.2010.054 http://dspace.nbuv.gov.ua/handle/123456789/146350 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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The aim of this paper is to construct horizontal Chern forms of a holomorphic vector bundle using complex Finsler structures. Also, some properties of these forms are studied. |
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Ida, C. |
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Ida, C. Horizontal Forms of Chern Type on Complex Finsler Bundles Symmetry, Integrability and Geometry: Methods and Applications |
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Ida, C. |
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Ida, C. |
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Horizontal Forms of Chern Type on Complex Finsler Bundles |
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Horizontal Forms of Chern Type on Complex Finsler Bundles |
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Horizontal Forms of Chern Type on Complex Finsler Bundles |
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Horizontal Forms of Chern Type on Complex Finsler Bundles |
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Horizontal Forms of Chern Type on Complex Finsler Bundles |
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horizontal forms of chern type on complex finsler bundles |
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Інститут математики НАН України |
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2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/146350 |
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Horizontal Forms of Chern Type on Complex Finsler Bundles / C. Ida // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 18 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT idac horizontalformsofcherntypeoncomplexfinslerbundles |
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2025-07-10T23:30:35Z |
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1837304638448074752 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 054, 7 pages
Horizontal Forms of Chern Type
on Complex Finsler Bundles
Cristian IDA
Department of Algebra, Geometry and Dif ferential Equations,
Transilvania University of Braşov, Str. Iuliu Maniu 50, Braşov 500091, România
E-mail: cristian.ida@unitbv.ro
Received October 28, 2009, in final form June 30, 2010; Published online July 09, 2010
doi:10.3842/SIGMA.2010.054
Abstract. The aim of this paper is to construct horizontal Chern forms of a holomorphic
vector bundle using complex Finsler structures. Also, some properties of these forms are
studied.
Key words: complex Finsler bundles; horizontal forms of Chern type
2010 Mathematics Subject Classification: 53B40; 57R20
1 Introduction and preliminaries
In [7] J.J. Faran posed an open question: Is it possible to define Chern forms of a holomorphic
vector bundle using complex Finsler structures? Recently, in [4, 5, 6] and [16] was studied the
Chern form of the associated hyperplane line bundle in terms of complex Finsler structures and
some of its applications.
The main purpose of this note is to obtain horizontal Chern forms of a complex Finsler
bundle, following the general construction of Chern forms [8, 10, 15]. Firstly, we define a h′′-
cohomology with respect to the horizontal conjugated differential operator and using the partial
Bott complex connection [3], we construct horizontal invariant forms of Chern type on the total
space. Also, we prove that the corresponding horizontal classes of Chern type do not depend on
the complex Finsler structure chosen in a family given by conformal changes of horizontal type.
Next, it is proved that these forms live on the associated projectivized bundle and are invariant
by a linear family of complex Finsler structures given by infinitesimal deformations.
Let π : E → M be a holomorphic vector bundle of rank m over a n-dimensional complex
manifold M . Let (U, (zk)), k = 1, . . . , n be a local chart on M and s = {sa}, a = 1, . . . ,m
a local frame for the sections of E over U . It is well known that this chart canonically induces
another one on E of the form (π−1(U), u = (zk, ηa)), k = 1, . . . , n, a = 1, . . . ,m, where s = ηasa
is a section on Ez = π−1(z), for all z ∈ M . If (π−1(U ′), u′ = (z′k, η′a)) is another local chart
on E, then the transition laws of these coordinates are
z′k = z′k(z), η′a = Ma
b (z)ηb,
where Ma
b (z), a, b = 1, . . . ,m are holomorphic functions on z and detMa
b 6= 0.
As we already know [3, 4], the total space E has a natural structure of m + n dimen-
sional complex manifold because the transition functions Ma
b are holomorphic. We consider
the complexified tangent bundle TCE of the real tangent bundle TRE and its decomposition
TCE = T ′E ⊕ T ′′E, where T ′E and T ′′E = T ′E are the holomorphic and antiholomorphic tan-
gent bundles of E, respectively. The vertical holomorphic tangent bundle V ′E = kerπ∗ is the
relative tangent bundle of the holomorphic projection π. A local frame field on V ′
uE is {∂/∂ηa},
a = 1, . . . ,m and V ′
uE is isomorphic to the sections module of E in u.
mailto:cristian.ida@unitbv.ro
http://dx.doi.org/10.3842/SIGMA.2010.054
2 C. Ida
A supplementary subbundle of V ′E in T ′E, i.e. T ′E = H ′E ⊕ V ′E is called a complex
nonlinear connection on E, briefly c.n.c. A local basis for the horizontal distribution H ′
uE,
called adapted for the c.n.c., is {δ/δzk = ∂/∂zk − Na
k ∂/∂η
a}, k = 1, . . . , n, where Na
k (z, η) are
the local coefficients of the c.n.c. In the sequel we consider the abreviate notations: ∂k = ∂/∂zk,
δk = δ/δzk and ∂̇a = ∂/∂ηa. The adapted basis denoted by {δk = δ/δzk} and {∂̇a = ∂/∂ηa}, for
H ′′E = H ′E and V ′′E = V ′E distributions are obtained respectively by conjugation anywhere.
Also, we note that the adapted cobases are given by {dzk}, {δηa = dηa + Na
k dz
k}, {dzk} and
{δηa = dηa +Na
k
dzk} which span the dual bundles H ′∗E, V ′∗E, H ′′∗E and V ′′∗E, respectively.
Definition 1 ([9]). A strictly pseudoconvex complex Finsler structure on E is a positive real
valued function F : E → R+ ∪ {0} which satisfies the following conditions:
1) F 2 is smooth on E∗ = E \ {0};
2) F (z, η) ≥ 0 and F (z, η) = 0 if and only if η = 0;
3) F (z, λη) = |λ|F (z, η) for any λ ∈ C∗ = C \ {0};
4) the complex hessian (hab) = (∂̇a∂̇b(F
2)) is positive definite and determines a hermitian
metric tensor on the fibers of the vertical bundle V ′E \ {zero section}.
Definition 2. The pair (E,F ) is called a complex Finsler bundle.
According to [1, 4, 12], a c.n.c. on (E,F ) depending only on the complex Finsler structure F
is the Chern–Finsler c.n.c., locally given by
CF
Na
k = hca∂k∂̇c
(
F 2
)
, (1)
where (hca) is the inverse of (hac).
An important property of the Chern–Finsler c.n.c. (see [4, 12]), is
[δj , δk] =
(
δk
CF
Na
j −δj
CF
Na
k
)
∂̇a = 0. (2)
Throughout this paper we consider adapted frames and coframes with respect to the Chern–
Finsler c.n.c.
According to [3], a partial connection ∇ of (1, 0) type on V ′E, defined by
∇XY = v′[X,Y ]
for any X ∈ Γ(H ′E) and Y ∈ Γ(V ′E), is called the partial Bott complex connection of (E,F ).
Here v′ is the natural vertical projector.
The (1, 0)-connection form ω = (ωa
b ) of ∇ is locally given by
ωa
b = La
bkdz
k, La
bk = ∂̇b
( CF
Na
k
)
(3)
and the (1, 1)-horizontal curvature form Ω = (Ωa
b ) of ∇ is locally given by
Ωa
b = Ωa
bjk
dzj ∧ dzk, Ωa
bjk
= −δk(L
a
bj). (4)
Remark 1. We notice that the partial Bott complex connection is in fact the classical Chern–
Rund connection, first introduced by H. Rund [14] in the case when E = T ′M is the holomorphic
tangent bundle of a complex manifold M .
Horizontal Forms of Chern Type on Complex Finsler Bundles 3
h′′-cohomology
Similar to [13], let us consider the set Ap,q,r,s(E) of all (p, q, r, s)-forms with complex values
on E, locally defined by
ϕ =
1
p!q!r!s!
ϕIpJqArBs
dzIp ∧ dzJq ∧ δηAr ∧ δηBs ,
where Ip = (i1 . . . ip), Jq = (j1 . . . jq), Ar = (a1 . . . ar), Bs = (b1 . . . bs) and these forms can be
nonzero only when they act on p vectors from Γ(H ′E), on q vectors from Γ(H ′′E), on r vectors
from Γ(V ′E) and on s vectors from Γ(V ′′E), respectively.
By (2), we get the following decomposition of the exterior differential
dAp,q,r,s(E) ⊂ Ap+1,q,r,s(E)⊕Ap,q+1,r,s(E)⊕Ap,q,r+1,s(E)⊕Ap,q,r,s+1(E)⊕
⊕Ap+1,q+1,r−1,s(E)⊕Ap+1,q,r−1,s+1(E)⊕Ap+1,q+1,r,s−1(E)⊕Ap,q+1,r+1,s−1(E)
which allows us to define eight morphisms of complex vector spaces if we consider different
components in the above decomposition.
If we consider the subset Ap,q,0,0(E) ⊂ Ap,q,r,s(E) of all horizontal forms of (p, q, 0, 0) type,
which can be nonzero only when they act on p vectors from Γ(H ′E) and q vectors from Γ(H ′′E),
then
dAp,q,0,0(E) ⊂ Ap+1,q,0,0(E)⊕Ap,q+1,0,0(E)⊕Ap,q,1,0(E)⊕Ap,q,0,1(E).
In particular, similar to [17, 18], the horizontal differential operators are defined by
d′h : Ap,q,0,0(E) → Ap+1,q,0,0(E), d′′h : Ap,q,0,0(E) → Ap,q+1,0,0(E)
where, for any ϕ = 1
p!q!ϕIpJq
dzIp ∧ dzJq ∈ Ap,q,0,0(E), we have
(d′hϕ)i1...ip+1Jq
=
p+1∑
k=1
(−1)k−1δik
(
ϕi1...îk...ip+1Jq
)
and
(d′′hϕ)Ipj1...jq+1
= (−1)p
q+1∑
k=1
(−1)k−1δjk
(
ϕ
Ipj1...ĵk...jq+1
)
. (5)
We note that by (2), we have d′′h ◦ d′′h = 0. Thus, with respect to the operator d′′h we can
define the h′′-cohomology groups of (E,F ) for (p, q, 0, 0)-forms by
Hq
(
E,Φp,0,0
)
=
ker{d′′h : Ap,q,0,0 → Ap,q+1,0,0}
Im {d′′h : Ap,q−1,0,0 → Ap,q,0,0}
, (6)
where Φp,0,0 is the sheaf of germs of (p, 0, 0, 0)-forms d′′h-closed.
2 Horizontal forms of Chern type
In this section, using the horizontal curvature form of partial Bott complex connection, we
construct horizontal forms of Chern type on a complex Finsler bundle.
With the notations from the previous section, we have ωa
b ∈ A1,0,0,0(E) and Ωa
b ∈ A1,1,0,0(E).
Also, by (4) and (5) we have Ω = d′′hω and taking into account the relation d′′h ◦ d′′h = 0, it
follows that Ω is a d′′h-closed differential form.
4 C. Ida
Let gl(m,C) ≈ Cm×m be the Lie algebra of the linear general group Gl(m,C). A symmetric
polynomial f ∈ Sj(Gl(m,C)) is invariant if
f
(
M−1X1M, . . . ,M−1XjM
)
= f(X1, . . . , Xj)
for every M ∈ Gl(m,C) and Xj ∈ gl(m,C).
Also, it is well known [8, 10, 15], that the algebra of invariant symmetric polynomials on
gl(m,C) is generated by the elementary symmetric functions fj ∈ Sj(Gl(m,C)) given by
det
(
Im − 1
2πi
X
)
=
m∑
j=0
fj(X) = 1− 1
2πi
trX + · · ·+
(
− 1
2πi
)m
detX. (7)
Next, we give a known result, but we express it in a form adapted to our study.
Proposition 1. At local changes on E, the horizontal curvature of partial Bott complex con-
nection changes by the rule
Ω′ = M−1ΩM,
where M = (Ma
b (z)) are the transition functions of the holomorphic bundle E.
Proof. By [12], we know that the change rules of the adapted frames and coframes are
δj =
∂z′k
∂zj
δ′k, ∂̇b = Ma
b ∂̇
′
a, dzj =
∂zj
∂z′k
dz′k, δηb = M b
aδη
′a (8)
and its conjugates.
On the other hand (see 7.1.9 in [12]), the local coefficients of the Chern–Finsler c.n.c. change
by the rule
∂z′k
∂zj
CF
N ′a
k = Ma
b
CF
N b
j −
∂Ma
b
∂zj
ηb. (9)
Finally, it follows by (3), (4), (8) and (9) that Ω′d
c = M b
c Ωa
bM
d
a . �
Now, we consider the (j, j, 0, 0)-invariant forms Ch
j (∇) defined by
det
(
Im − 1
2πi
Ω
)
=
m∑
j=0
Ch
j (∇).
Then, it follows by (7)
Ch
j (∇) =
(−1)j
(2πi)jj!
∑
a,b
δ
b1...bj
a1...ajΩ
a1
b1
∧ · · · ∧ Ωaj
bj
,
where δb1...bj
a1...aj are the Kronecker symbols.
Because d′′h satisfies the property
d′′h(ϕ ∧ ψ) = d′′hϕ ∧ ψ + (−1)deg ϕϕ ∧ d′′hψ
for any ϕ ∈ Ap,q,0,0(E) and ψ ∈ Ap′,q′,0,0(E), then by d′′hΩ = 0 we obtain that the differential
forms Ch
j (∇) are d′′h-closed.
Definition 3. Ch
j (∇) are called horizontal forms of Chern type of order j of the complex Finsler
bundle (E,F ).
Horizontal Forms of Chern Type on Complex Finsler Bundles 5
By (6) these forms define the horizontal cohomology classes
chj (∇) = [Ch
j (∇)] ∈ Hj
(
E,Φj,0,0
)
which are called horizontal classes of Chern type of order j of (E,F ). In particular, the first
horizontal class of Chern type is represented by the (1, 1, 0, 0)-form
Ch
1 (∇) = − 1
2πi
Ωa
ajk
dzj ∧ dzk.
Proposition 2. The first horizontal class of Chern type ch1(∇) is invariant by conformal change
of horizontal type: F 2 7→ eσ(z)F 2.
Proof. Let F 2 7→ F̃ 2 := eσ(z)F 2 be a conformal change of complex Finsler structures on E,
where σ(z) is a smooth function on M . Because h̃ab = eσ(z)hab, using (1) we get that the
Chern–Finsler c.n.c. changes by the rule
CF
Ña
k =
CF
Na
k +
∂σ
∂zk
ηa.
Thus, the (1, 0)-connection form of ∇ changes by the rule
ω̃ = ω + d′hσ ⊗ I, (10)
where I is the identity endomorphism of V ′E.
Applying d′′h in the (10) relation, we get the change rule of the (1, 1)-horizontal curvature
form of ∇, namely
Ω̃ = Ω + d′′hd′hσ ⊗ I
which ends the proof. �
In fact the above proposition remains valid for all higher horizontal Chern classes, namely:
Proposition 3. The horizontal classes of Chern type chj (∇) are invariant by conformal change
of horizontal type: F 2 7→ eσ(z)F 2 for any j = 1, . . . ,m.
Proof. It is classical that for a square matrix A of order p we have
det(A− λI) =
p∑
j=0
(−1)j∆p−j(A)λj ,
where
∆j(A) =
1
j!
∑
α,β
δ
β1...βj
α1...αja
α1
β1
· · · aαj
βj
is the sum of the principal minors of order j of the matrix A (see for instance [15, p. 235]).
Now, taking into account d′′h∆j(Ω) = 0, by direct calculations we have
Ch
j (∇̃) =
(−1)j
(2πi)j
∆j(Ω + d′′hd′hσ ⊗ I)
=
(−1)j
(2πi)j
(
∆j(Ω) +
1
j!
j∑
k=1
∆j−k(Ω) ∧ (d′′hd′hσ)k
)
= Ch
j (∇) + d′′h
(
(−1)j
(2πi)jj!
j∑
k=1
∆j−k(Ω) ∧ d′hσ ∧ (d′′hd′hσ)k−1
)
which says that Ch
j (∇̃) and Ch
j (∇) are in the same d′′h-cohomology class. �
6 C. Ida
It is well known that for every ϕ ∈ Ap,q(M), ϕ∗ = ϕ ◦ π ∈ Ap,q,0,0(E) is the natural lift of ϕ
to the total space E.
Remark 2. If the complex Finsler structure F comes from a hermitian structure h on E, namely
F 2(z, η) = h(η, η) = hab(z)η
aηb, then the coefficients of the partial Bott complex connection ∇
are independent of the fiber coordinates ηa, and these coincides with the classical connection
coefficients in hermitian bundles. In this case it is easy to see that Ch
j (∇) coincide with the
classical Chern forms of order j of hermitian bundle (E, h) lifted to the total space E.
In the sequel we suppose that the base manifold M is compact. Note that there is a natural
C∗ = C \ {0} action on E∗ = E \ {0} and the associated projectivized bundle is defined by
PE = E∗/C∗ with the projection p : PE → M . Each fiber Pz(E) = P (Ez) is isomorphic
to the (m − 1)-dimensional complex projective space Pm−1(C). The pull-back bundle p−1E is
a holomorphic vector bundle of rank m over PE. Thus, the local complex coordinates (z, η)
on E may be also considered as a local complex coordinates system for PE as long as η1, . . . , ηm
is considered as a homogeneous coordinate system for fibers. All the geometric objects on E
which are invariant after replacing η by λη, λ ∈ C∗ are valid on PE. The reason for working
on PE rather than on E is that PE is compact if M is compact (see [9, 11]).
Now, we simply denote by Ap,q(PE) the set of all horizontal forms of (p, q, 0, 0) type on E
whose coefficients are zero homogeneous with respect to fiber coordinates, namely ϕIpJq
(z, λη) =
ϕIpJq
(z, η), for any λ ∈ C∗.
By the homogeneity conditions of the complex Finsler structure F (see [3, 4, 12]), the local
coefficients of the Chern–Finsler c.n.c. are given by
CF
Na
k = La
bkη
b (11)
and, moreover
CF
Na
k (z, λη) = λ
CF
Na
k (z, η), ∀λ ∈ C∗. (12)
Then, it follows by (11) and (12)
La
bk(z, λη) = La
bk(z, η), ∀λ ∈ C∗
which says that the (1, 0)-connection forms ωa
b of ∇ live on PE. Similarly, by (4) we get
Ωa
bjk
(z, λη) = Ωa
bjk
(z, η) for any λ ∈ C∗ which says that the (1, 1)-curvature forms Ωa
b of ∇ live
on PE. Thus, Ch
j (∇) ∈ Aj,j(PE).
Finally, we note that properties of horizontal Chern forms in relation with basic properties of
classical Chern forms must be studied, as well as the independence of these forms with respect to
some families of complex Finsler structures on a holomorphic vector bundle. Another important
problem to solve is to describe the obstructions corresponding to these classes. Here we are able
to respond only partially to these questions.
Applying some results from [2], we get the invariance of horizontal Chern forms by a family
of complex Finsler structures given by infinitesimal deformations.
We consider a 1-parameter family {F 2
t }, t ∈ R of pseudoconvex complex Finsler structures
on a holomorphic vector bundle E. Let us put F 2
0 = F 2 and let ∇0 = ∇ its partial Bott complex
connection. The infinitesimal deformation V induced by F 2
t is defined by V :=
(
∂F 2
t
∂t
)
|t=0 and
its components with respect to a fixed frame field s = {sa}, a = 1, . . . ,m of E are given by
Vab =
(
∂ht ab
∂t
)
|t=0. We put V a
b = hcaVbc and we consider it as a section of the bundle End(V ′E).
If we consider F 2
t = F 2 + tV for sufficiently small t so that F 2
t remains pseudoconvex then
in Theorem 5.2 from [2] it is proved that if ∇V a
b = 0 then ∇t = ∇ and Ωt = Ω. Here
∇Z = (d′hZa + Zbωa
b )∂̇a, for any Z = Za∂̇a ∈ Γ(V ′E). Thus, we can conclude
Horizontal Forms of Chern Type on Complex Finsler Bundles 7
Proposition 4. If ∇V a
b = 0 then the horizontal Chern forms Ch
j (∇) of (E,F ) are invariant by
the linear family of complex Finsler structures given by F 2
t = F 2 + tV .
We notice that in Aikou’s paper [2], the partial connection is considered in the pull-back
bundle p−1E over PE and the calculations are similar as on V ′E.
Acknowledgements
The author is grateful to the anonymous referees and would like to thank them for generous sug-
gestions and comments. Also, I warmly thank Professor Gheorghe Pitiş for fruitful conversations
concerning this topics.
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http://dx.doi.org/10.1023/B:MAHU.0000038969.91179.e4
http://projecteuclid.org/getRecord?id=euclid.nmj/1118795367
http://dx.doi.org/10.1002/mana.19670340104
http://dx.doi.org/10.1007/BF02884722
http://dx.doi.org/10.1007/s11425-006-2055-8
1 Introduction and preliminaries
2 Horizontal forms of Chern type
References
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