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

Y. Chen and N. C. Leung (2015) “Deformability of Lie Algebra Bundles and Geometry of Rational Surfaces,”

International Mathematics Research Notices, rnv023, 12 pages.

doi:10.1093/imrn/rnv023

Deformability of Lie Algebra Bundles and Geometry of Rational Surfaces

Yunxia Chen

and Naichung Conan Leung

1

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

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

Correspondence to be sent to: e-mail: [email protected]

When X=Xn is a blowup of P² at npoints x1, . . . ,xn with n≤9, there is a canonical (affine) Lie algebra bundleE₀^En over it, where E9 is the affine E8. In this paper, we will give a detail study of the relationship between the geometry ofX_nand the deformability ofE₀^En.

1 Introduction

A rational surface is a surface birationally equivalent to the projective plane, its minimal model is the projective planeP²or the Hirzebruch surfaceFmform=0 orm≥2. In this paper, we considerX=X_n, a blowup ofP²atnpointsx₁, . . . ,x_nwithn≤9.

Whenn≤8,K_Xn^⊥⊂Pic(X_n)is isomorphic toΛEn, the root lattice of the simple Lie algebra E_n, then we have a root systemΦnof E_nand we can associate a Lie algebra bundleE₀^EnoverX_n[4,13,14],

E₀^En:=O_X^⊕n⊕

α∈Φn

O_X(α).

By restriction, we have an En-bundle over any anti-canonical curve Σ in Xn. Note that Σis always an elliptic curve. For a fixed elliptic curveΣ, the above construction gives a

Received August 13, 2014; Accepted January 13, 2015

c The Author(s) 2015. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected].

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bijection between del Pezzo surfaces containingΣandE_n-bundles overΣ[5,6,8,13,18].

Such an identification was predicted by the F-theory/string duality in physics [8]. This was generalized to all simple Lie algebras in [13,14].

When n=9, X₉ is not Fano and E₉= ˆE₈ is an affine Lie algebra. Corresponding results for theEˆ8-bundle overX9are obtained in [12].

In this paper, we explain how the geometry ofX9can be reflected by the deforma- bility of theEˆ8-bundleE₀^Eˆ8 over it. Similar results forXnandE₀^Enwithn≤8 can be easily deduced from this case. Among other things, we obtained the following results.

Theorem 1(Theorem 8). E₀^Eˆ8 is totally nondeformable if and only if the nine blowup

points inP²are in general position.

Theorem 2(Theorem 9). Suppose−K_X9 is nef, then

(i) X₉ admits an elliptic fibration with a multiple fiber of multiplicity m (m≥1) if and only if E₀^Eˆ8 is deformable in (−mK)-direction but not in (−m+1)K-direction.

(ii) X₉ has a(maximal) AD E curveC of typegif and only ifE₀^Eˆ8 is(maximal) Q1 g-deformable.

(iii) X9 has a (maximal) affine AD E curve C of type gˆ if and only if E0^Eˆ8 is

(maximal)g-deformable.ˆ

Here ADE means Lie algebras of types A_n,D_n, E₆, E₇andE₈, wherencan be any nature numbers.

The organization of this paper is as follows. Section 2 gives the construction of the (affine) AD E Lie algebra bundles over Xn. In Section 3, we state the definition of deformability and the main theorems. Section 4 studies the negative curves in X9. The proofs of the main theorems are in Section 5.

2 E_n-Bundle OverX_nwithn_≤9

The Picard group Pic(X_n)∼=H²(X_n,Z)is a rankn+1 lattice with generators h,l₁, . . . ,l_n, wherehis the class of lines inP² andl_i is the exceptional class of the blow-up at x_i. Soh²=1= −l_i² andh·l_i=0=l_i·l_j, i=j. Thus, H²(X_n,Z)∼=Z^1,n. The canonical class is K_Xn= −3h+l₁+ · · · +l_n. Denote

Φn:= {α∈H²(Xn,Z)|α²= −2, α·K=0}.

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Then Φn is a root system of type E_n when n≤8 and Φ9 is an affine real root system ofEˆ₈(also denoted as E₉). More explicitly,Φ_Eˆ8:=Φ9∪ {mK_X9|m=0,m∈Z}forms a root system of (untwisted) affine E₈-type (i.e., Eˆ₈-type) with Φ^re_Eˆ

8:=Φ9 the set of real roots andΦ_E^im_ˆ

8 := {mK_X9|m=0,m∈Z}the set of imaginary roots (see [10] or [12]). We have an Eˆ8-bundleE₀^Eˆ8 over X9:

E₀^Eˆ8=O^⊕9⊕

α∈Φ^re_Eˆ

8

O(α)

β∈Φ^im_Eˆ

8

O(β).

The Lie algebra structure onE₀^Eˆ8is explained in [12]. Whenn≤8,E₀^En=O^⊕n⊕

α∈ΦnO(α) is anE_n-bundle overX_n.

Remark 3. We can construct anEˆ8-bundle over a blowup ofFm(Hirzebruch surface)at

eight points similarly [3].

Definition 4. A curveC= ∪C_i in a surfaceXis called anAD E(respectively, affine AD E) curve of typeg(respectively,g) if eachˆ C_i is a smooth(−2)-curve in Xand the dual graph

ofC is a Dynkin diagram of the corresponding type.

Suppose C= ∪C_i is an (affine) AD E curve of type gin X_n, then C_i’s generates a subroot system Φ inside Φn sinceCi·K=0 for everyi. Therefore, the corresponding bundleE0^gis a Lie algebra subbundle ofE0^En.

SupposeE₀^g is ag-bundle over a surface X corresponding to a root systemΛg⊂ Pic(X)of typeg.

Definition 5. A Lie algebra subbundle F of E₀^g is called strict, if there exists a sub- root lattice Λof Λg such thatF is a direct sum of line bundles corresponding to the

roots inΛ.

In order to describeE0^Eˆ8 as a central extension of a loop Lie algebra bundle over X9, we pick any smooth(−1)-curvel inX9, then we have

E₀^Eˆ8∼=E₀^E8⊗

n∈Z

O(nK_X9)

⊕O,

whereE₀^E8 is the pull-back of theE₈-bundle overX₈viaπ:X₉→X₈, the blow down map ofl. The next proposition describes the converse.

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Proposition 6. WhenE₀^Eˆ8 is a central extension of a loop E₈-subbundle over Xfor some strictE₈-bundleF₀^E8 overX₉, that is,

E₀^Eˆ8∼=F₀^E8⊗

n∈Z

O(nKX9)

⊕O,

as a Lie algebra bundle isomorphism, then there is a unique(possibly reducible) (−1)- curvelin Xsuch thatF₀^E8 is constructed from thoseα∈Λ^resatisfyingα·l=0.

Proof. DenoteΔE8= {α1, . . . , α8}as a root base of the correspondingE₈Lie algebra from F₀^E8, we need to find a unique (−1)-curve l in X such that l·αi=0 for any αi in ΔE8. Since {±1} ×W(Eˆ₈)acts on the set of all root bases of Eˆ₈ simply transitively [11] and W(Eˆ₈)acts on the set of(−1)-curves [12], we only need to findl for one particular root base of any E₈ in Eˆ₈ and show that such a l is unique. For example, if we take α1= h−l1−l2−l3, αk=lk−1−lk for k=2, . . .8, then we can take l=l9 and by the condition thatl·αi=0,l²= −1=l·K, we know such al is unique.

3 Deformability of suchE₀^Eˆ8

In this section, we will describe relationships between the geometry of X9 and the deformability ofE₀^Eˆ8.

Recall when Pic(X)contains a latticeΛisomorphic to a root latticeΛg, then we have ag-bundleEover X[5,8,12–14].

E:=O^⊕r⊕

α∈Φ

O(α).

Infinitesimal deformations of holomorphic structures on E are parameterized by H¹(X,End(E)), and those which also preserve the Lie algebra structure are parame- terized by H¹(X,ad(E))=H¹(X,E)sincegis simple. Hence we introduce the following definitions.

Definition 7.

(i) E is called fully deformable if there exists a base Δ⊂Φ such that H¹(X,O(α))=0 for anyα∈Δ.

(ii) E is called h-deformable if there exists a strict h Lie algebra subbundle E^h⊆Ewhich is fully deformable.

(iii) Eis called deformable inα-direction forα∈Φ ifH¹(X,O(α))=0.

(iv) Eis called totally nondeformable if H¹(X,O(α))=0 for anyα∈Φ.

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After the definition of deformability, we state the main results of this paper in the following two theorems.

Theorem 8. E₀^Eˆ8 over X₉is totally nondeformable if and only if the nine blowup points

inP²are in general position.

Let us recall some facts about elliptic fibrations on X₉[15,17]. Any elliptic fibra- tion on X₉ must be relatively minimal, that is, there is no (−1)-curves in any of its fibrations, as there is no elliptic fibration onX₈, this is because the Euler characteristic of any elliptic surface is a multiple of 12 [7] and alsoχ(X₉)=12. There is at most one multiple fiber [9], say of multiplicitym. This happens precisely when there exists an irreducible pencil of degree 3minP²with nine base points, each of multiplicitymand X9is the blow up ofP²at these nine points. We can characterize the existence of such an elliptic fibration onX9in terms of deformability ofE0^Eˆ8along imaginary root directions.

For instance,X9with−KX9nef admits an elliptic fibration (without multiple fiber) if and only ifE₀^Eˆ8is deformable in(−mK)-direction for somem∈N(withm=1). Deformability ofE₀^Eˆ8can also detect the existence of AD E or affineAD E curves in X.

Theorem 9. Suppose−KX9 is nef, then

(i) X9 admits an elliptic fibration with a multiple fiber of multiplicity m (m≥1) if and only if E₀^Eˆ8 is deformable in (−mK)-direction but not in (−m+1)K-direction.

(ii) X₉has an(maximal) AD E curveC of typegif and only ifE₀^Eˆ8 is(maximal) g-deformable.

(iii) X₉ has a (maximal) affine AD E curve C of type gˆ if and only if E₀^Eˆ8 is

(maximal)g-deformable.ˆ

Here, we say an AD E or affine AD E curveC is maximal if it is not proper con- tained in another AD E or affineAD E curve. We sayE₀^Eˆ8 is maximalg(org) deformableˆ if there does not exist another fully deformable (affine) Lie algebra subbundle ofE₀^Eˆ8 containing thisg(org) bundle.ˆ

4 Negative Curves inX₉

In this section, we study negative rational curves inX₉. We can get corresponding results forXnwithn≤8 from thisn=9 case.

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A divisorDinXis called a(−m)-class ifD·D= −mandD·K=m−2. An effec- tive(−m)-class is called a(−m)-curve. Note when D=

n_iC_i is a(−m)-curve, we will also denote the corresponding curve∪C_i asD.

Use the notations in the above section, every effective divisorD=ah−9 i=1a_il_i∈ Pic(X9)must havea=D·h≥0. It is well known that all(−1)-classes are effective, and there are infinite number of them inX9. There are also infinite number of(−2)-classes, but whether they are effective or not depends on the positions of the nine blow-up points.

Definition 10. Let x₁, . . . ,x_n be ndistinct points in P². These npoints are said to be nonspecial with respect to Cremona transformations if for any Cremona transformation Twith centers withinxi’s, the pointsy1, . . . ,yncorresponding toxi’s underT are distinct points such that no three points amongy1, . . . ,ynare collinear.

Definition 11([12]). Letx₁, . . . ,x₉be nine points inP², we say they are in general position if they satisfy the following three conditions:

(i) they are distinct points inP²;

(ii) they are nonspecial with respect to Cremona transformations;

(iii) there is a unique cubic curve passing through all of them.

The conditions (i) and (ii) mean that any eight of these nine points are in general position. That is, no lines pass through three of them, no conics pass through six of them, and no cubic curves pass through eight of them with one of the eight points being a double point.

If the 9 blowing up points are in general position, then there is no effective (−2)-class in X9 [12]. In general, there are at most finite number of (−m)-curves with m≥3.

Lemma 12. Let D=ah−₉

i=1ailibe a(−m)-curve inX9withm≥3, then (i) m≤9;

(ii) 0≤a≤3;

(iii) −1≤ai≤2 for alli, and there exists some jwithaj=1;

(iv) there are finite number of such curves.

Proof. (i) SinceDis a(−m)-curve,D·D= −mandD·K=m−2, that is, a_i²=a²+m and

ai=3a+m−2.

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From the above two equations, we have (3a+m−2)²=

a_i ²≤9

a_i² =9(a²+m).

Thus,a≤^−m₆²₍⁺_m^13m₋₂₎⁻⁴, alsoa≥0 since Dis effective, hencem≤12.

Whenm≥10, we must havea=0, that means

a_i²=mand

a_i=m−2, hence a_i²−

a_i=2, which implies every a_i satisfies |a_i| ≤1 and there exists exactly one a_i witha_i= −1. But we also have

a_i=m−2≥8, which is impossible since we only have nineai’s.

(ii) Whenm≥4,a≤^−m₆²₍⁺_m^13m₋₂₎⁻⁴≤⁸₃<3. Whenm=3,a≤^−m₆²₍⁺_m^13m₋₂₎⁻⁴=¹³₃ <5. Hence we only need to prove there is no(−3)-curve witha=4.

Suppose not, then there exists ai’s such that

a_i²=19 and

ai=13. From a_i²−

ai=6, we know −2≤ai≤3. If there is anyai with ai=3, then the other ai’s can only be 0 or 1, but we have

a_i=13 and there is only ninea_i’s, which is impossible.

Hence−2≤a_i≤2, from

a_i²−

a_i=6, we can have at most threea_i’s equal to 2, which is also impossible since

a_i=13.

(iii) From

a_i²=a²+m,

a_i=3a+m−2 and 0≤a≤3, we have a_i=3a+m−2≥a²+m−2=

a²_i −2. Hence−1≤a_i≤2. And there are three cases:

Case1, oneai equal to 2, the others equal to 0 or 1;

Case2, oneai equal to−1, the others equal to 0 or 1;

Case3, allai’s are equal to 0 or 1.

By

ai=3a+m−2≥1, we know in case 2 and case 3, there must exist someai

withai=1. In case 1, if there is noai withai=1, thenD=ah−2lj. From

a²_i =a²+m, a_i=3a+m−2, we havea=0,m=4, henceD= −2l_j, which is not an effective divisor.

(iv) It is obvious from the above results.

From this lemma, we can easily obtain the following as a corollary.

Corollary 13. If there exists a (−m)-curve in X₉ with m≥3, then there also exists a (−m+1)-curve inX₉.

Proof. If D∈ |ah−

aili| is a (−m)-curve in X9 with m≥3, then there exists j with aj=1 by (iii) of Lemma 12. It is easy to check thatD+ljis a(−m+1)-curve inX9. If the nine blowing up points are in general position, then there is no(−2)-curve in X9, as a consequence, there is also no (−m)-curve in X9 with m≥3. The following

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result shows that this happens exactly whenX₉is almost Fano. We include a proof here as we could not find it in the literatures.

Lemma 14. X9has no(−m)-curve withm≥3 if and only if−KX9 is nef.

Proof. If−Kis nef, then fromC ·K⁻¹=2−m≥0 for any(−m)-curveC, we knowm≤2.

Conversely, assume X₉ has no(−m)-curve with m≥3. Since X₉ is a blowup of P²at nine points{x_i}⁹_i=1, we have an effective anti-canonical divisor D. Recall when D· Σ <0 for any irreducible curveΣ in X, Σ must be a component of D. So if D is an irreducible curve or a affine AD E curve, then Dis nef. We denote the image of DinP² asC, which is a cubic curve passing through these 9 blowing up points.

(i) IfC is smooth, then we are done as D∼=C and therefore irreducible.

(ii) IfC is reduced and irreducible, then it must be a nodal or cuspidal cubic.

If{x_i}⁹_i=1∩sing(C)=∅(sing(C)means the set of singular points onC), then D∼=C and we are done. Otherwise, sayx₁∈sing(C)and we write the strict and proper transformations ofC in Bl_x1(P²)asC₁andC₁+E,respectively.

Then the remaining x_i’s must have exactly 1 point(respectively, 7 points) lying on E (respectively, C₁) in order to avoid having (−m)-curve with m≥3. Thus, D is a affine AD E curve of type Aˆ₁ or I I I(Aˆ₁)for C being a nodal or cuspidal, respectively.

(iii) IfC is reduced and reducible, thenC=B∪H0or H1∪H2∪H3 with B and Hj’s are conic and distinct lines inP². As before, we must have exactly 6 xi’s on Band 3xi’s on each Hj and none on sing(C). Thus, D∼=C is a affine AD E curve of typeAˆ1, Aˆ2, I I I(Aˆ1),orV I(Aˆ2).

(iv) IfC is nonreduced,C=3H,Dmust have a(−m)-curve withm≥3.

HenceDis an irreducible curve or a affineAD E curve, we are done.

In the following two lemmas, we will use [1, Lemma 2.21] to give a criteria of a curve in X_nbeing an AD E or affine AD E curve. Lemma 2.21 can be reformulated as follows: ifC=_r

i=1Ci is a connected curve in a surfaceXsatisfying: (i)C_i²= −2 andCi· KX=0 for anyi; (ii)Ci·Cj≤1 for anyi=j; (iii)(Ci·Cj)r×r≤0. Then when(Ci·Cj)r×r<0, C is an AD Ecurve, otherwise, it is an affine AD E curve.

Lemma 15. Suppose −K_Xn (n≤8)is nef. LetC= ∪C_i be a connected curve in X_n. If C·

KXn=0, thenC is an AD Ecurve.

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Proof. Since −K_Xn is nef, C·K_Xn=0 implies C_i·K_Xn=0 for each i, that is, [C_i]∈ K^⊥∼=ΛEn. We haveC_i²<0 and(C_i+C_j)²<0 for anyiand j. Together with the genus for- mula, we haveC_i²= −2 andC_i·C_j≤1 fori=j. By [1, Lemma 2.21], we knowC is anAD E

curve.

Forn=9 case, we have the following lemma.

Lemma 16. Suppose−KX9 is nef. LetC= ∪Ci be a connected curve in X9. If C·KX9=0 andCi+KX9 is not effective for eachi, thenC is a smooth elliptic curve, an AD E curve

or an affineAD Ecurve.

Proof. Since −K_X9 is nef, C ·K_X9=0 implies C_i·K_X9=0 for each i, that is, [C_i]∈ K_X9^⊥∼=ΛE9. We haveC_i²≤0 and(C_i+C_j)²≤0 for anyiand j. Moreover, for any effec- tive divisor D∈ K_X9^⊥, if D²=0, then D∈ |mK_X9| for some nonzero integer m. From C_i²≤0 and genus formula, we haveC_i²= −2 or 0.

If there existsCi such that C_i²=0, then Ci∈ |mK| for some nonzero integerm.

SinceCi+KX9is not effective, we knowm= −1, that is,Ci∈ | −K|. IfCis not irreducible, then there existsCj which intersects Ci, which is impossible. SoC=Ci∈ | −K| is an elliptic curve or an affine A0curve by Lemma 14.

IfC_i²= −2 for anyi, thenC_i·C_j≤2 for anyi=j. If there existC_iandC_jsuch that C_i·C_j=2, then(C_i+C_j)²=0,C_i+C_j∈ |mK|for some integerm. HenceC =C_i∪C_jis an affineA₁curve, this is because ifC_kis another irreducible component ofC and assume it intersects withC_i, then it must be an irreducible component ofC_j, which contradicts to C_j being irreducible. Otherwise, we will have C_i²= −2 for each i andC_i·C_j≤1 for i=j. By [1, Lemma 2.21], we knowC is anAD E or affine AD Ecurve.

5 Proof of Theorems 8 and 9

Proof of Theorem 8. If the nine blowup points in P² are in general position, then for any α∈Φ9, we have h⁰(X,O(α))=0 [12]. Since K·K=0, we also have K−α∈Φ9 and therefore h²(X,O(α))=0 by Serre duality. However, the Riemann–Roch formula gives χ(X,O(α))=1+^α2−αK₂ =0 and thereforeh¹(X,O(α))=0. For the imaginary rootsmK’s, from [12, Lemma 4 and Proposition 11], we haveh⁰(X,O(mK))=0 andh⁰(X,O(−mK))= 1 form≥1. By Serre duality and Riemann–Roch formula, we haveh¹(X,O(mK))=0 for any imaginary rootmK. HenceE₀^Eˆ8 is totally nondeformable.

Conversely, if E₀^Eˆ8 is totally nondeformable, then X has no(possibly reducible) (−2)-curve, hence no (−n)-curve with n≥2. By [16, Proposition 10], this implies the

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nine blowup points are nonspecial with respect to Cremona transformations. Also from h¹(X,O(mK))=0 for any imaginary root mK, we obtain h⁰(X,O(−K))=1, we have a unique cubic curve in P² passing through all of the blow-up points. Hence, the nine

blow-up points inP²are in general position.

Proof of Theorem 9. (i) We have h¹(X,O(−mK))=h⁰(X,O(−mK))−1 for any m by Riemann–Roch formula. So E₀^Eˆ8 is deformable in (−mK)-direction if and only if h⁰(X,O(−mK))=2.

Let F0∈ | −K|, then by [2, Proposition 2.2], X admits an elliptic fibration with a multiple fiber of multiplicity m if and only if OF0(F0) is of order m in Pic(F0). But OF0(mF0)∼=OF0 if and only if h⁰(OF0(mF0))=1 as OF0(mF0) is topologically trivial. By the exact sequence

0−→OX−→OX(mF0)−→OF0(mF0)−→0

together with h¹(X,OX)=0, we know h⁰(OX(mF0))=1+h⁰(OF0(mF0)). So m=min {n:h⁰(OF0(nF0))=1} =min{n:h⁰(X,O(−nK))=2}.

(ii) If X has an AD E curve C of type g, we can use it to construct a fully deformable g-subbundle of E₀^Eˆ8 as in Section 3.2. When C is maximal, then this g-subbundle is not contained in any other fully deformable Lie algebra subbundle ofE₀^Eˆ8. Conversely, if E₀^Eˆ8 is maximal g-deformable, then we can find a baseΔ⊂ΦEˆ8 of gsuch thath¹(X,O(α))=0 for everyα∈Δ. Sinceχ(O(α))=1+^α2−α·₂ ^K=0, we must have h⁰(O(α))=0 orh²(O(α))=h⁰(O(K−α))=0, that is eitherαor K−αis effective. Hence, there must exist some integersm’s such thatα+mKis effective because−Kis effective, we denote the largest suchmasm_α.

We claim that for everyα∈Δ,C_α∈ |α+m_αK|is an irreducible(−2)-curve. If so, thenC=

α∈ΔC_αis a maximalAD Ecurve of typeg. If there exists reducibleC_α, we write C_α= ∪Di. Then each Di is perpendicular toK as−K is nef andC_α·K=0. SinceC_α+K is not effective, everyD_i+Kis also not effective and D_i∈ | −/ K|. HenceD_i²= −2 for any ias D²_i =0 will imply D_i∈ | −K|. We knowC_α is connected, this is because ifC_α is not connected, then one of its connected component must have self-intersection zero from C_α²= −2, which contradicts toC_α+K is not effective. HenceC=

α∈ΔC_α is an (affine) AD E curve by Lemma 16. It is obvious that this curve strictly contains ag-curve, which contradicts toE₀^Eˆ8 being maximalg-deformable.

(iii) The proof is similar to (ii).

Remark 17. If X₉ admits an elliptic fibration, then we can find m such that h¹(X9,O(−mK))=0. Conversely, ifh¹(X9,O(−mK))=0, we need to add the condition of

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−Kbeing nef to show thatXadmits an elliptic fibration. To see this, we takex₁, . . . ,x₅to be five points on a linel⊂P², and another four generic points(not onl)x₆, . . . ,x₉inP². Then we have an one parameter family of conicsC_t’s passing through these four points.

If we blow upP²at these nine points and denote the strict transforms ofl andC_twith same notations, thenl²= −4,C_t²=0. Moreover,Ct+l∈ | −K|andh⁰(X9,O(−K))=2. But

−Kis not nef as(−K)·l= −2, which implies that X9is not elliptic.

From the above, we can easily deduce similar results for the En-bundleE₀^Enover X_nwhenn≤8, namely

(i) E₀^Enis totally nondeformable if and only if thenblowup points inP²are in general position.

(ii) When−K_Xnnef,E₀^Enis maximalg-deformable if and only ifX_nhas a maximal gcurve.

Acknowledgement

We are grateful to R. Friedman, E. Looijenga, and J.J. Zhang for many useful comments and discussions.

Funding

The work of the second author was supported by a research grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (reference No. 401411).

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