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Bulletin de la SMF - Titles - 145 (2017) 1-27

Titles < 145

Hessian of the natural Hermitian form on twistor spaces
Guillaume Deschamps, Noël Le Du & Christophe Mourougane
Bulletin de la SMF 145, fascicule 1 (2017), 1-27

Télécharger cet article : Fichier PDF

Résumé :
Hessien de la forme hermitienne naturelle sur des espaces de twisteurs
Nous calculons le hessien id'd'' W de la forme hermitienne naturelle  successivement sur la famille de Calabi T (M,g,(I,J,K)) d'une variété hyperkählérienne (M,g,(I,J,K)), sur l'espace des twisteurs T (M,g) d'une variété riemannienne (M,g) de dimension 4 anti-auto duale et sur l'espace des twisteurs T (M,g,D) d'une variété quaternionique kähler (M,g,D). Nous montrons une propriété de convexité de la composante de l'espace des cycles de la famille de Calabi d'une variété hyperkählérienne, qui contient les droites twistorielles. Nous montrons aussi des propriétés de convexité de l'espace des 1-cycles de l'espace des twisteurs d'une variété d'Einstein de dimension 4 anti-auto duale à courbure scalaire négative et de l'espace des 1-cycles de l'espace des twisteurs d'une variété quaternionique kähler à courbure scalaire négative. Nous vérifions aussi qu'aucune variété fortement kählerienne avec torsion (KT) non kählérienne n'est obtenue par les constructions précédentes.

Abstract:
We compute the hessian id'd'' W of the natural Hermitian form  successively on the Calabi family (M,g,(I,J,K)) of a hyperkähler manifold (M,g,(I,J,K)), on the twistor space (M,g) of a 4-dimensional anti-self-dual Riemannian manifold (M,g) and on the twistor space (M,g,D) of a quaternionic Kähler manifold (M,g,D). We show a strong convexity property of the component of cycle space of the Calabi family of a hyperkähler manifold, that contains twistor lines. We also prove convexity properties of the 1-cycle space of the twistor space T (M,g) of a 4-dimensional anti-self-dual Einstein manifold (M,g) of non-positive scalar curvature and of the 1-cycle space of the twistor space T (M,g,D) of a quaternionic Kähler manifold (M,g,D) of non-positive scalar curvature. We check that no non-Kähler strong Kähler with torsion (KT) manifold occurs as such a twistor space.

Keywords: twistor space, 4-dimensional Riemannian manifold, quaternionic Kähler manifold, hyperkähler manifold, strong KT manifolds, Kobayashi Hyperbolicity

Class. math. : 53C28, 53C26, 32Q45


ISSN : 0037-9484
DOI : 10.24033/bsmf.2729
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1
Besse, A. L.
Einstein manifolds
Springer, 1987
Math Reviews MR867684
2
Kodaira, K.
A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds
Ann. of Math. 75 (1962) 146–162
Math Reviews MR0133841
3
Hitchin, N. J.
Kählerian twistor spaces
Proc. London Math. Soc. 43 (1981) 133–150
Math Reviews MR623721
4
Salamon, Simon
Quaternionic Kähler manifolds
Invent. math. 67 (1982) 143–171
Math Reviews MR664330
5
Campana, F.
An application of twistor theory to the nonhyperbolicity of certain compact symplectic Kähler manifolds
J. reine angew. Math. 425 (1992) 1–7
Math Reviews MR1151311 (93g:32036)
6
Campana, F.
Twistor spaces and nonhyperbolicity of certain symplectic Kähler manifolds
in Complex analysis (Wuppertal, 1991)
Aspects Math., E17 (1991) 64–69
Math Reviews MR1122161 (92j:32092)
7
Campana, F.
Orbifolds, special varieties and classification theory
Ann. Inst. Fourier (Grenoble) 54 (2004) 499–630
Math Reviews MR2097416 (2006c:14013)
8
Atiyah, M. F. and Hitchin, N. J. and Singer, I. M.
Self-duality in four-dimensional Riemannian geometry
Proc. Roy. Soc. London Ser. A 362 (1978) 425–461
Math Reviews MR506229
9
de Bartolomeis, Paulo and Nannicini, Antonella
Introduction to differential geometry of twistor spaces
in Geometric theory of singular phenomena in partial differential equations (Cortona, 1995)
Sympos. Math., XXXVIII (1998) 91–160
Math Reviews MR1702089 (2001j:53056)
10
Hitchin, N. J. and Karlhede, A. and Lindström, U. and Roček, M.
Hyper-Kähler metrics and supersymmetry
Comm. Math. Phys. 108 (1987) 535–589
Math Reviews MR877637 (88g:53048)
11
Lieberman, David I.
Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds
in Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975–1977)
Lecture Notes in Math. 670 (1978) 140–186
Math Reviews MR521918 (80h:32056)
12
Fujiki, Akira
Closedness of the Douady spaces of compact Kähler spaces
Publ. Res. Inst. Math. Sci. 14 (1978/79) 1–52
Math Reviews MR0486648 (58 \#6361)
13
Yau, Shing Tung
On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I
Comm. Pure Appl. Math. 31 (1978) 339–411
Math Reviews MR480350 (81d:53045)
14
Mori, Shigefumi and Mukai, Shigeru
The uniruledness of the moduli space of curves of genus 11
in Algebraic geometry (Tokyo/Kyoto, 1982)
Lecture Notes in Math. 1016 (1983) 334–353
Math Reviews MR726433 (85b:14033)
15
Huybrechts, Daniel
Birational symplectic manifolds and their deformations
J. Differential Geom. 45 (1997) 488–513
Math Reviews MR1472886 (98e:32031)
16
Göttsche, L. and Huybrechts, Daniel
Hodge numbers of moduli spaces of stable bundles on K3 surfaces
Internat. J. Math. 7 (1996) 359–372
Math Reviews MR1395935 (97d:14017)
17
Beauville, Arnaud
Variétés kähleriennes dont la première classe de Chern est nulle
J. Differential Geom. 18 (1983) 755–782
Math Reviews MR730926 (86c:32030)
18
O'Grady, Kieran G.
A new six-dimensional irreducible symplectic variety
J. Algebraic Geom. 12 (2003) 435–505
Math Reviews MR1966024 (2004c:14017)
19
O'Grady, Kieran G.
Desingularized moduli spaces of sheaves on a K3
J. reine angew. Math. 512 (1999) 49–117
Math Reviews MR1703077 (2000f:14066)
20
Voisin, Claire
Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes
in Complex projective geometry (Trieste, 1989/Bergen, 1989)
London Math. Soc. Lecture Note Ser. 179 (1992) 294–303
Math Reviewr MR1201391 (94b:32029)
21
Kawamata, Yujiro
On the length of an extremal rational curve
Invent. math. 105 (1991) 609–611
Math Reviews MR1117153 (92m:14026)
22
Huybrechts, Daniel
Compact hyperkähler manifolds
in Calabi-Yau manifolds and related geometries (Nordfjordeid, 2001)
Universitext (2003) 161–225
Math Reviews MR1963562
23
Huybrechts, Daniel
Moduli spaces of hyperkähler manifolds and mirror symmetry
in Intersection theory and moduli
ICTP Lect. Notes, XIX (2004) 185–247
Math Reviews MR2172498 (2006h:53039)
24
Barlet, Daniel
Convexité de l'espace des cycles
Bull. Soc. Math. France 106 (1978) 373–397
Math Reviews MR518045 (80m:32018)
25
Norguet, François and Siu, Yum Tong
Holomorphic convexity of spaces of analytic cycles
Bull. Soc. Math. France 105 (1977) 191–223
Math Reviews MR0590090 (58 \#28677)
26
Beauville, Arnaud
Variétés kähleriennes dont la première classe de Chern est nulle
J. Differential Geom. 18 (1983) 755–782
Math Reviews MR730926 (86c:32030)
27
Schoen, Richard
Conformal deformation of a Riemannian metric to constant scalar curvature
J. Differential Geom. 20 (1984) 479–495
Math Reviews MR788292 (86i:58137)
28
Hitchin, N. J.
Twistor spaces, Einstein metrics and isomonodromic deformations
J. Differential Geom. 42 (1995) 30–112
Math Reviews MR1350695 (96g:53057)
29
Kaledin, D. and Verbitsky, Misha
Non-Hermitian Yang-Mills connections
Selecta Math. (N.S.) 4 (1998) 279–320
Math Reviews MR1669956 (2000a:53041)
30
Berger, Marcel
Remarques sur les groupes d'holonomie des variétés riemanniennes
C. R. Acad. Sci. Paris Sér. A-B 262 (1966) A1316–A1318
Math Reviews MR0200860 (34 \#746)
31
Fornæss, John Erik and Narasimhan, Raghavan
The Levi problem on complex spaces with singularities
Math. Ann. 248 (1980) 47–72
Math Reviews MR569410 (81f:32020)
32
Lelong, Pierre
Intégration sur un ensemble analytique complexe
Bull. Soc. Math. France 85 (1957) 239–262
Math Reviews MR0095967 (20 \#2465)
33
Verbitsky, Misha
Ergodic complex structures on hyperkähler manifolds
Acta Math. 215 (2015) 161–182
Math Reviews MR3413979
34
Verbitsky, Misha
Rational curves and special metrics on twistor spaces
Geom. Topol. 18 (2014) 897–909
Math Reviews MR3190604
35
36
Fino, Anna and Parton, Maurizio and Salamon, Simon
Families of strong KT structures in six dimensions
Comment. Math. Helv. 79 (2004) 317–340
Math Reviews MR2059435 (2005f:53061)
37
Fino, Anna and Tomassini, Adriano
A survey on strong KT structures
Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52(100) (2009) 99–116
Math Reviews MR2521810 (2010h:53108)