Table Of Content

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 Marinõ-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 Marinõ-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 automorphism 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 