Holomorphic Deformations of Balanced Calabi-Yau ¯-Manifolds
Annales de l'Institut Fourier, to appear, 56 p.

Given a compact complex n-fold X satisfying the ¯-lemma and supposed to have a trivial canonical bundle K X and to admit a balanced (=semi-Kähler) Hermitian metric ω, we introduce the concept of deformations of X that are co-polarised by the balanced class [ω n-1 ]H n-1,n-1 (X,)H 2n-2 (X,) and show that the resulting theory of balanced co-polarised deformations is a natural extension of the classical theory of Kähler polarised deformations in the context of Calabi–Yau or holomorphic symplectic compact complex manifolds. The concept of Weil–Petersson metric still makes sense in this strictly more general, possibly non-Kähler context, while the Local Torelli Theorem still holds.

Soit X une variété complexe compacte lisse de dimension n, à fibré canonique trivial, qui satisfait le lemme du ¯ et possède une métrique hermitienne équilibrée (=semi-kählérienne) ω. Nous introduisons le concept de déformations de X co-polarisées par la classe équilibrée [ω n-1 ]H n-1,n-1 (X,)H 2n-2 (X,) et montrons que la théorie des déformations équilibrées co-polarisées est une extension naturelle de la théorie classique des déformations kählériennes polarisées dans le contexte des variétés complexes compactes lisses de Calabi–Yau et dans celui des variétés holomorphes symplectiques. La notion de métrique de Weil–Petersson a encore un sens dans ce contexte strictement plus général, non nécéssairement kählérien, tandis que le théorème de Torelli local est encore valable.

Received : 2017-09-14
Revised : 2018-02-06
Accepted : 2018-03-13
Classification:  32G05,  14C30,  14F43,  32Q25,  53C55
Keywords: co-polarisation by a balanced class, ¯-manifold, possibly non-Kähler Calabi–Yau manifold, deformations of complex structures, Weil–Petersson metric
     author = {Popovici, Dan},
     title = {Holomorphic Deformations of Balanced Calabi-Yau $\partial \bar{\partial }$-Manifolds},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
Popovici, Dan. Holomorphic Deformations of Balanced Calabi-Yau $\partial \bar{\partial }$-Manifolds. Annales de l'Institut Fourier, to appear, 56 p.

[1] Alessandrini, Lucia; Bassanelli, Giovanni Small Deformations of a Class of Compact Non-Kähler Manifolds, Proc. Am. Math. Soc., Tome 109 (1990) no. 4, pp. 1059-1062 | Zbl 0702.32017

[2] Alessandrini, Lucia; Bassanelli, Giovanni Metric Properties of Manifolds Bimeromorphic to Compact Kähler Spaces, J. Differ. Geom., Tome 37 (1993), pp. 95-121

[3] Alessandrini, Lucia; Bassanelli, Giovanni Modifications of Compact Balanced Manifolds, C. R. Math. Acad. Sci. Paris, Tome 320 (1995) no. 12, pp. 1517-1522

[4] Alessandrini, Lucia; Bassanelli, Giovanni The class of Compact Balanced Manifolds Is Invariant under Modifications, Complex Analysis and Geometry (Trento, 1993), Marcel Dekker (Lecture Notes in Pure and Applied Mathematics) Tome 173 (1996), pp. 1-17 | Zbl 0848.32021

[5] Campana, Frédéric The Class 𝒞 Is Not Stable by Small Deformations, Math. Ann., Tome 290 (1991), pp. 19-30

[6] Campana, Frédéric On Twistor Spaces of the Class 𝒞, J. Differ. Geom., Tome 33 (1991), pp. 541-549

[7] Campana, Frédéric Remarques sur les groupes de Kähler nilpotents, Ann. Sci. Éc. Norm. Supér., Tome 28 (1995) no. 3, pp. 307-316

[8] Campana, Frédéric Orbifolds, Special Varieties and Classification Theory, Ann. Inst. Fourier, Tome 54 (2004) no. 3, pp. 499-630

[9] Chiose, Ionuţ Obstructions to the Existence of Kähler Structures on Compact Complex Manifolds, Proc. Am. Math. Soc., Tome 142 (2014) no. 10, pp. 3561-3568 | Zbl 1310.32023

[10] Deligne, Pierre; Griffiths, Phillip; Morgan, John; Sullivan, Dennis Real Homotopy Theory of Kähler Manifolds, Invent. Math., Tome 29 (1975), pp. 245-274 | Zbl 0312.55011

[11] Demailly, Jean-Pierre Complex Analytic and Algebraic Geometry (http://www-fourier.ujf-grenoble.fr/~demailly/books.html )

[12] Fino, Anna; Otal, Antonio; Ugarte, Luis Six Dimensional Solvmanifolds with Holomorphically Trivial Canonical Bundle, Int. Math. Res. Not., Tome 2015 (2015) no. 24, pp. 13757-13799 | Zbl 1334.53079

[13] Friedman, Robert ¯-Lemma for General Clemens Manifolds (2017) (https://arxiv.org/abs/1708.00828v1 )

[14] Fu, Jixiang; Li, Jun; Yau, Shing-Tung Balanced Metrics on Non-Kähler Calabi-Yau Threefolds, J. Differ. Geom., Tome 90 (2012) no. 1, pp. 81-129 | Zbl 1264.32020

[15] Gauduchon, Paul Fibrés hermitiens à endomorphisme de Ricci non négatif, Bull. Soc. Math. Fr., Tome 105 (1977), pp. 113-140

[16] Gauduchon, Paul Structures de Weyl et théorèmes d’annulation sur une variété conforme autoduale, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Tome 18 (1991) no. 4, pp. 563-629 | Zbl 0763.53034

[17] Kasuya, Hisashi Techniques of Computations of Dolbeault Cohomology of Solvmanifolds, Math. Z., Tome 273 (2013) no. 1-2, pp. 437-447 | Zbl 1261.22009

[18] Kobayashi, Shoshichi; Wu, Hung-Hsi On Holomorphic Sections of Certain Hermitian Vector Bundles, Math. Ann., Tome 189 (1970), pp. 1-4 | Zbl 0189.52201

[19] Kodaira, Kunihiko; Spencer, Donald C. On Deformations of Complex Analytic Structures, III. Stability Theorems for Complex Structures, Ann. Math., Tome 71 (1960) no. 1, pp. 43-76 | Zbl 0128.16902

[20] Kuranishi, Masatake On the Locally Complete Families of Complex Analytic Structures, Ann. Math., Tome 75 (1962) no. 3, pp. 536-577 | Zbl 0106.15303

[21] Lebrun, Claude; Poon, Yat-Sun Twistors, Kähler Manifolds, and Bimeromorphic Geometry. II, J. Am. Math. Soc., Tome 5 (1992) no. 2, pp. 317-325 | Zbl 0766.5305

[22] Michelsohn, Marie-Louise On the Existence of Special Metrics in Complex Geometry, Acta Math., Tome 149 (1982) no. 3-4, pp. 261-295 | Zbl 0531.53053

[23] Popovici, Dan Deformation Openness and Closedness of Various Classes of Compact Complex Manifolds; Examples, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Tome 13 (2014) no. 2, pp. 255-305 | Zbl 1294.32006

[24] Popovici, Dan Aeppli Cohomology Classes Associated with Gauduchon Metrics on Compact Complex Manifolds, Bull. Soc. Math. Fr., Tome 143 (2015) no. 4, pp. 1-37 | Zbl 1347.53060

[25] Schweitzer, Michel Autour de la cohomologie de Bott-Chern (2007) (https://arxiv.org/abs/0709.3528v1 )

[26] Székelyhidi, Gábor; Tosatti, Valentino; Weinkove, Ben Gauduchon Metrics with Prescribed Volume Form, Acta Math., Tome 219 (2017) no. 1, pp. 181-211 | Zbl 1396.32010

[27] Tian, Gang Smoothness of the Universal Deformation Space of Compact Calabi-Yau Manifolds and Its Petersson-Weil Metric, Mathematical Aspects of String Theory (San Diego, 1986), World Scientific (Advanced Series in Mathematical Physics) Tome 1 (1987), pp. 629-646 | Zbl 0696.53040

[28] Todorov, Andrey N. The Weil-Petersson Geometry of the Moduli Space of SU(n3) (Calabi-Yau) Manifolds I, Commun. Math. Phys., Tome 126 (1989), pp. 325-346 | Zbl 0688.53030

[29] Tosatti, Valentino; Weinkove, Ben Hermitian Metrics, (n-1,n-1)-forms and Monge-Ampère Equations (2013) (https://arxiv.org/abs/1310.6326, to appear in J. Reine Angew. Math.)

[30] Voisin, Claire Hodge Theory and Complex Algebraic Geometry. I., Cambridge University Press, Cambridge Studies in Advanced Mathematics, Tome 76 (2002)

[31] Wu, Chun-Chun On the Geometry of Superstrings with Torsion, Harvard University (USA) (2006) (Ph. D. Thesis)

[32] Yoshioka, Kōta Moduli Spaces of Stable Sheaves on Abelian Surfaces, Math. Ann., Tome 321 (2001), pp. 817-884