Global topology of the Hitchin system - GEOM

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Symmetries of the SLn Hitchin fibration recall the spectral cover map π : Ca → C for an abelian variety A the dual Â := Pic0 (A) Definition For a ∈ A 0 reg the norm map Nm Ca/C : Pic 0 (Ca) → Pic 0 (C) is defined in any of the following three equivalent ways: 1 D divisor on Ca, Nm Ca/C (O(D)) = O(π∗D) 2 For L ∈ Pic 0 (Ca) define Nm Ca/C (L) = det(π∗(L)) ⊗ det −1 (π∗OCa ). 3 the norm map is the dual of the pull-back map π∗ a : Pic 0 (C) → Pic 0 (Ca), that is NmCa/C = ˇπ : Pic 0 (Ca) ∼ = ˇ Pic 0 (Ca) → Pic ˇ 0 (C) Pic 0 (C). the Prym variety Prym 0 (Ca) := ker(Nm Ca/C ) acts on Prym d (Ca) = ˇMa ❀ ˇMa is a torsor for ˇPa := Prym 0 (Ca). for PGLn: ˆMa is a torsor for ˆPa = Pic 0 (Ca)/ Pic 0 (C) ∼ = Prym 0 (Ca)/Γ ∼ = ˇPa/Γ 22 / 39
Duality of the Hitchin fibres short exact sequence of abelian varieties: 0 → Prym 0 (Ca) ↩→ Pic 0 (Ca) Nm Ca/C ↠ Pic(C) → 0 the dual sequence is 0 ← ˇ Prym 0 (Ca) ↞ Pic 0 (Ca) π ∗ ← Pic(C) ← 0 , ❀ ˇ Pa = Pic 0 (Ca)/ Pic(C) = ˆ Pa, ⇒ ˇ Pa and ˆ Pa are dual abelian varieties Theorem (Hausel-Thaddeus, 2003) For a regular a ∈ A 0 reg ˇMa and ˆMa are torsors for dual Abelian varieties (namely ˇ Pa and ˆ Pa). 23 / 39
Page 1 and 2: Global topology of the Hitchin syst
Page 3 and 4: Mirror Symmetry phenomenon first ar
Page 5 and 6: Strominger-Yau-Zaslow X CY 3-fold Y
Page 7 and 8: Hitchin system Hamiltonian system:
Page 9 and 10: The moduli space of vector bundles
Page 11 and 12: Cohomology of ˆ Ň The cohomologie
Page 13 and 14: Hitchin map for SLn and PGLn for SL
Page 15 and 16: Proper Hitchin map (E, φ) ∈ T
Page 17 and 18: PGLn Hitchin system over the same H
Page 19 and 20: Generic fibres of the Hitchin map a
Page 21: Symmetries of the GLn and PGLn Hitc
Page 25 and 26: E-polynomials (Deligne 1972) constr
Page 27 and 28: Topological mirror symmetry conject
Page 29 and 30: Example PGL2 γ ∈ Γ = Pic 0 (C)[
Page 31 and 32: Arithmetic technique to calculate E
Page 33 and 34: Character table of GL2(Fq) 33 / 39
Page 35 and 36: Betti - Topological Mirror Test for
Page 37 and 38: P = W conjecture recall Hitchin map
Page 39: Heuristics from TMS to FL let Γ be

Global topology of the Hitchin system - GEOM

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