A Lefschetz Fixed Point Formula
for Singular Arithmetic Schemes
with Smooth Generic Fibres
Shun Tang
Received: May 31, 2010 Revised: October 11, 2010 Communicated by Takeshi Saito
Abstract. In this article, we consider singular equivariant arith-metic schemes whose generic fibres are smooth. For such schemes, we prove a relative fixed point formula of Lefschetz type in the context of Arakelov geometry. This formula is an analog, in the arithmetic case, of the Lefschetz formula proved by R. W. Thomason in [31]. In par-ticular, our result implies a fixed point formula which was conjectured by V. Maillot and D. R¨ossler in [25].
2010 Mathematics Subject Classification: 14C40, 14G40, 14L30, 58J20, 58J52
Keywords and Phrases: fixed point formula, singular arithmetic scheme, Arakelov geometry
1 Introduction
It is the aim of this article to prove a singular Lefschetz fixed point formula for some schemes which admit the actions of a diagonalisable group scheme, in the context of Arakelov geometry. We first roughly describe the history of the study of such Lefschetz fixed point formulae and relative Lefschetz-Riemann-Roch problems.
A classical Lefschetz fixed point formula is to give an expression of the alter-nating sum of the traces ofHi(ϕ), as a sum of the contributions from the
com-ponents of the fixed point subvarietyXg. On the other hand, roughly speaking,
a Lefschetz-Riemann-Roch theorem is a commutative diagram in equivariant
K−theory which can be regarded as a Grothendieck type generalization of the Lefschetz fixed point formula. Indeed, when we choose the base variety in such a commutative diagram to be a point, we will get the ordinary Lefschetz fixed point formula. IfX is nonsingular, P. Donovan has proved such a theorem in [12] by using the results and some of the methods of the paper of A. Borel and J. P. Serre on the Grothendieck-Riemann-Roch theorem (cf. [10]). In [1], P. Baum, W. Fulton and G. Quart generalized Donovan’s theorem to singular varieties, the key step of their proof heavily relies on an elegant method called the deformation to the normal cone. Denote by G0(X, g) (resp. K0(X, g))
the Quillen’s algebraic K-group associated to the category of equivariant co-herent sheaves (resp. vector bundles of finite rank) on X, then K0(Pt, g) is
isomorphic to the group ringZ[k] andG0(X, g) (resp. K0(X, g)) has a natural
K0(Pt, g)-module (resp. K0(Pt, g)-algebra) structure. Let f be an
equivari-antly projective morphism between two equivariant varieties X and Y, then it is possible to define a push-forward morphismf∗ fromG0(X, g) toG0(Y, g)
in a rather standard way. LetRbe any flatK0(Pt, g)-algebra in which 1−ζ
is invertible for each non-trivialn-th root of unity ζ in k. The main result of Baum, Fulton and Quart reads: there exists a family of group homomorphisms
L.betweenK-groups making the following diagram
G0(X, g) L. //
f∗
G0(Xg, g)⊗Z[k]R
fg∗
G0(Y, g) L. //G0(Yg, g)⊗Z[k]R
commutative. IfZis a nonsingular equivariant variety such that there exists an equivariant closed immersion fromX toZ, then for every equivariant coherent sheafE onX the homomorphismL.is exactly given by the formula
L.(E) =λ−1−1(N∨
Z/Zg)·
X
j
(−1)jTorj
OZ(i∗E,OZg)
where NZ/Zg stands for the normal bundle of Zg in Z and λ−1(N
∨
Z/Zg) :=
P(
−1)j
∧jN∨
Z/Zg.
We would like to indicate that one can use the same method so called the defor-mation to the normal cone to extend Baum, Fulton and Quart’s result to general scheme case whereX andY are Noetherian, separated schemes endowed with projective actions of the diagonalisable group scheme µn associated to Z/nZ.
Here by a µn-action on X we understand a morphism mX : µn×X → X
which statisfies some compatibility properties. Denote by pX the projection
mean an isomorphismmE :p∗XE →m∗XE which satisfies certain associativity
properties. We refer to [20] and [21, Section 2] for the group scheme action theory we are talking about.
In [31], R. W. Thomason used another way to generalize Baum, Fulton and Quart’s result to the scheme case, and he removed the condition of projec-tivity. The strategy Thomason followed was to use Quillen’s localization sequence for higher equivariant K-groups to prove an algebraic concentra-tion theorem. Let D be an integral Noetherian ring, and let µn be the
diagonalisable group scheme over D associated to Z/nZ. Denote the ring
K0(D, µn) ∼= K0(D)[T]/(1−Tn) by R(µn). We consider the prime ideal ρ
in R(µn) with which the intersection of Z[Z/nZ] is exactly the kernel of the
canonical morphism Z[T]/(1−Tn) → Z[T]/(Φ
n) where Φn stands for then
-th cyclotomic polynomial (cf. [31, Lem. 1.6]). By construction -the elements 1−Tk fork= 1, . . . , n−1 are not contained inρ. LetX be aµ
n-equivariant
scheme overD, thenG0(X, µn) (resp. K0(X, µn)) has a naturalR(µn)-module
(resp. R(µn)-algebra) structure. Denote byi the inclusion from Xµn to X.
The algebraic concentration theorem reads: there exists a natural group ho-momorphism i∗ from G0(Xµn, µn)ρ to G0(X, µn)ρ which is an isomorphism.
Moreover, if X is regular, the inverse map ofi∗ is given by λ−1−1(NX/X∨ µn)·i
∗
whereNX/Xµn is the normal bundle ofXµn inX. This concentration theorem
can be used to prove a singular Lefschetz fixed point formula which is an exten-sion of Baum, Fulton and Quart’s result in general scheme case. Thomason’s approach has nothing to do with the construction of the deformation to the normal cone, and the localization he used is slightly weaker than Baum, Fulton and Quart’s in the sense that the complement of the idealρinR(µn) is not the
smallest algebra in which the elements 1−Tk (k= 1, . . . , n−1) are invertible. If one exactly chooses Rto be the complement of the ideal ρin R(µn), then
these two localizations are equal to each other.
In [21], K. K¨ohler and D. R¨ossler generalized the regular case of Baum, Fulton and Quart’s result to Arakelov geometry. To every regularµn-equivariant
arith-metic schemeX, they associate an equivariant arithmeticK0-groupKc0(X, µn)
which contains some smooth form class on Xµn(C) as analytic datum. Such
an equivariant arithmetic K0-group has a ring structure so that it is also an R(µn)-algebra. LetNX/Xµn be the normal bundle with respect to the regular
immersionXµn ֒→ X which is endowed with the quotient metric induced by
a chosen K¨ahler metric of X(C), then the main theorem in [21] reads: the element λ−1(N
∨
X/Xµn) is invertible inKc0(Xµn, µn)⊗R(µn)Rand we have the
following commutative diagram
c
K0(X, µn)
ΛR(f)
−1·τ
/
/
f∗
c
K0(Xµn, µn)⊗R(µn)R
fµn∗
c
K0(D, µn)
ι //
c
where ΛR(f) :=λ−1(N ∨
X/Xµn)·(1 +Rg(NX/Xµn)),τ stands for the restriction
map andι is the natural morphism from a ring or a module to its localization which sends an elementeto e1. HereRg(·) is the equivariantR-genus, the
def-inition of the two push-forward morphismsf∗ andfµn∗ involves an important
analytic datum which is called the equivariant analytic torsion. The strategy K¨ohler and R¨ossler followed to prove such an arithmetic Lefschetz-Riemann-Roch theorem was to use the construction of the deformation to the normal cone to prove an analog of this theorem for equivariant closed immersions. Af-ter that, they decompose the morphism f to a closed immersion h from X
to some projective space Pr
D followed by a smooth morphism p from PrD to
Spec(D). Then the theorem in general situation follows from an argument of investigating the behavior of the error term under the morphismshandp. Provided X. Ma’s extension of equivariant analytic torsion to higher equivariant analytic torsion form, it was conjectured by K¨ohler and R¨ossler in [22] that an analog of [21, Theorem 4.4] in relative setting holds. We have already proved this conjecture in [29]. Our method is similar to Thomason’s, we first show that there exists an arithmetic concentration theorem in Arakelov geometry and then deduce from it the relative Lefschetz fixed point formula. The same as Thomason’s approach, our method has nothing to do with the construction of the deformation to the normal cone, but unfortunately it only works for regular arithmetic schemes.
One may naturally asked that whether it is possible to construct a more general arithmetic Gc0-theory and prove a relative Lefschetz fixed point formula for
singular arithmetic schemes which is entirely an analog of Thomason’s singular Lefschetz formula in Arakelov geometry. The answer is Yes, and this is what we have done in this article. To do this, one needs aGc0-theoretic vanishing theorem
which can be viewed as an extension of K¨ohler and R¨ossler’s fixed point formula for closed immersions to the singular case. The proof of such a vanishing theorem occupies a lot of space in this article. LetX andY be two singular equivariant arithmetic schemes with smooth generic fibres, and letf :X→Y
be an equivariant morphism which is smooth on the complex numbers. Assume that theµn-action onY is trivial andf can be decomposed to beh◦iwherei
is an equivariant closed immersion fromX to some regular arithmetic scheme
Z and h : Z → Y is equivariant and smooth on the complex numbers. Let
η be an equivariant hermitian sheaf on X. Referring to Section 6.1 for the explanations of various notations, we announce that our main theorem in this article is the following equality which holds inGc0(Y, µn,∅)ρ:
f∗(η) =fµZn∗(i ∗
µn(λ
−1 −1(N
∨
Z/Zµn))·
X
k
(−1)kTorkOZ(i∗η,OZµn))
+
Z
Xg/Y
Td(T fg, ωX)chg(η)Tdfg(F, ωX)Td−1g (F)
−
Z
Xg/Y
+
Z
Xg/Y
f
Td(T fg, ωX, ωXZ)chg(η)Tdg(NZ/Zg)Td
−1
g (F).
The structure of this article is as follows. In Section 2, we recall some differential-geometric facts for the convenience of the reader. In Section 3, we formulate and prove a vanishing theorem for equivariant closed immersions in a purely analytic setting. In Section 4, we define the arithmticG0-groups with
respect to fixed wave front sets which are necessary for our later arguments. In Section 5 and Section 6, we formulate and prove the arithmetic concentration theorem and the relative Lefschetz fixed point formula for singular arithmetic schemes.
Acknowledgements. The author wishes to thank his thesis advisor Damian R¨ossler for providing such an interesting topic, also for his constant encourage-ment and for many fruitful discussions between them. The author is greatful to Xiaonan Ma, from whom he gets many meaningful comments and suggestions. Finally, thanks to the referee, for his/her excellent work.
2 Differential-geometric preliminaries
2.1 Equivariant Chern-Weil theory
LetGbe a compact Lie group and letM be a compact complex manifold which admits a holomorphicG-action. By an equivariant hermitian vector bundle on
M, we understand a hermitian vector bundle on M which admits aG-action compatible with the G-structure of M and whose metric isG-invariant. Let
g∈Gbe an automorphism ofM, we shall denote byMg={x∈M |g·x=x}
the fixed point submanifold. Mg is also a compact complex manifold.
Now let E be an equivariant hermitian vector bundle onM, it is well known that the restriction ofE toMgsplits as a direct sum
E|Mg=
M
ζ∈S1
Eζ
where the equivariant structure gE of E acts on E
ζ as multiplication by ζ.
We often write Eg for E1 and call it the 0-degree part of E |Mg. As usual,
Ap,q(M) stands for the space of (p, q)-forms Γ∞(M,ΛpT∗(1,0)M
∧ΛqT∗(0,1)M),
we define
e A(M) =
dimMM
p=0
(Ap,p(M)/(Im∂+ Im∂)).
We denote by ΩEζ ∈ A1,1(M
g) the curvature form associated to Eζ. Let
(φζ)ζ∈S1 be a family of GL(C)-invariant formal power series such that φζ ∈ C[[glrkE
ζ(C)]] where rkEζ stands for the rank ofEζ which is a locally constant
Definition 2.1. The way to associate a smooth form to an equivariant her-mitian vector bundleE by setting
φg(E) :=φ((φζ(−
ΩEζ
2πi))ζ∈S1)
is called ang-equivariant Chern-Weil theory associated to (φζ)ζ∈S1 andφ. The class ofφg(E) inAe(Mg) is independent of the metric.
Write ddcfor the differential operator2∂∂πi. The theory of equivariant secondary characteristic classes is described in the following theorem.
Theorem 2.2. To every short exact sequence ε: 0→E′ →E →E′′ →0 of equivariant hermitian vector bundles on M, there is a unique way to attach a class φeg(ε)∈Ae(Mg)which satisfies the following three conditions:
(i). φeg(ε)satisfies the differential equation
ddcφeg(ε) =φg(E
′
⊕E′′)−φg(E);
(ii). for every equivariant holomorphic map f :M′
→M,φeg(f∗ε) =fg∗φeg(ε);
(iii). φeg(ε) = 0 ifε is equivariantly and orthogonally split.
Proof. This is [21, Theorem 3.4].
We now give some examples of equivariant character forms and their corre-sponding secondary characteristic classes.
Example 2.3. (i). The equivariant Chern character form chg(E) :=
P
ζ∈S1ζch(Eζ).
(ii). The equivariant Todd form Tdg(E) :=
crkEg(Eg)
chg(Prkj=0E(−1)j∧jE
∨
). As in [18,
Thm. 10.1.1] one can show that
Tdg(E) = Td(Eg)
Y
ζ6=1
det( 1
1−ζ−1eΩEζ
2πi
).
(iii). Let ε : 0 → E′ → E → E′′ → 0 be a short exact sequence of hermi-tian vector bundles. The secondary Bott-Chern characteristic class is given by
e
chg(ε) =Pζ∈S1ζch(e εζ).
(iv). If the equivariant structure gε has the eigenvalues ζ
1,· · · , ζm, then the
secondary Todd class is given by
f
Tdg(ε) = m
X
i=1
i−1 Y
j=1
Tdg(Eζj)·Td(f εζi)·
m
Y
j=i+1
Tdg(E
′
ζj +E
′′
ζj).
LetE be an equivariant hermitian vector bundle with two different hermitian metrics h1 and h2, we shall write φeg(E, h1, h2) for the equivariant secondary
characteristic class associated to the exact sequence
0→(E, h1)→(E, h2)→0→0
where the map from (E, h1) to (E, h2) is the identity map.
2.2 Equivariant analytic torsion forms
In [7], J.-M. Bismut and K. K¨ohler extended the Ray-Singer analytic torsion to the higher analytic torsion form T for a holomorphic submersion. The purpose of making such an extension is that the differential equation on ddcT
gives a refinement of the Grothendieck-Riemann-Roch theorem. Later, in his article [23], X. Ma generalized J.-M. Bismut and K. K¨ohler’s results to the equivariant case. In this subsection, we shall briefly recall Ma’s construction of the equivariant analytic torsion form. This construction is not very important for understanding the rest of this article, but the equivariant analytic torsion form itself will be used to define a reasonable push-forward morphism between equivariant arithmeticG0-groups.
We first fix some notations and assumptions. Let f : M → B be a proper holomorphic submersion of complex manifolds, and let T M,T B be the holo-morphic tangent bundle onM,B. Denote byJT f the complex structure on the
real relative tangent bundleTRf, and assume thathT f is a hermitian metric on T f which induces a Riemannian metricgT f. LetTHM be a vector subbundle
ofT M such thatT M =THM
⊕T f, the following definition of K¨ahler fibration was given in [4, Def. 1.4].
Definition 2.5. The triple (f, hT f, THM) is said to define a K¨ahler fibration
if there exists a smooth real (1,1)−form ω which satisfies the following three conditions:
(i). ω is closed; (ii). TH
RM andTRf are orthogonal with respect toω;
(iii). ifX, Y ∈TRf, then ω(X, Y) =hX, JT fYigT f.
It was shown in [4, Thm. 1.5 and 1.7] that for a given K¨ahler fibration, the form ω is unique up to addition of a formf∗η where η is a real, closed (1,
1)-form on B. Moreover, for any real, closed (1,1)-formω on M such that the bilinear mapX, Y ∈TRf 7→ω(JT fX, Y)∈Rdefines a Riemannian metric and
hence a hermitian producthT f onT f, we can define a K¨ahler fibration whose
associated (1,1)-form is ω. In particular, for a givenf, a K¨ahler metric onM
defines a K¨ahler fibration if we chooseTHM to be the orthogonal complement
ofT f inT M andω to be the K¨ahler form associated to this metric.
We now recall the Bismut superconnection of a K¨ahler fibration. Let (ξ, hξ) be a hermitian complex vector bundle onM. Let∇T f,
hermitian connections on (T f, hT f) and (ξ, hξ). Let
∇Λ(T∗(0,1)f)
be the con-nection induced by ∇T f on Λ(T∗(0,1)f). Then we may define a connection on
Λ(T∗(0,1)f)
⊗ξby setting
∇Λ(T∗(0,1)f)⊗ξ =∇Λ(T∗(0,1)f)⊗1 + 1⊗ ∇ξ.
LetEbe the infinite-dimensional bundle onB whose fibre at each pointb∈B
consists of theC∞sections of Λ(T∗(0,1)f)
⊗ξ|f−1b. This bundleE is a smooth Z-graded bundle. We define a connection ∇E on E as follows. If U ∈TRB,
let UH be the lift of U in TH
R M so that f∗UH =U. Then for every smooth
sectionsofE overB, we set
∇E Us=∇
Λ(T∗(0,1)f)⊗ξ
UH s.
For b ∈ B, let ∂Zb
be the Dolbeault operator acting on Eb, and let ∂ Zb∗
be its formal adjoint with respect to the canonical hermitian product on Eb (cf.
[23, 1.2]). Let C(TRf) be the Clifford algebra of (TRf, hT f), then the bundle
Λ(T∗(0,1)f)⊗ξ has a natural C(TRf)-Clifford module structure. Actually, if U ∈T f, letU′
∈T∗(0,1)f correspond to U defined byU′(
·) =hT f(U,
·), then forU, V ∈T f we set
c(U) =√2U′∧, c(V) =−√2iV
where i(·) is the contraction operator (cf. [9, Definition 1.6]). Moreover, if U, V ∈TRB, we set T(UH, VH) =−PT f[UH, VH] where PT f stands for the
canonical projection fromT M toT f.
Definition 2.6. Lete1, . . . , e2m be a basis ofTRB, and lete1, . . . , e2mbe the
dual basis ofT∗
RB. Then the element c(T) =1
2
X
1≤α,β≤2m
eα∧eβ⊗bc(T(eHα, eHβ))
is a section of (f∗Λ(T∗
RB)⊗bEnd(Λ(T∗(0,1)f)⊗ξ))odd.
Definition2.7. Foru >0, the Bismut superconnection onEis the differential operator
Bu=∇E+√u(∂ Z
+∂Z∗)− 1 2√2uc(T)
onf∗(Λ(T∗
RB))⊗b(Λ(T∗(0,1)f)⊗ξ).
Definition2.8. LetNV be the number operator on Λ(T∗(0,1)f)⊗ξand onE,
namely NV acts as multiplication by pon Λp(T∗(0,1)f)⊗ξ. ForU, V ∈TRB,
set ωHH(U, V) = ωM(UH, VH) whereωM is the closed form in the definition
of K¨ahler fibration. Furthermore, foru >0, setNu=NV +iω
HH
We now turn to the equivariant case. LetGbe a compact Lie group, we shall assume that all complex manifolds, hermitian vector bundles and holomorphic morphisms considered above areG−equivariant and all metrics areG-invariant. We will additionally assume that the direct images Rkf
∗ξ are all locally free
so that the G-equivariant coherent sheaf R·f
∗ξ is locally free and hence a G
-equivariant vector bundle overB. [23, 1.2] gives aG-invariant hermitian metric (the L2-metric)hR·
f∗ξ on the vector bundle R·f ∗ξ.
Forg∈G, letMg={x∈M |g·x=x}andBg={b∈B|g·b=b}be the fixed
point submanifolds, then f induces a holomorphic submersionfg:Mg→Bg.
Let Φ be the homomorphismα7→(2iπ)−degα/2 of Λeven(T∗
called the equivariant analytic torsion form.
Theorem 2.11. The form Tg(ωM, hξ)lies inLp≥0Ap,p(Bg)and satisfies the
Proof. This is [23, Theorem 2.12]. We define a secondary characteristic class
such that it satisfies the following differential equation ddccheg(R·f∗ξ, hR
·
f∗ξ, h′R·
f∗ξ) = ch
g(R·f∗ξ, hR
· f∗ξ)
−chg(R·f∗ξ, h′R
· f∗ξ),
then the anomaly formula can be described as follows.
Theorem 2.12. (Anomaly formula) Let ω′ be the form associated to another K¨ahler fibration for f : M → B. Let h′T f be the metric on T f in this new
fibration and let h′ξ be another metric on ξ. The following identity holds in
e
A(Bg) :=Lp≥0(Ap,p(Bg)/(Im∂+ Im∂)):
Tg(ωM, hξ)−Tg(ω′M, h′ξ) =cheg(R·f∗ξ, hR
· f∗ξ, h′R·
f∗ξ)
−
Z
Mg/Bg
[Tdfg(T f, hT f, h′T f)chg(ξ, hξ)
+ Tdg(T f, h′T f)cheg(ξ, hξ, h′ξ)].
In particular, the class of Tg(ωM, hξ) inAe(Bg)only depends on (hT f, hξ).
Proof. This is [23, Theorem 2.13].
2.3 Equivariant Bott-Chern singular currents
The Bott-Chern singular current was defined by J.-M. Bismut, H. Gillet and C. Soul´e in [5] in order to generalize the usual Bott-Chern secondary char-acteristic class to the case where one considers the resolutions of hermitian vector bundles associated to the closed immersions of complex manifolds. In [2], J.-M. Bismut generalized this topic to the equivariant case. We shall recall Bismut’s construction of the equivariant Bott-Chern singular current in this subsection. Similar to the equivariant analytic torsion form, the construction itself is not very important for understanding our later arguments, we just re-call it for the convenience of the reader. Bismut’s construction was realized via some current valued zeta function which involves the supertraces of Quillen’s superconnections. This is similar to the non-equivariant case.
As before, let g be the automorphism corresponding to an element in a com-pact Lie group G. Let i : Y → X be an equivariant closed immersion of
G-equivariant K¨ahler manifolds, and let η be an equivariant hermitian vector bundle onY. Assume thatξ.is a complex (of homological type) of equivariant hermitian vector bundles on X which provides a resolution of i∗η. We denote
the differential of the complexξ. by v. Note thatξ. is acyclic outsideY and the homology sheaves of its restriction toY are locally free and hence they are all vector bundles. We write Hn = Hn(ξ. |Y) and define a Z-graded bundle
H =LnHn. For eachy∈Y andu∈T Xy, we denote by∂uv(y) the derivative
ofvatyin the directionuin any given holomorphic trivialization ofξ.neary. Then the map∂uv(y) acts onHy as a chain map, and this action only depends
on the imagezofuinNy whereN stands for the normal bundle ofi(Y) inX.
Let π be the projection from the normal bundle N to Y, then we have a canonical identification ofZ-graded chain complexes
(π∗H, ∂zv)∼= (π∗(∧N∨⊗η), iz).
For this, one can see [3, Section I. b]. Moreover, such an identification is an identification of G-bundles which induces a family of canonical isomorphisms
γn :Hn ∼=∧nN∨⊗η. Another way to describe these canonical isomorphisms
γn is applying [13, Exp. VII, Lemma 2.4 and Proposition 2.5]. These two
constructions coincide because they are both locally, on a suitable open covering {Uj}j∈J, determined by any complex morphism over the identity map ofη|Uj
from (ξ.|Uj, v) to the minimal resolution ofη|Uj (e.g. the Koszul resolution).
The advantage of using the construction given in [13] is that it remains valid for arithmetic varieties over any base instead of the complex numbers. Later in [2], for the use of normalization, J.-M. Bismut considered the automorphism ofN∨ defined by multiplying a constant−√−1, it induces an isomorphism of
chain complexes
(π∗(∧N∨⊗η), iz)∼= (π∗(∧N∨⊗η),
√ −1iz)
and hence
(π∗H, ∂zv)∼= (π∗(∧N∨⊗η),√−1iz).
This identification induces a family of isomorphismsfγn:Hn ∼=∧nN∨⊗η. By
finite dimensional Hodge theory, for each y ∈Y, there is a canonical isomor-phism
Hy∼={f ∈ξ.y|vf = 0, v∗f = 0}
wherev∗is the dual ofvwith respect to the metrics onξ.. This means thatH
can be regarded as a smoothZ-gradedG-equivariant subbundle ofξso that it carries an inducedG-invariant metric. On the other hand, we endow∧N∨⊗η
with the metric induced fromN andη. J.-M. Bismut introduced the following definition.
Definition 2.13. We say that the metrics on the complex of equivariant her-mitian vector bundles ξ. satisfy Bismut assumption (A) if the identification (π∗H, ∂
zv) ∼= (π∗(∧N∨⊗η),√−1iz) also identifies the metrics, it is
equiva-lent to the condition that the identification (π∗H, ∂
zv) ∼= (π∗(∧N∨⊗η), iz)
identifies the metrics.
Proposition2.14. There always existG-invariant metrics onξ.which satisfy Bismut assumption (A) with respect to the equivariant hermitian vector bundles N andη.
Proof. This is [2, Proposition 3.5].
From now on we always suppose that the metrics on a resolution satisfy Bismut assumption (A). Let∇ξ be the canonical hermitian holomorphic connection on
ξ., then for eachu >0, we may define aG-invariant superconnection
on theZ2-graded vector bundleξ. Moreover, let Φ be the mapα∈ ∧(TR∗Xg)→ (2πi)−degα/2α
∈ ∧(T∗
RXg) and denote
(Td−1g )′(N) :=
∂
∂b |b=0 (Tdg(b·Id−
ΩN
2πi) −1)
where ΩN is the curvature form associated to N. We recall as follows the
construction of the equivariant singular current given in [2, Section VI].
Lemma 2.15. Let NH be the number operator on the complex ξ. i.e. it acts on
ξj as multiplication byj, then fors∈Cand0<Re(s)< 12, the current valued zeta function
Zg(ξ.)(s) :=
1 Γ(s)
Z ∞
0
us−1[ΦTr
s(NHgexp(−Cu2)) + (Td−1g )′(N)chg(η)δYg]du
is well-defined on Xg and it has a meromorphic continuation to the complex
plane which is holomorphic at s= 0.
Definition 2.16. The equivariant singular Bott-Chern current onXg
associ-ated to the resolution ξ.is defined as
Tg(ξ.) := ∂
∂s |s=0Zg(ξ.)(s).
Theorem 2.17. The currentTg(ξ.)is a sum of (p, p)-currents and it satisfies
the differential equation
ddcTg(ξ.) =ig∗chg(η)Td−1g (N)−
X
k
(−1)kchg(ξk).
Moreover, the wave front set ofTg(ξ.) is contained inNg,∨R whereNg,∨R stands for the underlying real bundle of the dual of Ng.
Proof. This follows from [2, Theorem 6.7, Remark 6.8].
Finally, we recall a theorem concerning the relationship of equivariant Bott-Chern singular currents involved in a double complex. This theorem will be used to show that our definition of a general embedding morphism in equivari-ant arithmetic G0-theory is reasonable.
Theorem 2.18. Let
χ: 0→ηn → · · · →η1→η0→0
be an exact sequence of equivariant hermitian vector bundles on Y. Assume that we have the following double complex consisting of resolutions ofi∗χ such that all rows are exact sequences.
0 //ξ
n,· //
· · · //ξ
1,· //
ξ0,· //
0
0 //i∗η
For each k, we writeεk for the exact sequence
0→ξn,k → · · · →ξ1,k →ξ0,k→0.
Then we have the following equality inUe(Xg) :=Lp≥0(Dp,p(Xg)/(Im∂+Im∂)) n
X
j=0
(−1)jTg(ξj,·) =ig∗
e
chg(χ)
Tdg(N)−
X
k
(−1)kcheg(εk).
Here Dp,p(X
g)stands for the space of currents onXg of type(p, p).
Proof. This is [21, Theorem 3.14].
2.4 Bismut-Ma’s immersion formula
In this subsection, we shall recall Bismut-Ma’s immersion formula which reflects the behaviour of the equivariant analytic torsion forms of a K¨ahler fibration under composition of an immersion and a submersion. By translating to the equivariant arithmetic G0-theoretic language, such a formula can be used to
measure, in arithmeticG0-theory, the difference between a push-forward
mor-phism and the composition formed as an embedding mormor-phism followed by a push-forward morphism. Although Bismut-Ma’s immersion formula plays a very important role in our arguments, we shall not recall its proof since it is rather long and technical.
Let i : Y → X be an equivariant closed immersion of G-equivariant K¨ahler manifolds. Let S be a complex manifold with the trivial G-action, and let
f :Y →S, l :X →S be two equivariant holomorphic submersions such that
f =l◦i. Assume that η is an equivariant hermitian vector bundle onY and
ξ. provides a resolution ofi∗η onX whose metrics satisfy Bismut assumption
(A). Let ωY, ωX be the real, closed and G-invariant (1,1)-forms on Y, X
which induce the K¨ahler fibrations with respect to f and l respectively. We additionally assume thatωY is the pull-back ofωX so that the K¨ahler metric
onY is induced by the K¨ahler metric onX. As before, denote byN the normal bundle ofi(Y) inX. Consider the following exact sequence
N : 0→T f →T l|Y→N→0
whereN is endowed with the quotient metric, we shall writeTdfg(T f , T l|Y) for
f
Tdg(N) the equivariant Bott-Chern secondary characteristic class associated
to N. It satisfies the following differential equation
ddcTdfg(T f , T l|Y) = Tdg(T f, hT f)Tdg(N)−Tdg(T l|Y, hT l).
For simplicity, we shall suppose that in the resolution ξ., ξj are all l−acyclic
and moreoverη isf−acyclic. By an easy argument of long exact sequence, we have the following exact sequence
By the semi-continuity theorem, all the elements in the exact sequence above are vector bundles. In this case, we recall the definition of the L2-metrics on
direct images precisely as follows. We just takef∗hη as an example. Note that
the semi-continuity theorem implies that the natural map
(R0f∗η)s→H0(Ys, η|Ys)
is an isomorphism for every point s∈S where Ys stands for the fibre overs.
We may endowH0(Y
s, η|Ys) with aL
2-metric given by the formula
< u, v >L2:= 1 (2π)ds
Z
Ys
hη(u, v)ω
Y ds
ds!
where ds is the complex dimension of the fibre Ys. It can be shown that
these metrics depend onsin aC∞ manner (cf. [9, p.278]) and hence define a
hermitian metric onf∗η. We shall denote it byf∗hη.
In order to understand the statement of Bismut-Ma’s immersion formula, we still have to recall an important concept defined by J.-M. Bismut, the equiv-ariantR-genus. LetW be aG-equivariant complex manifold, and letEbe an equivariant hermitian vector bundle onW. Forζ∈S1 ands >1 consider the
zeta function
L(ζ, s) =
∞ X
k=1 ζk
ks
and its meromorphic continuation to the whole complex plane. Define the formal power series inx
e
R(ζ, x) :=
∞ X
n=0 ∂L
∂s(ζ,−n) +L(ζ,−n)
n
X
j=1
1 2j
xn
n!.
Definition2.19. The Bismut equivariantR-genus of an equivariant hermitian vector bundleE withE|Xg=
P
ζEζ is defined as
Rg(E) :=
X
ζ∈S1
TrRe(ζ,−Ω
Eζ
2πi)−TrRe(1/ζ,
ΩEζ
2πi)
where ΩEζ is the curvature form associated toE
ζ. Actually, the class ofRg(E)
inAe(Xg) is independent of the metric and we just writeRg(E) for it.
Further-more, the classRg(·) is additive.
above. Then the equality
m
X
i=0
(−1)iTg(ωX, hξi)−Tg(ωY, hη) +cheg(Ξ, hL
2 )
=
Z
Xg/S
Tdg(T l, hT l)Tg(ξ.) +
Z
Yg/S
f
Tdg(T f , T l|Y)
Tdg(N)
chg(η)
+
Z
Xg/S
Tdg(T l)Rg(T l) m
X
i=0
(−1)ich g(ξi)−
Z
Yg/S
Tdg(T f)Rg(T f)chg(η)
holds in Ae(S).
Proof. This is the combination of [8, Theorem 0.1 and 0.2], the main theorems in that paper.
3 A vanishing theorem for equivariant closed immersions
3.1 The statement
By a projective manifold we shall understand a compact complex manifold which is projective algebraic, that means a projective manifold is the complex analytic space X(C) associated to a smooth projective variety X over C(cf. [17, Appendix B]). Letµnbe the diagonalisable group variety overCassociated
to Z/nZ. We sayX isµn-equivariant if it admits aµn-projective action, this means the associated projective manifoldX(C) admits an action by the group of complexn-th roots of unity. Denote byXµnthe fixed point subscheme ofX,
by GAGA principle, Xµn(C) is equal to X(C)g whereg is the automorphism
onX(C) corresponding to a fixed primitiven-th root of unity. If no confusion arises, we shall not distinguish between X andX(C) as well as Xµ
n and Xg.
Since the classical arguments of locally free resolutions may not be compati-ble with the equivariant setting, we summarize some crucial facts we need as follows.
(i). Every equivariant coherent sheaf on an equivariantly projective scheme is an equivariant quotient of an equivariant locally free coherent sheaf.
(ii). Every equivariant coherent sheaf on an equivariantly projective scheme admits an equivariant locally free resolution. It is finite if the equivariant scheme is regular.
(iii). An exact sequence of equivariant coherent sheaves on an equivariantly projective scheme admits an exact sequence of equivariant locally free resolu-tions.
(iv). Any two equivariant locally free resolutions of an equivariant coherent sheaf on an equivariantly projective scheme can be dominated by a third one. Now leti:Y →Xbe aµn-equivariant closed immersion of projective manifolds
with normal bundle N. LetS be a projective manifold with the trivial µn
restrictionf :Y →S is an equivariant holomorphic submersion. According to our assumptions, we may define a K¨ahler fibration with respect tohby choosing a µn(C)-invariant K¨ahler formωX onX. By restrictingωX toY we obtain a
K¨ahler fibration with respect to f. The same thing goes tohg :Xg →S and
fg:Yg→S. Letη be an equivariant hermitian holomorphic vector bundle on
Y, assume that (ξ., v) is a complex of equivariant hermitian vector bundles on
X which provides a resolution ofi∗η, whose metrics satisfy Bismut assumption
(A).
Write Ng for the 0-degree part of N |Yg which is isomorphic to the normal
bundle of ig(Yg) in Xg and denote by F the orthogonal complement of Ng.
According to [13, Exp. VII, Lemma 2.4 and Proposition 2.5] we know that there exists a canonical isomorphism from the homology sheaf H(ξ. |Xg) to
ig∗(∧·F∨⊗η|Yg) which is equivariant. Then the restriction of (ξ., v) toXgcan
always split into a series of short exact sequences in the following way: (∗) : 0→Im→Ker→ig∗(∧·F∨⊗η|Yg)→0
and
(∗∗) : 0→Ker→ξ.|Xg→Im→0.
Suppose that ∧·F∨
⊗η |Yg and ξ. |Xg are all acyclic (higher direct images
vanish). Then according to an easy argument of long exact sequence, these short exact sequences (∗) and (∗∗) induce a series of short exact sequences of direct images:
H(∗) : 0→R0hg∗(Im)→R0hg∗(Ker)→R0fg∗(∧·F∨⊗η |Yg)→0
and
H(∗∗) : 0→R0hg∗(Ker)→R0hg∗(ξ.|Xg)→R
0h
g∗(Im)→0.
By semi-continuity theorem, all elements in the exact sequences above are vector bundles. We endow R0h
g∗(ξ. |Xg) and R
0f
g∗(∧·F∨⊗η |Yg) with the
L2-metrics which are induced by the metrics onξ., η andF. Here the normal
bundleN admits the quotient metric induced from the exact sequence 0→T f →T h|Y→N→0
and the bundle F admits the metric induced by the metric on N. Moreover, we endow R0h
g∗(Im) and R0hg∗(Ker) with the metrics induced by the L2
-metrics ofR0h
g∗(ξ.|Xg) so thatH(∗) andH(∗∗) become short exact sequences
of equivariant hermitian vector bundles. Denote by cheg(ξ., η) the alternating
sum of the equivariant secondary Bott-Chern characteristic classes ofH(∗) and H(∗∗) such that it satisfies the following differential equation
ddccheg(ξ., η) =
X
j
(−1)jch
g(R0fg∗(∧jF
∨
⊗η |Yg))
−X
j
Now the difference
δ(i, η, ξ.) :=cheg(ξ., η)−
X
k
(−1)kTg(ωYg, h∧
kF∨
⊗η|Yg)
+X
k
(−1)kT
g(ωXg, hξk|Xg)−
Z
Xg/S
Tg(ξ.)Td(T hg)
−
Z
Yg/S
Td(T fg)Td−1g (F)chg(η)R(Ng)
−
Z
Yg/S
chg(η)Td−1g (N)Td(f T fg, T hg|Yg)
makes sense and it is an element in Lp≥0Ap,p(S)/(Im∂+ Im∂). Here the
symbols Tg(·) in the summations stand for analytic torsion forms introduced
in Section 2.1, the symbol Tg(ξ.) in the integral is the equivariant Bott-Chern
singular current introduced in Section 2.2.
The vanishing theorem for equivariant closed immersions can be formulated as the following.
Theorem 3.1. Let i:Y →X be an equivariant closed immersion of projective manifolds, and letSbe a projective manifold with the trivialµn-action. Assume
that we are given two equivariant holomorphic submersions f : Y → S and h:X →S such thatf =h◦i. Then X admits an equivariant hermitian very ample invertible sheaf L relative to the morphism h, and for any equivariant hermitian resolution 0→ξm→ · · ·ξ1→ξ0→i∗η→0 we have
δ(i, η⊗i∗
L⊗n, ξ.⊗ L⊗n) = 0 for n≫0.
Here the metrics on the resolution are supposed to satisfy Bismut assumption (A).
3.2 Deformation to the normal cone
To prove the vanishing theorem for closed immersions, we use a geometric construction called the deformation to the normal cone which allows us to deform a resolution of hermitian vector bundle associated to a closed immersion of projective manifolds to a simpler one. Theδ-difference of this new simpler resolution is much easier to compute.
Let i : Y ֒→ X be a closed immersion of projective manifolds with normal bundleNX/Y. For a vector bundleE onX orY, the notationP(E) will stand
for the projective space bundle Proj(Sym(E∨)).
Definition 3.2. The deformation to the normal coneW(i) of the immersion
There are too many geometric objects and morphisms appearing in the con-struction of the deformation to the normal cone, we have to fix various notations in a clear way. We denote bypX (resp. pY) the projectionX×P1→X (resp.
Y ×P1→Y) and byπthe blow-down mapW →X×P1. We also denote by
qX (resp. qY) the projectionX×P1→P1(resp. Y×P1→P1) and byqW the
compositionqX◦π. It is well known that the mapqW is flat and for t∈P1,
we have
qW−1(t)∼=
X
× {t}, ift6=∞,
P∪X,e ift=∞,
whereXe is isomorphic to the blowing up ofX alongY andP is isomorphic to the projective completion of NX/Y i.e. the projective space bundleP(NX/Y ⊕
OY). Denote the canonical projection from P(NX/Y ⊕ OY) to Y byπP, then
the morphism OY → NX/Y ⊕ OY induces a canonical section i∞ : Y ֒→
P(NX/Y ⊕ OY) which is called the zero section embedding. Moreover, let
j:Y×P1→W be the canonical closed immersion induced byi×Id, then the
component Xe doesn’t meet j(Y ×P1) and the intersection of j(Y ×P1) with P is exactly the image ofY under the sectioni∞.
The advantage of the construction of the deformation to the normal cone is that we may control the rational equivalence class of the fibres q−1W(t). More precisely, in the language of line bundles, we have the isomorphisms O(X)∼=O(P+Xe)∼=O(P)⊗ O(Xe) which is an immediate consequence of the isomorphismO(0)∼=O(∞) onP1.
OnP =P(NX/Y ⊕ OY), there exists a tautological exact sequence 0→ O(−1)→π∗
P(NX/Y ⊕ OY)→Q→0
where Q is the tautological quotient bundle. This exact sequence and the inclusionOP →πP∗(NX/Y ⊕ OY) induce a sectionσ:OP →Qwhich vanishes
along the zero sectioni∞(Y). By duality we get a morphismQ∨ → OP, and
this morphism induces the following exact sequence
0→ ∧nQ∨→ · · · → ∧2Q∨→Q∨→ OP →i∞∗OY →0
where n is the rank ofQ. Note thati∞ is a section of πP i.e. πP ◦i∞ = Id,
the projection formula implies the following definition.
Definition3.3. For any vector bundleηonY, the following complex of vector bundles
0→ ∧nQ∨
⊗π∗
Pη→ · · · → ∧2Q∨⊗π∗Pη→Q∨⊗π∗Pη →πP∗η→0
provides a resolution of i∞∗η onP. This complex is called the Koszul
resolu-tion of i∞∗η and will be denoted byκ(η, NX/Y). If the normal bundle NX/Y
Now, assume that X is a µn-equivariant projective manifold and E is an
equivariant locally free sheaf on X. Then according to [20, (1.4) and (1.5)], P(E) admits a canonicalµn-equivariant structure such that the projection map P(E)→X is equivariant and the canonical bundleO(1) admits an equivariant structure. Moreover, letY →X be an equivariant closed immersion of projec-tive manifolds, according to [20, (1.6)] the action ofµn onX can be extended
to the blowing up BlYX such that the blow-down map is equivariant and the
canonical bundle O(1) admits an equivariant structure. So by endowing P1 with the trivialµn-action, the construction of the deformation to the normal
cone described above is compatible with the equivariant setting.
For the use of our later arguments, the K¨ahler metric chosen onW should be well controlled on the fibres of the deformation. For this purpose, it is necessary to introduce the following definition.
Definition3.4. (R¨ossler) A metrichonW is said to be normal to the defor-mation if
(a). it is invariant and K¨ahler;
(b). the restriction h |jg∗(Yg×P1) is a product h
′
×h′′, where h′ is a K¨ahler
metric onYg andh′′is a K¨ahler metric onP1;
(c). the intersections ofX× {0}withj∗(Y×P1) and ofP withj∗(Y ×P1) are
orthogonal at the fixed points.
Lemma 3.5. For any µn-invariant K¨ahler metric hX on X which induces an
invariant K¨ahler metric hY on Y, there exists a metric hW on W which is
normal to the deformation and the restriction of hW toX ∼=X
× {0} (resp. Y ∼=Y × {∞}) is exactly hX (resp. hY). Moreover, we may require that the
hermitian normal bundlesNY×P1/Y×{0} andNY×P1/Y×{∞} are both isometric
to the trivial bundles with trivial metrics.
Proof. The existence of the metric which is normal to the deformation is the content of [21, Lemma 6.13] and [28, Lemma 6.14], such a metric is constructed via the Grassmannian graph construction. Roughly speaking, according to an-other description of the deformation to the normal cone via the Grassmannian graph construction, we have an embeddingW →X×Pr×P1 and the metric
hW is the µ
n-average of the restriction of a product metric on X×Pr×P1
(cf. [28, Lemma 6.14]). When we endow X in the product with the metric
hX, the requirements on restrictions are automatically satisfied since hX is
µn-invariant. To fulfill the requirements on hermitian normal bundles, we may
just choose the Fubini-Study metric onP1.
We summarize some very important results about the application of the defor-mation to the normal cone as follows. Their proofs can be found in [21, Section 2 and 6.2].
(i). there exists an equivariant hermitian resolution of j∗p∗Y(η) on W, whose
metrics satisfy Bismut assumption (A) and whose restriction to Xe is equivari-antly and orthogonally split;
(ii). the natural morphism from the deformation to the normal cone W(ig)to
the fixed point submanifoldW(i)g is a closed immersion, this closed immersion
induces the closed immersionsP(NX
g/Yg⊕ OYg)→P(NX/Y ⊕ OY)g andXfg →
e Xg;
(iii). the fixed point submanifold of P(NX/Y ⊕ OY) is P(NX
g/Yg ⊕
OYg)
`
ζ6=1P((NX/Y)ζ);
(iv). the closed immersioni∞,g factors throughP(NXg/Yg⊕ OYg)and the other
components P((NX/Y)ζ)don’t meet Y. Hence the complex κ(OY, NX/Y)g,
ob-tained by taking the0-degree part of the Koszul resolution, provides a resolution of OYg onP(NX/Y ⊕ OY)g.
3.3 Proof of the vanishing theorem
We shall first prove the first part of the vanishing theorem for closed immersions i.e. the existence of an equivariant hermitian very ample invertible sheaf on
X which is relative to the morphismh:X →S. Generally speaking, such an invertible sheaf can be constructed rather easily sinceXadmits aµn-projective
action and theµn-action onSis supposed to be trivial, but for the whole proof
of the vanishing theorem we would like to construct a special one which is the pull-back of some equivariant hermitian very ample invertible sheaf on W(i) under the identificationX ∼=X× {0}. Our starting point is the following.
Definition 3.7. Let M be a µn-projective manifold, and let PnM be some
relative projective space over M. A µn-action on PnM arising from some µn
-action on the free sheafO⊕n+1
M via the functorial properties of the Proj symbol
will be called a globalµn-action.
The advantage of considering global µn-action is that on a projective space
which admits a globalµn-action the twisted line bundle O(1) is naturallyµn
-equivariant.
Lemma 3.8. The morphismh:X →S factors though some relative projective spacePr
S which admits a global µn-action.
Proof. By assumption,X admits aµn-projective action. Then [21, Lemma 2.4
and 2.5] imply that there exists an equivariant closed immersion from X to some projective spacePrendowed with a global action. By using the universal property of fibre product, we obtain a morphism fromX toPr
S =S×Prwhich
is equivariant. Moreover, this morphism is clearly a closed immersion. Since the action onSis trivial, the induced action on the fibre productS×Pris still global. So we are done.
Lemma3.9. Letl:W(i)→S be the compositionh◦pX◦π. ThenW(i)admits
Proof. By Lemma 3.8, h : X → S factors through some relative projective space Pr
S which admits a globalµn-action. So X admits an equivariant very
ample invertible sheaf relative toh. Since theµn-action onSis supposed to be
trivial,P1
X=X×P1∼=X×SP1Salso admits an equivariant very ample invertible
sheaf relative to the morphism h◦pX which is denoted byG. Moreover, by
construction, W(i) admits a very ample invertible sheaf OW(1)⊗π∗G⊗b for
someb≥0 which is relative to the blow-down map π(cf. [17, II. Proposition 7.10]). Assume that P1X×S PmS is the relative projective space associated to OW(1)⊗π∗G⊗b, and that PnS is the relative projective space associated to
G. Then the very ample invertible sheave on P1X×SPmS with respect to the embedding
P1
X×SPmS ֒→PnS×SPmS
can be written as G⊠OPm
S(1) whose restriction to W(i) is equal to OW(1)⊗
π∗G⊗b+1. Therefore,O
W(1)⊗π∗G⊗b+1is a very ample invertible sheaf onW(i)
relative tol:W(i)→S, this invertible sheaf is clearly equivariant.
From now on, we shall fix the equivariant very ample invertible sheaf L con-structed in Lemma 3.9. We also fix aµn-invariant hermitian metric onL, note
that this metric always exists according to an argument of partition of unity. When we deal with the tensor product of a coherent sheafF with some power L⊗n, we just write it as
F(n) for simplicity. Before we give the proof of the rest of the vanishing theorem, we shall recall the concept of equivariant standard complex and some technical results.
Definition 3.10. Let S be a projective manifold and let ξ. be a bounded complex of hermitian vector bundles onS. We sayξ.is a standard complex if the homology sheaves ofξ.are all locally free and they are endowed with some hermitian metrics. We shall write a standard complex as (ξ., hH) to emphasize
the choice of the metrics on the homology sheaves.
Definition3.11. LetSbe an equivariant projective manifold. An equivariant standard complex onS is a bounded complex of equivariant hermitian vector bundles on S whose restriction toSg is standard and the metrics on the
ho-mology sheaves areg-invariant. Again we shall write an equivariant standard complex as (ξ., hH) to emphasize the choice of the metrics on the homology
sheaves.
Due to [29, Theorem 5.9], to every equivariant standard complex (ξ., hH) on
an equivariant projective manifold S, there is a unique axiomatical way to associate an elementcheg(ξ., hH) inLp≥0Ap,p(Sg)/(Im∂+ Im∂) which satisfies
the differential equation ddccheg(ξ., hH) =
X
j
(−1)jchg(Hj(ξ.|Sg))−
X
j
(−1)jchg(ξj).
short exact sequence of standard complexes 0→ξ.′ |Sg→ξ.|Sg→ξ.
′′
|Sg→0.
Hence we obtain a long exact sequence of homology sheaves of these three standard complexes. We shall make a stronger assumption. Suppose that for any j ≥ 0, we have short exact sequence 0 → Hj(ξ.
Lemma 3.12. Let notations and assumptions be as above. The identity e
Proof. OnSg, every equivariant standard complex (ξ., hH) splits into a series of
short exact sequences of equivariant hermitian vector bundles in the following way
0→Im→Ker→H.→0 and
0→Ker→ξ.|Sg→Im→0.
According to the argument given after [29, Remark 5.10],cheg(ξ., hH) is equal
to the alternating sum of the equivariant Bott-Chern secondary characteristic classes of the short exact sequences above. Now since we have supposed that 0→Hj(ξ.
Then the identity in the statement of this lemma immediately follows from the construction ofcheg(ξ., hH) and the additivity property of the equivariant
Bott-Chern secondary characteristic classes.
Corollary 3.13. Let 0 → ξ.(m) → · · · → ξ.(1) → ξ.(0) → 0 be an exact sequence of equivariant standard complexes on S such that for every j ≥ 0,
0 →Hj(ξ.
Proof. We claim that for every 1≤k≤m, the kernel of the complex morphism
Sg and using an argument of long exact sequence, we know that the homology
sheaves of K |Sg are all equivariant hermitian vector bundles since for any
j ≥0 the bundle morphismHj(ξ.
(1)
|Sg)→Hj(ξ.
(0)
|Sg) is already surjective.
Therefore, the assumption of exactness on homologies implies that we can split 0 → ξ.(m) → · · · → ξ.(1) → ξ.(0) → 0 into a series of short exact sequences of equivariant standard complexes, so the identity in the statement of this corollary follows from Lemma 3.12.
Remark 3.14. A generalized version of Corollary 3.13, in which the exact sequence of (equivariant) standard complexes is replaced by an (equivariant) double standard complex was obtained in Xiaonan Ma’s Ph.D thesis (cf. [24]) where more discussions concerning spectral sequences were involved. Anyway, for arithmetical reason, we only need these special versions as in Lemma 3.12 and Corollary 3.13.
Now we turn back to our proof of the vanishing theorem. As before, let
W = W(i) be the deformation to the normal cone associated to an equiv-ariant closed immersion of projective manifolds i : Y → X. For simplicity, denote by P0
g the projective space bundle P(NXg/Yg ⊕ OYg). Moreover, given
an invariant K¨ahler metric onX, we fix an invariant K¨ahler metric onW which is constructed in Lemma 3.5. In this situation, all normal bundles appearing in the construction of the deformation to the normal cone will be endowed with the quotient metrics. We recall the following lemma.
Lemma 3.15. Over W(ig), there are hermitian metrics onO(Xg),O(Pg0)and
O(Xfg) such that the isometry O(Xg) =∼O(Pg0)⊗ O(Xfg) holds and such that
the restriction of O(Xg) toXg yields the metric of NW(ig)/Xg, the restriction
ofO(Xfg)toXfg yields the metric of NW(ig)/Xfg and the restriction ofO(Pg0)to
P0
g induces the metric ofNW(ig)/Pg0.
Proof. This is [21, Lemma 6.15].
Definition 3.16. Letη be an equivariant hermitian vector bundle on Y, we say that a resolution
Ξ : 0→ξem→ · · · →ξe0→j∗p∗Y(η)→0
satisfies the condition (T) if
(i). the metrics onξ.e satisfy Bismut assumption (A);
(ii). the restriction of Ξ to Xe is an equivariantly and orthogonally split exact sequence;
(iii). the restrictions of Ξ∇ toW(ig),Xg,Pg0,Xfg and Pg0∩Xfg are complexes
with l-acyclic elements and l-acyclic homologies, here Ξ∇ is the complex of
(iv). the tensor products Ξ∇ |W(ig) ⊗O(−Xg), Ξ∇ |W(ig) ⊗O(−P
0
g) and
Ξ∇ |W(ig) ⊗O(−Xfg) are complexes with l-acyclic elements and l-acyclic
ho-mologies.
From Theorem 3.6 (i), we already know that there always exists a resolution of j∗p∗Y(η) which satisfies the conditions (i) and (ii) in Definition 3.16. Let Ξ
be such a resolution, we have the following.
Proposition3.17. For n≫0,Ξ(n) satisfies the condition (T).
Proof. The reason is thatW(ig),Xg, Pg0, Xfg and Pg0∩Xfg are all closed
sub-manifolds ofW.
It is well known that both two squares in the following deformation diagram
Y × {0} s0 //
i
Y ×P1
j
Y × {∞}
s∞
o
o
i∞
X× {0} //W oo P(NX/Y ⊕NP1/∞)
are Tor-independent. Moreover, according to our choices of the K¨ahler metrics, we may identify Y × {0} with Y, X × {0} with X, Y × {∞} with Y and P(NX/Y ⊕NP1/∞) withP =P(NX/Y ⊕ OY). So if Ξ is a resolution ofj∗p∗Y(η) on W, then the restriction of Ξ to X (resp. P) provides a resolution of i∗η
(resp. i∞∗η). The following theorem is the kernel of the whole proof of the
vanishing theorem.
Theorem 3.18. (Deformation theorem) Let Ξ be a resolution of j∗p∗Y(η) on
W which satisfies the condition (T), then we have δ(Ξ|X) =δ(Ξ|P).
Proof. Consider the following tensor product of Ξ∇ |W(ig) with the Koszul
resolution associated to the immersionXg֒→W(ig)
0→Ξ∇|W(ig)⊗O(−Xg)→Ξ∇|W(ig)⊗OW(ig)→Ξ∇|W(ig)⊗iXg∗OXg →0.
We have to caution the reader that here the tensor product is not the usual tensor product of two complexes, precisely our resulting sequence is a double complex and we don’t take its total complex. Since we have assumed that Ξ satisfies the condition (T), this tensor product induces a short exact sequence of equivariant standard complexes onS by taking direct images. Forj≥0, its
j-th row is the following short exact sequence
εj : 0→R0lg0∗(O(−Xg)⊗ξej |W(ig))→R
0l0
g∗(ξej|W(ig))→R
0h
g∗(ξej|Xg)→0
wherel0
gis the composition of the inclusionW(ig)֒→W with the morphisml.
Note that thej-th homology of Ξ∇|W(ig)is equal tojg∗(∧
jFe∨
⊗p∗
Ygη|Yg)|W(ig)
immersion j. Actually jg factors through jg0 : Yg×P1 ֒→ W(ig), then the
j-th homology of Ξ∇ |W(ig) can be rewritten asj
0
g∗(∧jFe
∨
⊗p∗
Ygη |Yg). Write
Yg,0:=Yg× {0} for simplicity. Using the fact thatjg0
∗
O(−Xg) is isomorphic
to O(−Yg,0), we deduce from the short exact sequence
0→jg0∗(O(−Yg,0)⊗ ∧jFe ∨
⊗p∗Ygη |Yg)→j
0
g∗(OYg×P1⊗ ∧
jFe∨
⊗p∗Ygη|Yg)
→j0g∗(iYg∗OYg ⊗ ∧
jFe∨
⊗p∗Ygη|Yg)→0
that the j-th homologies of the induced short exact sequence of equivariant standard complexes form a short exact sequence
χj : 0→R0ug∗(O(−Yg,0)⊗ ∧jFe ∨
⊗p∗Ygη |Yg)→R
0u
g∗(∧jFe
∨
⊗p∗Ygη|Yg)
→R0fg∗(∧jF
∨
⊗η|Yg)→0
where ug is the composition of the inclusionYg×P1֒→W(ig) with the
mor-phisml0
g.
The main idea of this proof is that the equivariant Bott-Chern secondary char-acteristic class of the quotient term of the induced short exact sequence of equivariant standard complexes is nothing but cheg(Ξ∇ |X, hH) which appears
in the expression ofδ(Ξ|X) and the equivariant secondary characteristic classes
ofχj, εj can be computed by Bismut-Ma’s immersion formula.
Precisely, denote bygXg the Euler-Green current associated toXg which was
constructed by Bismut, Gillet and Soul´e in [6, Section 3. (f)], it satisfies the dif-ferential equation ddcgXg =δXg−c1(O(Xg)). We write Td(Xg) for Td(O(Xg)),
[6, Theorem 3.17] implies that Td−1(Xg)gXg is equal to the singular Bott-Chern
current of the Koszul resolution associated toXg֒→W(ig) modulo Im∂+ Im∂.
Moreover, writeξ. for the restriction Ξ∇|X. Then for anyj≥0, we compute
e
chg(εj) =Tg(ωXg, hξj|Xg)−Tg(ωW(ig), heξj|W(ig))
+Tg(ωW(ig), hO(−Xg)⊗ξej|W(ig))
+
Z
W(ig)/S
chg(ξej)Td(T lg0)Td
−1
(Xg)gXg
+
Z
Xg/S
chg(ξj)Td
−1
(NW(ig)/Xg)Td(f T hg, T l
0
g|Xg)
+
Z
Xg/S
chg(ξj)R(NW(ig)/Xg)Td(T hg).
Here, one should note that to simplify the last two terms in the right-hand side of Bismut-Ma’s immersion formula, we have used an Atiyah-Segal-Singer type formula for immersion
This formula is the content of [21, Theorem 6.16]. Similarly, for anyj≥0, we
duced short exact sequence of equivariant standard complexes. According to Lemma 3.12, we have
Similarly, we consider the tensor products of Ξ∇|W(ig)with the following three
Koszul resolutions
0→ O(−Xfg)→ OW(ig)→iXfg∗OXfg →0,
and
0→ O(−Xfg)⊗ O(−Pg0)→ O(−Xfg)⊕O(−Pg0)→ OW(ig)
→iXf
g∩Pg0∗OXfg∩Pg0→0.
We shall still denote byχ.(resp. ε.) the exact sequences consisting of homolo-gies (resp. elements) in the induced exact sequences of equivariant standard complexes.
For the first one, denote bygP0
g the Euler-Green current associated toP
0
g and
writeξ.∞for the restriction Ξ∇|P. Moreover, denote by Ω(−Pg0) the sub term
of the induced short exact sequence of equivariant standard complexes and denote bybgthe composition of the inclusionPg0֒→W(ig) with the morphism
l0
g. According to Lemma 3.12, we have
e
where F∞ is the non-zero degree part of the hermitian normal bundle N∞
For the second one, denote by gXf
g the Euler-Green current associated to Xfg
and denote by Ω(−Xfg) the sub term of the induced short exact sequence of
equivariant standard complexes. Since the restriction of Ξ to the componentXe
is equivariantly and orthogonally split, we know thatcheg(Ξ|Xfg, hH) is equal to
teristic class of the following short exact sequence
0→0→jg0
∗
O(−Xfg)→OYg×P1 →0.
We now consider the last one. This is also a Koszul resolution because Xfg
andP0
g intersect transversally. By [6, Theorem 3.20], the Euler-Green current
−X(−1)jTg(ωYg×P
1
, hO(−Yg,0)⊗∧jFe
∨
⊗p∗
Ygη|Yg)
−
Z
Yg×P1/S
X
(−1)jchg(∧jFe
∨
⊗p∗Ygη|Yg)Td(T ug)ch(Θ)e
+X(−1)jTg(ωW(ig), heξj|W(ig))
−X(−1)jT
g(ωW(ig), hO(−Xfg)⊗ξej|W(ig))
−X(−1)jTg(ωW(ig), hO(−P
0
g)⊗ξej|W(ig)) +X(−1)jTg(ωW(ig), hO(−Xg)⊗ξej|W(ig))
−
Z
W(ig)/S
{X(−1)jchg(ξej)Td(T l0g)Td−1(Xfg)Td−1(Pg0)
·[c1(O(Pg0))gXfg+δXfggP
0
g]}.
(4)
Here the elementch(Θ) is the equivariant secondary characteristic class of thee following short exact sequence
Θ : 0→ O(−Yg,0)→jg0
∗
O(−Xfg)⊕ O(−Yg,∞)→OYg×P1→0.
Since s0 : Y × {0} → Y ×P1 and s∞ : Y × {∞} →Y ×P1 are sections of
smooth morphism, the normal sequences
0→T fg→T ug|Yg,0→NYg×P1/Yg,0→0 and
0→T fg→T ug|Yg,∞→NYg×P1/Yg,∞→0
are orthogonally split so that Td(f T fg, T ug |Yg,0) and Td(f T fg, T ug |Yg,∞) are
both equal to 0. Moreover, the normal bundles NYg×P1/Yg,0 and NYg×P1/Yg,∞
are isomorphic to trivial bundles so that R(NYg×P1/Yg,0) and R(NYg×P1/Yg,∞)
are both equal to 0. Furthermore, we may drop all the terms where an integral is taken overXfg because P(−1)jchg(ξej) vanishes onXfg.
Now, we compute (1)−(2)−(3)+(4) which is
e
chg(Ξ∇|X, hH)−cheg(Ξ∇|P0
g, h
H) +X(
−1)jTg(ωXg, hξj|Xg)
−X(−1)jT g(ωP
0
g, hξ
∞
j |P0
g)−X(−1)jT
g(ωYg, h∧
jF∨
⊗η|Yg)
+X(−1)jT
g(ωYg, h∧
jF∨ ∞⊗η|Yg)
=
Z
Yg×P1/S
{X(−1)jchg(∧jFe
∨
⊗p∗Ygη|Yg)Td(T ug)·[Td
−1(Y
g,0)gYg,0 −Td−1(Yg,∞)gYg,∞+ch(e j
0
g
∗
−
may use the Atiyah-Segal-Singer type formula for immersions and the projec-tion formula in cohomology to compute
iX g∗
vanishes. An entirely analogous reasoning implies that
iP g∗
Thus, we are left with the equality
−Td−1(Xfg)gXfg+ Td−1(Xfg)Td−1(Pg0)c1(O(Pg0))gXfg]}
Using the differential equation whichTg(ξ.e) satisfies, we compute
−
Here we have used the equation
Td−1(Xg)c1(O(Xg))−Td−1(Pg0)c1(O(Pg0))−Td
−1
(Xfg)c1(O(Xfg))
+Td−1(Xfg)Td−1(Pg0)c1(O(Pg0))c1(O(Xfg)) = 0 (6)
which is [21, (23)].
Again using the fact that ξ.e is equivariantly and orthogonally split on Xe, the first integral in the right-hand side of (5) is equal to
So we get
g)δYg are both equal to 0. Combining these computations above
We now compute the second integral in the right-hand side of (5). According
Furthermore, by replacing all tangent bundles by relative tangent bundles, on can carry through the proof given in [21, P. 378-379] to show that
f
Td(T ug, T l0g|Yg×P1)|Yg,0=Td(f T fg, T hg|Yg)
and
f
Td(T ug, T lg0|Yg×P1)|Yg,∞=Td(f T fg, T bg|Yg).
So combining with the equation (6), we get
+ At last, using the fact that the intersections in the deformation diagram are transversal and the fact that j0
g(Yg×P1) has no intersection with Xfg, we can Gathering (5), (7), (8) and (9) we finally get
e
On the other hand, by definition, we have
−
Z
Yg/S
chg(η)Td−1g (N∞)Td(f T fg, T b′g|Yg)
where b′ : P
→ S is the composition of the inclusion P ֒→ W(i) and the morphisml. Note thatP0
g is an open and closed submanifold ofPg andξ.
∞
is orthogonally split on the other components since they all belong to Xeg, then
we can rewriteδ(Ξ|P) as
δ(Ξ|P) =cheg(Ξ∇|P0
g, h
H)
−X
k
(−1)kTg(ωYg, h∧
kF∨ ∞⊗η|Yg)
+X
k
(−1)kTg(ωP
0
g, hξ
∞
k|P0
g)
−
Z
Yg/S
Td(T fg)Td−1g (F)chg(η)R(Ng)
−
Z
P0
g/S
Tg(ξ.
∞
)Td(T bg)
−
Z
Yg/S
chg(η)Td−1g (N∞)Td(f T fg, T bg|Yg).
Comparing with the definition ofδ(Ξ|X), the equality (10) implies that
δ(Ξ|X)−δ(Ξ|P) = 0
which completes the proof of this deformation theorem.
Now we consider the zero section imbeddingi∞:Y →P =P(N∞⊕ OY). Here
we use the fact thatN∞is isomorphic toNX/Y, we caution the reader that this
is not necessarily an isometry sinceN∞carries the quotient metric induced by
the K¨ahler metric on P butNX/Y carries the quotient metric induced by the
K¨ahler metric onX. We recall that onP we have a tautological exact sequence 0→ O(−1)→π∗
P(N∞⊕ OY)→Q→0.
The equivariant section σ: OP →π∗P(N∞⊕ OY)→Q induces the following
Koszul resolution
0→ ∧rkQQ∨
→ · · · →Q∨→ OP →i∞∗OY →0.
Sinceσ is equivariant, the image ofOPg underσ|Pg is contained inQg. This
means thatσ|Pg induces a Koszul resolution onPg of the following form
0→ ∧rkQgQ∨
g → · · · →Q∨g → OPg →i∞,g∗OYg →0.
