Oscar Garc a-Prada ICMAT-CSIC, Madrid Hamburg, 10{14 ...€¦ · Oscar Garc a-Prada ICMAT-CSIC,...

Higgs bundles and higher Teichmuller spaces

Oscar Garcıa-PradaICMAT-CSIC, Madrid

Hamburg, 10–14 September 2018

Oscar Garcıa-Prada ICMAT-CSIC, Madrid Higgs bundles and higher Teichmuller spaces

1. Moduli space of representations

S oriented smooth compact surface of genus g ≥ 2

π1(S) fundamental group of S

G connected semisimple Lie group (real or complex)

A representation of π1(S) in G

is a homomorphism

ρ : π1(S)→ G

Hom(π1(S),G ) is an analytic variety, which is algebraic if G isalgebraic

G acts on Hom(π1(S),G ) by conjugation:

(g · ρ)(γ) = gρ(γ)g−1 for g ∈ G , ρ ∈ Hom(π1(S),G )


ρ is a reductive representation if composed with the adjointrepresentation in the Lie algebra of G , decomposes as a sumof irreducible representations

Hom+(π1(S),G ): set of reductive representations

Moduli space of representations or character variety

The moduli space of representations of π1(S) in G is defined to bethe orbit space

R(S ,G ) = Hom+(π1(S),G )/G

R(S ,G ) is an analytic variety (algebraic if G is algebraic)

Interested in the topology and geometry of R(S ,G )

Complex algebraic geometry approach: Higgs bundles


2. Higgs bundles

X compact Riemann surface

H ⊂ G maximal compact subgroup of G

θ Cartan involution of g, Lie algebra of G , defining theCartan decomposition:

g = h + m

where h is the Lie algebra of HWe have [m,m] ⊂ h, [m, h] ⊂ h

The Cartan decomposition is orthogonal with respect to theKilling form of g

Complexification of isotropy representation

ι : HC → GL(mC)


A G -Higgs bundle on X

is a pair (E , ϕ) consisting of

E a holomorphic principal HC-bundle over X

ϕ a holomorphic section of E (mC)⊗ K ,where E (mC) is the associated vector bundle with fibre mC viathe complexified isotropy representationand K is the canonical line bundle of X

To define stability for G -Higgs bundles, consider for s ∈ ih:

Parabolic subgroupPs = g ∈ HC : etsge−ts is bounded as t →∞Character χs : ps → C defined by s (ps Lie algebra of Ps)Subspace ms = Y ∈ mC : ι(ets)Y is bounded as t →∞For σ a reduction of E to Ps

deg(E )(σ, s) :=i

2π

∫Xχs(F ).

F : curvature of a connection on the Ps -bundle defined by σ


Stability of G -Higgs bundles

(E , ϕ) is:

(semi)stable if

deg(E )(σ, s) > 0(≥ 0)

for any s ∈ ih and any holomorphic reductionσ ∈ Γ(E (HC/Ps)) such that ϕ ∈ H0(X ,Eσ(ms)⊗ K )

polystable if (E , ϕ) can be reduced to a G ′-Higgs bundle,with G ′ ⊂ G reductive and (E , ϕ) stable as a G ′-Higgs bundle

The moduli space of polystable G -Higgs bundles

M(X ,G )

is the set of isomorphism classes of polystable G -Higgs bundlesM(X ,G ) is as complex algebraic variety (GIT construction,Schmitt, 2008)


G = SL(n,C)

When G is a clasical group we can formulate the theory interms of vector bundles

In this case H = SU(n), HC = SL(n,C) and mC = sl(n,C)Hence, an SL(n,C)-Higgs bundle is equivalent to a pair(V , ϕ)V rank n holomorphic vector bundle with detV = Oϕ : V → V ⊗ K with Trϕ = 0

(V , ϕ) is stable:deg(V ′) < 0 for every V ′ ⊂ V such that ϕ(V ′) ⊂ V ′ ⊗ K(V , ϕ) is polystable:(V , ϕ) = ⊕(Vi , ϕi ) with degVi = 0 and (Vi , ϕi ) stable

We recover the original notions introduced by Hitchin (1987)


G = SU(p, q)

In this case H = S(U(p)× U(q)),HC = S(GL(p,C)× GL(q,C)), andmC = Hom(Cq,Cp)⊕ Hom(Cp,Cq)Hence, an SU(p, q)-Higgs bundle is equivalent to a tuple(V ,W , β, γ)V and W are rank p and q holomorphic vector bundles,respectively, with detV ⊗ detW = Oβ : W → V ⊗ K and γ : V →W ⊗ K(V ,W , β, γ) is stable:deg(V ′) + deg(W ′) < 0 for every V ′ ⊂ V and W ′ ⊂W suchthat β(W ′) ⊂ V ′ ⊗ K and γ(V ′) ⊂W ′ ⊗ K(V ,W , β, γ) is polystable if the associatedSL(p + q,C)-Higgs bundle

V ⊕W and ϕ =

(0 βγ 0

)is polystable


G = Sp(2n,R)

In this case H = U(n), HC = GL(n,C), andmC = S2(Cn)⊕ S2(Cn)∗

Hence, an Sp(2n,R)-Higgs bundle is equivalent to a triple(V , β, γ)V is rank n holomorphic vector bundles,β : V ∗ → V ⊗ K and γ : V → V ∗ ⊗ K symmetric

stability of (V , β, γ) is a condition on certain two stepfiltrations V1 ⊂ V2 ⊂ V (not enough to check the conditionfor subbundles of V )(V , β, γ) is polystable if the associated SL(p + q,C)-Higgsbundle

V ⊕ V ∗ and ϕ =

(0 βγ 0

)is polystable


Theorem

A G -Higgs (E , ϕ) is polystable if and only if there exists areduction h of the structure group of E from HC to H, such that

Fh − [ϕ, τh(ϕ)] = 0 (Hitchin equation)

τh : Ω1,0(E (mC))→ Ω0,1(E (mC)) is the combination of theanti-holomorphic involution in E (mC) defined by the compactreal form at each point determined by h and the conjugationof 1-forms

Fh is the curvature of the unique H-connection compatiblewith the holomorphic structure of E

Proved by:

Hitchin (1987): G = SL(2,C)

Simpson (1988): G complex

Bradlow–G–Mundet (2003) & G–Gothen–Mundet (2009):G real


Non-abelian Hodge correspondence

Let S be a smooth compact surface and J be a complex structureon S . Let X = (S , J). There is a homeomorphism

R(S ,G ) ∼=M(X ,G )

Let (E , ϕ) be a polystable G -Higgs bundle and h a solution toHitchin equations

∇ = ∂E − τh(∂E ) + ϕ− τh(ϕ)

is a flat G -connection and the holonomy representation ρ isreductive

Converse: Existence of a harmonic metric on a reductiveflat G -bundle. Proved by Donaldson (1987) for G = SL(2,C)and Corlette (1988) for real reductive G


Theorem

Let ρ be a representation of π1(X ) in G with corresponding flatG -bundle Eρ. Let Eρ(G/H) be the associated G/H-bundle. Thenthe existence of a harmonic section of Eρ(G/H) is equivalent tothe reductiveness of ρ.

Recall that a section of Eρ(G/H)→ X is equivalent to aπ1(X )-equivariant map

f : X → G/H

where π1(X ) acts on X by Deck transformations and on G/Hvia ρ

The harmonicity of the section is equivalent to solving Hitchinequations


3. Topological invariants

Given ρ : π1(S)→ G , there is an associated flat G -bundle onS , defined as Eρ = S ×ρ G (S : universal cover of S):

Hom(π1(S),G )/G ∼= H1(S ,G ) = iso. classes of flat G -bundles

Let G be the universal covering group of G . We have anexact sequence

1→ π1(G )→ G → G → 1

which gives rise to the (pointed sets) cohomology sequence

H1(S , G )→ H1(S ,G )c→ H2(S , π1(G ))

topological invariant of ρ:c(ρ) := c(Eρ) ∈ H2(X , π1(G )) ∼= π1(G )

We can define the subvariety

Rc(S ,G ) := ρ ∈ R(S ,G ) : c(ρ) = c


Similarly, we can define a topological invariant of a G -Higgsbundle (E , ϕ) over X as the topological class of theHC-bundle E (recall H ⊂ G is a maximal compact subgroup)

H1(X ,HC) = isomorphisms classes of HC-bundlesWe have

H1(X , HC)→ H1(X ,HC)c→ H2(X , π1(HC))

topological invariant of (E , ϕ):

c(E , ϕ) ∈ H2(X , π1(HC)) ∼= π1(HC)

We can define the subvariety

Mc(X ,G ) := (E , ϕ) ∈M(X ,G ) : c(E , ϕ) = c


Recall π1(G ) ∼= π1(H) ∼= π1(HC)

For c ∈ π1(G ) ∼= π1(HC) we have de homeomorphism

Rc(S ,G ) ∼=Mc(X ,G )

Theorem

If G is compact (Ramanathan, 1975) or complex (J. Li, 1993;G-Oliveira, 2017)

π0(R(S ,G )) = π0(M(X ,G )) ∼= π1(G )

The story is very different for non-compact real Lie groups(non-complex). The map

π0(R(S ,G )) = π0(M(X ,G ))→ π1(G )

may be neither surjective nor injective!


4. G = SL(2,R)

The topological invariant of ρ ∈ R(S , SL(2,R)) in this case isan integer (basically the Euler class) d ∈ Z ∼= π1(G )

Rd := ρ ∈ R(S , SL(2,R)) : with Euler class d

Theorem (Milnor, 1958)

Rd is empty unless|d | ≤ g − 1

An SL(2,R)-Higgs bundle is a tuple (L, β, γ)L line bundle over X β ∈ H0(X , L2K ) and γ ∈ H0(X , L−2K )Equivalently it can be seen as an SL(2,C)-Higgs bundle(V , ϕ) with V = L⊕ L−1 and

ϕ =

(0 βγ 0

)Milnor’s inequality follows from the stability of (V , ϕ)(Hitchin, 1987)


Theorem (Goldman, 1988; Hitchin 1987)

Rd is connected if |d | < g − 1

Rd has 22g connected components if |d | = g − 1

Let Rmax := Rd for |d | = g − 1

Each connected component of Rmax consists entirely ofFuchsian representations (discrete and faithful) and can beidentified with Teichmuller space (Goldman, 1980)

Question: Are there other (non compact) groups with similarfeatures to those of SL(2,R)?

Split real groups (SL(2,R) is the split real form of SL(2,C))

Groups of Hermitian type (Poincare discD = SL(2,R)/SO(2))

Orthogonal groups SO(p, q) (PSL(2,R) ∼= SO0(1, 2))


5. Split real forms

Split real form = in the Cartan decomposition g = h⊕m, thespace m contains a maximal abelian subalgebra of g

Every complex semisimple Lie group has a split real formExamples: SL(n,R), Sp(2n,R), SO(n, n), SO(n, n + 1)

Consider G = SL(n,R)A basis for the invariant polynomials on sl(n,C) is providedby the coefficients of the characteristic polynomial of atrace-free matrix,

det(x − A) = xn + p1(A)xn−2 + . . .+ pn−1(A),

where deg(pi ) = i + 1.


Consider the Hitchin map

p :M(X , SL(n,C))→n−1⊕i=1

H0(K i+1)

defined byp(E , ϕ) = (p1(ϕ), . . . , pn−1(ϕ)),

Hitchin (1992) constructed a section of this map giving an

isomorphism between the vector spacen−1⊕i=1

H0(K i+1) and a

connected component of the moduli spaceM(X , SL(n,R)) ⊂M(X , SL(n,C))

This is called a Hitchin component (coincides with aTeichmuller component ∼= H0(X ,K 2) when G = SL(2,R))

Hitchin (1992) gives a general construction for any split realform


The Hitchin component is essentially unique if G is a splitform of adjoint type (i.e. without centre)

Every representation in the Hitchin component can bedeformed to a representation factoring asπ1(S)→ SL(2,R)→ G , where π1(S)→ SL(2,R) is in aTeichmuller component and SL(2,R)→ G is a principalrepresentation

A Hitchin component consists entirely of discrete andfaithful representations (Labourie 2006)

For G = SL(3,R), the Hitchin component parametrizesconvex projective structures on the compact surface(Choi–Goldman 1997)

The Hitchin component parameterizes certain type ofgeometric structures (Guichard–Wienhard 2008)


6. The Hitchin map and the Hitchin–Kostant–Rallis section

G semisimple real Lie group

H ⊂ G maximal compact subgroup of G

θ Cartan involution of g, Lie algebra of G , defining theCartan decomposition:

g = h + m

a ⊂ m maximal abelian subalgebra

gC =⊕

λ∈Λ(aC)

gλ,

Λ(aC) ⊂ (aC)∗ = restricted roots of gC with respect to aC

W (aC): restricted Weyl group of gC associated to aC

(group of automorphisms of aC generated by reflections onthe hyperplanes defined by the restricted roots Λ(aC))


Chevalley:

C[mC]HC ∼= C[aC]W (aC)

Hence we have an algebraic morphism

mC mC HC ∼= aC/W (aC)

This map is C∗-equivariant. In particular, it induces amorphism

h : mC ⊗ K → aC ⊗ K/W (aC).

The map is also HC-equivariant, thus defining a morphism

h :M(X ,G )→ B(X ,G ) := H0(X , aC ⊗ K/W (aC))

h is called the Hitchin map

B(X ,G ) is called the Hitchin base


Kostant–Rallis constructed g ⊂ g: maximal splitsubalgebra of g

g is a θ-invariant subalgebra of g generated by a and aprincipal normal three dimensional subalgebra (∼= sl(2,C)) ofgC invariant by the conjugation defining g ⊂ gC

G ⊂ G : maximal split subgroup (analytic subgroupcorresponding to g)

H ⊂ G maximal compact subgroup, g = h + mWe have

C[mC]HC ∼= C[aC]W (aC) ∼= C[mC]H

C.

HenceB(X ,G ) = B(X , G )

If G is complex G = Gsplit, the split real form of G . Inparticular B(X ,G ) = B(X ,Gsplit)


Let G be a real form of a complex semisimple Lie group GC:

G is quasi-split if cg(a) is abelian (split real forms, SU(p, p),SU(p, p + 1), SO(p, p + 2), and E 2

6 )

x ∈ mC is said to be regular if dim cmC(x) = dim aC, wherecmC(x) = y ∈ mC : [y , x ] = 0.G is quasi-split if and only if mC ∩ gCreg = mC

reg.

There is an exact sequence

1→ G → NGC(G )→ Q → 1,

where Q is finite


Theorem (G-Peon–Ramanan 2017)

The choice of a square root of K determines |Q| inequivalentsections of the Hitchin map

h :M(X ,G )→ B(X ,G ).

Each such section sG satisfies

1. If G is quasi-split, sG (B(X ,G )) is contained in the stablelocus of M(X ,G ).

2. If G is not quasi-split, the image of the section is contained inthe strictly polystable locus.

3. For arbitrary groups, the Higgs field is everywhere regular.

4. The section factors through M(X , G ).

5. If G = Gsplit is the split real form sG is the factorization of theHitchin section through M(X ,Gsplit) and sG (B(X ,Gsplit)) is atopological component of M(X ,Gsplit) (Hitchincomponent).


7. Non-compact real forms of Hermitian type

G of Hermitian type means that G/H admits a complexstructure compatible with the Riemannian structure of G/H,making G/H a Kahler manifold

If G is simple the centre of h is one-dimensional and thealmost complex structure on G/H is defined by a generatingelement in J ∈ Z (h)

This complex structure defines a decomposition

mC = m+ ⊕m−,

where m+ and m− are the (1, 0) and the (0, 1) part of mC

respectively

Classical connected simple groups of Hermitian type:SU(p, q), Sp(2n,R), SO∗(2n), SO0(2, n)

Two exceptional real forms E−146 and E−25

7


Let (E , ϕ) be a G -Higgs bundle over X .The decomposition mC = m+ ⊕m− gives a vector bundledecomposition

E (mC) = E (m+)⊕ E (m−)

Hence

ϕ = (ϕ+, ϕ−) ∈ H0(X ,E (m+)⊗ K )⊕ H0(X ,E (m−)⊗ K )

The torsion-free part of π1(H) is isomorphic to Z (most of thetime π1(H) ∼= Z) and hence the topological invariant of eithera representation of π1(X ) in G , or of a G -Higgs bundle, isessentialy given by an integer d ∈ Z. This is related to theso-called Toledo invariant.


The Toledo character χT : hC → C is defined, for Y ∈ hC interms of the Killing form, by

χT (Y ) =1

N〈−iJ,Y 〉,

where N is the dual Coxeter number.

The Toledo character χT defines a symmetric Kahler form onG/H by

ω(Y ,Z ) = iχT ([Y ,Z ]), for Y ,Z ∈ m,

with minimal holomorphic sectional curvature −1.

Define oJ to be the order of e2πJ and ` = |ZC0 ∩ [HC,HC]|,

where ZC0 is the identity component of Z (HC). For q ∈ Q,

the character qχT lifts to a character HC → C∗ if and only ifq is an integral multiple of

qT =`N

oJ dimm


Let (E , ϕ) be a G -Higgs bundle. Maybe up to an integermultiple, χT lifts to a character χT of HC. Let E (χT ) be theline bundle associated to E via the character χT . We definethe Toledo invariant τ of (E , ϕ)

τ = τ(E ) := deg(E (χT )).

If χT is not defined, but only χqT , one must replace the

definition by 1q deg E (χq

T ).Let ρ : π1(X )→ G be reductive and let (E , ϕ) be thecorresponding polystable G -Higgs bundle. Let f : X → G/Hbe the corresponding harmonic metric. Then

τ(E ) =1

2π

∫Xf ∗ω,

In particular, τ(E ) = τ(ρ) — the Toledo invariant of ρ.Let d ∈ Z be the projection on the torsion-free part of theclass c(E , ϕ) = c(ρ). Then d is related to τ by

τ =d

qTOscar Garcıa-Prada ICMAT-CSIC, Madrid Higgs bundles and higher Teichmuller spaces

Theorem (Milnor–Wood inequality)

Let (E , ϕ) be semistable. The Toledo invariant τ satisfies

|τ | ≤ rank(G/H)(2g − 2)

Proved for the classical groups for representations byDomic–Toledo (1987) and for Higgs bundles byBradlow–G–Gothen (2001)

General proof for representations by Burger–Iozzi–Wienhard(2010), and for Higgs bundles by Biquard–G–Rubio (2017)

For G = SU(p, q), G -Higgs bundle: (V ,W , β, γ)Toledo invariant = τ = 2 degV (degW = − degV )Milnor–Wood inequality: |τ | ≤ minp, q(2g − 2)

For G = Sp(2n,R), G -Higgs bundle: (V , β, γ)Toledo invariant = τ = 2 degVMilnor–Wood inequality: |τ | ≤ n(2g − 2)


G/H can be realized as a bounded symmetric domain D inm+, say (Cartan for the classical groups andHarish-Chandra in general)

D is called of tube type if it is biholomorphic to a tube TΩ

over a cone Ω: TΩ = Rn + iΩ, for Ω ⊂ Rn

The Poincare disc, the domain corresponding toG = SU(1, 1), is of tube type. The tube is the upper-halfplane and the biholomorphism is the Cayley transform

The Shilov boundary of D is defined as the smallest closedsubset S of the topological boundary ∂D for which everyfunction f continuous on D and holomorphic on D satisfiesthat

|f (z)| ≤ maxw∈S|f (w)| for every z ∈ D

D is of tube type if and only if S is a compact symmetricspace of the form H/H ′. In this case Ω = G ′/H ′ is itsnon-compact dual symmetric space


The symmetric spaces defined by Sp(2n,R), SO0(2, n) are oftube type.

The symmetric space defined by SU(p, q) is of tube type ifand only if p = q.

The symmetric space defined by SO∗(2n) is of tube type ifand only if n is even.

The E7 Hermitian real form is of tube type

The E6 Hermitian real form is not of tube type

Every bounded symmetric domain has a maximal tubesubdomain


Want to study the Maximal Toledo invariant moduli spacein the tube case (the non-tube reduces to the tube case)

Mmax(X ,G ) :=Mτ (X ,G ) for |τ | = rank(G/H)(2g − 2)

Assume that G is of tube type

From Chevally theorem on invariant polynomials, one candeduce the existence of a homogeneous polynomial functiondet : m+ → C, satisfying for h ∈ HC and x ∈ m+

det(Ad(h)x) = χT (h) det(x)

Definem+

reg := x ∈ m+ det(x) 6= 0

Thenm+

reg∼= HC/H ′C

(recall that S = H/H ′)


Let G ′/H ′ be the non-compact dual of the Shilov boundaryH/H ′, and g′ = h′ + m′ be the corresponding Cartandecomposition. Any element x ∈ m+

reg defines an

H ′C-equivariant isomorphism

ad(x) : m− → m′C

Let (E , ϕ+, ϕ−) ∈Mmax(X ,G ) withτ = − rank(G/H)(2g − 2). Then ϕ+(x) ∈ m+

reg∼= HC/H ′C

for every x ∈ X and if κ is an oJ -root of K , ϕ+ defines areduction of the HC-bundle E ⊗ κ to an H ′C-bundle E ′.

We also have an isomorphism

ad(ϕ+) : E ′(m−)→ E ′(m′C)

So that we can define a Higgs field

ϕ′ := ad(ϕ+)(ϕ−) = [ϕ+, ϕ−] ∈ H0(X ,E ′(m′C)⊗ K 2).

The pair (E ′, ϕ′) is a K 2-twisted G ′-Higgs bundle.


Theorem (Cayley Correspondence)

Let G be a such G/H is a Hermitian symmetric space of tubetype, and let Ω = G ′/H ′ be the non-compact dual of the Shilovboundary S = H/H ′ of G/H. Assume that oJ divides 2g − 2.Then the correspondence (E , ϕ) 7→ (E ′, ϕ′) defines an isomorphism

Mmax(X ,G ) ∼=MK2(X ,G ′),

where MK2(X ,G ′) is the moduli space of K 2-twisted G ′-Higgsbundles

Proved for the classical groups by Bradlow–G–Gothen(2006) G–Gothen–Mundet (2013) (G = Sp(2n,R))

General case proved by Biquard–G–Rubio (2017)


The connected components of M(X ,G ) are not fullydistinguished by the usual topological invariants. The dualgroup G ′ detects new hidden invariants (for example forG = Sp(2n,R), S = H/H ′, G ′ = GL(n,R), —Stiefel–Whitney classes

Rmax(S ,G ) consists entirely of discrete and faithfulrepresentations (Burger–Iozzi–Labourie–Wienhard, 2006)

The mapping class group of S acts properly on Rmax(S ,G )(Wienhard, 2006)

All common features with Hitchin components

G = Sp(2n,R) is both split and Hermitian and

MHitchin(X ,G ) ⊂Mmax(X ,G )

A higher Teichmuller component of R(S ,G ) or M(X ,G )is defined as one that has this kind of properties

Question: Are there other groups besides split and hermitianreal forms for which higher Teichmuller components exist?


8. Higher Teichmuller components for G = SO(p, q)

Joint work with M. Aparicio, S. Bradlow, B. Collier, P.Gothen and A. Oliveira (arXiv:1801.08561 andarxiv:1802.08093)An SO(p, q)-Higgs bundle is defined by a triple (V ,W , η)where V and W are respectively rank p and rank q vectorbundles with orthogonal structures such thatdet(W ) ' det(V ), and η is a holomorphic bundle mapη : W → V ⊗ KFor p > 2, rank p orthogonal bundles on X are classifiedtopologically by their first and second Stiefel–Whitneyclasses, sw1 ∈ H1(X ,Z2) and sw2 ∈ H2(X ,Z2)Since det(W ) ' det(V ) sw1(V ) = sw1(W )The components of the moduli space M(SO(p, q)) are thuspartially labeled by triples (a, b, c) ∈ Z2g

2 × Z2 × Z2, wherea = sw1(V ) ∈ H1(X ,Z2), b = sw2(V ) ∈ H2(X ,Z2), andc = sw2(W ) ∈ H2(X ,Z2)

M(SO(p, q)) =∐

(a,b,c)∈Z2g2 ×Z2×Z2

Ma,b,c(SO(p, q))


Proposition

Assume that 2 < p ≤ q. For every (a, b, c) ∈ Z2g2 × Z2 × Z2 the

space Ma,b,c(SO(p, q)) has a non-empty connected component

denoted by Ma,b,ctop (SO(p, q))

Define

Mtop(SO(p, q)) =∐a,b,c

Ma,b,ctop (SO(p, q))

Our main result shows that the moduli space M(SO(p, q))has additional exotic components disjoint from thecomponents of Mtop(SO(p, q))

We identify these exotic components as products of modulispaces of L-twisted Higgs bundles, where in each factor L isa positive power of the canonical bundle K


Theorem (Generalized Cayley Correspondence)

Fix integers (p, q) such that 2 < p < q − 1. For each choice ofa ∈ Z2g

2 and c ∈ Z2, the moduli space M(SO(p, q)) has aconnected component disjoint from Mtop(SO(p, q)). Thiscomponent is isomorphic to

Ma,cKp (SO(1, q−p+1))×MK2(SO0(1, 1))×· · ·×MK2p−2(SO0(1, 1))

The case of SO(2, q) is special since the group is of Hermitiantype

The existence of exotic components for SO(p, p + 1) for p > 2is proved by Collier (2017)

Conjecture: These exotic components are higher Teichmullercomponents (i.e. consist entirely of discrete and faithfulrepresentations)


Evidence: Notion of positivity recently introduced byGuichard–Wienhard (2016)

This generalizes Lusztig positivity for split real forms(related to Fock–Goncharov positivity)

It generalizes also a notion of positivity for Hermitian groups(related to causality)

Conjecture (Guichard–Wienhard): The only groups for whichthere are higher Teichmuller components are those admittinga positive structure

The only classical groups admitting positive structures are:split groups, hermitian groups of tube type and SO(p, q)!

Sp(2n,R) admits two different positive structures

The exceptional groups admitting positive structures are realforms of F4, E6, E7 and E8, whose restricted root system is oftype F4

F 44 is such an example: Existence of exotic components is

proved by Bradlow–Collier–G–Gothen–Oliveira (work inprogress). This group has also two different positive structures


9. Prehomogeneous vector spacesTheory introduced by Sato (Sato–Kimura, 1977)G Complex reductive Lie group

A prehomogeneous vector space for G is a complex finitedimensional vector space V together with a holomorphicrepresentation of G on V such that G has an open and denseorbit in V . It turns out that such an orbit is unique and dense.

Let V be a prehomogeneous vector space for G withrepresentation ρ. A non-constant holomorphic functionf : V → C is called a relative invariant for the action of G ifthere exists a character χ : G → C∗ such that

f (ρ(g)x) = χ(g)f (x) for every g ∈ G and x ∈ V .

Let Ω be the open orbit in V and S := V − Ω be the singularset in V . The prehomogeneous vector space V is calledregular if the isotropy subgroup G x is reductive for any x ∈ Ω.


A relative invariant is, up to constant multiple, uniquelydetermined by its corresponding character. In particular anyrelative invariant is a homogeneous function.

The existence of a relative invariant is equivalent to S havingan irreducible component of codimension one.

The prehomogeneous vector space V is regular if and only if Sis a hypersurface.

Let S = ∪i=li=1Si be the decomposition of S in irreducible

components. If V is regular, then for every Si (which is ahypersurface) there is a character χi and a correspondingrelative invariant fi such that Si = x ∈ V | fi (x) = 0.The number l coincides with the number of G -irreduciblesummands in V . In particular, if V is irreducible l = 1.


Assume that V is regular

Let x ∈ Ω. Let χ be a character of G and f be thecorresponding relative invariant. We define the rank of xwith respect to χ as

rankχ(x) = deg(f ).

We can actually define the rank of any point x ∈ V . To dothis, we polarize f to get an r -linear map q on V such thatq(x , . . . , x) = f (x); then the rank of x is the maximal integerr ′ such that the (r − r ′)-form q(x , . . . , x , ·, . . . , ·) is notidentically zero.

In fact, rank can be defined for any point of V even if V isnot regular


Let g be a complex semisimple Lie algebra. A Z-grading of gis a decomposition

g =⊕k∈Z

gk

such that [gj , gk ] = gj+k .

A Z-grading of g defines a parabolic subalgebra p ⊂ gC, givenby

p =⊕i≥0

gi

And in fact any Z-grading is defined by a parabolic subalgebra.

Let G be a complex semisimple Lie group with a graded Liealgebra g =

⊕k∈Z gk , and let G0 be the analytic subgroup of

G with Lie algebra g0. Then g1 is a prehomogeneous space forthe adjoint action of G0. In fact g1 has only a finite numberof G0-orbits (hence one of them must be open).The prehomogeneus vector space g1 is said to be of parabolictype.


Example: The Hermitian case

The representation of HC on m+, where G/H is Hermitian isa prehomogeneous vector space of parabolic type:Z-grading:

gC = m− ⊕ hC ⊕m+

m+ is regular if and only if G/H is of tube type

If G is simple and of tube type, then m+ is irreducible and detis the basic relative invariant for the action of HC on m+,corresponding of course to the Toledo character of HC

The same applies to m−


Let G be a real form of a semisimple complex Lie group GC,i.e. G = (GC)σ, where σ is a conjugation of GC. The groupG is said to be of Hodge type if σ = τ Int(g), where τ is acompact conjugation and g ∈ GC

Let G be Lie group of Hodge type and H ⊂ G be a maximalcompact subgroup. A period domain is a G -homogeneousspace G/H0 equipped with a homogeneous complex structure,where H0 ⊂ H is the H-centralizer of a compact torus T0 ⊂ H

A period domain defines a canonical Z-gradinggC =

⊕k∈Z g

Ck , satifying HC

0 is the complex subgroup of GC

corresponding to gC0 , and

hC =⊕

k=even

gCk mC =⊕

k=odd

gCk .

We will refer to the structure defined above as a Hodge structureon G .


Some examples of groups of Hodge type and Hodge structures

Hermitian groups are of Hodge type. The symmetric spaceG/H is itself a period domain (H0 = H)

G = SO(2p, q), H = SO(2p)× SO(q), H0 = U(p)× SO(q)

G = Sp(p, q), H = Sp(p)× Sp(q), H0 = U(p)× Sp(q)

Let GC be a semisimple complex Lie groups andgC =

⊕k∈Z g

Ck be a Z-grading. Then there exists a real form

G of GC of Hodge type and a Hodge structure on G suchthat gC2i ⊂ hC and gC2i+1 ⊂ mC

Let’s go back to Higgs bundles!


10. Hodges bundles and variations of Hodge structureG : real form of a semisimple complex Lie group GC

Consider a Z-grading of gC

gC =⊕k∈Z

gCk , gCk = hCk ⊕mCk

A G -Higgs bundle (E , ϕ) in M(X ,G ) is called a Hodgebundle if E admits a reduction of structure group to HC

0 , theLie group corresponding to hC0 , and ϕ ∈ H0(X ,E (mC

1 )⊗ K )Consider the natural action of C∗ on the moduli spaceM(X ,G ) of G -Higgs bundles given for λ ∈ C∗ by

λ · (E , ϕ) := (E , λϕ)

A point (E , ϕ) in M(X ,G ) is fixed under the C∗-action if andonly if it is a Hodge bundle (Simpson, 1992)There is a good reason to call this a Hodge bundle: if (E , ϕ)is a Hodge bundle then there exists a real form of Hodge typeG ′ ⊂ GC such that the grading of gC comes from a perioddomain G ′/H0


Assume that G is of Hodge type anf let (E , ϕ) in M(X ,G ).Let ρ : π1(X )→ G be the corresponding representation. Then(E , ϕ) is a Hodge bundle coming from a Hodge structure onG if and only if the equivariant harmonic map f : X → G/Hlifts to an equivariant horizontal holomorphic mapf : X → G/H0

The pair (ρ, f ) is called a variation of Hodge structure(VHS)

Solving Hitchin equations for a polystable G -Higgs bundle oneobtains a function

f :M(X ,G )→ R

defined by f (E , ϕ) = ‖ϕ‖2L2 , called Hitchin function

A smooth point (E , ϕ) is a critical point of the Hitchinfunction if and only it is a Hodge bundle


Assume that G is of Hodge type and G is equipped with aHodge structure: H0 ⊂ H, Z-grading of gC, etc. Then gC1 is aprehomogeneous space for the action of HC

0

Assume that gC1 is regular

Milnor–Arakelov inequality (Biquard–Collier–G–Toledo, 2018)

Let (E , ϕ) be a semistable Hodge bundle. Let χ be a character ofHC

0 . Then|τχ| ≤ rankχ(ϕ)(2g − 2),

where rankχ(ϕ) is the rank of ϕ(x) for a generic x ∈ X . Inparticular

|τχ| ≤ rχ(2g − 2),

where rχ = deg(fχ) and fχ is the homogeneous functioncorresponding to χ


There is a particular character χT of HC0 (coming from the

complex structure of G/H0) for which more is actually trueLet τ := τχT

. Of course: |τ | ≤ rχT(2g − 2)

A Hodge bundle is called maximal if |τ | = rχT(2g − 2)

Cayley correspondence (Biquard–Collier–G–Toledo, 2018)

Let (E , ϕ) be a Hodge bundle with |τ | = rχT(2g − 2). We have

the following:(1) rankϕ(x) = rχT

for every x ∈ X(2) The Higgs field ϕ defines a reduction of E ⊗ κ to a H ′C0 -bundleE ′ over X , where H ′C0 is the isotropy subgroup of an element in theopen orbit of gC1 , and every H ′C0 -bundle comes from a Hodgebundle with |τ | = rχT

(2g − 2)(3) Moreover, (E , ϕ) is (poly,semi)stable if and only if E ′ is(poly,semi)stable


Suppose that a maximal Hodge bundle (EHC0, ϕ1) is a

minimum of the Hitchin function, where recall ϕ1 takesvalues in gC1 . Then consider a Higgs bundle of the form

(E , ϕ) = (EHC0, ϕ1,

⊕k<0

ϕ2k+1)

with ϕ2k+1 ∈ H0(X , (EHC0

(gC2k+1)⊗ K )

There is a Cayley correspondence for these objects whichsends (E , ϕ) to a pair (E ′, ϕ′) where E ′ is an H ′0

C-bundle and

ϕ′ =⊕k<0

ϕ′2k+1 := [ϕ1,⊕k<0

ϕ2k+1]

(EHC0, ϕ1) is a minimum is equivalent (Bradlow–G–Gothen,

2003) to having (H ′0-equivariant) isomorphisms

ad(ϕ1) : E (gC2k)⊗ K∼=−→ E (gC2k+1), k < −1


This implies that

ϕ′2k+1 ∈ H0(X , (EH′0C(gC2k)⊗ K 1−k) fork < −1

On the other hand ϕ′1 ∈ H0(X , (EH′0C(V )⊗ K 2 where

V ⊕ h′C0 = gC0

Claim (Bradlow–Collier–G–Gothen–Oliveira, work in progress

(1) If the dimension of the moduli space of objects(EHC

0, ϕ1,

⊕k<0 ϕ2k+1) equals the dimension of M(X ,G ), then

this moduli space is a union of higher Teichmuller components ofM(X ,G ). The Cayley partner detects new topological invariants(2) Moreover, the parabolic subgroup P ⊂ G defining positivity onG in the sense of Guichard–Wienhard can be identified from thestructure of the Cayley parter (the rigidity given by being amaximal minimum!). This identification will use the solution to theHitchin equation, that is a reduction of the bundle E to anH-bundle.


Oscar Garc a-Prada ICMAT-CSIC, Madrid Hamburg, 10{14 ...€¦ · Oscar Garc a-Prada ICMAT-CSIC,...

Documents

Transcript of Oscar Garc a-Prada ICMAT-CSIC, Madrid Hamburg, 10{14 ...€¦ · Oscar Garc a-Prada ICMAT-CSIC,...