LA DIASTASIS DI EUGENIO CALABI. webpage: loi.sc.unica.it

Transcription

1 LA DIASTASIS DI EUGENIO CALABI webpage: loi.sc.unica.it 1

2 (V,ω) varietà di Kähler reale analitica ω U = i 2 Φ, U V Φ:U R potenziale Kähleriano di ω i 2 Φ = i 2 Ψ = Φ=Ψ+ϕ + ϕ, ϕ Hol(U) Φ reale analitico = Φ :W U U C Φ(p, q) Φ(p) = Φ(p, p) Φ(p, q) = Φ(q, p) 2

3 Diastasis (Calabi, Ann. of Math. 1953) Supponiamo W = U U D : U U R D(p, q) = Φ(p, p)+ Φ(q, q) Φ(p, q) Φ(q, p) D(p, q) =D(q, p) D(p, p) =0 i 2 D(p, ) =ω Proprietà fondamentale della diastasis f :(V,ω) (W, Ω) olomorfa f (Ω) = ω D V (p, q) =D W (f(p),f(q)) 3

4 DIASTASIS DEGLI SPAZI DI FORME Esempio 1. (C n,ω = 2 i nj=1 dz j d z j ) ω = i 2 z 2 = Φ(z) = z 2 = Φ(z, w) =z w D(z,w) = z 2 + w 2 z w w z = z w 2 Esempio 2. (CP n b,ω FS), b>0 p 0 =[1, 0,...,0] U 0 = {Z 0 0} D(p 0,z)= 1 b log(1 + b z 2 ),z j = Z j Z 0 Esempio 3. (CH n b,ω hyp), b<0 CH n b = {z Cn z 2 < 1 b } D(0,z)= 1 b log(1 + b z 2 ) Esempio 4. Spazi di Fubini F (n, b),n,b R F (n, 0) = C n,f(n, b) =CP n b, CHn b. 4

5 One could formulate several conjectures on the behaviour of the diastasis in the large, which in the author s opinion would furnish ample material for future study (Calabi 1953) The diastasis has been rarely used up to now but it might still have a rich future (Berger 2000) 5

6 RIGIDITY SPACE FORMS EINSTEIN METRICS SYMMETRIC SPACES DIASTASIS GEODESICS BALLS BALANCED METRICS QUANTIZATION EXPANSION T Y Z 6

7 RIGIDITA Teorema (Calabi, Ann. of Math. 1953) f :(V,ω) F (n, b), g :(V,ω) F (n,b) full n = n g = U f. Teorema (M. Green, J. Diff. Geom. 1978) f :(V,ω) (W, Ω), g :(V,ω) (W, Ω) non degeneri U Aut(W ) Symp(W, Ω) tale che g = U f, 7

8 SPAZI DI FORME COMPLESSI Teorema (Calabi, Ann. of Math. 1953) CH k b Cn C k CP n b CH k b CP n b Teorema (Calabi, Ann. of Math. 1953) F (n, b) f F (n,b ) = F (n, b) f F (n,b ) Teorema (Umehara, Tokyo J. Math. 1987) (V,ω), (V,ω) C n (V,ω) CH n b (V,ω), (V,ω) C n (V,ω) CP n b (V,ω), (V,ω) CH n b (V,ω) CP n b 8

9 METRICHE DI EINSTEIN Teorema Le uniche sottovarietàcompatte di Kähler- -Einstein V n di CP n+2 (n 2) sono: Q n CP n+1 CP n+2 (S. Chern, J. Differ. Geom. 1, 1967) CP n CP n+2 (K. Tsukada, Math. Ann. 274, 1986) Teorema (D. Hulin, J. Geom. Anal. 10, 2000) KE compatta CP n b = c 1 (KE) > 0 Teorema (M. Umehara, Tôhoku Math. J., 1987) KE f C n (oppure KE f CH n b ) = f tot. geod. 9

10 DOMANDE SULLE METRICHE DI EINSTEIN I Teorema (D. Hulin) KE compatta CP n b = c 1 (KE) > 0 D1 KE compatta CP n b = KE = S(simmetrico)? D2 KE CP n b, scal > 0= KE V compatta? 10

11 DOMANDE SULLE METRICHE DI EINSTEIN II Ingrediente fondamentale nel teorema di Hulin: Coordinate di Bochner (Bochner 1947) p V (z 1,...,z k )inu V,z j (p) =0 D(p, q) = z(q) 2 + j,k a j,k z j (q) z k (q), a j,0 = a 0,j =0. D(p, q) =(ρ(p, q)) 2 + O((ρ(p, q)) 4 ) D3 Quale èillegame tra le coordinate di Bochner di una varietà compatta e c 1 (KE)? Teorema (Arezzo-Loi, Sem. der Univ. Hamburg 74, 2004) Supponiamo che le le coordinate di Bochner (z 1,...z k )intorno ad un punto p di una varietà dike compatta si estendano a funzioni olomorfe (f 1,...f k ) su un aperto U KE tali che i k U 2 kdf 1 d f 1 df k d f k <. Allora c 1 (KE) > 0. 11

12 SPAZI SIMMETRICI I Teorema (H. Nakagawa and R. Takagi, J. Math. Soc. Japan 28, 1976) LocS f C n (oppure LocS f CH n b ) = f tot. geod. Teorema (M. Takeuchi, Japan J. Math 4, 1978) LocS completo f CPb n f = LocS = S + CP n b Teorema (H. Tasaki, Osaka J. Math. 22, 1985) {q S + D(p, ) non è definita} = Cutlocus(p) Teorema (M. Engliš, preprint 2005) Sia (V,ω) una varietà di Kähler reale analitica. p (e D(p,q) )= q (e D(p,q) ) (V,ω) =LocS 12

13 SPAZI SIMMETRICI II Teorema (A. Loi, Diff. Geom. Appl. 2005) Sia (W, Ω) una varietà di Kähler almost projective-like, i.e. e D(x, ) è definita in W x W. Allora LocS completo f (W, Ω) = LocS = S f (W, Ω) Corollario 1 LocS completo f Ŝ = LocS = S f Ŝ Corollario 2 LocS completo f F (N,b) = LocS = S f F (N,b) Teorema (A. Loi, Diff. Geom. Appl. 2005) S i S j,i j, i, j =, 0, + 13

14 DOMANDE SUGLI SPAZI SIMMETRICI Teorema (M. Umehara) KE f C n (oppure KE f CH n b ) = f tot. geod. D1 KE f S = f tot. geod.? Teorema (Umehara) (V,ω), (V,ω) C n (V,ω) CH n b (V,ω), (V,ω) C n (V,ω) CP n b (V,ω), (V,ω) CH n b (V,ω) CP n b D2 (V,ω), (V,ω) C n (V,ω) S? D3 (V,ω), (V,ω) C n (V,ω) S +? D4 (V,ω), (V,ω) S (V,ω) S +? 14

15 GEODETICHE Teorema (H. Tasaki) M p = {q S + D(p, ) non è definita} = Cutlocus(p) Problema Studiare i legami tra il Cutlocus(p) diun punto p (V,ω) em p. Piùingenerale, studiare i legami tra la diastasis e l applicazione esponenziale. Teorema (A. Loi, Diff. Geom. Appl. Luglio 2005) Sia (V,ω,g) una varietà reale analitica. Esiste U TV e un embedding tale che: ν : U TV : (x, v) (x, ν x (v)) ν x : T x V U T x V ν(u) D(x, exp x (ν x (v))) = g x (v, v), (x, v) U 15

16 BIBLIOGRAFIA Diastasis e immersioni isometriche S. Bochner, Curvature in Hermitian metric, Bull. Amer. Math. Soc. 53 (1947), E. Calabi, Isometric Imbeddings of Complex Manifolds, Ann. of Math. 58 (1953), M. Green, Metric rigidity of holomorphic maps to Kähler manifolds, J. Diff. Geom. 13 (1978), M. Umehara, Kähler Submanifolds of Complex Space Forms, Tokyo J. Math. vol. 10 (1987), Diastasis e metriche di Kähler-Einstein C. Arezzo and A. Loi, A note on Kaehler Einstein metrics and Bochner s coordinates, Abh. Math. Sem. der Univ. Hamburg 74 (2004), D. Hulin, Sous-variétés complexes d Einstein de l espace projectif, Bull. Soc. math. France 124 (1996), D. Hulin, Kähler-Einstein metrics and projective embeddings, J. Geom. Anal. 10 (2000), Y. Matsuyama, On a 2-dimensional Einstein Kähler submanifold of a complex space form Proc. Amer. Math. Soc. 95 (1985), K. Tsukada, Einstein-Kähler Submanifolds with codimension two in a Complex Space Form, Math. Ann. 274 (1986), M. Umehara, Einstein Kähler submanifolds of complex linear or hyperbolic space, Tôhoku Math. J. (1987),

17 Diastasis e spazi simmetrici M. Takeuchi, Homogeneous Kähler submanifolds in projective spaces, Japan J. Math 4 (1978), H. Nakagawa and R. Takagi, On locally symmetric Kähler submanifolds in a complex projective space, J.Math. Soc. Japan 28 no. 4 (1976), M. Engliš, A characterization of symmetric domains, preprint 2005, available in: englis /papers.html - no. 47. Y. Taniguchi, On congruent holomorphic mappings into a hermitian symmetric space, Osaka J. Math. 36 (1999), A. Loi, Calabi s diastasis function for Hermitian symmetric spaces, to appear in Diff. Geom. Appl. H. Tasaki, The cut locus and the diastasis of a Hermitian symmetric space of compact type, Osaka J. Math. 22 (1985), Diastasis, quantizzazione e sviluppo asintotico di T Y Z C. Arezzo and A. Loi, Quantization of Kähler manifolds and the asymptotic expansion of Tian Yau Zelditch, J.Geom. Phys. 47 (2003), F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Quantum mechanics as a deformation of classical mechanics, Lett. Math. Phys. 1 (1977), F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization, part I, Ann. of Phys. 111 (1978),

18 F.A. Berezin, Quantization, Math. USSR Izvestija 8 (1974), M. Cahen, S. Gutt, J. H. Rawnsley, Quantization of Kähler manifolds I: Geometric interpretation of Berezin s quantization, JGP. 7 (1990), M. Cahen, S. Gutt, J. H. Rawnsley, Quantization of Kähler manifolds II, Trans. Amer. Math. Soc. 337 (1993), M. Cahen, S. Gutt, J. H. Rawnsley, Quantization of Kähler manifolds III, Lett. Math. Phys. 30 (1994), M. Cahen, S. Gutt, J. H. Rawnsley, Quantization of Kähler manifolds IV, Lett. Math. Phys. 34 (1995), A. Karabegov, M. Schlichenmaier, Identification of Berezin Toeplitz quantization, J. Reine Angew. Math. 540 (2001), A. Loi, The Tian Yau Zelditch asymptotic expansion for real analytic Kähler metrics, Intern. J. of Geometric Methods in Modern Physics v. 1 No 3 (2004), A. Loi, ALaplace integral, the T-Y-Z expansion and Berezin s transform on a Kaehler manifold to appear in Intern. J. of Geometric Methods in Modern Physics. M. Schlichenmaier, Deformation quantization of compat Kähler manifolds by Berezin Toeplitz quantization, Math. Phys. Stud. 22 (2000), Estensioni del concetto di diastasis A. Romero, An extension of Calabi s rigidity theorem to complex submanifolds of indefinite complex space forms, Advances in Maths 13 (1974),

19 M. Umehara, Diastases and real analytic functions on complex manifolds, J. Math. Soc. Japan 40 (1988), Diastasis e geodetiche A. Loi, A Laplace integral on a Kähler manifold and Calabi s diastasis function, to appear in Diff. Geom. Appl. 19