New Moduli Spaces

as well as a reduction R : X → P/Q_G(λ), we nd for every choice of two rep-resentatives of s^j(x^j)Q_G(aⁱ) and R(x^j)Q_G(λ) a g ∈ Q_G(λ)\G/Q_G(aⁱ) that maps one to the other - s^j(x^j)g = R(x^j).⁴⁸ In 2.7 we will dene the weight func-tion µ(λ, s^j) = −hλ, g⁻¹a^jgi. Since the value of µ depends only on the class of g ∈ Q_G(λ)\G/Q_G(a^j) we may assume w. l. o. g. that gλ(z)g⁻¹ ∈T_a^j. Let(vⁱ_aj)_i be the weight vectors to T_a^j with weights (χⁱ_aj)_i. Then

gλ(z)g⁻¹v_aⁱj =σ(gλ(z)g⁻¹, v_aⁱj) =χⁱ_aj(gλ(z)g⁻¹)v_aⁱj

⇒λ(z)g⁻¹v_aⁱj =χⁱ_aj(gλ(z)g⁻¹)g⁻¹v_aⁱj,

i. e. v^ij_λ :=g⁻¹vⁱ_aj is a weight vector of Tλ with weight χ^ij_λ =χⁱ_aj(g·. . .·g⁻¹). In particular there is an i₀ such thatχⁱ_a⁰j =χ_a^j and thus

χⁱ_λ^0j(λ) = χ_a^j(gλg⁻¹) = hλ, g⁻¹a^jgi.

We have seen above, that using a trivialization φ associated to R, s^j(x^j) maps to g⁻¹v^j whileR(x^j) maps tov^j -v^j weight vector to χ_a^j. Hence

−min





 χ^ij_λ(λ)

a^ij 6= 0, φ(s^j(x^j)) =

dim(Vσ)

X

k=1

a^kjv^kj_λ







=−χⁱ_λ^0j(λ)

=−hλ, g⁻¹a^jgi.

Remark. In 2.7 it can be seen, that the right-hand term is constant on the class Q_G(λ)\G/Q_G(a^j), i. e. independent of the chosen trivializations used to dene s^j(x^j) and R(x^j).

a tuple of representations σ_a : Gl(C^ra) ,→ Gl(V_σa). The weights of σ_a and σ are connected as usual (cf. 1.5). We denote the weights as in the classical parabolic case by(β^ij)i[r]j[|S|] and (β_a^ij)a[|A|]i[r_a]j[|S|].

1.72. Denition. Choose for every puncture x^j ∈ S a tuple of representations σ^j : Gl(C^ra)a→Gl(Vσ)as above and denote by σ^j_athe resulting representations of Gl(C^ra). Denote by (β^ij)_i[r], β^1j ≥. . .≥β^rj the maximal weight ofσ^j.

Consider a tuple((E_a, (s^j_a)j[|S|])a[|A|], ϕ, L)fors^j_a∈

×

a∈AP(E_σa)(cf. 1.2). Recall that every proper ltration (F^k, α^k)k[m], α^k ∈ Q⁺ as in 1.6 comes from a one-parameter subgroup λ: C^∗ →Sl(C^ra)_a resp. one-parameter subgroups λ_a : C^∗ → Gl(C^ra). We call a tuple (κ_a, ξ_a, δ, ε^j)−(semi)stable if

M^κ,ξ(Fⁱ, αⁱ) +δµ(Fⁱ, αⁱ, ϕ) +ε^j X

j:x^j∈S

µ^σj(λ, s^j) (≥) 0 holds for all weighted ltrations(Fⁱ, αⁱ)_i[r] (as in 1.6) and

M_par^κ,ξ(Fⁱ, αⁱ) =

r

X

i=1

αⁱ·(deg(E) rk(Fⁱ)−deg(Fⁱ) rk(E)

+X

a∈A

ξ_a(rk(E_a) rk(Fⁱ)−rk(F_aⁱ) rk(E)))

µ(Fⁱ, αⁱ, ϕ) :=−min ( _u

X

j=1

γ^ij

(ij)_j[u] ∈ {1, . . . , r}^u :ϕ|

(^Nuj=1F^ij)^⊕v 6≡0 )

,

where µ^σj(λ, s^j) is dened as in (WF) with a trivialization like the one of 1.69.

Remark. Using 1.66 the following construction extends as in 1.65 to tuples ((E_a)a[|A|], ϕ, (s^j)j[|S|])with s^j ∈P(E_σa)|_x^j.

1.73. Recall that in 1.16 we constructed a parameter space X and then added Graÿmannian varieties G^ij_a to parametrize parabolic ltrations. Now we know that there are u^σaj, v^σaj, w^σja ∈ Z such that E_σ^j_a ⊂ (E^⊗uσ

aj

a )^⊕vσ

aj

⊗det(E_a)^⊗wσ

ja,

and as before we nd sheavesFj^σaj = V^⊗uσ

ja

a

^⊕vσ

ja

⊗π_X^∗(O_X(−u^σja·n))|_x^j, Kj^σja = det(EQa)^⊗wσ

j

a|_x^j, X^σaj =P(H om(πQ,∗(F_j^σaj), πQ,∗(K_j^σaj))) −→^π Q. We have again a tautological morphism

ψX^σja : (π×id_X)^∗(Fj^σaj)→(π×id_X)^∗(Kj^σja)⊗π^∗

X^σja(O

X^σaj(1)).

We may pull these morphisms back to X ×T×

×

1≤j≤|S|

a∈A

X^σaj and nd a closed subscheme G_par ⊂

×

1≤j≤|S|

a∈A

X^σja such that

×

a∈Aψ

X^σja splits over E^T×Gpar,σ. The universal properties are proved as in the classical case.

For the Gieseker morphism we replace the Plücker embedding in the last com-ponent by the identity. Then all proofs and calculations of the classical case apply in this situation as well, with one exception; namely the admissibil-ity condition. It may happen that given a ltration (V^k)_k[p] of V as before, q_uσj, v^σj|

(Vθ)^⊕vσj = 0, V_θ = Nu^σj

i=1V^θ(i) even if the induced map on E_σ|_x^j is non-zero on the subbundles generated by (Vθ)^⊕vσ resp. their intersection with Eσ|_x^j. Thus we have to bind −(χⁱ_λ^0j(γ_r^rjk) − χⁱ_λ^0j(γ^r

jk

rcoh)) where r^jk = dim(F^k)|_x^j and r^jk_coh = dim(F^k,coh)|_x^j.⁴⁹ This term is bounded by rPr^jk

i=r_coh^jk β_i^ij₀ and therefore

−(χⁱ_λ^0j(γ_r^rjk)−χⁱ_λ^0j(γ^r

jk

rcoh)) ≤ β^1j(r^jk −r_coh^jk )r. Hence we call the stability param-eters ε^j admissible if they are positive, decreasing and ε^jβ^1j < 1 or equivalently if ε^jPs^j

i=1δ^ij < 1. If σ is not irreducible, i. e. decomposes σ^j = ⊕^m_t=1σ_t^j, the admissibility condition becomes ε^jmax{β_t^1j : 1 ≤ t ≤ m} < 1 for (β_t^ij)_i[r] the maximal weight of σ_t^j.

49χⁱ_λ^0jis the character ofTλto the weight vectorv^i0jwith non-zero coecienta^i0jand minimal weight function. Let(β_i^ij

0)i be the corresponding weight. In the notation of 1.52−(χⁱ_λ^0j(γ_r^rjk)− χⁱ_λ^0j(γ^r

jk

rcoh))corresponds toPs^j i=1δ_i^ij

0(f^ijk−f_coh^ijk)r.

2 The Moduli space of Projective Parabolic Higgs Bundles

The second chapter studies projective parabolicς-Higgs bundles and their moduli space.

2.1. Principal Bundles. An algebraic (resp. holomorphic) principal G-bundle on the Riemann surface X is a C-scheme (resp. complex space) P with a right action σ : P ×G → P and a G-invariant projection π : P → X such that P is locally trivial in the étale topology (resp. strong topology). For algebraic G -bundles we may equivalently choose a trivialization in the fppf-topology or that locally X admits an unramied cover V → U ⊂ X such that the local pullback of P is trivial, i. e. P ×_X V ' V ×G ([Mi80] 4.10, [Sch08] p. 101f). Note that the category of holomorphic G-bundles (with G-equivariant holomorphic maps) onX is equivalent to the category of algebraic G-bundles (withG-equivariant X -morphisms).¹

We are mostly concerned with connected reductive algebraic groups, for which the trivialization may be chosen in the Zariski topology ([Sch08] 2.1.1.17) on X. More generally, for a scheme of nite typeY a principalG-bundle with connected reductive structure group onY ×X is trivial w. r. t. the product of the étale and the Zariski topology on Y ×X. Bundles with respect to not necessarily connected reductive algebraic groups may however occur when we consider H-bundles for H ⊂Ga subgroup.

Given a parabolic subgroup P^j ⊂ G for every punctures x^j ∈ S a parabolic (principal) G-bundle is a pair (P, (s^j)_j[|S|]) with P a principal G-bundle and s^j : {x^j} →P ×_X {x^j}/P^j reductions.

2.2. Projective Higgs Bundles. Let W be a vector space and P(W) the corresponding projective space. Let P be a principal G-bundle of xed topolog-ical type on X and φ ∈ H⁰(X,P(P_ς)) for a xed homogeneous² representation

1see for example [GAGA] and [Ser58].

2ς homogeneous :⇔ς homogeneous in 2.12. See as well 2.14.

ς : G →Gl(W) and P_ς =P ×_ς W, P(P_ς) ' P ×_ςP(W).³ To give φ is equivalent to the choice of a line bundle L and a surjection ϕ : P_ς → L ([Ha77], II.7.12).

In order for our construction to work, we will allow ϕ to be arbitrary non-trivial for now. Once a (projective) parameter scheme is constructed parametrizing non-trivial homomorphisms ϕ : P_ς → L, the surjective ϕ will form an open invariant subset thereof.

The triple (P, ϕ, L) is called a projective ς-Higgs bundle. A projective parabolic ς-Higgs bundle is a quadruple (P, (s^j)_j[|S|], ϕ, L) with additional reductions s^j for every puncture.

2.1. The Semistability Concept of Parabolic G-Bundles

In this rst section we will dene a semistability concept for projective parabolic ς-Higgs bundles. We will then rewrite the semistability criterion in terms of an associated parabolic Higgs tuple.

2.3. Let P be a principal G-bundle on X. Fix a faithful representation ι : G→ Gl(U), U a vector space. Denote by P_ι the principal Gl(U)-bundle induced by ι. LetP^j, 1≤j ≤ |S|be a tuple of parabolic subgroups of G- one for each puncture x^j ∈S - and choose reductions s^j :{x^j} →P ×_X {x^j}/P^j.

We follow the approach by [Bra91] (see as well [HS10].4) to dene the concept of (semi)stability for tuples (P, (s^j)_j[|S|], ϕ).

For a one-parameter subgroup λ∈Hom(C^∗, G)denote P_G(λ) := {g ∈G|lim

z→0λ(z)gλ(z)⁻¹ exists in G}, Q_G(λ) := P_G(−λ).⁴ LetTι ⊂Gl(U)be a maximal torus corresponding to a basis(uⁱ)_i[dim(U)] and denote ( ·, · ) the symmetricQ−bilinear map

Tˆι ×Tˆι →Q, Tˆι = Hom(C^∗, Tι)⊗_ZQ. induced by

Z^dim(U)×Z^dim(U) 3((aⁱ)_i[dim(U)],(b^j)_j[dim(U)])7→

dim(U)

X

i=1

aⁱbⁱ ∈Z.

Furthermore we nd the dual pairing h · , · i : ˆT_ι ×Tˇ_ι → Q (cf. 1.34) for Tˇι := Hom(Tι,C^∗)⊗_Z Q the rational character group. Hence for every rational one-parameter subgroupλ_Gl ∈Tˆ_ι there is a rational character χ_λ,Gl ∈Tˇ_ι such that

(λ⁰, λ_Gl) =hλ⁰, χ_λ,Gli, ∀λ⁰ ∈Tˆ_ι.

3In abuse of notation we wroteς for both the action onW and the induced action onP(W).

4−λdenotes the inverse element ofλin the groupTˆι, i. e. z7→λ(z)⁻¹.

In factλ_Gl denes a character in Hom(Q_Gl(λ_Gl),C^∗)⊗_ZQ, which we call in abuse of notation χ_λ,Gl, too. Further if T_ι is the extension of a maximal torus T ⊂ G and λ_Gl =ι◦λ, λ ∈Hom(C^∗, G)⊗_ZQ, then the pairing

h · , · i: ˆT ×Tˇ→Q, Tˆ := Hom(C^∗, T)⊗_ZQ, Tˇ:= Hom(T,C^∗)⊗_ZQ is induced by the canonical pairing for the group Gl(U). Observe that this map is independent of the chosen extension Tι of T.⁵ Analogously we nd χλ and we haveQ_G(λ) =Q_Gl(λ_Gl)∩Gas well as χ_λ,Gl|_QG(λ) =χ_λ.

2.4. Denition. A characterχ:Q_Ad →C^∗, Q_Ad⊂Ad(G)⁶ a parabolic subgroup, is called anti-dominant if the line bundle PQ_Ad(χAd) is ample. Here PQ_Ad denotes the Q_Ad-bundle Ad(G) → Ad(G)/Q_Ad and P_QAd(χ_Ad) the line bundle associated byχ_Ad.

If Q ⊂ G is a parabolic subgroup and QAd ⊂ Ad(G) is the induced parabolic subgroup, thenχ:Q→C^∗ is called anti-dominant, ifχ= Ad◦χ_Ad, Ad :Q→Q_Ad holds for an anti-dominant character χ_Ad of Q_Ad.

IfG is semisimple, χ is anti-dominant, ifPQ(χ) is ample.

2.5. Proposition. Let G be a semi-simple linear algebraic group. The map Gˆ 3 λ → (P_G(λ), χ−λ) into the set of pairs of a parabolic subgroup and a dominant character χ_−λ is surjective.⁷

Gˆ 3 λ → (Q_G(λ), χ_λ) into the set of pairs of a parabolic subgroup and an anti-dominant character χ_λ is surjective, too.

Every parabolic subgroup of a (connected) reductive group is of the formQ_G(λ)for some one-parameter subgroup λ of G.

Proof. [GLSS08], section 3.2 or [Sch04], Example 2.1.8. The last statement is proven in Springer [Sp81], Proposition 8.4.5.

Remark. For future reference note that ifGis generally reductive a (anti-)dominant character vanishes on the radical Rad(G) ([Ram96i], 2.14).

2.6. Forλ :C^∗ →Sl(U)with strictly ascending weights γ¹, . . . , γ^m we get QGl(λ) = {diag(A¹, . . . , A^m) +N : A^j ∈Gl(r^j −r^j−1,C),

N a strictly block upper triangular matrix}.

Then Qm

j=1det(A^j)^γj is an anti-dominant character ([Sch08], 2.4.9).

5Lemma 2.8 in Chapter II of [MFK].

6Recall thatAd(G)is semisimple forGreductive.

7For the denition of a dominant character see for example [Ram96i], 2.14.

2.7. Let P^j be a parabolic subgroup and T^j ⊂ P^j a maximal torus.⁸ Let τ^j ∈ Tˆ₊^j = {τ ∈ Tˆ^j| Q_G(τ) = P^j}⁹ and s^j : {x^j} → P|_x^j/Q_G(τ^j). Choose a stability parameter (τ^j)j[|S|], τ^j ∈ Tˆ₊^j. Let λ : C → G be a one-parameter subgroup and (Q_G(λ), χ_λ) the corresponding pair of a parabolic subgroup and a character.

Let R^j = R|_x^j : {x^j} → P|_x^j/Q_G(λ) be a reduction of the structure group.¹⁰ We will write R_rep^j (x^j), s^j_rep(x^j) for (a choice of) representatives in P|_x^j, i. e.

[Rrep^j (x^j)] = R^j(x^j) ∈ P|_x^j/Q_G(λ) and [s^j_rep(x^j)] = s^j(x^j) ∈ P|_x^j/Q_G(τ^j). Then we nd an element g_j ∈ G : Rrep^j (x^j)g_j = s^j_rep(x^j). Now we may shift the orbit R_rep^j (x^j)Q_G(λ) by g_j, so that it intersects with s^j(x^j)_repQ_G(τ^j). The intersection of two Borel (and hence of two parabolic) subgroups always contains a maximal torus. Denote such a torus byT^j ⊂Q_G(λ)∩g_j⁻¹Q_G(τ^j)g_j. Then we nd elements h_j ∈ Q_G(τ^j), h ∈ Q_G(λ) such that g_jh_jτ^j(C^∗)h⁻¹_j g_j⁻¹, hλ(C^∗)h⁻¹ ⊂ T^j. Let τ^sjrep = g_jh_jτ^jh⁻¹_j g_j⁻¹ and λ^Rrepj = hλh⁻¹ be the corresponding one-parameter subgroups of T^j. Now we may dene hτ^sjrep, λ^Rrepj i. Observe that hτ^sjrep, λ^Rrepj i is independent of the choices made. In fact if N (T^j) denotes the normalizer of T^j, thenhis unique up to an element ofN (T^j)∩Q_G(λ)and analogously forh^j. Using the faithful representation ι the Z^dim(U)-elements corresponding to τ^sjrep, λ^Rrepj are left invariant when conjugating with one of the available permutation matrices.

Thus hτ^sj, χ^R_λ^ji = hτ^sj, λ^Rji := hτ^sjrep, λ^Rrepj i is well-dened and depends only on the classg_j ∈Q_G(τ^j)\G/Q_G(λ).¹¹

2.8. Proposition. Fix a one-parameter subgroup τ^j as well as τ_Gl^j = ι◦τ^j for every x^j ∈S. Let (P, (s^j)j[|S|]) be a principal G-bundle and (Pι, (s^j_Gl)j) with

s^j_Gl :{x^j}−→^sj P ×_X {x^j}/Q(τ^j),→P_ι×_X {x^j}/Q(τ_Gl^j ), R_Gl^j :{x^j}−→^sj P ×_X {x^j}/Q(λ),→P_ι×_X {x^j}/Q(λ_Gl) for a one-parameter subgroup λ:C^∗ →G. Then

hτ_Gl^sj, χ^R_ι◦λ,Gl^j i=hτ^sj, χ^R_λ^ji, ∀1≤j ≤ |S|.

Proof. Obvious by denition of the inner product. See as well [HS10], 5.1.2.

2.9. Let λ : C^∗ → G be a one-parameter subgroup and χ_λ the associated anti-dominant character. Consider the principal Q_G(λ)-bundle P → P/Q_G(λ) and

8By Borel, [Bo91] IV.11.3 Corollary, we know that maximal tori in G coincide with the maximal tori in the various Borel subgroups, and by IV.11.17 that every parabolic subgroup is conjugated to exactly one-parabolic subgroup containing a given Borel subgroupB.

9See [HS10], section 4.1 for an equivalent denition ofTˆ₊^j.

10For an equivalent denition of reductions of the structure group see e. g. [KN63], I.5, ber bundles.

11χ^R_λ^j denotes the character toλ^Rj.

P_QG(λ)(χ_λ)the χ_λ-associated line bundle onP/Q_G(λ). Let R :X →P/Q_G(λ) be a reduction and P_QG(χ_λ,R) = R^∗(P_QG(λ)(χ_λ)).

Observe, that R extends to a reduction

RGl :X→P/QG(λ),→Pι/QGl(ι◦λ)

and that every parabolic subgroup Q_Gl(ι◦λ) ⊂ Gl(U) by denition stabilizes a ag. Let (F^j)_j[r] be the ag of rank (r^j)_j[m] subbundles of E = Pι12 induced by λ_Gl =ι◦λ and (γ^j)_j[r] resp. (α^j)_j[r] the corresponding weights. Note that (F^j)_j[r]

depends on the reductionRGl. We get the following relation degPQ(χλ,R) = degPQ_Gl(χλ_Gl,RGl) =

m−1

X

j=1

α^j(deg(E) rk(F^j)−deg(F^j) rk(E)).

Proof. Since we have a reduction of the structure group to Q_G(λ) we nd Q_G(λ) -valued transition function (g^ij)_ij of our principal G-bundle P ([KN63] Pro. 5.3 and Pro. 5.6.). Ifιis our embedding ofG ,→Sl(U)we get the transition functions of R^∗(P_QG(λ)(χ_λ)) as (χ_λ(ι◦g^ij))_ij w. r. t. the induced trivializations. ι◦g^ij ∈ Q_Sl(U)(ι◦λ) is a block upper triangular matrix of the form

ι◦g^ij =







h^ij₁ ∗ ...

0 h^ij_m





. Hence we have

χ_λ(ι◦g^ij) =

m

Y

k=1

det(h^ij_k)^γk. On the other hand consider the vector bundle Lm

k=1(E ⊗(F^k))^αkr, where F^k is the subbundle with transition functions

H_k^ij =







h^ij₁ ∗ ...

0 h^ij_k





. The determinant of Lm

k=1(E⊗(F^k))^αkr has thus transition functions

m

Y

k=1

(det(g^ij)^rk ·det(((H_k^ij)^t)⁻¹)^r)^αkr =

m

Y

k=1 m

Y

l=1

det(h^ij_l )^αkrrk·

k

Y

l=1

(det(h^ij_l ))^−r2αk

!

=

m

Y

l=1

(det(h^ij_l ))^{rPmk=1αkrk−r2Pmk=lαk}.

12More precisely: E the vector bundle corresponding to theGl(U)−bundleP_ι.

Using ^Pmj=1γj(r_r^j−rj−1) = 0, r⁰ = 0 from 1.3 we see that rPm−1

k=1 α^kr^k = γ^mr. Furthermore −r²Pm−1

k=l α^k = −r(γ^m −γ^l) = −γ^mr+γ^lr. Putting both formu-las together we get Qm

l=1(det(h^ij_l ))^γlr. Therefore det(Lm

k=1(E ⊗ (F^k))^αkr) and R^∗(P_QG(λ)(χ_λ))^⊗r are isomorphic bundles and hence have the same degree, i. e.

deg(R^∗(P_QG(λ)(χ_λ))) =

m

X

k=1

α^kdeg(E⊗(F^k)) =

m

X

k=1

α^k(deg(E)r^k−rdeg(F^k)).

Remark. See as well [HS10] 5.1 or [GS05] by Tomás Gómez and Ignacio Sols, Lemma 5.6. for a proof in the case of a higher dimensional base variety.

Fritzsche, Grauert [FG02] or Kobayashi, Nomizu [KN63] give an excellent account of the connection between ber bundles and transition functions on a Riemann surface X. An algebraic disussion of this relation is given for example in [Mi80].

As transition functions are particularly easy to work with, we will use this descrip-tion again in secdescrip-tion 3.5 as well as chapter 4. It should be mendescrip-tioned however that some of our results can be proved without using cocycles.

Furthermore by the calculation in 1.38, 2.6, 2.8 and an embeddingιinto Sl(U)we see that

m−1

X

k=1

α^k

s^j

X

i=1

β^ij (r^ij −r^i−1,j)r^k−(r^ijk−r^i−1,j,k)r

=−hτ^sj, χ^R_λ^ji, δ^ij =rαⁱ(τ_Gl^sj) = rαⁱ(τ^j), R^j =R|{x^j}13

is the parabolic contribution. More precisely, by 2.8 we get¹⁴ hτ^sj, χ^R_λ^ji=

s^j−1

X

i=1

αⁱ(τ^sj)·r·

m

X

k=1

γ^k(r^k−r^k−1−(r^ij,k−r^ij,k−1))

=−

s^j−1

X

i=1

αⁱ(τ^sj)·r·

m

X

k=1

α^k·r(r^k−r^ij,k)

=−

s^j−1

X

i=1

αⁱ(τ^sj)·r·

m

X

k=1

α^k·(r(r^k−r^ij,k)−r^k(r−r^ij))

=−

s^j−1

X

i=1

δ^ij ·

m

X

k=1

α^k·(r^ijr^k−r^ij,kr).

13αⁱ(λ^sj)is theα-weight of the one-parameter subgroupλ^sj. Further note that the weightsαⁱ are left invariant when conjugating the corresponding one-parameter subgroup, i. e. αⁱ(τ_Gl^sj) = αⁱ(τ^j).

14SetVk =F^k|x^j/F^k−1|x^j,χk =γ^k andVⁱ=E^ij in 1.38.

Finally we use 1.8 for the transition to (β^ij)_i[s^j_]. Putting both results together we receive

degP_Q(χ_λ,R)− X

j:x^j∈S

hτ^sj, χ^R_λ^ji

=

m−1

X

k=1

α^k(par-deg(E) rk(F^k)−par-deg(F^k) rk(E)),

where λcorresponds to the ltration (F^k)_k[m] plus the weights(α^k)_k[m] and τ^sj to the ltration(E^ij)_i[s^j_] plus the weights (δ^ij/r)_i[s^j_] as above.

Remark. Note that occasionally in the literature τ is replaced by −τ.

2.10. Denition. A stability parameter τ^j ∈Tˆ₊^j is called ι-admissible if the cor-responding weights rαⁱ(τ^j) are admissible, i. e. rPs^j

i=1αⁱ(τ^j)<1holds for every 1 ≤j ≤ |S|. The denition extends to arbitrary representations G →Gl(W) for some vector spaceW.

2.11. Denition. A parabolic principal G-bundle (P, (s^j)j[|S|]) over the marked surface(X, S)is calledτ-semistable, if for every one-parameter subgroupλ:C^∗ → G and every reductionR :X →P/Q_G(λ)

degP_Q(χ_λ,R)− X

j:x^j∈S

hτ^sj, χ^R_λ^ji ≥0 holds.

Before we dene a weight function for the Higgs eldϕ:P_ς →Lwe should state a few general facts about the representations used. Consequentially we will be able to express the intrinsic denition of semistability in terms of the associated vector bundle and an associated homomorphism.

2.12. LetGbe a reductive algebraic group. Then there is a representationι:G→ Gl(U) for a vector space U s. t. ι is a closed embedding (Borel, [Bo91], Corollary 1.4). Furthermore if ς : G → Gl(W), W vector space is another representation, then we nd representations ς : Gl(U)→Gl(W)and ς˜:G→Gl( ˜U), W =U ⊕U˜ such that ς◦ι=ς ⊕ς˜([KP00], 5.4, Prop. 1).

Observe, that we can modifyιtoι⁰ :=ι⊕(det⁻¹◦ι) :G→Sl(U⊕C)⊂Gl(U⊕C) which is still faithful.

2.13. Lemma. Let ι : G → Gl(U) be a faithful representation, then there is a decomposition of U into G-modules U_a, a ∈ A nite, s. t. ι(Rad(G)) ⊂ Z (

×

a∈AGl(U_a)), i. e. the radical maps to the center.

Proof. Rad(G) is a torus and hence induces a decomposition (U_a)a[|A|] into eigenspaces to characters χ_a,Rad(G) : Rad(G) → C^∗ ([Bo91], Proposition before Denition 11.22). Since Rad(G) ⊂ Z(G) we have for all r ∈ Rad(G), ∀g ∈ G, ∀u_a ∈U_a

ι(r, ι(g, u_a)) =ι(rg, u_a) =ι(gr, u_a) = ι(g, ι(r, u_a))

=ι(g, χ_a,Rad(r)u_a) =χ_a,Rad(r)ι(g, u_a).

Thereforeι(g, u_a)∈U_a, i. e. GpreservesU_aand we have a decomposition ofU into G-modulesU_a. By denitionι(Rad(G))⊂Z (

×

a∈AGl(U_a))([Sch08], 2.6.1).

Notation. From now on let ι denote a faithful representation G ,→ Gl(U_a)_a[|A|]∩ Sl(U), U :=L

a∈AU_a (see 2.12 and 2.13).

2.14. Denition. A representation ς : H → Gl(W), H = Gl(U),

×

a∈AGl(Ua) is called polynomial, if the matrix coecients ς^ij are polynomial functions. It is called rational if det^r·ς^ij is polynomial for some r. ς is called homogeneous of degree r if ς(z ·idU) = z^r ·idW resp. ς(z ·id

×

a∈AUa) = z^r·idW. In particular homogeneous representations are rational.

Remark to 2.14. (i) When we talk about representations without further speci-cation, we refer to rational representations.

(ii) The standard representation of Gl(U) on U^⊗u for a vector space U and an integer u is polynomial.

(iii) The denition is independent of the chosen basis of W.

(iv) The determinant representationdet^⊗w : Gl(U)→C^∗is polynomial forw≥0. (v) The tensor product, the direct sum, exterior powers, symmetric powers, sub-representations and quotient sub-representations of polynomial (resp. rational) representations are polynomial (resp. rational).

(vi) The dual representation of a rational representation is rational. Every irre-ducible representation is homogeneous.

(vii) The representation ς in 2.12 is rational by (ii)-(vi).

For more details see [KP00] sections 5.1 and 5.2.

2.15. Proposition. (i) For every representation ς : Gl(U)→ Gl(W) there are integers u^j, v, w, 1≤j ≤v such that ς is direct summand of the standard representation

Gl(U)→Gl





v

M

j=1

U^⊗uj

!

⊗





dimU

^ U

!^⊗w





. If ς is homogeneous, u:=u_j, ∀1≤j ≤v.

(ii) Fix κ_a ∈ N+, a ∈ A. For every representation ς :

×

a∈AGl(U_a) → Gl(W) there are integers u^j, v, w, 1 ≤ j ≤ v such that ς is direct summand of the standard representation

×

a∈AGl(U_a)→Gl

Lv

j=1(U(κ_a))^⊗uj

⊗

VdimU(κa)

U(κ_a)⊗w , U(κ_a) :=M

a∈A

U_a^⊕κa. If ς is homogeneous, u:=u_j, ∀1≤j ≤v.

Proof. (i) is proved by the proposition in [KP00] 5.3 as well as in [CMS], Theorem 14.3. (ii) is precisely the statement of [KP00], 5.4, Proposition 1 already used in 2.12. Note that 2.12 and the remark to 2.14 provide us with a representation ς : Gl(U(κ_a))→ Gl(W) such that ς ◦ι =ς ⊕ς˜for some suitable representation ς˜ and ι:

×

a∈AGl(U_a),→Gl(U(κ_a))an embedding.

The special property of homogeneous representations follows directly from the denition (of homogeneity).

2.16. Higgs Field. We still need to dene a semistability condition for the Higgs eld. Let ι be our faithful representation (cf. 2.12, 2.13) and ς the corresponding homogeneous representation such thatς◦ι=ς⊕ς˜holds for some representationς˜ (cf. 2.12, remark to 2.14). NowE_ς =Pς◦ι =P_ς⊕P_˜ς andE =L

a∈AE_athe tuple of vector bundles associated by ι. Consequentially the morphismϕ:P_ς →Linduces a morphismϕ◦pr₁ :Eς →L. We call(E, ϕ, L): the pseudo (ς ◦ι)-Higgs bundle induced from(P, ϕ, L). Now we may extendϕby 2.15 to(E^⊗u)^⊕v⊗(detE^⊗w)= E_ς⊕E_ςˆfor yet another representation ˆς.

2.17. Note that given a one-parameter subgroup λ : C^∗ → G, ι as before, R : X → P/Q_G(λ) a reduction and π : P → X the bundle projection, we can pull back the Q_G(λ)-bundle with projectionπ_R to a Q_G(λ)-bundle Q_R over X

Q_R ^//

P

π

{{

π_R

X R //P/QG(λ).

Observe that (Q_R)ς|_QG(λ) 'P_ς .¹⁵ Now Q_Gl(W)(ς ◦λ) induces a ltration (F_ς^k)_k[m]

of P_ς. Asϕ6= 0

µ(λ,R, ϕ) :=−min{γ^j| ϕ|_F^j

ς 6= 0, 1≤j ≤m},

15P admits local trivializations withQG(λ)-valued transition functions.

is well-dened.

On the other hand λ induces a one-parameter subgroup ι◦λ : C^∗ → Gl(U_a)_a :=

×

a∈AGl(U_a) with associated ltration (F^j)_j[m], F^j ⊂ E. Hence using 2.15 we obtain a ltration (Nu

j F^ij)^⊕v of (E^⊗u)^⊕v. The two ltrations ((Nu

j F^ij)^⊕v∩P_ς)_i and (F_ς^k)k identify under the identication(E^⊗u)^⊕v⊗(detE^⊗w) =Pς⊕P˜ς⊕Eˆς. Thus µ(F^j, α^j, pr₁◦ϕ) = µ(λ, R, ϕ). To simplify notation in future we will usually omit the projection pr₁.

2.18. We may further include the choice of a characterξofGinto the semistability concept as follows: a character ofGinduces a character of the radicalRad(G). Ifι as in 2.16 maps the radicalRad(G)toZ(

×

a∈AGl(Ua)), thenξ|_Rad(G)comes from a character ofZ(

×

a∈AGl(U_a))⊂

×

a∈AGl(U_a), therefore from a choice of rational numbers ξ_a with P

a∈Aξ_ar_a = 0.¹⁶ The identical calculation as in the parabolic case shows that for every one-parameter subgroupλofGwith associated weighted ag (F^k, α^k)_k, rk(F^k) =r^k, rk(F_a^k) =r_a^k: hλ, ξi=Pm

k=1α^kP

a∈Aξ_a(r_ar^k−rr_a^k) =

−Pm

k=1α^kPr

a=1ξ_arr^k_a.

2.19. Denition. Let Y be a scheme of nite type over C, P^l → Jac^l×X a Poincaré line bundle and τ^j xed parabolic weights to given parabolic subgroups P^j ⊂G. AY-family of projectiveς-Higgs bundles (of given topological type(ϑ, l)) is a tuple (PY, (s^j_Y)j[|S|], ϕ_Y, v_Y, HY) where

1. PY is principal Gbundle (of topological type ϑ) over every point{y}. 2. v_Y :Y →Jac^l is a morphism, HY →Y a line bundle.

3. ϕ_Y :PY,ς →(v_Y ×id_X)^∗(P^l)⊗π^∗_Y(HY) is a homomorphism non-trivial on bers over y∈Y.

4. s^j_Y :Y × {x^j} →PY ×X (Y × {x^j})/QG(τ^j) for all x^j ∈S.

An isomorphism of projectiveY-families is an isomorphism of the underlying prin-cipalG-bundles that extends in the natural way to the associated objects such that it commutes with an isomorphism of the line bundles HY.

2.20. Denition. A parabolic principal ς-Higgs bundle (P, (s^j)_j[|S|], ϕ, L) over the marked surface (X, S) is called (ξ, τ, δ)-(semi)stable, if for every one-parameter subgroup λ:C^∗ →G and every reductionR :X →P/Q_G(λ)

degP_Q(χ_λ,R)− X

j:x^j∈S

hτ^sj, χ^R_λ^ji+δµ(λ,R, ϕ) +hλ, ξi (≥) 0.

16ι(Rad(G)) is a torus, hence identies with (C^∗)^m, thusι looks component-by-component as Q

iz_i^aij. Now nding a character that extendsξ equals solving an inhomogeneous system of linear equations with a highest rank matrixA= (aij)_i[m]j[m].

Given a faithful representation ι : G ,→ Gl(U_a)_a as before a parabolic principal ς-Higgs bundle (P, (s^j)_j[|S|], ϕ, L) is (ξ, τ, δ)-semistable if and only if for every one-parameter subgroup λ:C^∗ →G and every reductionR :X →P/Q_G(λ)

m−1

X

j=1

α^j(par-deg(E) rk(F^j)−par-deg(F^j) rk(E)) +δ·µ(F^j, α^j,pr₁◦ϕ)−

−

m

X

k=1

α^k

r

X

a=1

ξark(E) rk(F_a^k)≥0.

Im Dokument Moduli spaces of parabolic twisted generalized Higgs bundles (Seite 66-79)