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The Even Clifford Structure of a Hermitian Symmetric Space

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The Even Clifford Structure of a Hermitian Symmetric Space The even Clifford structure of a Hermitian symmetric space Paolo Piccinni (Sapienza Universit`adi Roma) INdAM Workshop New Perspectives in Differential Geometry: Special Metrics and Quaternionic Geometry Roma, November 16-20, 2015 1 / 40 Results from MathSciNet: Mathematical Reviews on the Web c Copyright American Mathematical Society 2015 MR0663019 (83g:32002) 32-06 53-06 58E20 FSymposia Mathematica. Vol. XXVI. Conference on Invariant Metrics, Harmonic Maps and Related Questions held at the Istituto Nazionale di Alta Matematica Francesco Severi (INDAM), Rome, May 26–29, 1980. Academic Press, Inc.[Harcourt Brace Jovanovich, Publishers], London-New York, 1982. 247 pp. Vol. XXV has been reviewed [MR 82e:6201]. { } Contents: Wilhelm Kaup, Bounded symmetric domains in complex Hilbert spaces (pp. 11–21); Luc Lemaire, Harmonic maps of finite energy from a complete surface to a compact manifold (pp. 23–26); T. J. Willmore, k-Stein Riemannian metrics on Lie groups (pp. 27–35); Michael A. O’Connor, Affinely invariant metrics on cones (pp. 37–67); G´abor T´oth,On harmonic maps into locally symmetric Riemannian manifolds (pp. 69–94); Jean-Pierre Vigu´e,Domaines born´essym´etriquesdans un espace de Banach complexe [Bounded symmetric domains in a complex Banach space] (pp. 95–104); Andr´e Lichnerowicz, Tenseurs holomorphes sur une vari´et´ek¨ahlerienne[Holomorphic tensors on a K¨ahlermanifold] (pp. 105–115); Wilhelm Klingenberg, The Morse complex (pp. 117–122); J. Eells and J. C. Wood, The existence and construction of certain harmonic maps (pp. 123–138); S. M. Salamon, Quaternionic manifolds (pp. 139–151); Shoshichi Kobayashi, Invariant distances for projective structures (pp. 153–161); K. Goebel, Uniform convexity of Carath´eodory’s metric on the Hilbert ball and its consequences (pp. 163–179); Alfred Gray, Comparison theorems for the volumes of half-tubes as generalizations of the formulas of Steiner (pp. 181–195); J. H. Sampson, On harmonic mappings (pp. 197–210); Edoardo Vesentini, Complex geodesics and holomorphic maps (pp. 211–230); Graziano Gentili, A class of invariant distances on convex cones (pp. 231–243). The papers are being reviewed individually. { } Concerning this room ... 2 / 40 Results from MathSciNet: Mathematical Reviews on the Web c Copyright American Mathematical Society 2015 MR0663019 (83g:32002) 32-06 53-06 58E20 FSymposia Mathematica. Vol. XXVI. Conference on Invariant Metrics, Harmonic Maps and Related Questions held at the Istituto Nazionale di Alta Matematica Francesco Severi (INDAM), Rome, May 26–29, 1980. Academic Press, Inc.[Harcourt Brace Jovanovich, Publishers], London-New York, 1982. 247 pp. Vol. XXV has been reviewed [MR 82e:6201]. { } Contents: Wilhelm Kaup, Bounded symmetric domains in complex Hilbert spaces (pp. 11–21); Luc Lemaire, Harmonic maps of finite energy from a complete surface to a compact manifold (pp. 23–26); T. J. Willmore, k-Stein Riemannian metrics on Lie groups (pp. 27–35); Michael A. O’Connor, Affinely invariant metrics on cones (pp. 37–67); G´abor T´oth,On harmonic maps into locally symmetric Riemannian manifolds (pp. 69–94); Jean-Pierre Vigu´e,Domaines born´essym´etriquesdans un espace de Banach complexe [Bounded symmetric domains in a complex Banach space] (pp. 95–104); Andr´e Lichnerowicz, Tenseurs holomorphes sur une vari´et´ek¨ahlerienne[Holomorphic tensors on a K¨ahlermanifold] (pp. 105–115); Wilhelm Klingenberg, The Morse complex (pp. 117–122); J. Eells and J. C. Wood, The existence and construction of certain harmonic maps (pp. 123–138); S. M. Salamon, Quaternionic manifolds (pp. 139–151); Shoshichi Kobayashi, Invariant distances for projective structures (pp. 153–161); K. Goebel, Uniform convexity of Carath´eodory’s metric on the Hilbert ball and its consequences (pp. 163–179); Alfred Gray, Comparison theorems for the volumes of half-tubes as generalizations of the formulas of Steiner (pp. 181–195); J. H. Sampson, On harmonic mappings (pp. 197–210); Edoardo Vesentini, Complex geodesics and holomorphic maps (pp. 211–230); Graziano Gentili, A class of invariant distances on convex cones (pp. 231–243). The papers are being reviewed individually. { } Concerning this room ... 2 / 40 All K¨ahlerwith positive c1. Therefore projective algebraic Fano manifolds. The projective algebraic description of the one I want to talk about E EIII = 6 Spin(10) U(1) × 26 is the so-called fourth Severi variety, smooth projective in CP . [F. Zak, 1983] A Hermitian symmetric space and Severi varieties Hermitian symmetric spaces of compact type A property Hermitian symmetric spaces of compact type: k+m SU(k + m) SO(m + 2) m+1 Grk (C ) = ; Qm = CP S(U(k) U(m)) SO(m) SO(2) ⊂ × × SO(2m) Sp(k) E E ; ; EIII = 6 ; EVII = 7 U(m) U(k) Spin(10) U(1) E U(1) · 6 · 3 / 40 The projective algebraic description of the one I want to talk about E EIII = 6 Spin(10) U(1) × 26 is the so-called fourth Severi variety, smooth projective in CP . [F. Zak, 1983] A Hermitian symmetric space and Severi varieties Hermitian symmetric spaces of compact type A property Hermitian symmetric spaces of compact type: k+m SU(k + m) SO(m + 2) m+1 Grk (C ) = ; Qm = CP S(U(k) U(m)) SO(m) SO(2) ⊂ × × SO(2m) Sp(k) E E ; ; EIII = 6 ; EVII = 7 U(m) U(k) Spin(10) U(1) E U(1) · 6 · All K¨ahlerwith positive c1. Therefore projective algebraic Fano manifolds. 3 / 40 A Hermitian symmetric space and Severi varieties Hermitian symmetric spaces of compact type A property Hermitian symmetric spaces of compact type: k+m SU(k + m) SO(m + 2) m+1 Grk (C ) = ; Qm = CP S(U(k) U(m)) SO(m) SO(2) ⊂ × × SO(2m) Sp(k) E E ; ; EIII = 6 ; EVII = 7 U(m) U(k) Spin(10) U(1) E U(1) · 6 · All K¨ahlerwith positive c1. Therefore projective algebraic Fano manifolds. The projective algebraic description of the one I want to talk about E EIII = 6 Spin(10) U(1) × 26 is the so-called fourth Severi variety, smooth projective in CP . [F. Zak, 1983] 3 / 40 4 5 Then, up to projective equivalence, V2 is the Veronese surface V2 CP , 2 ⊂ i.e. the image of CP under the Veronese map 2 5 CP , CP ; ! 2 2 2 (x0; x1; x2) (z0 = x ; z1 = x0x1; z2 = x0x2; z3 = x ; z4 = x1x2; z5 = x ): ! 0 1 2 [F. Severi, Rend. Circolo Mat. Palermo, 1901]. A Hermitian symmetric space and Severi varieties The first result: F. Severi, 1901 4 5 The Veronese surface V2 CP ⊂ Theorem (F. Severi, 1901) 5 Let V2 be a smooth surface in CP , not contained in a hyperplane. Assume that the chord variety Chord V2 = variety of all the secant and tangent lines 5 does not fill CP . 4 / 40 2 i.e. the image of CP under the Veronese map 2 5 CP , CP ; ! 2 2 2 (x0; x1; x2) (z0 = x ; z1 = x0x1; z2 = x0x2; z3 = x ; z4 = x1x2; z5 = x ): ! 0 1 2 [F. Severi, Rend. Circolo Mat. Palermo, 1901]. A Hermitian symmetric space and Severi varieties The first result: F. Severi, 1901 4 5 The Veronese surface V2 CP ⊂ Theorem (F. Severi, 1901) 5 Let V2 be a smooth surface in CP , not contained in a hyperplane. Assume that the chord variety Chord V2 = variety of all the secant and tangent lines 5 does not fill CP . 4 5 Then, up to projective equivalence, V2 is the Veronese surface V2 CP , ⊂ 4 / 40 A Hermitian symmetric space and Severi varieties The first result: F. Severi, 1901 4 5 The Veronese surface V2 CP ⊂ Theorem (F. Severi, 1901) 5 Let V2 be a smooth surface in CP , not contained in a hyperplane. Assume that the chord variety Chord V2 = variety of all the secant and tangent lines 5 does not fill CP . 4 5 Then, up to projective equivalence, V2 is the Veronese surface V2 CP , 2 ⊂ i.e. the image of CP under the Veronese map 2 5 CP , CP ; ! 2 2 2 (x0; x1; x2) (z0 = x ; z1 = x0x1; z2 = x0x2; z3 = x ; z4 = x1x2; z5 = x ): ! 0 1 2 [F. Severi, Rend. Circolo Mat. Palermo, 1901]. 4 / 40 the Veronese surface V 4 is given by the vanishing of all 2 2 minors, i. e. 2 × Z 2 = (trace Z)Z: Then (e.g. exercise in J. Harris, Algebraic Geometry, a First Course) 4 Chord V2 : det Z = 0: Thus: 4 2 V2 = doubly degenerate conics in CP ; f g 4 2 Chord V2 = all degenerate conics in CP : f g A Hermitian symmetric space and Severi varieties The first result: F. Severi, 1901 A symmetric matrix Computation 5 By arranging coordinates of CP in a symmetric matrix z0 z1 z2 Z = z1 z3 z4 z2 z4 z5 5 / 40 Then (e.g. exercise in J. Harris, Algebraic Geometry, a First Course) 4 Chord V2 : det Z = 0: Thus: 4 2 V2 = doubly degenerate conics in CP ; f g 4 2 Chord V2 = all degenerate conics in CP : f g A Hermitian symmetric space and Severi varieties The first result: F. Severi, 1901 A symmetric matrix Computation 5 By arranging coordinates of CP in a symmetric matrix z0 z1 z2 Z = z1 z3 z4 z2 z4 z5 the Veronese surface V 4 is given by the vanishing of all 2 2 minors, i. e. 2 × Z 2 = (trace Z)Z: 5 / 40 Thus: 4 2 V2 = doubly degenerate conics in CP ; f g 4 2 Chord V2 = all degenerate conics in CP : f g A Hermitian symmetric space and Severi varieties The first result: F. Severi, 1901 A symmetric matrix Computation 5 By arranging coordinates of CP in a symmetric matrix z0 z1 z2 Z = z1 z3 z4 z2 z4 z5 the Veronese surface V 4 is given by the vanishing of all 2 2 minors, i.
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