Deformations of twisted harmonic maps and Higgs bundles

Introduction First order deformations Second order deformations
Deformations of twisted harmonic maps
Marco Spinaci
Institut Fourier, Grenoble
November 25, 2013
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Table of contents
1 Background and preliminary results
Harmonic maps
Higgs bundles
The universal twisted harmonic map
2 First order deformations
Equivariant and harmonic deformations
First variation of the energy
3 Second order deformations
Variation of the energy
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Harmonic maps
Twisted harmonic maps
X: closed connected Riemannian (later: Kähler) manifold;
Γ = π1(X, x0): its fundental group;
ρ: Γ → G: representation to an algebraic reductive group
G (usually: G = GL(r, C));
N = G/K: symmetric space of the non-compact type
(K = U(r), N: positive deﬁnite Hermitian matrices);
˜X → X: universal cover.
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Harmonic maps
Twisted harmonic maps
X: closed connected Riemannian (later: Kähler) manifold;
Γ = π1(X, x0): its fundental group;
ρ: Γ → G: representation to an algebraic reductive group
G (usually: G = GL(r, C));
N = G/K: symmetric space of the non-compact type
(K = U(r), N: positive deﬁnite Hermitian matrices);
˜X → X: universal cover.
Deﬁnition
A map f : ˜X → N is ρ-equivariant (or: twisted) if
f (γ˜x) = ρ(γ)f (˜x) for γ ∈ Γ, ˜x ∈ ˜X.
It is harmonic if it minimizes
E(f ) =
1
2 X
df
2
dVolg.
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Harmonic maps
Harmonic metrics
Deﬁnition
A representation ρ: Γ → G is semi-simple if ρ ∼= i ρi, each ρi
being irreducible.
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Harmonic maps
Harmonic metrics
Deﬁnition
being irreducible.
Theorem (Donaldson ’87, Corlette ’88)
A ρ-equivariant harmonic map exists if and only if ρ is
semi-simple. It is unique up to multiplication by an element of
H = ZG(Image(ρ)).
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Harmonic maps
Harmonic metrics
Definition
being irreducible.
Theorem (Donaldson ’87, Corlette ’88)
A ρ-equivariant harmonic map exists if and only if ρ is
semi-simple. It is unique up to multiplication by an element of
H = ZG(Image(ρ)).
Equivariant map
f : ˜X → GL(r, C)/U(r)
←→
Hermitian positive definite metric
on (V, D) = ˜X × Cr /Γ, d .
Definition
The Hermitian metric is harmonic if f is.
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Higgs bundles
Maurer-Cartan form
Write d = dcan
+ β, where β is a g = Lie(G)-valued 1-form and:
dcan
is a metric connection;
β is self-adjoint. We call it the “Maurer-Cartan” 1-form.
Then β ∼= df through the identiﬁcation:
ϑTN : N × g ⊇ [p]
∼
−→ TN
(n, ξ) −→ ∂
∂t exp(tξ) · n
t=0
.
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Higgs bundles
Maurer-Cartan form
Write d = dcan
dcan
ϑTN : N × g ⊇ [p]
∼
−→ TN
(n, ξ) −→ ∂
∂t exp(tξ) · n
t=0
.
Fact: The map f is harmonic ⇐⇒ dcan∗
β = 0.
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Higgs bundles
Maurer-Cartan form
Write d = dcan
dcan
ϑTN : N × g ⊇ [p]
∼
−→ TN
(n, ξ) −→ ∂
∂t exp(tξ) · n
t=0
.
Fact: The map f is harmonic ⇐⇒ dcan∗
β = 0.
Theorem (Siu ’80, Sampson ’86)
If X is Kähler, decompose (1, 0) and (0, 1) parts:
dcan
= ∂ + ¯∂; β = θ + θ∗
.
Then ¯∂2 = 0, ¯∂(θ) = 0 and θ ∧ θ = 0, that is, (V, ¯∂, θ) is a Higgs
bundle.
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Higgs bundles
Moduli spaces
Theorem (Hitchin ’87, Simpson ’94)
Assume further that X is projective. There exist homeomorphic
moduli spaces:
MB(X, G) = Hom(Γ, G)//G
∼
−→ MDol(X, G).
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Higgs bundles
Moduli spaces
Theorem (Hitchin ’87, Simpson ’94)
Assume further that X is projective. There exist homeomorphic
moduli spaces:
MB(X, G) = Hom(Γ, G)//G
∼
−→ MDol(X, G).
Deﬁnition
Energy functional E : Hom(Γ, G) → R:
E(ρ) = inf E(f ) =
1
2 X
df 2
f is ρ-equivariant, smooth .
Energy functional E : MDol(X, G) → R:
E(E, θ) = θ
2
L2 .
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Higgs bundles
Motivation
Let Σ be a Riemann surface, X be Kähler.
Hitchin ’87: Topology of the connected components of
MB(Σ, PSL(2, R)).
Hitchin ’92: “Teichmüller” component of MB(Σ, PSL(n, R)).
Bradlow–García-Prada–Gothen ’03: Connected components of
MB(Σ, PU(p, q)).
García-Prada–Gothen–Mundet-Riera ’13: Connected
components of MB(Σ, Sp(2n, R)).
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Higgs bundles
Motivation
Let Σ be a Riemann surface, X be Kähler.
Hitchin ’87: Topology of the connected components of
MB(Σ, PSL(2, R)).
Hitchin ’92: “Teichmüller” component of MB(Σ, PSL(n, R)).
Bradlow–García-Prada–Gothen ’03: Connected components of
MB(Σ, PU(p, q)).
García-Prada–Gothen–Mundet-Riera ’13: Connected
components of MB(Σ, Sp(2n, R)).
Toledo ’12: Plurisubharmonicity of the energy on the
Teichmüller space (i.e. ﬁxed ρ but varying complex structure on
Σ).
Biswas-Schumacher ’06: Computation of ∂ ¯∂E on the moduli
spaces of stable Higgs bundles on X (i.e. where H is trivial).
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Universal map
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Universal map
Deﬁnition
˜x0 ∈ ˜X; Y =
(n, ρ) ∈ N × Hom(Γ, G) ∃f(n,ρ) : ˜X → N,
ρ-equivariant and harmonic s.t. f (˜x0) = n
.
Deﬁne the universal twisted harmonic map H : Y × ˜X → N by
H (n, ρ, ˜x) = f(n,ρ)(˜x).
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Universal map
Deﬁnition
˜x0 ∈ ˜X; Y =
(n, ρ) ∈ N × Hom(Γ, G) ∃f(n,ρ) : ˜X → N,
.
H (n, ρ, ˜x) = f(n,ρ)(˜x).
Proposition
The subset Y ⊆ N × Hom(Γ, G) is closed; the map H is
continuous.
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Universal map
Deﬁnition
˜x0 ∈ ˜X; Y =
(n, ρ) ∈ N × Hom(Γ, G) ∃f(n,ρ) : ˜X → N,
.
H (n, ρ, ˜x) = f(n,ρ)(˜x).
Proposition
The subset Y ⊆ N × Hom(Γ, G) is closed; the map H is
continuous.
Proposition
The energy functional E : Hom(Γ, G) → R is continuous.
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Basic deﬁnitions
Deﬁnition
Deformation of ρ0 is c ∈ Z1(Γ, g); write ρ
(1)
t = (ρ0, c).
Deformation of f is v ∈ C∞(f ∗TN).
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Basic definitions
Definition
Deformation of ρ0 is c ∈ Z1(Γ, g); write ρ
(1)
t = (ρ0, c).
Deformation of f is v ∈ C∞(f ∗TN).
Definition
A first order deformation v ∈ C∞(f ∗TN) is said:
ρ
(1)
t -equivariant if v(γ˜x) = ρ0(γ)v(˜x) + ϑTN f (γ˜x), c(γ) ;
harmonic if J (v) = 0, where locally
J (v) = −
j,k
gjk D
∂xj
D
∂xk
v + RN ∂f
∂xk
, v
∂f
∂xj
.
Fact: They are the “natural” definitions.
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Construction of equivariant harmonic deformations
Let ω ∈ H1(X, Ad(ρ0)) be harmonic, representing c.
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Construction of equivariant harmonic deformations
Let ω ∈ H1(X, Ad(ρ0)) be harmonic, representing c.
Theorem A
Let P = F : ˜X → g dF = ω, F(γ˜x) = Adρ0(γ)F(˜x) + c(γ) .
Then P is an aﬃne space over h, the Lie algebra of
H = ZG(Image(ρ0)), and
ϑTN : P v ∈ C∞
(f ∗
TN) ρ
(1)
t -equivariant and harmonic .
The latter space is aﬃne over h ∩ p (in particular, it is always
non-empty).
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Critical points as C-VHS
Corollary
If v = ∂ft
∂t is harmonic and ρ
(1)
t -equivariant, with ω as above:
∂E(ft)
∂t t=0
=
X
ω, β dVol.
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Corollary
If v = ∂ft
(1)
∂E(ft)
∂t t=0
=
X
ω, β dVol.
Theorem B
Suppose X to be Kähler. Then, the critical points of the energy
coincide with the complex variations of Hodge structure.
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Corollary
If v = ∂ft
(1)
∂E(ft)
∂t t=0
=
X
ω, β dVol.
Theorem B
Suppose X to be Kähler. Then, the critical points of the energy
coincide with the complex variations of Hodge structure.
Proof.
⊇ S1-action: ϕ: S1 → G. Let γ = ∂ϕ(eiθ)
∂θ θ=0
.Then β = Dcγ.
⊆ ω = θ + D η; 0 = X ω, θ + θ∗ = X ω 2.
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Deﬁnitions
Deﬁnition
A deformation of (ρ0, c) is a (c, k) ∈ Z1 Γ, Ad(ρ0, c) ; write
ρ
(2)
t = (ρ0, c, k): Γ → J2G.
A deformation of (f , v) is w ∈ C∞(f ∗TN).
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Definitions
Definition
A deformation of (ρ0, c) is a (c, k) ∈ Z1 Γ, Ad(ρ0, c) ; write
ρ
(2)
t = (ρ0, c, k): Γ → J2G.
A deformation of (f , v) is w ∈ C∞(f ∗TN).
Definition
D2 =
d 0
ad(ω) d
; D2,∗ =
d∗
0
ω∗¬ d∗ .
A map (F, F2): ˜X → g × g is equivariant of harmonic type if
D2,∗D2
F
F2
= 0; (f , F, F2) is ρ
(2)
t -equivariant.
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Existence of (F, F2)
Lemma
There is a ϑJ2N : N × g × g → TN × TN sending equivariant
(F, F2) of harmonic type to equivariant harmonic (v, w).
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Lemma
Deﬁnition
If (F, F2) is equivariant of harmonic type:
ω
ψ
:= D2
F
F2
; then
dψ = −[ω, ω];
d∗
ψ = −ω∗ ¬ ω.
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Lemma
Deﬁnition
If (F, F2) is equivariant of harmonic type:
ω
ψ
:= D2
F
F2
; then
dψ = −[ω, ω];
d∗
ψ = −ω∗ ¬ ω.
Theorem (Goldman-Millson ’88)
The second order obstruction of extending a ﬁrst order
deformation ρ
(1)
t of a semi-simple ρ0 is the cohomology class of
[ω, ω] ∈ Z2(Γ, Ad(ρ0)). If Γ is a Kähler group, then this is the
only obstruction.
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The construction theorem
Theorem C
Let G be complex. The following are equivalent:
1 There exists an equivariant (F, F2) of harmonic type;
2 There exists a ψ such that dψ = −[ω, ω] and
d∗
ψ = −ω∗ ¬ ω;
3 ω is a minimum for · L2 in its H-orbit, where
H = ZG(Image(ρ0)) acts on H1(M, Ad(ρ0)) by conjugation;
4 There exist two harmonic deformations (v, w) and (v , w ),
one (ρ0, c, k)-equivariant and the other
(ρ0, ic, −k)-equivariant.
Any of these is true for all harmonic metrics f if and only if
H0(X, Ad(ρ
(1)
t )) is a ﬂat R[t]/(t2)-module.
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Plurisubharmonicity
Corollary
∂2E(ft)
∂t2 t=0
=
X
ψ, β + ω[p] 2
dVol.
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Plurisubharmonicity
Corollary
∂2E(ft)
∂t2 t=0
=
X
ψ, β + ω[p] 2
dVol.
Theorem D
Let G be complex, X Riemannian. The energy functional is
strictly plurisubharmonic with respect to the Betti complex
structure JB. More precisely:
∂2
∂t2
+ JB
∂
∂t
2
E(ft) =
X
ω
2
dVol.
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Positivity of the Hessian
Theorem E
Suppose that X is Kähler and ρ0 is the monodromy of a
C-VHS. Write ˙A = ω[k] 0,1
and ˙Φ = ω[p] 1,0
. Then:
∂2E(ρt)
∂t2 t=0
= 2
X p
− p ˙A−p,p 2
+ (1 − p) ˙Φ−p,p 2
dVol,
where ξ = p ξ−p,p is C-VHS of weight 0 on End(V) = ˜X ×Γ g.
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Theorem E
and ˙Φ = ω[p] 1,0
. Then:
∂2E(ρt)
∂t2 t=0
= 2
X p
− p ˙A−p,p 2
+ (1 − p) ˙Φ−p,p 2
dVol,
Corollary
If ρt : Γ → G0 = Image(ρ0), the Hodge-Deligne decomposition is
a decomposition in eigenspaces of Hess(E).
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Theorem E
and ˙Φ = ω[p] 1,0
. Then:
∂2E(ρt)
∂t2 t=0
= 2
X p
− p ˙A−p,p 2
+ (1 − p) ˙Φ−p,p 2
dVol,
Corollary
If ρt : Γ → G0 = Image(ρ0), the Hodge-Deligne decomposition is
a decomposition in eigenspaces of Hess(E).
Corollary
If furthermore G0/K0 is Hermitian symmetric, then
Hess(E) ≥ 0, vanishing along C-VHS only.
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Deformations of twisted harmonic maps and Higgs bundles

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