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 and*p**g*=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*
in*Y.*

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 *E**n*and del Pezzo surfaces *X*of degree 9−*n*for 1≤*n≤*8. The root lattice of *E**n*

can be identified with*K*_{X}^{⊥}, the orthogonal complement to*K**X*in Pic(X). Furthermore, all
the lines in *X* form a representation of *E**n*. Using the configuration of these lines, we
can construct an*E** _{n}*Lie 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 containΣ*and the moduli space of

*E*

*-bundles over*

_{n}*Σ*. 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*:*Y*→*X*

is the minimal resolution of a compact complex surface*X*with a rational double point.

Then the dual graph of the exceptional divisor_{n}

*i*=1*C**i* in*Y*is an *AD E* Dynkin diagram.

From this, we have an*AD E*root system*Φ*:= {α=

*a** _{i}*[C

*]|α*

_{i}^{2}= −2}and we can construct an

*AD E*Lie algebra bundle over

*Y:*

*E*_{0}^{g}:=*O*_{Y}^{⊕}* ^{n}*⊕

*α∈Φ*

*O*_{Y}*(α).*

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

**Theorem 1**(Propositions 6, 7, Theorem 9, and Lemma 10). Assume that*Y*is the minimal
resolution of a surface *X* with a rational double point at *p*of type gand*C*=*Σ**i** ^{n}*=1

*C*

*i*is the exceptional divisor. If

*p*

*g*

*(X)*=0

*,*then

(1) given any *(ϕ**C**i**)*^{n}_{i}_{=}_{1}∈*Ω*^{0,1}*(Y,*_{n}

*i*=1*O(C**i**))* with *∂ϕ*¯ *C**i*=0 for every *i, it can be*
extended to *ϕ*=*(ϕ*_{α}*)*_{α∈Φ}^{+}∈*Ω*^{0}^{,}^{1}*(Y,*

*α∈Φ*^{+}*O(α))*such that*∂*¯* _{ϕ}*:= ¯

*∂*+ad(ϕ) is a holomorphic structure on

*E*

_{0}

^{g}. We denote this new holomorphic bundle as

*E*

_{ϕ}^{g}; (2) such a

*∂*¯

*is compatible with the Lie algebra structure;*

_{ϕ}(3) *E*_{ϕ}^{g} is trivial on*C** _{i}* if and only if [

*ϕ*

*C*

*i*|

*C*

*i*]=0∈

*H*

^{1}

*(C*

_{i}*,O*

_{C}

_{i}*(C*

_{i}*))*∼=C; (4) there exists [

*ϕ*

*C*

*i*]∈

*H*

^{1}

*(Y,O(C*

_{i}*))*such that [

*ϕ*

*C*

*i*|

*C*

*i*]=0;

(5) such a*E*_{ϕ}^{g}can descend to*X*if and only if [*ϕ**C**i*|*C**i*]=0 for every*i.*

**Remark 2.** Infinitesimal deformations of holomorphic bundle structures on *E*_{0}^{g} are
parameterized by *H*^{1}*(Y,*End*(E*_{0}^{g}*))*, and those that also preserve the Lie algebra struc-
ture are parameterized by *H*^{1}*(Y,*ad*(E*_{0}^{g}*))*=*H*^{1}*(Y,E*_{0}^{g}*)*, since gis semi-simple. If *p*_{g}*(X)*=
*q(X)*=0, for example, rational surface, then for any *α*∈*Φ*^{−}, *H*^{1}*(Y,O(α))*=0. Hence,
*H*^{1}*(Y,E*_{0}^{g}*)*=*H*^{1}*(Y,*

*α∈Φ*^{+}*O(α))*.

This generalizes the work of Friedman–Morgan [8], in which they considered*E** _{n}*
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 in

*Y.*

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Here is an outline of our results. We first study*(−*1*)*-curves in*Y*which are (pos-
sibly reducible) rational curves with self-intersection−1. If there exists a*(−*1*)*-curve*C*_{0}
in*X*passing through *p*with minuscule multiplicity*C** _{k}*(Definition 15), then

*(−*1

*)*-curves

*l*’s in

*Y*with

*π(l)*=

*C*

_{0}form the minuscule representation

*V*of g corresponding to

*C*

*(Proposition 21). (Here*

_{k}*V*is the lowest weight representation with lowest weight dual to

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

L^{(}_{0}^{g}^{,}^{V}* ^{)}*:=

*l:**(−*1*)−*curve
*π(**l**)=**C*0

*O**Y**(l),*

over*Y*constructed from these*(−*1*)*-curves*l’s. This bundle cannot descend to* *X*as it is
not trivial over each*C**i*. (Unless specified otherwise,*C**i* always refers to an irreducible
component of*C*, that is,*i*=0.)

**Theorem 3**(Theorems 23 and 24). For the bundleL^{(}_{0}^{g}^{,}^{V}* ^{)}* with the corresponding minus-
cule representation

*ρ*:g−→End

*(V)*,

(1) there exists *ϕ*=*(ϕ**α**)**α∈Φ*^{+}∈*Ω*^{0}^{,}^{1}*(Y,*

*α∈Φ*^{+}*O(α))* such that*∂*¯*ϕ*:= ¯*∂*0+*ρ(ϕ)*is a
holomorphic structure onL^{(}_{0}^{g}^{,}^{V}* ^{)}*. We denote this new holomorphic bundle as
L

^{(}

_{ϕ}^{g}

*;*

^{,V)}(2) L^{(}_{ϕ}^{g}* ^{,V)}*is trivial on

*C*

*i*if and only if [

*ϕ*

*C*

*i*|

*C*

*i*]=0∈

*H*

^{1}

*(Y,O*

*C*

*i*

*(C*

*i*

*))*;

(3) when*V*is the standard representation ofg, there exists a holomorphic fiber-
wise symmetric multi-linear form

*f*:
*r*

L^{(}_{ϕ}^{g}^{,}^{V}* ^{)}*−→

*O*

*Y*

*(D),*

with *r*=0,2,3,4 when g=*A**n**,D**n**,E*6*,E*7, respectively, such that *E*_{ϕ}^{g}∼=

aut0*(*L^{(}_{ϕ}^{g}^{,}^{V}^{)}*,* *f).*

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

*l**O(l)*such that the action of *E*_{ϕ}^{g} on this bundle is holomorphic
and it can descend to*X*as well.

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**Example 4.** When we blowup two distinct points, we have a surface*Y*with two*(−*1*)*-
curves*l*_{1}and*l*_{2}as exceptional curves.L_{0}:=*O*_{Y}*(l*_{1}*)*⊕*O*_{Y}*(l*_{2}*)*is aC^{2}-bundle and the bundle
*ζ*_{0}^{A}^{1}of its symmetries is an sl*(*2*)*- or*A*_{1}-bundle over*Y.*

When the two points become infinitesimally close, then *C*_{1}=*l*_{2}−*l*_{1} is effective,
namely a *(−*2*)*-curve in *Y. If we blow down* *C*1 in *Y, we get a surface* *X* with an *A*1

singularity.L0cannot descend to*X*asL0|*C*1∼=*O*_{P}^{1}*(−*1*)*⊕*O*_{P}^{1}*(*1*)*. Using the Euler sequence
0→*O*_{P}^{1}*(−1)*→*O*_{P}^{⊕2}1 →*O*_{P}^{1}*(1)*→0, we deformL_{0}|*C*1 to become trivial and using *p**g*=0 to
lift this deformation to*Y. The resulting bundles*L* _{ϕ}*and

*ζ*

_{ϕ}

^{A}^{1}do descend to

*X.*

For every *AD E* case with *V* the standard representation, we have L^{(}_{0}^{g}^{,}^{V}* ^{)}*|

*C*

*i*∼=

*O*

_{P}

^{⊕}1

*+*

^{m}*(O*

_{P}

^{1}

*(*1

*)*+

*O*

_{P}

^{1}

*(−*1

*))*

^{⊕}

*. For*

^{n}*A*

*cases, our arguments are similar to the above*

_{n}*A*

_{1}case. For

*D*

*cases, further arguments are needed as the pairs of*

_{n}*O*

_{P}

^{1}

*(±*1

*)*inL

^{(}_{0}

^{D}

^{n}

^{,C}^{2n}

*|*

^{)}*C*

*i*

are in different locations comparing with the *A** _{n}*cases, and we also need to check that
the holomorphic structure

*∂*¯

*ϕ*onL

^{(}_{0}

^{D}

^{n}

^{,C}^{2n}

*preserves the natural quadratic form*

^{)}*q. For the*

*E*

_{6}(respectively,

*E*

_{7}) case, since the cubic form

*c*(respectively, quartic form

*t) is more*complicated than the quadratic form

*q*in

*D*

*n*cases, the calculations are more involved.

The*E*8case is rather different and we handle it by reductions to *A*7and*D*7cases.

The organization of this paper is as follows. Section 2 gives the construction of
*AD E*Lie algebra bundles over*Y*directly. 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 to*X* in *A** _{n}*,

*D*

*, and*

_{n}*E*

*(n=8) cases separately in Sections 4–6.*

_{n}The proofs of the main theorems in this paper are given in Section 7.

Notation: For a holomorphic bundle *(E*_{0}*,∂*¯0*)*, if we construct a new holomorhic
structure*∂*¯*ϕ*on *E*_{0}, then we denote the resulting bundle as*E** _{ϕ}*.

2 **AD ELie Algebra Bundles**
2.1 **AD E singularities**

A rational double point*p*in a surface*X*can be described locally as a quotient singularity
C^{2}*/Γ* with *Γ* a finite subgroup of SL*(*2*,*C). It is also called a Kleinian singularity or
*AD E* singularity [2]. We can writeC^{2}*/Γ* as zeros of a polynomial*F(X,Y,Z)*inC^{3}, where
*F(X,Y,Z)*is*X** ^{n}*+

*Y Z,X*

^{n}^{+}

^{1}+

*XY*

^{2}+

*Z*

^{2},

*X*

^{4}+

*Y*

^{3}+

*Z*

^{2},

*X*

^{3}

*Y*+

*Y*

^{3}+

*Z*

^{2}, or

*X*

^{5}+

*Y*

^{3}+

*Z*

^{2}and the corresponding singularity is called of type

*A*

*,*

_{n}*D*

*,*

_{n}*E*

_{6},

*E*

_{7}, or

*E*

_{8}, respectively. The reason is if we consider the minimal resolution

*Y*of

*X, then every irreducible component*of the exceptional divisor

*C*=

_{n}*i*=1*C**i* is a smooth rational curve with normal bundle

*O*_{P}^{1}*(−*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

*H*^{2}*(Y,*Z)=*H*^{2}*(X,*Z)⊕*Λ,*
where*Λ*= {

*a**i*[C*i*]|a*i*∈Z}. The set*Φ*:= {α∈*Λ|α*^{2}= −2}is a simply laced (i.e., *AD E) root*
system of a simple Lie algebragand*Δ*= {[C*i*]}is a base of*Φ*. For any*α*∈*Φ*, there exists a
unique divisor*D*=

*a**i**C**i*with*α*=[*D], and we define a line bundleO(α)*:=*O(D)*over*Y.*

2.2 **Lie algebra bundles**

We define a Lie algebra bundle of typegover*Y*as follows:

*E*_{0}^{g}:=*O*^{⊕n}⊕

*α∈Φ*

*O(α).*

For every open chart*U*of*Y, we takex*_{α}* ^{U}*to be a nonvanishing holomorphic section
of

*O*

*U*

*(α)*and

*h*

^{U}*(i=1, . . . ,*

_{i}*n*) nonvanishing holomorphic sections of

*O*

_{U}^{⊕n}. Define a Lie algebra structure [,] on

*E*

_{0}

^{g}such that{x

_{α}*’s,*

^{U}*h*

^{U}*’s}is the Chevalley basis [12], that is,*

_{i}(1) [h^{U}_{i}*,h*^{U}* _{j}*]=0, 1≤

*i,*

*j*≤

*n;*

(2) [h^{U}_{i}*,x*^{U}* _{α}*]= α,

*C*

_{i}*x*

_{α}*, 1≤*

^{U}*i*≤

*n*,

*α*∈

*Φ*;

(3) [x_{α}^{U}*,x*^{U}_{−α}]=*h*^{U}* _{α}* is aZ-linear combination of

*h*

^{U}*;*

_{i}(4) if*α*,*β* are independent roots, and*β*−*rα, . . . , β*+*qα*is the*α*-string through
*β*, then [x_{α}^{U}*,x*_{β}* ^{U}*]=0 if

*q*=0; otherwise [x

^{U}

_{α}*,x*

_{β}*]= ±(*

^{U}*r*+1

*)x*

_{α+β}*.*

^{U}Since g is simply laced, all its roots have the same length, we have that any
*α*-string through*β* is of length at most 2. So (4) can be written as [x^{U}_{α}*,* *x*_{β}* ^{U}*]=

*n*

_{α,β}*x*

^{U}*, where*

_{α+β}*n*

*= ±1 if*

_{α,β}*α*+

*β*∈

*Φ*, otherwise

*n*

*=0. From the Jacobi identity, we have for any*

_{α,β}*α, β, γ*∈

*Φ*,

*n*

_{α,β}*n*

*+*

_{α+β,γ}*n*

_{β,γ}*n*

*+*

_{β+γ,α}*n*

_{γ,α}*n*

_{γ}_{+α,β}=0. This Lie algebra structure is com- patible with different trivializations of

*E*

_{0}

^{g}[15].

By Friedman–Morgan [8], a bundle over *Y* can descend to *X* if and only if its
restriction to each irreducible component *C** _{i}* of the exceptional divisor is trivial. But

*E*

_{0}

^{g}|

*C*

*i*is not trivial as

*O(*[C

*]*

_{i}*)|*

*C*

*i*∼=

*O*

_{P}

^{1}

*(−*2

*)*. We will construct a new holomorphic structure on

*E*

_{0}

^{g}, which preserves the Lie algebra structure, and therefore, the resulting bundle

*E*

_{ϕ}^{g}can descend to

*X.*

As we have fixed a base*Δ*of*Φ*, we have a decomposition*Φ*=*Φ*^{+}∪*Φ*^{−}into posi-
tive and negative roots.

**Definition 5.** Given any*ϕ*=*(ϕ**α**)**α∈Φ*^{+}∈*Ω*^{0}^{,}^{1}*(Y,*

*α∈Φ*^{+}*O(α))*, we define*∂*¯*ϕ*:*Ω*^{0}^{,}^{0}*(Y,E*_{0}^{g}*)*−→

*Ω*^{0}^{,}^{1}*(Y,E*_{0}^{g}*)*by

*∂*¯* _{ϕ}*:= ¯

*∂*0+ad(ϕ):= ¯

*∂*0+

*α∈Φ*^{+}

ad(ϕ_{α}*),*

where*∂*¯0is the standard holomorphic structure of*E*_{0}^{g}. More explicitly, if we write*ϕ** _{α}*=

*c*

^{U}

_{α}*x*

_{α}*locally for some one form*

^{U}*c*

^{U}*, then ad(ϕ*

_{α}

_{α}*)*=

*c*

^{U}*ad(x*

_{α}

_{α}

^{U}*).*

**Proposition 6.** *∂*¯* _{ϕ}* is compatible with the Lie algebra structure, that is,

*∂*¯

*[,]=0.*

_{ϕ}**Proof.** This follows directly from the Jacobi identity.

For*∂*¯* _{ϕ}*to define a holomorphic structure, we need

0= ¯*∂*_{ϕ}^{2}=

*α∈Φ*^{+}

⎛

⎝*∂*¯0*c*^{U}* _{α}* +

*β+γ*=α

*(n*_{β,γ}*c*_{β}^{U}*c*^{U}_{γ}*)*

⎞

⎠ad*(x*^{U}_{α}*),*

that is,*∂*¯0*ϕ** _{α}*+

*β+γ*=α*(n*_{β,γ}*ϕ*_{β}*ϕ*_{γ}*)*=0 for any*α*∈*Φ*^{+}. Explicitly:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

*∂*¯0*ϕ**C**i*=0*,* *i*=1*,*2*. . . ,n,*

*∂*¯0*ϕ**C**i*+*C**j*=*n**C**i**,**C**j**ϕ**C**i**ϕ**C**j**,* if*C**i*+*C**j*∈*Φ*^{+}*,*
*...*

**Proposition 7.** Given any*(ϕ**C**i**)*^{n}* _{i=1}*∈

*Ω*

^{0}

^{,}^{1}

*(Y,*

*n*

*i=1**O(C*_{i}*))*with*∂*¯0*ϕ**C**i*=0 for every*i, it can*
be extended to *ϕ*=*(ϕ**α**)**α∈Φ*^{+}∈*Ω*^{0}^{,}^{1}*(Y,*

*α∈Φ*^{+}*O(α))* such that *∂*¯_{ϕ}^{2}=0. Namely, we have a

holomorphic vector bundle*E*_{ϕ}^{g} over*Y.*

To prove this proposition, we need the following lemma. For any*α*=*n*

*i=1**a*_{i}*C** _{i}*∈

*Φ*

^{+}, we define ht

*(α)*:=

_{n}*i*=1*a**i*.

**Lemma 8.** For any*α*∈*Φ*^{+},*H*^{2}*(Y,O(α))*=0.

**Proof.** If ht*(α)*=1, that is,*α*=*C** _{i}*,

*H*

^{2}

*(Y,O(C*

_{i}*))*=0 follows from the long exact sequence associated to 0→

*O*

*→*

_{Y}*O*

_{Y}*(C*

_{i}*)*→

*O*

_{C}

_{i}*(C*

_{i}*)*→0 and

*p*

*=0.*

_{g}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
some*C**i* such that *α*·*C**i*= −1, that is, *β*:=*α*−*C**i*∈*Φ*^{+} with ht*(β)*=*m. Using the long*

exact sequence associated to 0→*O*_{Y}*(β)*→*O*_{Y}*(α)*→*O*_{C}_{i}*(α)*→0, *O*_{C}_{i}*(α)*∼=*O*_{P}^{1}*(−*1*),* and
*H*^{2}*(Y,O(β))*=0 by induction, we have *H*^{2}*(Y,O(α))*=0.

**Proof of Proposition 7.** We solve the equations *∂*¯0*ϕ** _{α}*=

*β+γ*=α*n*_{β,γ}*ϕ*_{β}*ϕ** _{γ}* for

*ϕ*

*∈*

_{α}*Ω*

^{0}

^{,}^{1}

*(Y,O(α))*inductively on ht(α).

For ht*(α)*=2, that is, *α*=*C** _{i}*+

*C*

*with*

_{j}*C*

*·*

_{i}*C*

*=1, since [*

_{j}*ϕ*

*C*

*i*

*ϕ*

*C*

*j*]∈

*H*

^{2}

*(Y,O(C*

*+*

_{i}*C*

_{j}*))*=0, we can find

*ϕ*

*C*

*i*+

*C*

*j*satisfying

*∂*¯0

*ϕ*

*C*

*i*+

*C*

*j*= ±ϕ

*C*

*i*

*ϕ*

*C*

*j*.

Suppose that we have solved the equations for all*ϕ**β*’s with ht*(β)*≤*m. For*

*∂*¯0*ϕ**α*=

*β+γ*=α

*n*_{β,γ}*ϕ**β**ϕ**γ**,*

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

*β+γ*=α*n*_{β,γ}*ϕ*_{β}*ϕ*_{γ}*)*=

*δ+λ+μ=α*

*(n*_{δ,λ}*n** _{δ+λ,μ}*+

*n*

_{λ,μ}*n*

*+*

_{λ+μ,δ}*n*

_{μ,δ}*n*

_{μ+δ,λ}*)ϕ*

_{δ}*ϕ*

_{λ}*ϕ*

*=0, [*

_{μ}*β+γ*=α*n*_{β,γ}*ϕ*_{β}*ϕ** _{γ}*]∈

*H*

^{2}

*(Y,O(α))*=0, we can

solve for*ϕ** _{α}*.

Denote

*Ψ**Y*

*ϕ*=*(ϕ*_{α}*)*_{α∈Φ}^{+}∈*Ω*^{0}^{,}^{1}

*Y,*

*α∈Φ*^{+}

*O(α)*

|¯*∂*_{ϕ}^{2}=0

and

*Ψ**X*{ϕ∈*Ψ**Y*|[*ϕ**C**i*|*C**i*]=0 for*i*=1*,*2*, . . . ,n*}.

**Theorem 9.** *E*_{ϕ}^{g}is trivial on*C** _{i}* if and only if [

*ϕ*

*C*

*i*|

*C*

*i*]=0∈

*H*

^{1}

*(Y,O*

_{C}

_{i}*(C*

_{i}*))*.

**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 any*C**i*, there always exists*ϕ**C**i*∈*Ω*^{0,1}*(Y,* *O(C**i**))*
such that 0=[ϕ*C**i*|*C**i*]∈*H*^{1}*(Y,O**C**i**(C**i**))*∼=C.

**Lemma 10.** For any *C**i* in *Y, the restriction homomorphism* *H*^{1}*(Y,* *O**Y**(C**i**))*→*H*^{1}*(Y,*

*O**C**i**(C**i**))*is surjective.

**Proof.** The above restriction homomorphism is part of a long exact sequence induced
by 0→*O**Y*→*O**Y**(C**i**)*→*O**C**i**(C**i**)*→0. The lemma follows directly from*p**g**(Y)*=0.

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

For *ADE* Lie algebras, *A** _{n}*=sl

*(n*+1

*)*is the space of tracefree endomorphisms ofC

^{n}^{+}

^{1}and

*D*

*n*=

*o(*2n

*)*is the space of infinitesimal automorphisms of C

^{2n}which preserve a nondegenerate quadratic form

*q*on C

^{2n}. In fact,

*E*6 (respectively,

*E*7) is the space of infinitesimal automorphisms ofC

^{27}(respectively,C

^{56}) which preserve a particular cubic form

*c*on C

^{27}(respectively, quartic form

*t*on C

^{56}) [1]. We call the above representation the

*standard representation*ofg, that is,

g Standard representation

*A**n*=sl(n+1) C^{n+1}*D**n*=*o(2n)* C^{2n}

*E*6 C^{27}

*E*_{7} C^{56}

Note that all these standard representations are the fundamental representations cor-
responding to the left nodes (i.e.*C*_{1}) in the corresponding Dynkin diagrams (Figures1–3)
and they are minuscule representations.

**Fig. 1.** The Dynkin diagram of*A**n*.

**Fig. 2.** The Dynkin diagram of*D**n*.

**Fig. 3.** The Dynkin diagram of*E**n*.

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
*A** _{n}*=sl

*(n*+1

*)*∧

*C*

^{k}

^{n}^{+}

^{1}for

*k*=1

*,*2

*, . . . ,n*

*D*

*=*

_{n}*o(*2n

*)*C

^{2n}

*,S*

^{+}

*,S*

^{−}

*E*_{6} C^{27}*,*C¯^{27}

*E*7 C^{56}

Note that*E*_{8}has no minuscule representation.

3.3 **Configurations of****(−1)-curves**

In this subsection, we describe*(−1)-curves inX*and*Y.*

**Definition 12.** A*(−1)-curve in a surfaceY*is a genus-zero*(possibly reducible)*curve*l*

in*Y*with*l*·*l*= −1.

**Remark 13.** The genus-zero condition can be replaced by*l*·*K** _{Y}*= −1 by the genus for-

mula, where*K** _{Y}*is the canonical divisor of

*Y.*

Let*C*_{0}be a curve in*X*passing through *p.*

**Definition 14.** (1)*C*0is called a*(−*1*)*-curve in *X*if there exists a*(−*1*)*-curve*l* in*Y*such
that*π(l)*=*C*0, or equivalently, the strict transform of*C*0 is a*(−1)-curveC*˜0in*Y. (2) The*
multiplicity of*C*0at *p*is defined to be_{n}

*i*=1*a**i*[C*i*]∈*Λ, wherea**i*= ˜*C*0·*C**i*.
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.) If*C*_{0}⊂*X*has multiplicity
*C** _{k}*at

*p*whose dual weight determines a minuscule representation

*V, then we useC*

_{0}

*to denote*

^{k}*C*˜0. The construction of such

*X’s andC*0’s can be found in Appendix.

**Definition 15.** (1) We call*C*_{0}has minuscule multiplicity*C** _{k}*∈

*Λ*at

*p*if

*C*

_{0}has multiplicity

*C*

*and the dual weight of−*

_{k}*C*

*determines a minuscule representation*

_{k}*V. (2) In this case,*we denote

*I*

^{(}^{g}

^{,}

^{V}*= {*

^{)}*l*:

*(−*1

*)*-curve in

*Y*|π(

*l)*=

*C*

_{0}}. If there is no ambiguity, we will simply write

*I*

^{(}^{g}

^{,}

^{V}*as*

^{)}*I*. Note that

*I*⊂

*C*

_{0}

*+*

^{k}*Λ*≥0, where

*Λ*≥0= {

*a**i*[C*i*] :*a**i*≥0}.

**Lemma 16.** In the above situation, the cardinality of*I* is given by|*I*| =dim*V*.

**Proof.** By the genus formula and every*C**i*∼=P^{1}being a*(−*2*)*-curve, we have*C**i*·*K**Y*=0.

Since *C*_{0}* ^{k}*·

*K*

*Y*= −1, each

*(−*1

*)*-curve has the form

*l*=

*C*

_{0}

*+*

^{k}*a**i**C**i* with *a**i*’s nonnegative
integers. From*l·l*= −1,we can determine{a*i*}^{}s for*l*to be a*(−1)-curve by direct compu-*

tations.

**Remark 17.** The intersection product is negative definite on the sublattice of Pic(X)
generated by*C*_{0}^{k}*,C*1*, . . . ,C**n*and we use its negative as an inner product.

**Lemma 18.** In the above situation, for any*l*∈*I*,*α*∈*Φ*, we have|l·*α| ≤*1.

**Proof.** We claim that for any*v*∈*C*_{0}* ^{k}*+

*Λ, we havev*·

*v*≤ −1. We prove the claim by direct computations. In

*(A*

*n*

*,*∧

*C*

^{k}

^{n}^{+}

^{1}

*)*case:

*C*_{0}* ^{k}*+

*a*

*i*

*C*

*i*

2

= −1+2a*k*−*(a*_{1}^{2}+*(a*1−*a*2*)*^{2}+ · · · +*(a**k*−1−*a**k**)*^{2}*)*−*((a**k*−*a**k*+1*)*^{2}+ · · · +*a*^{2}_{n}*)*

≤ −1*.*

The other cases can be proved similarly.

Since *l,l*+*α,* *l*−*α*∈*C*_{0}* ^{k}*+

*Λ*by assumptions, we have

*l*·

*l*= −1≥

*(l*+

*α)*·

*(l*+

*α)*, and hence

*l*·

*α*≤1. Also

*l*·

*l*= −1≥

*(l*−

*α)*·

*(l*−

*α)*, and hence

*l*·

*α*≥ −1.

**Lemma 19.** In the above situation, for any*l*∈*I* that is not*C*^{k}_{0}, there exists*C** _{i}* such that

*l*·*C** _{i}*= −1.

**Proof.** From *l*=*C*_{0}* ^{k}*+

*a*_{i}*C** _{i}*=

*C*

_{0}

^{k}*(a*

*≥0*

_{i}*)*, we have

*a*

*≥1. From*

_{k}*l*·

*l*= −1, we have

*(*

*a*_{i}*C*_{i}*)*^{2}= −2a* _{k}*. If there does not exist such an

*i*with

*l*·

*C*

*= −1, then by Lemma 18,*

_{i}*l*·

*C*

*≥0 for every*

_{i}*i,l*·

*(*

*a*_{i}*C*_{i}*)*≥0. But*l*·*(*

*a*_{i}*C*_{i}*)*=*a** _{k}*+

*(*

*a*_{i}*C*_{i}*)*^{2}= −*a** _{k}*≤ −1 leads to a

contradiction.

**Lemma 20.** In the above situation, for any*l,l*^{}∈*I*, *H*^{2}*(Y,O(l*−*l*^{}*))*=0.

**Proof.** Firstly, we prove *H*^{2}*(Y,O(C*_{0}* ^{k}*−

*l))*=0 for any

*l*=

*C*

_{0}

*+*

^{k}*a*_{i}*C** _{i}*∈

*I*inductively on ht

*(l)*:=

*a** _{i}*. If ht

*(l)*=0, that is,

*l*is

*C*

_{0}

*, the claim follows from*

^{k}*p*

*=0. Suppose that the claim is true for any*

_{g}*l*

^{}∈

*I*with ht

*(l*

^{}

*)*≤

*m*−1. Then for any

*l*∈

*I*with ht

*(l)*=

*m, by*Lemma 19, there exists

*i*such that

*l*·

*C*

*= −1. This implies*

_{i}*(l*−

*C*

_{i}*)*∈

*I*with ht

*(l*−

*C*

_{i}*)*=

*m*−1

*,*and therefore,

*H*

^{2}

*(Y,O(C*

_{0}

*−*

^{k}*(l*−

*C*

_{i}*)))*=0 by induction hypothesis. Using the long exact sequence induced from

0→*O**Y**(C*_{0}* ^{k}*−

*l)*→

*O*

*Y*

*(C*

_{0}

*−*

^{k}*(l*−

*C*

*i*

*))*→

*O*

*C*

*i*

*(C*

^{k}_{0}−

*(l*−

*C*

*i*

*))*→0, and

*O*

_{C}

_{i}*(C*

^{k}_{0}−

*(l*−

*C*

_{i}*))*∼=

*O*

_{P}

^{1}

*(−*1

*)*or

*O*

_{P}

^{1}, we have the claim.

If *H*^{2}*(Y,O(l*−*l*^{}*))*=0, then there exists a section *s*∈*H*^{0}*(Y,K*_{Y}*(l*^{}−*l))* by Serre
duality. Since there exists a nonzero section *t*∈*H*^{0}*(Y,O(l*−*C*_{0}^{k}*))*, we have *st*∈
*H*^{0}*(Y,K**Y**(l*^{}−*C*_{0}^{k}*))*∼=*H*^{2}*(Y,O(C*_{0}* ^{k}*−

*l*

^{}

*))*=0, which is a contradiction.

3.4 **Minuscule representations from****(−1)-curves**

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

*α∈Φ*g* _{α}*. As before, we use {

*x*

*’s,*

_{α}*h*

*’s} to denote its Chevalley basis. If*

_{i}*C*

_{0}has minuscule multiplicity

*C*

*, we denote*

_{k}*V*0:=C* ^{I}* =

*l*∈*I*

Cv*l*,

where *v**l* is the base vector of *V*0 generated by *l. Then we define a bilinear map [,*] :
g⊗*V*0→*V*0(possibly up to±signs) as follows:

[x*, v**l*]=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

α,*l* if*x*=*h*_{α}

±v*l*+α if*x*=*x*_{α}*,* *l*+*α* ∈*I,*
0 if*x*=*x*_{α}*,* *l*+*α /*∈*I.*

**Proposition 21.** The signs in the above bilinear mapg⊗*V*_{0}→*V*_{0}can be chosen so that
it defines an action ofgon*V*_{0}. Moreover,*V*_{0}is isomorphic to the minuscule representa-

tion*V*.

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

For the second part, since [x_{α}*, v**C*_{0}* ^{k}*]=0 for any

*α*∈

*Φ*

^{−},

*v*

*C*

_{0}

*is the lowest weight vector of*

^{k}*V*0 with weight corresponding to −C

*k*. Also we know the fundamental representation

*V*corresponding to−

*C*

*has the same dimension with*

_{k}*V*

_{0}by Lemma 16.

Hence,*V*_{0}is isomorphic to the minuscule representation*V.*

Here, we show how to determine the signs. Take any*l*∈*I*,*v**l*is a weight vector of
the above action. For*x*=*x** _{α}*and

*v*

*l*with weight

*w*, we define [x

*, v*

*l*]=

*n*

_{α,w}*v*

*l+α*, where

*n*

*=*

_{α,w}±1 if*l*+*α*∈*I*, otherwise*n** _{α,w}*=0. By [[x

*,y], v*

*l*]=[x

*,*[y

*, v*

*l*]]−[y

*,*[x

*, v*

*l*]], we have

*n*

_{α,β}*n*

*−*

_{α+β,w}*n*

_{β,w}*n*

_{α,β}_{+w}+

*n*

_{α,w}*n*

*=0.*

_{β,α+w}**Remark 22.** Recall that for any*l*=*C*_{0}* ^{k}*+

*a**i**C**i*∈*I*, we define ht*(l)*:=

*a**i*. Using this,
we can define a filtered structure for*I*:*I*=*I*0⊃*I*1⊃ · · · ⊃*I**m*, where*m*=max*l*∈*I*ht*(l)*,*I**i*=
{*l*∈*I*|ht*(l)*≤*m*−*i*}and*I**i*\*I**i*+1= {*l*∈*I*|ht*(l)*=*m*−*i*}. This ht*(l)*also enables us to define a
partial order of*I*. Say|I| =*N, we denotel**N*:=*C*_{0}* ^{k}*since it is the only element with ht=0.

Similarly,*l**N*−1:=*C*_{0}* ^{k}*+

*C*

*k*. 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 in*Y*can be used to construct representation bundles of*E*_{ϕ}^{g}
for every minuscule representation ofg. The proofs of theorems in this subsection will
be given in Section 7.

When*C*0⊂*X*has minuscule multiplicity*C**k*at *p*with 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 *X*to construct bundles.)

L^{(}_{0}^{g}* ^{,V)}*:=

*l*∈*I*^{(g,}^{V}^{)}

*O(l).*

L^{(}_{0}^{g}^{,}^{V}* ^{)}* has a natural filtration

*F*

^{•}: L

^{(}_{0}

^{g}

^{,}

^{V}*=*

^{)}*F*

^{0}L⊃

*F*

^{1}L⊃ · · · ⊃

*F*

*L, induced from the flittered structure on*

^{m}*I*, namely

*F*

*L*

^{i}

^{(}_{0}

^{g}

^{,}

^{V}*=*

^{)}*l*∈*I**i* *O(l).*

L^{(}_{0}^{g}^{,}^{V}* ^{)}* cannot descend to

*X*as

*O*

_{C}

_{k}*(C*

^{k}_{0}

*)*∼=

*O*

_{P}

^{1}

*(*1

*)*(because

*C*

*·*

_{k}*C*

_{0}

*=1 by the definition of the minuscule multiplicity). For any*

^{k}*C*

*and any*

_{i}*l*∈

*I*, we have

*O*

_{C}

_{i}*(l)*∼=

*O*

_{P}

^{1}

*(±*1

*)*or

*O*

_{P}

^{1}by Lemma 18. For every fixed

*C*

*, if there is a*

_{i}*l*∈

*I*such that

*O*

_{C}

_{i}*(l)*∼=

*O*

_{P}

^{1}

*(*1

*)*, then

*(l*+

*C*

_{i}*)*

^{2}= −1=

*(l*+

*C*

_{i}*)*·

*K*

*, that is,*

_{Y}*l*+

*C*

*∈*

_{i}*I*, also

*O*

_{C}

_{i}*(l*+

*C*

_{i}*)*∼=

*O*

_{P}

^{1}

*(−*1

*)*. That means that among the direct summands of L

^{(}_{0}

^{g}

^{,}

^{V}*|*

^{)}*C*

*i*,

*O*

_{P}

^{1}

*(*1

*)*and

*O*

_{P}

^{1}

*(−*1

*)*occur in pairs, and each pair is given by two

*(−*1

*)*-curves in

*I*whose difference is

*C*

*. This gives us a chance to deformL*

_{i}

^{(}_{0}

^{g}

*to get another bundle that can descend to*

^{,V)}*X.*

**Theorem 23.** If there exists a *(−*1*)*-curve*C*_{0} in *X* with minuscule multiplicity*C** _{k}* at

*p*and

*ρ*:g−→End

*(V)*is the corresponding representation, then

L^{(}_{ϕ}^{g}^{,}^{V}* ^{)}*:=

*l*∈*I*

*O(l),* *∂*¯*ϕ*:= ¯*∂*0+*ρ(ϕ)*

*,*

with*ϕ*∈*Ψ**Y* is a holomorphic bundle over*Y*which preserves the filtration onL^{(}_{0}^{g}^{,}^{V}* ^{)}*and
it is a holomorphic representation bundle of

*E*

_{ϕ}^{g}. Moreover,L

^{(}

_{ϕ}^{g}

^{,}

^{V}*is trivial on*

^{)}*C*

*if and only if [*

_{i}*ϕ*

*C*

*i*|

*C*

*i*]=0∈

*H*

^{1}

*(Y,O*

_{C}

_{i}*(C*

_{i}*))*. For

*C*

*with*

_{k}*k*=1, the corresponding minuscule representation

*V*is the standard representation ofg. Wheng=

*A*

*, it is simply sl*

_{n}*(n*+1

*)*=aut

_{0}

*(V)*. Wheng=

*D*

*(respec- tively,*

_{n}*E*

_{6}and

*E*

_{7}), there exists a quadratic (respectively, cubic and quartic) form

*f*on

*V*such thatg=aut

*(V,f)*. The next theorem tells us that we can globalize this construction over

*Y*to recover the Lie algebra bundle

*E*

_{ϕ}^{g}over

*Y. But this does not work forE*

_{ϕ}

^{E}^{8}as

*E*

_{8}has no standard representation.

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

*f*:
*r*

L^{(}_{ϕ}^{g}^{,}^{V}* ^{)}*−→

*O*

*Y*

*(D),*

with*r*=0*,*2*,*3*,*4 wheng=*A**n**,D**n**,E*6*,E*7, respectively, such that*E*_{ϕ}^{g}∼=aut0*(*L^{(}_{ϕ}^{g}^{,V)}*,f)*.
It is obvious that*E*_{ϕ}^{g} does not depend on the existence of the *(−*1*)*-curve*C*_{0}, for
the minuscule representation bundles, we have the following results.

**Theorem 25.** There exists a divisor*B*in*Y*and an integer*k, such that the bundle*L^{(}_{ϕ}^{g}^{,}^{V}* ^{)}*:=

*S*

*L*

^{k}

^{(}

_{ϕ}^{g}

^{,}

^{V}*⊗*

^{)}*O(−B)*with

*ϕ*∈

*Ψ*

*X*can descend to

*X*and does not depend on the existence of

*C*0.

3.6 **Outline of proofs forg**=**E****8**

Wheng=*E*_{8}, there exists a natural symmetric tensor *f* on its standard representation
*V*such thatg=aut_{0}*(V,f)*. The set*I*^{(}^{g}^{,}^{V}* ^{)}*of

*(−*1

*)*-curves has cardinality

*N*=dim

*V. Given*

*η*:=(η

*i,*

*j*

*)*

*N×N*with

*η*

*i,*

*j*∈

*Ω*

^{0}

^{,}^{1}

*(Y,O(l*

*−*

_{i}*l*

_{j}*))*for every

*l*

*=*

_{i}*l*

*∈*

_{j}*I*

^{(}^{g}

^{,}

^{V}*, we consider the operator*

^{)}*∂*¯*η*:= ¯*∂*0+*η*onL^{(}_{0}^{g}^{,}^{V}* ^{)}*:=

*l∈I*^{(g,V)}*O*_{Y}*(l)*. We will look for*η*that satisfies:

(1) (filtration)*η**i,**j*=0 for*i>j*for the partial ordering introduced in Section 3.4;

(2) (holomorphic structure)*(∂*¯0+*η)*^{2}=0;

(3) (Lie algebra structure)*∂*¯*η**f*=0;

(4) (descendent) for every*C** _{k}*, if

*l*

*−*

_{i}*l*

*=*

_{j}*C*

*, then 0=[*

_{k}*η*

*i,*

*j*|

*C*

*k*]∈

*H*

^{1}

*(Y,O*

_{C}

_{k}*(C*

_{k}*))*.

**Remark 26.** Property *(2)*implies that we can define a new holomorphic structure on
L^{(}_{0}^{g}^{,}^{V}* ^{)}*. Properties

*(1)*and

*(3)*require that for any

*η*

*i*

*,*

*j*=0,

*η*

*i*

*,*

*j*∈

*Ω*

^{0}

^{,}^{1}

*(Y,O(α))*for some

*α*∈

*Φ*

^{+}. We will show that if

*η*satisfies (1)–(3), then (4) is equivalent toL

^{(}

_{η}^{g}

^{,}

^{V}*being trivial on every*

^{)}*C*

*, that is,L*

_{k}

^{(}

_{η}^{g}

^{,}

^{V}*can descend to*

^{)}*X.*

Denote

*Ξ*_{Y}^{g}{η=(η*i**,**j**)**N*×*N*|ηsatisfies (1)–(3)}

and

*Ξ**X*^{g}{η∈Ξ*Y*^{g}|ηsatisfies (4)};

then each*η*in*Ξ*_{Y}^{g}determines a filtered holomorphic bundleL^{(}_{η}^{g}^{,}^{V}* ^{)}*over

*Y*together with a holomorphic tensor

*f*on it. It can descend to

*X*if

*η*∈

*Ξ*

*X*

^{g}.

Since g=aut*(V,* *f)*, for any *η*∈Ξ*Y*^{g}, we have a holomorphic Lie algebra bundle
*ζ*_{η}^{g}:=aut(L^{(}_{η}^{g}^{,V)}*,* *f)*over*Y*of typeg, andL^{(}_{η}^{g}* ^{,V)}*is automatically a representation bundle
of

*ζ*

_{η}^{g}. Furthermore, if

*η*∈Ξ

_{X}^{g}, then

*ζ*

_{η}^{g}can descend to

*X.*

For a general minuscule representation ofg, given any*η*∈Ξ_{Y}^{g}, we show that there
exists a unique holomorphic structure onL^{(}_{0}^{g}^{,}^{V}* ^{)}*, such that the action of

*ζ*

_{η}^{g}on the new holomorphic bundleL

^{(}

_{η}^{g}

^{,}

^{V}*is holomorphic. Furthermore, if*

^{)}*η*∈Ξ

^{g}

*, thenL*

_{X}

^{(}

_{η}^{g}

^{,}

^{V}*can descend to*

^{)}*X.*

4 **A****n****Case**

We recall that *A**n*=sl*(n*+1*,*C)=aut0*(C*^{n+1}*)* (where aut0 means tracefree endomor-
phisms). The standard representation of *A**n* is C* ^{n+1}* and minuscule representations of

*A*

*n*are∧

*C*

^{k}

^{n}^{+}

^{1},

*k*=1,2, . . . ,

*n.*

4.1 **A****n****standard representation bundleL**^{(A}_{η}^{n}^{,C}^{n+1}^{)}

We consider a surface*X*with an *A** _{n}*singularity

*p*and a

*(−*1

*)*-curve

*C*

_{0}passing through

*p*with multiplicity

*C*

_{1}; then

*I*

^{(}

^{A}

^{n}

^{,C}

^{n}^{+}

^{1}

*= {*

^{)}*C*

_{0}

^{1}+

*k*

*i=1**C** _{i}*|0≤

*k*≤

*n*}has cardinality

*n*+1. We order these

*(−*1

*)*-curves:

*l*

*k*=

*C*

_{0}

^{1}+

_{n}_{+}

_{1}

_{−}

_{k}*i*=1 *C**i* for 1≤*k*≤*n*+1. For any*l**i*=*l**j*∈*I*,*l**i*·*l**j*=0.