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Advance Access Publication April 12, 2013 doi:10.1093/imrn/rnt065

AD E Bundles over Surfaces with AD E Singularities

Yunxia Chen and Naichung Conan Leung

Department of Mathematics, The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

Correspondence to be sent to: yxchen76@gmail.com

Given a compact complex surface X with an AD E singularity andpg=0, we construct AD E bundles over X and its minimal resolution Y. Furthermore, we describe their minuscule representation bundles in terms of configurations of (reducible)(−1)-curves inY.

1 Introduction

It has long been known that there are deep connections between Lie theory and the geometry of surfaces. A famous example is an amazing connection between Lie groups of type Enand del Pezzo surfaces Xof degree 9−nfor 1≤n≤8. The root lattice of En

can be identified withKX, the orthogonal complement toKXin Pic(X). Furthermore, all the lines in X form a representation of En. Using the configuration of these lines, we can construct anEnLie algebra bundle over X[15]. If we restrict it to the anti-canonical curve in X, which is an elliptic curveΣ, then we obtain an isomorphism between the moduli space of degree 9−ndel Pezzo surfaces which containΣand the moduli space of En-bundles overΣ. This work is motivated from string/F-theory duality, and it has been studied extensively by Friedman–Morgan–Witten [8–10], Donagi [3–5, 7], Leung–Zhang [14–16], and others [6,13,17,18].

Received August 9, 2012; Accepted March 21, 2013

c The Author(s) 2013. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com.

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In this paper, we study the relationships between simply laced, or AD E, Lie theory and rational double points of surfaces. Suppose

π:YX

is the minimal resolution of a compact complex surfaceXwith a rational double point.

Then the dual graph of the exceptional divisorn

i=1Ci inYis an AD E Dynkin diagram.

From this, we have anAD Eroot systemΦ:= {α=

ai[Ci]|α2= −2}and we can construct an AD ELie algebra bundle overY:

E0g:=OYn

α∈Φ

OY(α).

Even though this bundle cannot descend to X, we show that it can be deformed to one that can descend toX,provided thatpg(X)=0.

Theorem 1(Propositions 6, 7, Theorem 9, and Lemma 10). Assume thatYis the minimal resolution of a surface X with a rational double point at pof type gandC=Σin=1Ci is the exceptional divisor. Ifpg(X)=0,then

(1) given any Ci)ni=1Ω0,1(Y,n

i=1O(Ci)) with ∂ϕ¯ Ci=0 for every i, it can be extended to ϕ=α)α∈Φ+Ω0,1(Y,

α∈Φ+O(α))such that¯ϕ:= ¯+ad(ϕ) is a holomorphic structure onE0g. We denote this new holomorphic bundle asEϕg; (2) such a¯ϕis compatible with the Lie algebra structure;

(3) Eϕg is trivial onCi if and only if [ϕCi|Ci]=0∈H1(Ci,OCi(Ci))∼=C; (4) there exists [ϕCi]∈H1(Y,O(Ci))such that [ϕCi|Ci]=0;

(5) such aEϕgcan descend toXif and only if [ϕCi|Ci]=0 for everyi.

Remark 2. Infinitesimal deformations of holomorphic bundle structures on E0g are parameterized by H1(Y,End(E0g)), and those that also preserve the Lie algebra struc- ture are parameterized by H1(Y,ad(E0g))=H1(Y,E0g), since gis semi-simple. If pg(X)= q(X)=0, for example, rational surface, then for any αΦ, H1(Y,O(α))=0. Hence, H1(Y,E0g)=H1(Y,

α∈Φ+O(α)).

This generalizes the work of Friedman–Morgan [8], in which they consideredEn bundles over generalized del Pezzo surfaces. In this paper, we will also describe the minuscule representation bundles of these Lie algebra bundles in terms of(−1)-curves inY.

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Here is an outline of our results. We first study(−1)-curves inYwhich are (pos- sibly reducible) rational curves with self-intersection−1. If there exists a(−1)-curveC0 inXpassing through pwith minuscule multiplicityCk(Definition 15), then(−1)-curves l’s in Y with π(l)=C0 form the minuscule representation V of g corresponding to Ck (Proposition 21). (HereVis the lowest weight representation with lowest weight dual to

Ck, that is,V is dual to the highest weight representation with highest weight dual to Ck.) WhenV is the standard representation ofg, the configuration of these(−1)-curves determines a symmetric tensor f onV such thatgis the space of infinitesimal symme- tries of(V, f). We consider the bundle

L(0g,V):=

l:(−1)−curve π(l)=C0

OY(l),

overYconstructed from these(−1)-curvesl’s. This bundle cannot descend to Xas it is not trivial over eachCi. (Unless specified otherwise,Ci always refers to an irreducible component ofC, that is,i=0.)

Theorem 3(Theorems 23 and 24). For the bundleL(0g,V) with the corresponding minus- cule representationρ:g−→End(V),

(1) there exists ϕ=α)α∈Φ+Ω0,1(Y,

α∈Φ+O(α)) such that¯ϕ:= ¯0+ρ(ϕ)is a holomorphic structure onL(0g,V). We denote this new holomorphic bundle as L(ϕg,V);

(2) L(ϕg,V)is trivial onCi if and only if [ϕCi|Ci]=0∈H1(Y,OCi(Ci));

(3) whenVis the standard representation ofg, there exists a holomorphic fiber- wise symmetric multi-linear form

f: r

L(ϕg,V)−→OY(D),

with r=0,2,3,4 when g=An,Dn,E6,E7, respectively, such that Eϕg∼=

aut0(L(ϕg,V), f).

When V is a minuscule representation ofg, there exists a unique holomorphic structure onL(0g,V):=

lO(l)such that the action of Eϕg on this bundle is holomorphic and it can descend toXas well.

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Example 4. When we blowup two distinct points, we have a surfaceYwith two(−1)- curvesl1andl2as exceptional curves.L0:=OY(l1)OY(l2)is aC2-bundle and the bundle ζ0A1of its symmetries is an sl(2)- orA1-bundle overY.

When the two points become infinitesimally close, then C1=l2l1 is effective, namely a (−2)-curve in Y. If we blow down C1 in Y, we get a surface X with an A1

singularity.L0cannot descend toXasL0|C1∼=OP1(−1)OP1(1). Using the Euler sequence 0→OP1(−1)OP⊕21OP1(1)→0, we deformL0|C1 to become trivial and using pg=0 to lift this deformation toY. The resulting bundlesLϕandζϕA1do descend to X.

For every AD E case with V the standard representation, we have L(0g,V)|Ci∼= OP1m+(OP1(1)+OP1(−1))n. For Ancases, our arguments are similar to the above A1 case. For Dncases, further arguments are needed as the pairs of OP11)inL(0Dn,C2n)|Ci

are in different locations comparing with the Ancases, and we also need to check that the holomorphic structure¯ϕonL(0Dn,C2n)preserves the natural quadratic formq. For the E6 (respectively, E7) case, since the cubic form c(respectively, quartic formt) is more complicated than the quadratic formq inDncases, the calculations are more involved.

TheE8case is rather different and we handle it by reductions to A7andD7cases.

The organization of this paper is as follows. Section 2 gives the construction of AD ELie algebra bundles overYdirectly. In Section 3, we review the definition of minus- cule representations and construct all minuscule representations using(−1)-curves in Y. Using these, we construct the Lie algebra bundles and minuscule representation bun- dles which can descend toX in An, Dn, and En(n=8) cases separately in Sections 4–6.

The proofs of the main theorems in this paper are given in Section 7.

Notation: For a holomorphic bundle (E0,∂¯0), if we construct a new holomorhic structure¯ϕon E0, then we denote the resulting bundle asEϕ.

2 AD ELie Algebra Bundles 2.1 AD E singularities

A rational double pointpin a surfaceXcan be described locally as a quotient singularity C2 with Γ a finite subgroup of SL(2,C). It is also called a Kleinian singularity or AD E singularity [2]. We can writeC2 as zeros of a polynomialF(X,Y,Z)inC3, where F(X,Y,Z)isXn+Y Z,Xn+1+XY2+Z2,X4+Y3+Z2,X3Y+Y3+Z2, orX5+Y3+Z2and the corresponding singularity is called of type An, Dn, E6, E7, or E8, respectively. The reason is if we consider the minimal resolutionYofX, then every irreducible component of the exceptional divisor C=n

i=1Ci is a smooth rational curve with normal bundle

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OP1(−2), that is, a(−2)-curve, and the dual graph of the exceptional divisor is an AD E Dynkin diagram.

There is a natural decomposition

H2(Y,Z)=H2(X,Z)⊕Λ, whereΛ= {

ai[Ci]|ai∈Z}. The setΦ:= {α∈Λ|α2= −2}is a simply laced (i.e., AD E) root system of a simple Lie algebragandΔ= {[Ci]}is a base ofΦ. For anyαΦ, there exists a unique divisorD=

aiCiwithα=[D], and we define a line bundleO(α):=O(D)overY.

2.2 Lie algebra bundles

We define a Lie algebra bundle of typegoverYas follows:

E0g:=O⊕n

α∈Φ

O(α).

For every open chartUofY, we takexαUto be a nonvanishing holomorphic section of OU(α) and hUi (i=1, . . . ,n) nonvanishing holomorphic sections of OU⊕n. Define a Lie algebra structure [,] onE0g such that{xαU’s,hUi ’s}is the Chevalley basis [12], that is,

(1) [hUi ,hUj]=0, 1≤i, jn;

(2) [hUi ,xUα]= α,CixαU, 1≤in,αΦ;

(3) [xαU,xU−α]=hUα is aZ-linear combination ofhUi ;

(4) ifα,β are independent roots, andβrα, . . . , β+is theα-string through β, then [xαU,xβU]=0 ifq=0; otherwise [xUα,xβU]= ±(r+1)xα+βU .

Since g is simply laced, all its roots have the same length, we have that any α-string throughβ is of length at most 2. So (4) can be written as [xUα, xβU]=nα,βxUα+β, where nα,β= ±1 if α+βΦ, otherwisenα,β=0. From the Jacobi identity, we have for anyα, β, γΦ,nα,βnα+β,γ +nβ,γnβ+γ,α+nγ,αnγ+α,β=0. This Lie algebra structure is com- patible with different trivializations ofE0g[15].

By Friedman–Morgan [8], a bundle over Y can descend to X if and only if its restriction to each irreducible component Ci of the exceptional divisor is trivial. But E0g|Ci is not trivial asO([Ci])|Ci ∼=OP1(−2). We will construct a new holomorphic structure onE0g, which preserves the Lie algebra structure, and therefore, the resulting bundleEϕg can descend toX.

As we have fixed a baseΔofΦ, we have a decompositionΦ=Φ+Φinto posi- tive and negative roots.

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Definition 5. Given anyϕ=α)α∈Φ+Ω0,1(Y,

α∈Φ+O(α)), we define¯ϕ:Ω0,0(Y,E0g)−→

Ω0,1(Y,E0g)by

¯ϕ:= ¯0+ad(ϕ):= ¯0+

α∈Φ+

ad(ϕα),

where¯0is the standard holomorphic structure ofE0g. More explicitly, if we writeϕα= cUαxαU locally for some one formcUα, then ad(ϕα)=cUαad(xαU).

Proposition 6. ¯ϕ is compatible with the Lie algebra structure, that is,¯ϕ[,]=0.

Proof. This follows directly from the Jacobi identity.

For¯ϕto define a holomorphic structure, we need

0= ¯ϕ2=

α∈Φ+

¯0cUα +

β+γ

(nβ,γcβUcUγ)

⎠ad(xUα),

that is,¯0ϕα+

β+γ(nβ,γϕβϕγ)=0 for anyαΦ+. Explicitly:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

¯0ϕCi=0, i=1,2. . . ,n,

¯0ϕCi+Cj=nCi,CjϕCiϕCj, ifCi+CjΦ+, ...

Proposition 7. Given anyCi)ni=1Ω0,1(Y,n

i=1O(Ci))with¯0ϕCi=0 for everyi, it can be extended to ϕ=α)α∈Φ+Ω0,1(Y,

α∈Φ+O(α)) such that ¯ϕ2=0. Namely, we have a

holomorphic vector bundleEϕg overY.

To prove this proposition, we need the following lemma. For anyα=n

i=1aiCiΦ+, we define ht(α):=n

i=1ai.

Lemma 8. For anyαΦ+,H2(Y,O(α))=0.

Proof. If ht(α)=1, that is,α=Ci,H2(Y,O(Ci))=0 follows from the long exact sequence associated to 0→OYOY(Ci)OCi(Ci)→0 and pg=0.

By induction, suppose that the lemma is true for every β with ht(β)=m. Given any α with ht(α)=m+1, by Humphreys [12, Section 10.2, Lemma A], there exists someCi such that α·Ci= −1, that is, β:=αCiΦ+ with ht(β)=m. Using the long

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exact sequence associated to 0→OY(β)OY(α)OCi(α)→0, OCi(α)∼=OP1(−1), and H2(Y,O(β))=0 by induction, we have H2(Y,O(α))=0.

Proof of Proposition 7. We solve the equations ¯0ϕα=

β+γnβ,γϕβϕγ for ϕαΩ0,1(Y,O(α))inductively on ht(α).

For ht(α)=2, that is, α=Ci+Cj with Ci·Cj=1, since [ϕCiϕCj]∈H2(Y,O(Ci+ Cj))=0, we can findϕCi+Cj satisfying¯0ϕCi+Cj= ±ϕCiϕCj.

Suppose that we have solved the equations for allϕβ’s with ht(β)m. For

¯0ϕα=

β+γ

nβ,γϕβϕγ,

with ht(α)=m+1, we have ht(β),ht(γ )≤m. Using ¯0(

β+γnβ,γϕβϕγ)=

δ+λ+μ=α

(nδ,λnδ+λ,μ+nλ,μnλ+μ,δ+nμ,δnμ+δ,λδϕλϕμ=0, [

β+γnβ,γϕβϕγ]∈H2(Y,O(α))=0, we can

solve forϕα.

Denote

ΨY

ϕ=α)α∈Φ+Ω0,1

Y,

α∈Φ+

O(α)

ϕ2=0

and

ΨX{ϕ∈ΨY|[ϕCi|Ci]=0 fori=1,2, . . . ,n}.

Theorem 9. Eϕgis trivial onCi if and only if [ϕCi|Ci]=0∈H1(Y,OCi(Ci)).

Proof. We will discuss the AD E cases separately in Sections 4–6 and the proof will be

completed in Section 7.

The next lemma says that given anyCi, there always existsϕCiΩ0,1(Y, O(Ci)) such that 0=[ϕCi|Ci]∈H1(Y,OCi(Ci))∼=C.

Lemma 10. For any Ci in Y, the restriction homomorphism H1(Y, OY(Ci))H1(Y,

OCi(Ci))is surjective.

Proof. The above restriction homomorphism is part of a long exact sequence induced by 0→OYOY(Ci)OCi(Ci)→0. The lemma follows directly frompg(Y)=0.

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3 Minuscule Representations and(−1)-Curves 3.1 Standard representations

For ADE Lie algebras, An=sl(n+1) is the space of tracefree endomorphisms ofCn+1 and Dn=o(2n) is the space of infinitesimal automorphisms of C2n which preserve a nondegenerate quadratic form q on C2n. In fact, E6 (respectively, E7) is the space of infinitesimal automorphisms ofC27(respectively,C56) which preserve a particular cubic formcon C27 (respectively, quartic formton C56) [1]. We call the above representation thestandard representationofg, that is,

g Standard representation

An=sl(n+1) Cn+1 Dn=o(2n) C2n

E6 C27

E7 C56

Note that all these standard representations are the fundamental representations cor- responding to the left nodes (i.e.C1) in the corresponding Dynkin diagrams (Figures1–3) and they are minuscule representations.

Fig. 1. The Dynkin diagram ofAn.

Fig. 2. The Dynkin diagram ofDn.

Fig. 3. The Dynkin diagram ofEn.

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3.2 Minuscule representations

Definition 11. A minuscule (respectively, quasi-minuscule) representation of a semi-simple Lie algebra is an irreducible representation such that the Weyl group acts transitively on all the weights(respectively, nonzero weights). Minuscule representations are always fundamental representations and quasi- minuscule representations are either minuscule or adjoint representations.

g Miniscule representations An=sl(n+1)kCn+1fork=1,2, . . . ,n Dn=o(2n) C2n,S+,S

E6 C27,27

E7 C56

Note thatE8has no minuscule representation.

3.3 Configurations of(−1)-curves

In this subsection, we describe(−1)-curves inXandY.

Definition 12. A(−1)-curve in a surfaceYis a genus-zero(possibly reducible)curvel

inYwithl·l= −1.

Remark 13. The genus-zero condition can be replaced byl·KY= −1 by the genus for-

mula, whereKYis the canonical divisor ofY.

LetC0be a curve inXpassing through p.

Definition 14. (1)C0is called a(−1)-curve in Xif there exists a(−1)-curvel inYsuch thatπ(l)=C0, or equivalently, the strict transform ofC0 is a(−1)-curveC˜0inY. (2) The multiplicity ofC0at pis defined to ben

i=1ai[Ci]∈Λ, whereai= ˜C0·Ci. Recall from Lie theory that any irreducible representation of a simple Lie alge- bra is determined by its lowest weight. The fundamental representations are those irre- ducible representations whose lowest weight is dual to the negative of some base root.

(The usual definition for fundamental representations uses highest weight. But in this paper, we will use lowest weight for simplicity of notations.) IfC0Xhas multiplicity Ckat pwhose dual weight determines a minuscule representationV, then we useC0kto denoteC˜0. The construction of suchX’s andC0’s can be found in Appendix.

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Definition 15. (1) We callC0has minuscule multiplicityCkΛatpifC0has multiplicity Ckand the dual weight of−Ckdetermines a minuscule representationV. (2) In this case, we denoteI(g,V)= {l:(−1)-curve inY|π(l)=C0}. If there is no ambiguity, we will simply write I(g,V)as I. Note thatIC0k+Λ≥0, whereΛ0= {

ai[Ci] :ai≥0}.

Lemma 16. In the above situation, the cardinality ofI is given by|I| =dimV.

Proof. By the genus formula and everyCi∼=P1being a(−2)-curve, we haveCi·KY=0.

Since C0k·KY= −1, each(−1)-curve has the form l=C0k+

aiCi with ai’s nonnegative integers. Froml·l= −1,we can determine{ai}s forlto be a(−1)-curve by direct compu-

tations.

Remark 17. The intersection product is negative definite on the sublattice of Pic(X) generated byC0k,C1, . . . ,Cnand we use its negative as an inner product.

Lemma 18. In the above situation, for anylI,αΦ, we have|l·α| ≤1.

Proof. We claim that for anyvC0k+Λ, we havev·v≤ −1. We prove the claim by direct computations. In(An,kCn+1)case:

C0k+ aiCi

2

= −1+2ak(a12+(a1a2)2+ · · · +(ak1ak)2)((akak+1)2+ · · · +a2n)

≤ −1.

The other cases can be proved similarly.

Since l,l+α, lαC0k+Λ by assumptions, we havel·l= −1≥(l+α)·(l+α), and hencel·α≤1. Alsol·l= −1≥(lα)·(lα), and hencel·α≥ −1.

Lemma 19. In the above situation, for anylI that is notCk0, there existsCi such that

l·Ci= −1.

Proof. From l=C0k+

aiCi=C0k (ai≥0), we have ak≥1. From l·l= −1, we have (

aiCi)2= −2ak. If there does not exist such an i with l·Ci= −1, then by Lemma 18, l·Ci≥0 for everyi,l·(

aiCi)≥0. Butl·(

aiCi)=ak+(

aiCi)2= −ak≤ −1 leads to a

contradiction.

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Lemma 20. In the above situation, for anyl,lI, H2(Y,O(ll))=0.

Proof. Firstly, we prove H2(Y,O(C0kl))=0 for anyl=C0k+

aiCiI inductively on ht(l):=

ai. If ht(l)=0, that is, l is C0k, the claim follows from pg=0. Suppose that the claim is true for anylI with ht(l)m−1. Then for any lI with ht(l)=m, by Lemma 19, there existsi such thatl·Ci= −1. This implies(lCi)I with ht(lCi)= m−1,and therefore,H2(Y,O(C0k(lCi)))=0 by induction hypothesis. Using the long exact sequence induced from

0→OY(C0kl)OY(C0k(lCi))OCi(Ck0(lCi))→0, andOCi(Ck0(lCi))∼=OP1(−1)orOP1, we have the claim.

If H2(Y,O(ll))=0, then there exists a section sH0(Y,KY(ll)) by Serre duality. Since there exists a nonzero section tH0(Y,O(lC0k)), we have stH0(Y,KY(lC0k))∼=H2(Y,O(C0kl))=0, which is a contradiction.

3.4 Minuscule representations from(−1)-curves

Recall from the AD E root systemΦ that we can recover the corresponding Lie algebra g=h⊕

α∈Φgα. As before, we use {xα’s, hi’s} to denote its Chevalley basis. If C0 has minuscule multiplicityCk, we denote

V0:=CI =

lI

Cvl,

where vl is the base vector of V0 generated by l. Then we define a bilinear map [,] : g⊗V0V0(possibly up to±signs) as follows:

[x, vl]=

⎧⎪

⎪⎪

⎪⎪

⎪⎩

α,l ifx=hα

±vl ifx=xα, l+αI, 0 ifx=xα, l+α /I.

Proposition 21. The signs in the above bilinear mapg⊗V0V0can be chosen so that it defines an action ofgonV0. Moreover,V0is isomorphic to the minuscule representa-

tionV.

Proof. For the first part, similar to [19], we use Lemma 18 to show [[x,y], vl]=[x,[y, vl]]− [y,[x, vl]].

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For the second part, since [xα, vC0k]=0 for any αΦ, vC0k is the lowest weight vector of V0 with weight corresponding to −Ck. Also we know the fundamental representationV corresponding to−Ckhas the same dimension withV0 by Lemma 16.

Hence,V0is isomorphic to the minuscule representationV.

Here, we show how to determine the signs. Take anylI,vlis a weight vector of the above action. Forx=xαandvlwith weightw, we define [x, vl]=nα,wvl+α, wherenα,w=

±1 ifl+αI, otherwisenα,w=0. By [[x,y], vl]=[x,[y, vl]]−[y,[x, vl]], we havenα,βnα+β,wnβ,wnα,β+w+nα,wnβ,α+w=0.

Remark 22. Recall that for anyl=C0k+

aiCiI, we define ht(l):=

ai. Using this, we can define a filtered structure forI:I=I0I1⊃ · · · ⊃Im, wherem=maxlIht(l),Ii= {lI|ht(l)mi}andIi\Ii+1= {lI|ht(l)=mi}. This ht(l)also enables us to define a partial order ofI. Say|I| =N, we denotelN:=C0ksince it is the only element with ht=0.

Similarly,lN1:=C0k+Ck. Of course, there are some ambiguity of this ordering, and if so, we will just make a choice to order these(−1)-curves.

3.5 Bundles from(−1)-curves

The geometry of(−1)-curves inYcan be used to construct representation bundles ofEϕg for every minuscule representation ofg. The proofs of theorems in this subsection will be given in Section 7.

WhenC0Xhas minuscule multiplicityCkat pwith the corresponding minus- cule representation V, we define (When X is a del Pezzo surface, we use lines in X to construct bundles [8]. So here we use(−1)-curves in Xto construct bundles.)

L(0g,V):=

lI(g,V)

O(l).

L(0g,V) has a natural filtration F: L(0g,V)=F0L⊃F1L⊃ · · · ⊃FmL, induced from the flittered structure onI, namely FiL(0g,V)=

lIi O(l).

L(0g,V) cannot descend to X as OCk(Ck0)∼=OP1(1) (because Ck·C0k=1 by the definition of the minuscule multiplicity). For any Ci and any lI, we have OCi(l)∼= OP11) or OP1 by Lemma 18. For every fixed Ci, if there is a lI such that OCi(l)∼= OP1(1), then (l+Ci)2= −1=(l+Ci)·KY, that is, l+CiI, also OCi(l+Ci)∼=OP1(−1). That means that among the direct summands of L(0g,V)|Ci, OP1(1)and OP1(−1)occur in pairs, and each pair is given by two(−1)-curves inI whose difference isCi. This gives us a chance to deformL(0g,V)to get another bundle that can descend toX.

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Theorem 23. If there exists a (−1)-curveC0 in X with minuscule multiplicityCk at p andρ:g−→End(V)is the corresponding representation, then

L(ϕg,V):=

lI

O(l), ¯ϕ:= ¯0+ρ(ϕ)

,

withϕΨY is a holomorphic bundle overYwhich preserves the filtration onL(0g,V)and it is a holomorphic representation bundle ofEϕg. Moreover,L(ϕg,V) is trivial onCi if and only if [ϕCi|Ci]=0∈H1(Y,OCi(Ci)). ForCkwithk=1, the corresponding minuscule representationVis the standard representation ofg. Wheng=An, it is simply sl(n+1)=aut0(V). Wheng=Dn(respec- tively,E6andE7), there exists a quadratic (respectively, cubic and quartic) form f onV such thatg=aut(V,f). The next theorem tells us that we can globalize this construction overYto recover the Lie algebra bundleEϕg overY. But this does not work forEϕE8 asE8 has no standard representation.

Theorem 24. Under the same assumptions as in Theorem 23 withk=1, there exists a holomorphic fiberwise symmetric multi-linear form

f: r

L(ϕg,V)−→OY(D),

withr=0,2,3,4 wheng=An,Dn,E6,E7, respectively, such thatEϕg∼=aut0(L(ϕg,V),f). It is obvious thatEϕg does not depend on the existence of the (−1)-curveC0, for the minuscule representation bundles, we have the following results.

Theorem 25. There exists a divisorBinYand an integerk, such that the bundleL(ϕg,V):= SkL(ϕg,V)O(−B)withϕΨX can descend to X and does not depend on the existence of

C0.

3.6 Outline of proofs forg=E8

Wheng=E8, there exists a natural symmetric tensor f on its standard representation Vsuch thatg=aut0(V,f). The setI(g,V)of(−1)-curves has cardinalityN=dimV. Given η:=(ηi,j)N×Nwithηi,jΩ0,1(Y,O(lilj))for everyli=ljI(g,V), we consider the operator

¯η:= ¯0+ηonL(0g,V):=

l∈I(g,V)OY(l). We will look forηthat satisfies:

(1) (filtration)ηi,j=0 fori>jfor the partial ordering introduced in Section 3.4;

(2) (holomorphic structure)(∂¯0+η)2=0;

at The Chinese University of Hong Kong on October 4, 2014http://imrn.oxfordjournals.org/Downloaded from

(14)

(3) (Lie algebra structure)¯ηf=0;

(4) (descendent) for everyCk, iflilj=Ck, then 0=[ηi,j|Ck]∈H1(Y,OCk(Ck)).

Remark 26. Property (2)implies that we can define a new holomorphic structure on L(0g,V). Properties (1) and (3) require that for any ηi,j=0, ηi,jΩ0,1(Y,O(α)) for some αΦ+. We will show that ifηsatisfies (1)–(3), then (4) is equivalent toL(ηg,V)being trivial on everyCk, that is,L(ηg,V)can descend toX.

Denote

ΞYg{η=(ηi,j)N×N|ηsatisfies (1)–(3)}

and

ΞXg{η∈ΞYg|ηsatisfies (4)};

then eachηinΞYgdetermines a filtered holomorphic bundleL(ηg,V)overYtogether with a holomorphic tensor fon it. It can descend to XifηΞXg.

Since g=aut(V, f), for any η∈ΞYg, we have a holomorphic Lie algebra bundle ζηg:=aut(L(ηg,V), f)overYof typeg, andL(ηg,V)is automatically a representation bundle ofζηg. Furthermore, ifη∈ΞXg, thenζηgcan descend toX.

For a general minuscule representation ofg, given anyη∈ΞYg, we show that there exists a unique holomorphic structure onL(0g,V), such that the action of ζηg on the new holomorphic bundleL(ηg,V)is holomorphic. Furthermore, ifη∈ΞgX, thenL(ηg,V)can descend toX.

4 AnCase

We recall that An=sl(n+1,C)=aut0(Cn+1) (where aut0 means tracefree endomor- phisms). The standard representation of An is Cn+1 and minuscule representations of Anare∧kCn+1,k=1,2, . . . ,n.

4.1 Anstandard representation bundleL(Aη n,Cn+1)

We consider a surfaceXwith an Ansingularity pand a(−1)-curveC0 passing through pwith multiplicityC1; thenI(An,Cn+1)= {C01+k

i=1Ci|0≤kn}has cardinalityn+1. We order these(−1)-curves:lk=C01+n+1k

i=1 Ci for 1≤kn+1. For anyli=ljI,li·lj=0.

at The Chinese University of Hong Kong on October 4, 2014http://imrn.oxfordjournals.org/Downloaded from

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