Affine ADE bundles over surfaces with p

Affine ADE bundles over surfaces with p

= 0

Yunxia Chen¹ ·Naichung Conan Leung²

Received: 5 June 2014 / Accepted: 7 February 2016

© Springer-Verlag Berlin Heidelberg 2016

Abstract Given any Kodaira curveCin a complex surfaceX, we construct a simply-laced affine Lie algebra bundleE overX. Whenpg(X)=0, we construct deformations of holo- morphic structures onEsuch that the new bundle is trivial over anyADEcurveCinsideC and therefore descends to the singular surface obtained by contractingC.

1 Introduction

LetXbe a complex surface and⊂Pic(X)be a sublattice. Ifis isomorphic to the root latticegof a simple Lie algebrag, then we have a root systemofgand we can associate a Lie algebra bundleE₀^goverX[6,10,11]:

E₀^g:=O^⊕r_X ⊕

α∈

OX(α).

This can be generalized to the affine Lie algebra^g[9].

There are many instances when this happens. Here we list the following three cases as examples:

(1) WhenXn is a del Pezzo surface, namely a blowup ofP² atn ≤8 points in general position (orP¹×P¹),

KXn

_⊥

⊂Pic(Xn)is isomorphic toEn. Thus we have anEn- bundle overXn. By restriction, we have anEn-bundle over any anti-canonical curve inX_n. Notice thatis always a genus one curve. For a fixed elliptic curve, the above

B

[email protected] Naichung Conan Leung [email protected]

1 School of Science, East China University of Science and Technology, Meilong Road 130, Shanghai, China

2 Department of Mathematics, The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

construction gives a bijection between del Pezzo surfaces containingandE_n-bundles over[4,5,7,10,12,14]. Such an identification was predicted by the F-theory/string duality in physics [7]. This was generalized to all simple Lie algebras in [10,11]. When n=9,X9is not Fano andE9= ˆE8is an affine Lie algebra. Corresponding results for theEˆ8-bundle overX9are obtained in [9].

(2) WhenXis the canonical resolution of a surfaceXwith a rational double point of type g, then the corresponding exceptional curveC =

C_i is anADEcurve of typeg. Therefore all theseC_i span a sublattice of Pic(X)which is isomorphic to g, thus giving ag-bundleE₀^goverX. When pg(X)= 0, there exists a deformationEϕ^g ofE₀^g such thatE_ϕ^gis trivial over eachCi, thus it can descend to the singular surfaceX[2].

(3) WhenXis a relatively minimal elliptic surface, Kodaira classified all possible singular fibers (see e.g., [1]) and we call such a curveC =

C_ia Kodaira curve. Its irreducible componentsC_ispan a sublattice of Pic(X)which is isomorphic to the root lattice of an affine root systemgand therefore we can construct an affine Lie algebra bundleE₀^g overX.

The motivations of this paper have three aspects. One is to generalize the results forADE bundles in [2] to affineADEbundles (see Theorem 1below). The second is the natural question: if the del Pezzo surfaceXn (resp. X9) has a rational double point, does theEn- bundle (resp.Eˆ8-bundle) still exist? For this question, Friedman and Morgan gave a positive answer for del Pezzo surfaces [6]. In this paper, the authors will give a positive answer for both cases using a very different method (see Remark22). The third is the following question:

for a complex surfaceXwithpg(X)=0 and containing a Kodaira curveC, there is a natural affineADEbundle of the corresponding type over it, can we deform this bundle such that it can descend to the singular surface obtained by contracting anyADEcurveCinsideC (Remark23)?

Theorem 1 (Lemma13, Proposition17and Theorem21)Let X be a complex surface with pg=0. If X has a Kodaira curve C=_r

i=0Ci of typeg, then (i) given any

(ϕCi)^r_i=0∈^0,1

X, r

i=0

O(Ci)

with∂ϕC_i =0for every i , it can be extended to ϕ=(ϕα)_α∈⁺_g ∈^0,1

X,

α∈⁺_g

O(α)

such that∂_ϕ:=∂+ad(ϕ)is a holomorphic structure onE₀^g. We denote the new bundle as_Eϕ^g;

(ii) the new holomorphic structure∂ϕis compatible with the Lie algebra structure on_E0^g; (iii) the new bundleE_ϕ^gis trivial on Ciif and only if

[ϕCi|Ci] =0∈H¹(Ci,OCi(Ci))∼=C; (iv) there exists[ϕCi] ∈H¹(X,O(C_i))such that[ϕCi|Ci] =0.

The organization of this paper is as follows. Section2gives the construction of the (affine) ADELie algebra bundles directly from (affine)ADEcurves. In Sect.3, we assumep_g(X)=0.

We construct deformations of the holomorphic structures on these bundles such that the new bundles are trivial over irreducible components of the curve.

Notationfor a holomorphic bundle(E0, ∂0)withE0 =

iO(D_i), ∂0means the∂-operator for the direct sum holomorphic structure. If we construct a new holomorphic structure∂ϕon E0, we denote the resulting bundle asEϕ.

2 AffineADEbundles from affineADEcurves

2.1ADEand affineADEcurves Definition 2 A curveC=

Ciin a surfaceXis called anADE(resp. affineADE) curve of typeg(resp.^g) if eachC_i is a smooth(−2)-curve inXand the dual graph ofCis a Dynkin diagram of the corresponding type.

It is known thatCis anADEcurve if and only ifCcan be contracted to a rational double point. In this case, the intersection matrix(C_i·C_j)is negative definite [1].

IfCis an affineADEcurve, then the intersection matrix(C_i·C_j)is non positive definite and there existsni (these are unique if we askni to be positive integers without common integers) such thatF:=

niCi satisfiesF·F=0. Dynkin diagrams of affineADEtypes are drawn as follows and the correspondingniCi are labelled in the pictures.ADEDynkin diagrams can be obtained by removing the node corresponding toC₀(Fig.1).

Remark 3 We will also call a nodal or cuspidal rational curve with trivial normal bundle an A0curve.

Remark 4 By Kodaira’s classification of singular fibers of relative minimal elliptic surfaces, every singular fiber is an affine ADEcurve unless it is rational with a cusp, tacnode or triplepoint (corresponding to typeIIorIII(A₁)orVI(A₂)in Kodaira’s notation), which can also be regarded as a degenerated affineADEcurve of type A₀, A₁or A₂ respectively. In this paper, we will not distinguish affineADEcurves from their degenerated forms since they have the same intersection matrices. We also call the affineADEcurves as Kodaira curves.

Definition 5 A bundleEis called anADE(resp. affineADE) bundle of typeg(resp.^g) ifE has a fiberwise Lie algebra structure of the corresponding type.

In the following two subsections, we will recall an explicit construction of the Lie algebra g-bundle, loop Lie algebraLg-bundle and the affine Lie algebra^g-bundle from (affine)ADE curves inX.

2.2ADEbundles SupposeC =_r

i=1Ciis anADEcurve of typeginX. We will construct the corresponding ADEbundleE₀^goverXas follows [2].

Note the rank r of gequals the number of C_i. We set := {α = _r

i=1a_iC_i

∈ H²(X,Z)|α²= −2}. Thenis a simply-laced root system ofgwith a base := {[Ci]|i= 1,2, . . . ,r}. We have a decomposition=⁺∪⁻into positive and negative roots. We define a bundle_E0^(g,)overXas follows:

E₀^(g,):=O^⊕r⊕

α∈

O(α).

HereO(α)=O_r

i=1a_iC_i

whereα=_r

i=1a_iC_i

. There is an inner product,on defined byα, β := −α·β, negative of the intersection form.

For every open chartUofX, we takex_α^Uto be a nonvanishing section of_OU(α)andh^U_i (1≤i ≤r) nonvanishing sections of_O^⊕r_U . Define a Lie algebra structure[,]on_E0^(g,) such that{xα, α∈;hi,1≤i ≤r}is the Chevalley basis [8], i.e.,

(a) [h^U_i ,h^U_j]=0,1≤i, j≤r.

(b) [h^U_i ,x^U_α]= α, C_ix_α^U,1≤i≤r, α∈. (c) [x^U_α,x^U_−α]=h^U_α is aZ-linear combination ofh^U_i .

(d) Ifα, βare independent roots, andβ−pα, . . . , β+qαis theα-string throughβ, then [x^U_α,x^U_β]=0 ifq=0, otherwise[x_α^U,x_β^U]= ±(p+1)x_α+β^U .

Eˆ6: t t t t t t

t

1C1 2C2 3C3 2C4 1C5

2C6

1C0

Eˆ7: t t t t t t t t

1C1 2C2 3C3 4C4 3C5 2C6

2C7

1C0

Eˆ8: t t t t t t t t t

1C0 2C1 3C2 4C3 5C4 6C5 4C6

3C8

2C7

Aˆn: t t t t t t

HH HH HH H r r r

1C₁ 1C₂ 1C_n−21C_n−11C_n 1C0

Dˆ_n: t tr r r t t t

1C₁ 2C₂ 2C_n−32C_n−21C_n−1 1C_n 1C₀

Fig. 1 Dynkin diagrams of affineADEtypes

Sincegis a simply-laced Lie algebra, all the roots forghave the same length, we have any α-string throughβis of length at most 2. So (d) can be written as[x_α^U,x_β^U]=n_α,βx_α+β^U , wheren_α,β= ±1 ifα+β∈, otherwisen_α,β=0. It is easy to check that these Lie algebra structures are compatible with different trivializations ofE₀^(g,)(see page 10 of [10] for more details). HenceE₀^(g,)is a Lie algebra bundle of typegoverX.

2.3 AffineADEbundles SupposeC =_r

i=0Ci is an affineADEcurve of typeginX. We will construct the corre- sponding affineADEbundleE₀^gof typegoverXas follows.

First, we choose an extended root of^g, sayC₀, thengis corresponding to the Dynkin diagram consists of thoseC_iwithi=0, i.e.,

:=

⎧⎨

⎩α=

⎡

⎣

i=0

aiCi

⎤

⎦∈H²(X,Z)|α²= −2

⎫⎬

⎭

is the root system ofg. As above, we have ag-bundle E₀^(g,)=O^⊕r⊕

α∈

O(α).

We define

E₀^(Lg,):=

n∈Z

E₀^(g,)⊗O(n F)

and

E₀^(g,):=

n∈Z

(E₀^(g,)⊗O(n F))⊕O.

We know

g:= {α+n F|α∈,n∈Z}

{n F|n∈Z,n=0}

is an affine root system and it decomposes into the union of positive and negative roots, i.e., g=⁺_g ∪⁻_g, where

⁺_g =

aiCi ∈g|ai ≥0 f or all i

= α+n F|α∈⁺,n∈Z_≥0

!∪ α+n F|α∈⁻,n∈Z_≥1

!∪ n F|n∈Z_≥1

! and⁻_g = −⁺_g.

To describe the Lie algebra structures, we proceed as before, for every open chartUofX, we take a local basise^U_i ofE₀^(g,)|U(e^U_i is justh^U_j orx_α^Uas above),e^U_{n F}ofO(n F)|U,e^U_c of O|U, compatible with the tensor product, for example,e^U_{n F}⊗e^U_{m F} =e^U_(n+m)F. Then define

"

e^U_i e^U_{n F},e^U_je_{m F}^U

#

Lg,:= [e^U_i ,e^U_j]e^U_(n+m)F, (1)

"

e^U_i e^U_{n F}+λe^U_c,e^U_je^U_{m F}+μe^U_c#

g,:="

e^U_i ,e^U_j

#

e^U_(n+m)F+nδn+m,0k

e^U_i ,e^U_j

e^U_c. (2)

Here[,]is the Lie bracket onE₀^(g,)andk(x,y)=Tr(ad(x)ad(y))is the Killing form ong.

Lemma 6 The above(1)[resp.(2)]defines a fiberwise loop(resp. affine)Lie algebra struc- ture which is compatible with any trivialization ofE₀^(Lg,)(resp.E₀^(g,)).

Proof See Proposition 23 of [9].

From the above lemma, we have the following result.

Proposition 7 If C is an affine ADE curve of typegin X , then_E0^(Lg,)(resp._E0^(g,))is a loop(resp. affine)Lie algebra bundle of type Lg(resp.g)over X .

Note anyCiwithni =1 can be chosen as the extended root.

Proposition 8 The loop Lie algebra bundle(E₀^(Lg,),[,]Lg,)does not depend on the choice of the extended root.

Proof SupposeCk(k=0)is another root withnk=1. We set

=

⎧⎨

⎩β=

⎡

⎣

i=k

biCi

⎤

⎦∈H²(X,Z)|β²= −2

⎫⎬

⎭.

Thenis a root system ofg. As before, we construct the Lie algebra bundleE₀^(g,)and E₀^(Lg,)from.

We denote α0 :=

i=0n_iC_i = F − C₀, the longest root in . For any α =

i=0ai(α)Ci ∈ ,ak(α)can only be 0,±1. Hence there is a bijection betweenand given byα → β = α−a_k(α)F. Then from the definition ofE₀^(Lg,)andE₀^(Lg,), we know they are the same as holomorphic vector bundles.

We compare the Lie brackets on them. We choose a local basis of_E0^(Lg,)compatible with those ofE₀^(Lg,)and define[,]Lg,similarly as[,]Lg,, i.e.,

(i) whenβ=α∈∩, we takex_β =x_α; (ii) whenβ=α+F∈⁺\, we takex_β=x_αe_F; (iii) whenβ=α−F∈⁻\, we takex_β=x_αe_−F;

(iv) take h_i(i = 0,k) as before, take h₀ = −h_α0 as we want [x_C0,x_−C0]Lg, = [x−α0+F,x_α0−F]Lg,.

It is obvious[,]Lg,= [,]Lg,onE₀^(Lg,)∼=E₀^(Lg,). For the affine case, we recall that the Killing form ofgis the symmetric bilinear map k:g×g→Cdefined byk(x,y) = Tr(ad(x)ad(y)). It is ad-invariant, that is forx,y,z ∈ g,k([x,y],z)=k(x,[y,z]).

Lemma 9 For any simple simply-laced Lie algebra g with a Chevalley basis {x_α, α ∈

;hi,1≤i≤r}and m^∗(g)the dual Coxeter number ofg, we have (i) k(h_i,x_α)=0for any i andα;

(ii) k(xα,x_β)=0for anyα+β=0;

(iii) k(hi,hj)=2m^∗(g)Ci,Cj; (iv) k(x_α,x_−α)=2m^∗(g)for anyα.

Proof Directly from the Killing formkbeing ad-invariant or see [13].

Proposition 10 The affine Lie algebra bundle(E₀^(g,),[,]g,)does not depend on the choice of the extended root.

Proof Follow the notation in Proposition8, but we will take h₀= −h_α0+2m^∗(^g)e_c. We will check that[,]g,= [,]g,onE₀^(g,)=E₀^(g,):

(a) whenβ1=α1+F, β2=α2+F∈⁺\, α1, α2∈⁻\we have [h_β1e_{n F},h_β2e_{m F}]g,=nδn+m,0k(h_β1,h_β2)e_c, which is the same with

[h_−α1e_{n F},h_−α2e_{m F}]g,=nδ_n+m,0k(h_α1,h_α2)ec,

sincek(h_β1,h_β2)=2m^∗(g)β1, β2 =2m^∗(g)F−α1,F−α2 =k(h_α1,h_α2).

(b) For[hie_{n F},x_αe_{m F}]g,, automatically fromk(hi,x_α)=0 and loop case.

(c) Whenβ=α+F∈⁺\, α∈⁻\,

[x_βe_{n F},x_−βe_{m F}]g, =h_βe_(n+m)F+nδ_n+m,0k(x_β,x_−β)ec, which is the same with

[x−αe₍n+1)F,x_αe₍m−1)F]g,= −hαe₍n+m)F+(n+1)δn+m,0k(xα,x_−α)ec, by consideringm+n=0 andm+n=0 separately.

(d) For[xα1en F,x_α2em F]g,withα1+α2 =0, automatically fromk(xα1,x_α2) =0 and

loop case.

For simplicity, we will omitin(g, ), (Lg, )and(g, )when there is no confusion.

3 Trivialization ofE₀^goverC_i after deformations IfC =

Ci is an affineADEcurve in X, then the corresponding F =

niCi satisfies F·F=0, i.e.,OF(F)is a topologically trivial bundle. IfOF(F)is trivial holomorphically andq(X)=0, then from the long exact sequence of cohomologies induced by 0→OX→ OX(F)→OF(F)→0, we know H⁰(X,OX(F))∼=C². HenceFis a fiber of an elliptic fibration onX.

SupposeXis an elliptic surface, i.e., there is a smooth curveBand a surjective morphism π:X → Bwhose generic fiber F_b (b ∈ B) is an elliptic curve. Assumeπ is singular at b₀∈ BandF_b0 =

n_iC_i is a singular fiber of typeg. Hence, we have ag-bundleE₀^gover X. The restriction ofE₀^gto any fiberF_b, other thanF_b0, is trivial becauseF_b∩C_i =∅for anyi. However,E₀^g|F_b0 is not trivial, for instanceO(−Ci)|C_i ∼= O_P¹(2). Nevertheless, we will show that after deformations of holomorphic structures,E₀^gwill become trivial on every irreducible component ofF_b0.

3.1 Review ofADEcases

In our earlier paper [2], we showed how to take successive extensions to make theg-bundle E₀^gtrivial on every componentCiof theADEcurveC =_r

i=1Ci. Definition 11 Given any ϕ = (ϕα)_α∈⁺ ∈ ^0,1(X,

α∈⁺O(α)), we define ∂ϕ:^0,0 (X,E₀^g)−→^0,1(X,E₀^g)by

∂_ϕ:=∂0+ad(ϕ):=∂0+

α∈⁺

ad(ϕ_α) ,

More explicitly, if we writeϕ_α = c^U_αx_α^U locally for some one formc^U_α, then ad(ϕ_α)= c^U_α ad(x_α^U). It is easy to check that∂ϕis well-defined and compatible with the Lie algebra structure, i.e.,∂ϕ[,]=0. For∂ϕto define a holomorphic structure, we need

0=∂²_ϕ=

α∈⁺

⎛

⎝∂0c^U_α +

β+γ=α

n_β,γc^U_β ∧c_γ^U ⎞

⎠ad(x_α^U).

That is∂0ϕ_α+

β+γ=α(n_β,γϕ_β∧ϕ_γ)=0 for anyα∈⁺. Explicitly:

⎧⎪

⎪⎨

⎪⎪

⎩

∂0ϕC_i =0 i∈ {1,2, . . . ,r}

∂0ϕC_i+C_j =nC_i,C_jϕC_i ∧ϕC_j ifCi+Cj ∈⁺ ...

Recall{C_i}^r_i=1⊂⁺is a base.

Proposition 12 Given any(ϕCi)^r_i=1 ∈ ^0,1(X,_r

i=1O(C_i))with∂ϕCi =0for any i , it can be extended toϕ=(ϕα)_α∈⁺ ∈^0,1(X,

α∈⁺O(α))satisfying∂²_ϕ =0, so that we have a holomorphicg-bundleE_ϕ^gover X .

The proof of this proposition uses the following lemma.

Lemma 13 If p_g(X)=0, then

(i) for anyα∈⁺,H²(X,O(α))=0.

(ii) the restriction homomorphism H¹(X,OX(Ci))→H¹(X,OC_i(Ci))is surjective.

Theorem 14 For any given i , the holomorphicg-bundleE_ϕ^g over X is trivial on C_i if and only if[ϕCi|Ci] =0.

The proof of this theorem can be found in Theorem 9 of [2]. Note that part (ii) of Lemma 13says that suchϕC_i can always be found.

3.2 Trivializations in loopADEcases

Definition 15 Given anyϕ =(ϕα)_α∈⁺_g ∈^0,1(X,

α∈⁺_gO(α)), we define∂(ϕ,):^0,0 (X,E₀^Lg)−→^0,1(X,E₀^Lg)by∂_(ϕ,):=∂0+ad(ϕ).

More explicitly, if we writeϕα = c^U_αx^U_α locally for some one formc^U_α, then by the decomposition of⁺_g in Sect.2.3, we have (here we omit the local chartU for simplicity):

∂(ϕ,):=∂0+

n∈Z≥0

α∈⁺

c_{α+n F}ad(xαen F)+c_−α+(n+1)Fad

x_−αe_(n+1)F

+

n∈Z≥0

r i=1

c_(n+1)Fⁱ ad

hie₍n+1)F

.

Proposition 16 ∂(ϕ,)is compatible with the Lie algebra structure onE₀^Lg.

Proof ∂_(ϕ,)[,]Lg,=0 follows directly from the Jacobi identity.

For∂_(ϕ,)to define a holomorphic structure, we need∂²_(ϕ,)=0, which is equivalent to the following equations:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

∂0ϕ_{n F}ⁱ =

p+q=n

α∈⁺±ai(h_α)ϕ_α+p F∧ϕ_−α+q F,

∂0ϕα+n F =

p+q=n

α1+α2=α±ϕα1+p F∧ϕα2+q F

+

p+q=n_r

i=1α,Ciϕα+p F∧ϕ_{q F}ⁱ ,

∂0ϕ−α+n F =

p+q=n

α2−α1=α±ϕα1+p F∧ϕ−α2+q F

+

p+q=n

_r

i=1−α,C_iϕ_{−α+p F}∧ϕ_{q F}ⁱ , wherea_i(h_α)is the coefficient ofh_i inh_α.

Proposition 17 Given any(ϕCi)^r_i=0 ∈^0,1(X,_r

i=0O(C_i))with∂ϕCi =0for every i , it can be extended toϕ=(ϕα)_α∈⁺_g ∈^0,1(X,

α∈⁺_g O(α))satisfying∂²_ϕ =0. Namely we have a holomorphic Lg-bundleE_ϕ^Lgover X .

In order to prove this proposition, we need the following lemma.

Lemma 18 If pg(X)=0, then for anyα∈⁺,n∈Z_≥0,H²(X,O(n F)), H²(X,O(α+ n F))and H²(X,O(−α+(n+1)F))are zero.

Proof Since F is an effective divisor and H⁰(X,KX) = 0, we have for any n ≥ 0,H⁰(X,K_X(−n F)) = 0. This is equivalent to H²(X,O(n F)) = 0 by Serre duality.

Similarly, H²(X,O(α+n F)) =0 follows from H⁰(X,K_X(−α)) ∼= H²(X,O(α))= 0 (Lemma13). The proof ofH²(X,O(−α+(n+1)F))=0 uses the fact thatF−αis an

effective divisor for anyα∈⁺.

Proof of Proposition17 The equation∂²_(ϕ,)=0 can be rewritten as follows:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

∂0ϕCi =0 fori∈ {1,2, . . . ,r},

∂0ϕ_α =

α1+α2=α

±ϕ_α1∧ϕ_α2 ,

∂0ϕ−α0+F=∂0ϕC0=0,

∂0ϕ−α+F =

α2−α1=α

±ϕα1∧ϕ−α2+F ,

∂0ϕⁱ_F=

α∈⁺(±ai(h_α)ϕ_α∧ϕ_−α+F) , ...

whereα0=F−C₀is the longest root in.

Firstly, we can solve for all theϕα(α ∈⁺)fromH²(X,O(α))=0 (Proposition12).

Secondly, we get all theϕ−α+F(α∈⁺)fromH²(X,O(−α+F))=0. Thirdly, since we have all theϕαandϕ−α+F, we can solve for all theϕⁱ_Ffor 1≤i ≤rfromH²(X,O(F))=0.

Do this process forϕα+n F, ϕ−α+(n+1)Fandϕ_(n+1)Fⁱ inductively onn.

By Lemma13, there always exists ϕC_i ∈ ^0,1(X,O(Ci))such that 0 = [ϕC_i|C_i] ∈ H¹(X,OCi(C_i))∼=Cfor eachi=0,1, . . . ,r.

Theorem 19 For any given i , the holomorphic Lg-bundle_Eϕ^Lgover X is trivial on C_iif and only if[ϕCi|Ci] =0.

Proof The proof will be given in Sects.3.4and3.5. In Sect.3.4, we deal with all the loop ADEcases except loopE₈case which will be analyzed in Sect.3.5.

3.3 Trivializations in affineADEcases

Follow the notation in Sect.3.2, we define∂(ϕ,) := ∂0+ad(ϕ)on_E0^g. Note the adjoint action here is defined using the affine Lie bracket.

Proposition 20 ∂_(ϕ,)is compatible with the Lie algebra structure onE₀^g.

Proof ∂(ϕ,)[,]g,=0 follows directly from the Jacobi identity and the Killing form being

invariant under the adjoint action.

It is easy to see that∂²_(ϕ,)=0 in the affine case is equivalent to∂²_(ϕ,)=0 in the loop case. Hence we have a new holomorphic structure∂_(ϕ,)onE₀^g.

Theorem 21 For any given i , the holomorphic^g-bundleE_ϕ^g over X is trivial on C_i if and only if[ϕCi|Ci] =0.

Proof This follows from Theorem19, 0 →O → Eϕ^g →Eϕ^Lg →0 andE xt_P¹₁(O,O)=

H¹(P¹,O)=0.

From the construction of∂_ϕin Sect.3.1and∂_(ϕ,)above, we have the following obser- vation: letXbe a complex surface withpg(X)=0. If⊂Pic(X)is isomorphic to the root latticeg(resp.g) ofADEtype (resp. affineADEtype) andC =

C_i is anADEcurve of typehwith each irreducible curveC_ifrom the corresponding root systemg(resp.g), then we can deform the Lie algebra bundleE₀^g(resp.E₀^g) such that its deformationE_ϕ^g(resp.

Eϕ^g) is trivial over everyCi. To show this, we will describe the corresponding holomorphic structure∂_ϕ(resp.∂_(ϕ,)) in detail. We choose theseCi as basis ofhand extend it to the basis ofg(resp.g), then construct∂_ϕ(resp.∂_(ϕ,)) as follows:

(1) forα∈⁺_g\⁺_h, takeϕ_α=0;

(2) forCi∈⁺_h, takeϕCi such that[ϕCi|Ci] =0;

(3) forα∈⁺_h, α=C_i, takeϕαsuch that∂(ϕ,h):=∂0+

α∈⁺_had(ϕα)satisfy∂²_(ϕ,h)=0.

It obviously that suchϕ_αexist and the corresponding∂_ϕ(resp.∂_(ϕ,)) satisfy the integra- bility condition. And from the above theorem, the new bundleE_ϕ^g(resp.E_ϕ^g) is trivial over everyC_i.

Remark 22 In particular, if the del Pezzo surfaceX_n(resp.X₉) has a rational double point, then we can construct anE_n-bundle (resp. E₈-bundle) on its minimal resolution such that its restriction to each irreducible component of the exceptional locus is trivial, then thisE_n- bundle (resp.E8-bundle) can descend to the singular surfaceXn(resp.X9). Therefore for a del Pezzo surfaceXn(resp.X9) with a rational double point, theEn-bundle (resp.E8-bundle) still exists. The relationship between the deformability of theE₈-bundle and the geometry ofX₉is shown in [3].

Remark 23 For a complex surfaceXwithp_g(X)=0 and containing anADEcurve (resp.

Kodaira curve)C, we have a corresponding typeADEbundle (resp. affineADEbundle). If we contract anyADEcurveCinsideC, then we will get a singular surface with a rational double point. By the above observation, we can deform this bundle such that it can descend to this singular surface.

3.4 Proof (except the loopE8case)

In this subsection, we use the symmetry of the affineADEDynkin diagram (exceptE₈) to show thatE_ϕ^Lgis trivial onC_iif and only if[ϕCi|Ci] =0.

Recall that_Eϕ^Lgand_E0^Lghave the same underlyingC^∞-vector bundle, but with a holo- morphic structure∂(ϕ,)of the following upper triangular block shape:

∂ϕ =

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

... ... ... ... ...

... ∂_E(g,)

ϕ ⊗O((n+1)F) ∗ ∗ ...

... O ∂_E(g,)

ϕ ⊗O(n F) ∗ ...

... O O ∂_E(g,)

ϕ ⊗O((n−1)F) ...

... ... ... ... ...

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠ .

i.e.,Eϕ^Lgis constructed from successive extensions of theseEϕ^(g,)⊗O(n F)(n∈Z).

Note∂_(ϕ,)|_E(g,)

ϕ = ∂0+

α∈⁺ad(ϕ_α). By Theorem14, for everyi =0,E_ϕ^(g,)is trivial onCiif and only if[ϕC_i|C_i] =0. We also knowO(F)|C_i is trivial for everyibecause F·Ci =0. Thus, wheni =0,Eϕ^Lg|Ci is constructed from successive extensions of trivial vector bundles overCi ∼=P¹. This implies thatE_ϕ^Lg|C_i is trivial if and only if[ϕC_i|C_i] =0 asE xt_P¹₁(O,O)= H¹(P¹,O)=0.

Now we consideri =0. Sinceg=E8, the affine Dynkin diagram always admits a diagram automorphism, that means we can writeE₀^Lgas

n∈Z(E₀^(g,)⊗O(n F))(see Proposition8).

Suppose the extended root corresponding toisC_k, and the longest root inisβ0. We will rewrite the holomorphic structure∂(ϕ,) in terms of the root system. Note

∂(ϕ,)is determined by the loop Lie algebra structure which is independent of the choice of the extended root. We choose a local base ofE₀^(g,)as in Proposition8and define∂_(ψ,)to be the same with∂_(ϕ,), then obviouslyψD=ϕDwhenD=n F.

Because(E_ϕ^(Lg,), ∂(ϕ,)) =(E_ψ^(Lg,), ∂(ψ,))as a holomorphic vector bundle, similar to the arguments in(E_ϕ^(Lg,), ∂_(ϕ,))case, we have wheni=k,E_ϕ^Lgis trivial onCi if and only if[ψCi|Ci] =0. NoteψC0=ϕ_−α0+F =ϕC0. So we have Theorem19wheng=E₈.