Computing Hodge integrals with one lambda-class

0.31 MB·English

by  Yon-Seo Kim

COMPUTING HODGE INTEGRALS WITH ONE λ-CLASS
YON-SEO KIM
Department of Mathematics, UCLA
5
0
0 Abstract. AllHodgeintegralswithat-mostoneλ-classcanbe expressedaspoly-
2 nomials in terms of lower dimensional Hodge integrals with at-most one λ-class.

n Algorithm to compute any given Hodge integral with at-most one λ-class is dis-
a
cussed and some examples are presented.
J

9

1
v
1. Introduction
8
1
Let denote the Deligne-Mumford moduli stack of stable curves of genus g
0 g,n
M
1 with n marked points. A Hodge integral is an integral of the form
0
5
ψj1 ψjnλk1 λkg
0 1 ··· n 1 ··· g
/ ZMg,n
h
p where ψ is the first Chern class of the cotangent line bundle at the i-th marked
i
-
point, and λ , ,λ are the Chern classes of the Hodge bundle. Hodge integrals
h 1 g
···
t arise naturally in the calculations of Gromov-Witten invariants by localization tech-
a
m niques. Their explicit evaluations are difficult problems. The famous Witten’s con-
: jecture/Kontsevich’s theorem [21],[10] gives a recursive relation of Hodge integrals
v
i involving ψ classes only;
X
ar (1) ψj1 ψjn
1 ··· n
ZMg,n
and some of them can be computed recursively through String Equation and KdV
hierarchy. In [4], C.Faber developed an algorithm to compute intersection numbers
of type (1). Also, E.Getzler obtained recursion relations [3] for the case of g = 2, one
of which is given as;
τ (x) = τ (x)γ γa + τ (x)γ τ (γa)
k+2 2 k+1 a 0 2 k a 0 1 2
hh ii hh ii hh ii hh ii hh ii
7
τ (x)γ γaγ γb + τ (x)γ γ γa γb
k a 0 b 0 2 k a b 0 1 1
−hh ii hh ii hh ii 10hh ii hh ii hh ii
1 1
+ τ (x)γ γ γaγb τ (x)γ γaγ γb
k a b 0 1 k a 1 b 0
10hh ii hh ii − 240hh ii hh ii
13 1
+ τ (x)γ γaγ γb + τ (x)γ γaγ γb for k 0
k a b 0 1 k a b 0
240hh ii hh ii 960hh ii ≥
When the λ-classes are involved, the computation of Hodge integrals is not easy.
There was λ -conjecture which computes for the case of one top-degree λ-class as [5];
g
1
2 YON-SEOKIM
2g +n 3 22g−1 1 B
(2) ψk1 ψknλ = − − | 2g|
1 ··· n g k , ,k 22g−1 (2g)!
ZMg,n (cid:18) 1 ··· n(cid:19)
where B areBernoulli numbers and k + +k = 2g 3+n. By using Marin˜o-Vafa
2g 1 n
··· −
formula [18][14], the case of λ with one marked point can be computed as;
g−1
2g−1
1 1 (2g 1)!(2g 1)!
(3) ψ2g−1λ = b 1 − 2 − b b
1 g−1 g i − 2 (2g 1)! g1 g2
ZMg,1 Xi=1 g1+Xg2=g −
andthe case ofmore thanonemarked points canbe computed by repeatedly applying
the Cut-and-Join equation:
∂Ω √ 1λ ∂2Ω ∂Ω ∂Ω ∂Ω
(4) = − ijp +ijp +(i+j)p p
i+j i+j i j
∂τ 2 ∂p ∂p ∂p ∂p ∂p
i j i j i+j
iX,j≥1(cid:16) (cid:17)
which was used in the proof of Marin˜o-Vafa formula, and proven to be an effective
tool in studying Hodge integrals.
The moduli space of relative stable morphisms admits a natural S1-action induced
from the S1-action on the target space. And as a result of the localization formula
applied to it, the following convolution formula is obtained;
Theorem 6.1. For any partition µ and e with e < µ +l(µ) χ, we have
| | | | −
λl(µ)−χ Φ• ( λ)z • (λ) = 0
µ,ν − νDν,e
|ν|=|µ|
(cid:2) (cid:3) X
where the sum is taken over all partitions ν of the same size as µ.
Here χ is the prescribed Euler number of domain curves, λa means taking the
coefficient of λa, Φ•(λ) is a generating series of Double Hurwitz Numbers, and •(λ)
(cid:2) (cid:3) D
is a certain generating series of Hodge integrals. This formula gives many relations
between Hodge integrals with at-most one λ-class, and it is enough to consider the
special case of µ = (d) for some positive integers d to compute all Hodge integrals
with at-mostone λ-class. More precisely, thefollowing theorem isproved inSection 7;
Theorem 7.2. Any given Hodge integral with one λ-class:
ψk1 ψknλ
1 ··· n j
ZMg,n
where k , ,k N 0 , j 0,1,2, ,g , is explicitly expressed as a polynomial
1 n
··· ∈ ∪{ } ∈ { ··· }
in terms of lower-dimensional Hodge integrals with one λ-class. Therefore it computes
all Hodge integrals with one λ-class.
The rest of the paper is organized as follows: In Section 2, we summarize various
versions of localization formulas which will be used in computing Hodge integrals. In
Section 3, the Relative Moduli Space is defined and the natural S1-action on it is
COMPUTING HODGE INTEGRALS WITH ONE λ-CLASS 3
introduced. Also the fixed locus of the S1-action and their corresponding description
interms ofgraphsisdiscussed. InSection 4, we compute theEuler classof thenormal
bundle of the fixed locus of the S1-action in the relative moduli space. In Section 5,
the Double Hurwitz Numbers and its description in terms of Hodge integrals over a
certain moduli space is discussed. In Section 6, the Recursion Formula which gives
relations between Hodge integrals with at-most one λ-class is proved. In Section 7,
the Recursion Formula is used prove that all Hodge integrals with at-most one λ-class
is explicitly expressed as a polynomial in terms of lower dimensional Hodge integrals
with at-most one λ-class. In Section 8, an algorithm to implement the Recursion
Formula and to compute each Hodge integral with at-most one λ-class is discussed.
In Section 9, some examples of the algorithm in section 8 are presented.
Acknowledgements. I would like to thank my advisor, Professor Kefeng Liu
for his constant support and encouragements as well as many inspiring discussions.
I would also like to thank Xiaowei Wang for helpful discussions and friendship.
2. Localization Formula
In this section, I will summarize various versions of localization formulas.
2.1. Equivariant Cohomology. Let G be a compact Lie group acting on M. The
equivariant cohomology of M is defined as the ordinary cohomology of the space M
G
obtained from a fixed universal G-bundle EG, by the mixing construction
M = EG M
G G
×
Here, G acts on the right of EG and on the left of M, and the notation means that
we identify (pg,q) ∼ (p,gq) for p EG, q M, g G. Hence M is the bundle with
G
∈ ∈ ∈
fibre M over theclassifying space BG associated tothe universal bundle EG BG.
−→
We have natural projection map π : M BG and σ : M M/G, which fits
G G
−→ −→
into the mixing diagram of Cartan and Borel:
EG oo EG M // M
×
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
BG oo π E M σ // M/G
G
×
If G acts smoothly on M, then we have M = M/G. This is not true in general but
G ∼
it turns out that M is a better functorial construction and the proper homotopy
G
theoretic quotient of M by G. In any case, the equivariant cohomology, denoted by
H∗(M), is defined by
G
H∗(M) = H∗(M )
G G
and constitutes a contravariant functor from G-spaces to modules over the base ring
H∗ := H∗(pt) = H∗(BG). The map σ defines a natural map σ∗ : H∗(M/G)
G G −→
H∗(M) which is an isomorphism if G acts freely. The inclusion i : M M induces
G −→ G
a natural map i∗ : H∗(M) H∗(M).
G −→
4 YON-SEOKIM
2.2. Atiyah-Bott Localization Formula. Let i : V ֒ M be a map of compact
→
manifolds. The tubularneighborhoodofV inside M canbeidentified withthenormal
bundle of V. On the total space of the normal bundle, there is the Thom form
Φ which has compact support in the fibres and integrates to one in each fiber.
V
Extending this form by zero gives a form in M, and multiplying by Φ provides
V
a map H∗(V) = H∗+k(M,M V) H∗(M). In particular, the cohomology class
∼
\ −→
1 H0(V) is sent to the Thom class and this class restricts to be the Euler class of
∈
the normal bundle of V in M, . Hence, we see that
V/M
N
i∗i 1 = e( )
∗ V/M
N
This also holds in equivariant cohomology by same argument applied to V ,M . The
G G
theorem of Atiyah and Bott says that an inverse of the Euler class of the normal
bundle always exists along the fixed locus of a group action. Precisely, i∗/e( ) is
V/M
N
the inverse of i in equivariant cohomology, i.e. for any equivariant class φ,
∗
i i∗φ
∗
φ =
e )
( F/M
F N
X
holds where F runs over the fixed locus of the group action. In the integrated form,
we have
i∗φ
φ =
e( )
ZM F ZF NF/M
X
2.3. Functorial Localization Formula. Let X and Y be T-manifolds. Assume
f : X Y is a T-equivariant map, j : E ֒ Y is a fixed component in Y, and
E
−→ →
i : F ֒ f−1(E) is a fixed component in X. For any equivariant class ω H∗(X),
F → ∈ T
we have the diagrams;
F iF // X i∗F(ω) oo i∗F ω
eT(F/X)
g=f|F f (cid:15)(cid:15) g! f!
E(cid:15)(cid:15) jE // Y(cid:15)(cid:15) g! eTi∗F(F(ω/X) ) oo jE∗ f!((cid:15)(cid:15)ω)
h i
Applying the Atiyah-Bott Localizatio Formula with the naturality relation f(ω
!
·
f∗α) = fω α, we obtain the Functorial Localization Formula:
!
·
i∗(ω) j∗f(ω)
(5) g F = E !
!
e (F/X) e (E/Y)
T T
h i
COMPUTING HODGE INTEGRALS WITH ONE λ-CLASS 5
2.4. Virtual Functorial Localization Formula. The above Functorial Localiza-
tion Formula is also valid in the case where X and F are virtual fundamental classes.
In this paper, we will use • (P1,µ) vir for X, and F vir for F. Hence for any
Mχ,n Γ
equivariant class ω, we have:
(cid:2) (cid:3) (cid:2) (cid:3)
i∗(ω)
(6) ω = Γ
Z M•χ,n(P1,µ) vir XFΓ Z FΓ vir eT(FΓ/M•χ(P1,µ))
(cid:2) (cid:3) (cid:2) (cid:3)
3. Relative Moduli Space and S1-action
3.1. Moduli space of relative morphisms. For any non-negative integer m, let
P1[m] = P1 P1 P1
(0) ∪ (1) ∪···∪ (m)
be a chain of m + 1 copies of P1 such that P1 is glued to P1 at p(l) for 0 l
(l) (l+1) 1 ≤ ≤
m 1. The irreducible component P1 is referred to as the root component, and
− (0)
(l) (l+1)
other irreducible components are called bubble components. Two points p = p
in P1 are fixed. Denote by π[m] : P1[m] P1 the map which is id1en6 tity1 on
(l) −→
the root component and contracts all bubble components to P(0). Also denote by
1
P1(m) = P1 P1 the union of bubble components of P1[m].
(1) ∪···∪ (m)
For a fixed partition µ of a positive integer d, let • (P1,µ) be the moduli space of
Mχ,n
relative morphisms f : C;x , ,x ,z , ,z P1[m],p(m) such that
1 ··· l(µ) 1 ··· n −→ 1
(1) C;x1, ,xl(µ)(cid:0),z1, ,zn isapossibly-di(cid:1)sconne(cid:0)ctedpresta(cid:1)blecurveofEuler
··· ···
number χwithl(µ)+nmarked points. Here, the marked points areunordered.
(cid:0) (cid:1)
(2) f−1(p(m)) = l(µ)µ x as Cartier divisors and deg(π[m] f) = µ .
1 i=1 i i ◦ | |
(3) The preimage of each node in P1[m] consists of nodes of C. If f(y) = p(l) and
P 1
C and C are two irreducible components of C which intersects at y, then
1 2
(l)
f and f have the same contact order to p at y.
|C1 |C2 1
(4) Theautomorphismgroupoff isfinite. Here, anautomorphismoff consistsof
an automorphism of the domain curve and and automorphism of the pointed
curve P1(m),p(0),p(m) .
1 1
Following [11](cid:0)[12], • (P1,µ)(cid:1)is a separated, proper Deligne-Mumford stack with a
Mχ,n
perfect obstruction theory of virtual dimension r = χ+ µ + l(µ) +n, and hence
− | |
has a virtual fundamental class of degree r.
3.2. Torus Action. Consider the C∗-action t [z0 : z1] = [tz0 : z1] on P1. There are
·
two fixed points p = [0 : 1] and p = [1 : 0]. Extend this action to the action on
0 1
P1[m] by identifying the root component with P1 and giving trivial actions on bubble
components. Then this extended action on P1[m] induces an action on • (P1,µ).
Mχ,n
6 YON-SEOKIM
3.3. Fixed Locus. The connected components of the fixed points set of • (P1,µ)
Mχ,n
under the induced torus action can be parametrized by labeled graphs. For any
f : C;x , ,x ,z , ,z P1[m] representing a fixed point of the C∗-action
1 l(µ) 1 n
··· ··· −→
on • (P1,µ),therestrictionoffˆ:= π[m] f : C P1 toanirreduciblecomponent
of CM(cid:0) iχs,neither a constant map(cid:1)to one of th◦e C∗ -fi−x→ed points p ,p , or a covering of
0 1
P1 fully ramified over p and p . Associate a labeled graph Γ to the C∗-fixed point
0 1
f : C;x , ,x ,z , ,z P1[m] as follows:
1 l(µ) 1 n
··· ··· −→
(1) For each connected component C of fˆ−1( p ,p ), assign a vertex v, a label
(cid:2) (cid:0) (cid:1) v (cid:3) 0 1
{ }
ˆ
0, if f(C ) = p
v 0
g(v)whichisthearithmeticgenusofC ,,andalabeli(v) = .
v ˆ
1, if f(C ) = p
( v 1
Denote by V(Γ)(k) the set of vertices v with i(v) = k, for k = 0,1. The set
V(Γ) of vertices of the graph Γ will then be the disjoint union of V(Γ)(0) and
V(Γ)(1).
(2) Assign an edge e to each rational irreducible component C of C such that
e
ˆ ˆ
f is not a constant map. Then f is fully ramified over p and p with
|Ce |Ce 0 1
degree d(e). Let E(Γ) denote the set of edges of Γ.
(3) The set of flags is given by F(Γ) = (v,e) : v V(Γ),e E(Γ),C C =
v e
{ ∈ ∈ ∩ 6 ∅}
(4) For each v V(Γ), define d(v) = d(e) and let ν(v) be the partition
∈ (v,e)∈F(Γ)
of d(v) determined by d(e) : (v,e) F(Γ) . In case m > 0, we assign
{ P ∈ }
an additional label for each v V(Γ)(1): let µ(v) be the partition of d(v)
∈
determined by the ramification of f : C P1(m) over p(m).
|Cv v −→ 1
LetG (P1,µ)bethesetofallthegraphsassociatedtotheC∗-fixedpointsin • (P1,µ).
χ Mχ,n
We now describe the set of fixed points associated to a given graph Γ G (P1,µ).
χ
∈
Case m = 0: Any C∗-fixed point in • (P1,µ) which is represented by a
• Mχ,n
morphism to P1[0] = P1 is associated to a graph Γ G0(P1,µ) such that
0 ∈ χ
V(Γ )(0) = v0, ,v0 , g(v0) = g , k N, (2 2g ) = χ
0 { 1 ··· k} i i ∈ − i
i
X
V(Γ )(1) = v∞, ,v∞ , g(v∞) = = g(v∞ ) = 0,
0 { 1 ··· l(µ)} 1 ··· l(µ)
E(Γ ) = e , ,e , d(e ) = µ for i = 1, ,l(µ)
0 1 l(µ) i i
{ ··· } ···
j(v) = number of z ’s mapped to v, j(v) = n
i
V(XΓ0)(0)
The two end-points of the edge e are v0 and v∞ for some 1 j k. Let
i j i ≤ ≤
µ(v0) = µ e has v0 as an endpoint . Define
i { j| j i }
=
MΓ0 Mgi,l(µ(vi0))+j(vi0)
1≤i≤k
Y
where we take = = = pt , then there is a morphism
0,1 0,2 1,0
i : M• (P1,Mµ) whoseMimage i{s th}e fixed locus F associated
Γ0 MΓ0 −→ Mχ,n Γ0
to Γ . Hence i induces an isomorphism /A = F where A is the
0 Γ0 MΓ0 Γ0 ∼ Γ0 Γ0
COMPUTING HODGE INTEGRALS WITH ONE λ-CLASS 7
automorphism group of any morphism associated to the graph Γ , which can
0
be obtained from the short exact sequence
l(µ)
1 Z A Aut (Γ ) 1
−→ µi −→ Γ0 −→ 0 −→
i=1
Y
The virtual dimension of F with only stable vertices is
Γ0
3
d = χ+l(µ)+n
Γ0 −2
For any vertex v V(Γ )(0), introduce multiplicity of v, m(v), as follows:
0
∈
m(v) = w V(Γ )(0) g(v) = g(w), µ(v) = µ(w), j(v) = j(w)
0
|{ ∈ | }|
Pick one representatives from each group of identical vertices v , ,v so
1 l
{ ··· }
that m(v ) = V(Γ )(0) . Denote by m(v ) = m and now we can find the
k 0 k k
order of automor|phism gro|up of any Γ G0 (P1,µ) to be:
P 0 ∈ χ,n
(7) Aut Γ = m ! j(v )! Aut µ(v ) mk
0 k k k
| | | |
k
Y
Case m > 0: For any given f : (C;x , ,x ,z , ,z ) P1[m], con-
1 l(µ) 1 n
• ··· ··· −→
sider f and f which are defined as follows:
0 ∞
– f : Let C = f−1(p(0)), y , ,y = f−1(p(0)), and z , ,z
0 0 0 { 1 ··· l(ν)} 1 { 1 ··· n0}
mapped to C . Then f : (C ;y , ,y ,z , ,z ) P1 corre-
0 0 0 1 ··· l(ν) 1 ··· n0 −→
sponds to the case of m = 0.
– f : Let C = f−1(P1(m)) and f = f , then f corresponds to an
∞ ∞ ∞ |C∞ ∞
(1)
element of corresponding to the graph Γ described below.
MΓ
Classify the vertices of the graph Γ as follows:
VI(Γ)(0) = v V(Γ)(0) : r (v) = 1 ,
0
{ ∈ − }
VII(Γ)(i) = v V(Γ)(i) : r (v) = 0 , for i = 0,1
i
{ ∈ }
VS(Γ)(i) = v V(Γ)(i) : r (v) > 0 , for i = 0,1
i
{ ∈ }
where r (v) = 2g(v) 2+val(v)+j(v), for v V(Γ)(0),
0
− ∈
r (v) = 2g(v) 2+l(µ(v))+l(ν(v))+j(v), for v V(Γ)(1)
1
− ∈
(0) (1) (0)
Define = , where = and
MΓ MΓ × MΓ MΓ v∈VS(Γ)(0) Mg(v),val(v)+j(v)
(1) is the moduli space of morphisms fˆ: Cˆ P1(m) such that
MΓ −Q→
(1) Cˆ = C
v∈V(Γ)(1) v
(2) Foreachv V(Γ)(1), C ;x , ,x ,y , ,y ,z , ,z
F ∈ v v,1 ··· v,l(µ(v)) v,1 ··· v,l(ν(v)) 1 ··· j(v)
is a prestable curve of genus g(v) with l(µ(v)) + l(ν(v)) + j(v) ordered
(cid:0) (cid:1)
marked points.
8 YON-SEOKIM
(3) As Cartier divisors,
(fˆ )−1(p(0)) = l(ν(v))ν(v) y , (fˆ )−1(p(m)) = l(µ(v))µ(v) x
|Cv 1 j=1 j v,j |Cv 1 i=1 i v,i
The morphism (fˆ )−1(E) E is of degree d(v) for each irreducible
component E of PP1|C(mv ). −→ P
ˆ ˆ
(4) The automorphism group of f is finite. Here, an automorphism of f
consists ofanautomorphismofthedomaincurveCˆ andanautomorphism
of the pointed curve (P1(m),p(0),p(m)) , which is an element of (C∗)m.
1 1
There is a morphism i : • (P1,µ) whose image is the fixed locus
Γ MΓ −→ Mχ,n
F associated to the graph Γ. Hence i induces an isomorphism /A = F
Γ Γ Γ Γ ∼ Γ
M
where A is the automorphism group of any morphism associated to the graph
Γ
Γ, which can be obtained from the short exact sequence
1 Z A Aut(Γ) 1
d(e) Γ
−→ −→ −→ −→
e∈E(Γ)
Y
(1) (1)
The virtual dimension of is given by d = r (v) 1. Use
MΓ Γ v∈V(Γ)(1) 1 −
the identities (cid:16) (cid:17)
P
χ = χ χ +2l(ν), g = g(v) V(Γ) + E(Γ) +1
0 ∞
− − − − | | | |
v∈V(Γ)
X
to get the virtual dimension of F :
Γ
3
(0) (1)
d =d +d = χ +l(ν) χ +l(µ)+l(ν) 1+n
Γ Γ Γ −2 0 − ∞ − ∞
= 3g(v) 3+val(v) + r (v) 1+n
1 ∞
− −
v∈VXS(Γ)(0)(cid:0) (cid:1) (cid:0)v∈VX(Γ)(1) (cid:1)
=2g 3+l(µ)+ g(v) 1 + VI(Γ)(0) +n
∞
− − | |
v∈VXS(Γ)(0)(cid:0) (cid:1)
1
= χ χ+l(µ) 1+n
0 ∞
− 2 − −
By similar observation as in the case of m = 0, we find the order of automor-
phism group of any given Γ G∞(P1,µ) to be:
∈ χ
(8) Aut Γ = Aut Γ Aut µ n ! j(v)! Aut µ(v) Aut ν(v)
0 k
| | | | | | | | | |
(cid:16)Y (cid:17)(cid:16)VY(Γ)(1) (cid:17)
In this case, n is the multiplicity of vertices in V(Γ)(1) with identical µ(v),
k
ν(v), j(v), and g(v). The presence of automorphism group of µ is due to the
fact that we can exchange two marked points with same ramification type µ
i
without changing the type of corresponding graph.
COMPUTING HODGE INTEGRALS WITH ONE λ-CLASS 9
4. Computation of e ( vir)
T NΓ
In this section, I will summarize the computations of e ( vir) in [14] which will be
T NΓ
needed for localization computation. Denote by (ω) the 1-dimensional representation
of C∗ given by λ z = λωz for λ C∗, z C. For a given graph Γ G (P1,µ), let
g,n
· ∈ ∈ ∈
(9) f : (C,x , ,x ,z , ,z ) P1[m]
1 l(µ) 1 n
··· ··· −→
be a fixed point of th(cid:2)e C∗-action on • (P1,µ) associated to Γ.(cid:3)Given a flag (v,e)
Mχ,n ∈
F(Γ), denote by q C the node at which C and C intersect. Also let ψ
(v,e) v e (v,e)
∈
denote the first Chern class of the cotangent line bundles over , i.e. the fiber at
Γ
M
the fixed point (9) is given by T∗ C . The Euler class e ( vir) is given by;
q(v,e) v T NΓ
1 e (Tˆ2)
T
=
eT(NΓvir) eT(Tˆ1)
where Vˆ denotes the moving part of any vector bundle V, and T1, T2 are the tangent
space and the obstruction space of • (P1,µ), respectively. Here, T1 and T2 can be
Mχ,n
computed through the following two exact sequences [12]:
0 Ext0(Ω (D), ) H0(D•) T1 Ext1(Ω (D), ) H1(D•) T2 0
C C C C
→ O → → → O → → →
0 // H0(C,f∗(ω (logp(m)))∨) // H0(D•) // m−1H0(R•)
P1[m] 1 l=0 et l
L
// H1(C,f∗(ω (logp(m)))∨) // H1(D•) // m−1H1(R•) // 0
P1[m] 1 l=0 et
where ω is the dualizing sheaf of P1[m], D = x + +Lx is the branch divisor,
P1[m] 1 l(µ)
···
(l)
and for n = the number of nodes over p ;
l 1
H0(R•) = T f−1(P1 ) = C⊕nl
et l ∼ q (l) ∼
q∈fM−1(p1(l)) (cid:0) (cid:1)
He1t(R•l) ∼= Tp1(l)P1(l) ⊗Tp1(l)P1(l+1) ⊕(nl−1)
(cid:0) (cid:1)
Recall the map π[m] : P1[m] P1, and observe that for fˆ= π[m] f we have
−→ ◦
f∗ ωP1[m](logp1(m)) ∨ ∼= fˆ∗OP1(1)
Let F be the set of fixed po(cid:0)ints associated t(cid:1)o Γ G (P1,µ) and assume that
Γ χ,n
∈
f : (C,x , ,x ,z , ,z ) P1[m] F • (P1,µ)
1 ··· l(µ) 1 ··· n −→ ∈ Γ ⊂ Mχ,n
(cid:2) (cid:3)
10 YON-SEOKIM
The C∗-action on • (P1,µ) induces C∗-actions on
Mχ,n
m−1
Ext0(Ω (D), ) , H0(C,fˆ∗ (1)) , H0(R•)
C OC OP1 et l
l=0
M
m−1
Ext1(Ω (D), ) , H1(C,fˆ∗ (1)) , H1(R•)
C OC OP1 et l
l=0
M
The moving part of each of these groups form vector bundles over . We will use
Γ
M
the same notationˆto denote the induced vector bundles. In particular,
·
\
m−1
H0(R•) = 0, and
et l
l=0
M
\
m−1 0, if m = 0
H1(R•) =
et l H1(R•) = T P1 T P1 ⊕(n0−1), if m > 0
Ml=0 ( et 0 p1(0) (0) ⊗ p1(0) (1)
(cid:0) (cid:1)
Hence we have;
\ \ \
1 = eT(Tˆ2) = eT Ext0(ΩC(D),OC) eT H1(C,fˆ∗OP1(1)) eT ml=−01He1t(R•l)
eT(NΓvir) eT(Tˆ1) (cid:0) eT H0(C\,fˆ(cid:1)∗ P(cid:0)1(1)) eT Ext1(Ω\C(cid:1)(D(cid:0)),LC) (cid:1)
O O
(cid:0) (cid:1) (cid:0) (cid:1)
Case m = 0:
For each v V(Γ ) and µ , ,µ the ramification type in the vertex v, we
0 v,1 v,l(µ(v))
∈ ···
have under the convention to write µ = when g(v) = 0,l(µ(v)) = l(e(v)) = 1;
v,2
∞
\
m−1
H1(R•) = 0
et l v
l=0
M
\ 1 , if v I
Ext0(Ω (D), ) = µv,1 ∈
C OC v (0(cid:16), (cid:17) if v II or S
∈
0, if v I
∈
\
Ext1(Ω (D), ) =  1 + 1 , if v II
C OC v  µv,1 µv,2 ∈
(cid:16) l(µ(v))T (cid:17) C T C , if v S
i=1 q(v,ev,i) v ⊗ q(v,ev,i) ev,i ∈
L
