Higgs bundles over elliptic curves - ICMAT

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[Si3]C. T. Simpson, Moduli of representations of the fundamental group of a smoothprojective variety II, Publ. Math., Inst. Hautes Etud. Sci. 80 (1995), 5–79.[Tu] L. Tu, Semistable bundles over an elliptic curve, Adv. Math. 98 (1993) 1-26.186
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Higgs bundles over elliptic curvesE
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It may be normal, darling; but I’
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Grazie anche a Matteo per la loro m
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7 U ∗ (2m) and GL(n, R)-Higgs bun
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Weyl, tenemos queM(G) ∼ = (X ⊗
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Gracias a la equivalencia entre la
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En el capítulo 4 damos una descrip
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de Higgs poliestables con la propie
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Chapter 2IntroductionIn this thesis
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surfaces of genus g ≥ 2, and ther
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fibres over a non-generic point are
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5.4.11 and 5.4.22]. We give a bijec
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In Section 9.4 we see that all the
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Chapter 3The moduli problem3.1 Some
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If we have another scheme M ′ and
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is an isomorphism. Furthermore, if
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gr(E, Φ). If our definition of S-e
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M(GL(n, C)) 0 of zero degree Higgs
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Two Sp(2m, C)-Higgs bundles, (E, Ω
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As happens in the vectorial case, t
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Once we have set up this notation,
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is an isomorphism.Let us denote by
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We can give a description of Sym h
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Now ⊕ sj=1 F h jhas a unique subb
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Let us study the non-coprime case.
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where D λ1 ,...,λ nis the diagona
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An arbitrary point of B (n,d) is gi
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Proof. Set n ′ = n/h and d ′ =
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Consider the following mapˆp n : (
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Lemma 4.4.9. Let λ arb be as in (4
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field which is paremetrized by ker(
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Proposition 4.5.7. There is a surje
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gcd(h/r, m 1 /r, . . . , m l /r) =
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Chapter 5Moduli spaces of symplecti
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Proposition 5.1.3. There are no sta
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Proof. By definition (E, Θ, Φ) is
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5.2 Moduli spaces of Sp(2m, C)-Higg
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Proof. Since Z m /Γ m = Sym m (T
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Furthermore the diagramM(Sp(2m, C))
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We have a surjective morphismX× .
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where V is a family of line bundles
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Lemma 5.3.8. The following diagram
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Remark 5.3.13. Combining (5.15), Le
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Proposition 5.4.1. Let n > 2. Let (
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is a section of O X×T such that T
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We have that {q 2m+1,2 , . . . , q
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Remark 5.4.15. Recall the family E
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Lemma 5.4.18. The following diagram
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Proof. The family E n,ω2 induces a
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Chapter 6U(p, q)-Higgs bundles over
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We denote by M st (U(p, q)) (a,b) t
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Corollary 6.2.5. There are no polys
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where ı 1 (E x 0(r,r,d,d) ) and ı
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We see that the quotient of T ∗ X
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Let us recall that, using the invar
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By Proposition 6.3.5 νĔ(p,q,a,b)
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The notions of stability, semistabi
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We say that the locally graded fami
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with (L i , φ i ) a Higgs bundle o
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Remark 7.3.3. Since by Proposition
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We can also consider the natural pr
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Corollary 7.3.13. The generic fibre
Page 135 and 136: Theorem 7.4.3. There exists a modul
Page 137 and 138: We take ¨T n,l to be an irreducibl
Page 139 and 140: Lemma 7.4.11. We have the following
Page 141: Part IIIHiggs bundles for complex r
Page 144 and 145: Since the Lie algebra z is the univ
Page 146 and 147: the Lie algebra, and let h be a max
Page 148 and 149: 1. If G = SU(n) (resp. G = SL(n, C)
Page 150 and 151: If G is compact or complex reductiv
Page 152 and 153: the semisimple part. Note that sinc
Page 154 and 155: Write Z ′ = Z G ([s ′ 1, d 1 ]
Page 156 and 157: Recall the group Γ R defined in (8
Page 158 and 159: Chapter 9G-Higgs bundles over an el
Page 160 and 161: We say that a family E → X × T o
Page 162 and 163: Proof. By Corollary 9.2.3 E is poly
Page 164 and 165: By Proposition 9.2.6 z g (ρ) = z g
Page 166 and 167: Proposition 9.3.6. Let (E, Φ) be a
Page 168 and 169: We denote by E x 0(n,d)the underlyi
Page 170 and 171: By Proposition 9.1.2 (E L , Φ L )
Page 172 and 173: Equivalently, if we have g ′ ∈
Page 174 and 175: Remark 9.4.9. Since M(G) d is a nor
Page 176 and 177: If (E, Φ) is the polystable repres
Page 178 and 179: Proposition 9.5.3. Let s ∈ (C ⊗
Page 180 and 181: Corollary 9.5.5. Let us take s a
Page 183 and 184: Bibliography[ALR][AG]A. Adem, J. Le
Page 185: [Hi2] N. J. Hitchin, Stable bundles
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