Geometry & Topology

A family of compact complex and symplectic Calabi–Yau manifolds that are non-Kähler

Lizhen Qin and Botong Wang

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We construct a family of 6 –dimensional compact manifolds M ( A ) which are simultaneously diffeomorphic to complex Calabi–Yau manifolds and symplectic Calabi–Yau manifolds. They have fundamental groups , their odd-degree Betti numbers are even, they satisfy the hard Lefschetz property, and their real homotopy types are formal. However, M ( A ) × Y is never homotopy equivalent to a compact Kähler manifold for any topological space Y . The main ingredient to show the non-Kählerness is a structure theorem of cohomology jump loci due to the second author.

Article information

Geom. Topol., Volume 22, Number 4 (2018), 2115-2144.

Received: 17 June 2016
Revised: 15 April 2017
Accepted: 15 June 2017
First available in Project Euclid: 13 April 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32J27: Compact Kähler manifolds: generalizations, classification 53D05: Symplectic manifolds, general

Kähler manifolds Calabi-Yau manifolds


Qin, Lizhen; Wang, Botong. A family of compact complex and symplectic Calabi–Yau manifolds that are non-Kähler. Geom. Topol. 22 (2018), no. 4, 2115--2144. doi:10.2140/gt.2018.22.2115.

Export citation


  • A Akhmedov, Symplectic Calabi–Yau $6$–manifolds, Adv. Math. 262 (2014) 115–125
  • J Amorós, M Burger, K Corlette, D Kotschick, D Toledo, Fundamental groups of compact Kähler manifolds, Mathematical Surveys and Monographs 44, Amer. Math. Soc., Providence, RI (1996)
  • D Arapura, Geometry of cohomology support loci for local systems, I, J. Algebraic Geom. 6 (1997) 563–597
  • I K Babenko, I A Taĭmanov, On nonformal simply connected symplectic manifolds, Sibirsk. Mat. Zh. 41 (2000) 253–269 In Russian; translated in Sib. Math. J. 41 (2000) 204–217
  • S Baldridge, P Kirk, Coisotropic Luttinger surgery and some new examples of symplectic Calabi–Yau $6$–manifolds, Indiana Univ. Math. J. 62 (2013) 1457–1471
  • W Barth, C Peters, A Van de Ven, Compact complex surfaces, Ergeb. Math. Grenzgeb. (3) 4, Springer (1984)
  • F A Bogomolov, On Guan's examples of simply connected non-Kähler compact complex manifolds, Amer. J. Math. 118 (1996) 1037–1046
  • R Bott, L W Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer (1982)
  • M Burger, Fundamental groups of Kähler manifolds and geometric group theory, from “Séminaire Bourbaki, 2009/2010”, Astérisque 339, Soc. Math. France (2011) exposé 1022, 305–321
  • L A Cordero, M Fernández, A Gray, Symplectic manifolds with no Kähler structure, Topology 25 (1986) 375–380
  • P Deligne, P Griffiths, J Morgan, D Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975) 245–274
  • A Dimca, Sheaves in topology, Springer (2004)
  • M Fernández, M de León, M Saralegui, A six-dimensional compact symplectic solvmanifold without Kähler structures, Osaka J. Math. 33 (1996) 19–35
  • M Fernández, V Muñoz, Formality of Donaldson submanifolds, Math. Z. 250 (2005) 149–175
  • M Fernández, V Muñoz, An $8$–dimensional nonformal, simply connected, symplectic manifold, Ann. of Math. 167 (2008) 1045–1054
  • J Fine, D Panov, Symplectic Calabi–Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold, J. Differential Geom. 82 (2009) 155–205
  • J Fine, D Panov, Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle, Geom. Topol. 14 (2010) 1723–1763
  • R Friedman, On threefolds with trivial canonical bundle, from “Complex geometry and Lie theory” (J A Carlson, C H Clemens, D R Morrison, editors), Proc. Sympos. Pure Math. 53, Amer. Math. Soc., Providence, RI (1991) 103–134
  • E Goldstein, S Prokushkin, Geometric model for complex non-Kähler manifolds with ${\rm SU}(3)$ structure, Comm. Math. Phys. 251 (2004) 65–78
  • R E Gompf, A new construction of symplectic manifolds, Ann. of Math. 142 (1995) 527–595
  • G Grantcharov, Geometry of compact complex homogeneous spaces with vanishing first Chern class, Adv. Math. 226 (2011) 3136–3159
  • P Griffiths, J Harris, Principles of algebraic geometry, Wiley, New York (1978)
  • D Guan, Examples of compact holomorphic symplectic manifolds which are not Kählerian, II, Invent. Math. 121 (1995) 135–145
  • D Guan, Examples of compact holomorphic symplectic manifolds which are not Kählerian, III, Internat. J. Math. 6 (1995) 709–718
  • J Gutowski, S Ivanov, G Papadopoulos, Deformations of generalized calibrations and compact non-Kähler manifolds with vanishing first Chern class, Asian J. Math. 7 (2003) 39–79
  • S Halperin, Lectures on minimal models, Mém. Soc. Math. France 9–10, Soc. Math. France, Paris (1983)
  • P Lu, G Tian, The complex structures on connected sums of $S^3\times S^3$, from “Manifolds and geometry” (P de Bartolomeis, F Tricerri, E Vesentini, editors), Sympos. Math. XXXVI, Cambridge Univ. Press (1996) 284–293
  • G \TH Magnússon, Automorphisms and examples of compact non-Kähler manifolds, preprint (2012)
  • D McDuff, Examples of simply-connected symplectic non-Kählerian manifolds, J. Differential Geom. 20 (1984) 267–277
  • J W Milnor, J D Stasheff, Characteristic classes, Annals of Mathematics Studies, Princeton Univ. Press (1974)
  • J Morrow, K Kodaira, Complex manifolds, Holt, Rinehart and Winston, New York (1971) Reprinted (with errata) by AMS Chelsea Publishing, Providence, RI (2006)
  • S Papadima, A Suciu, Geometric and algebraic aspects of $1$–formality, Bull. Math. Soc. Sci. Math. Roumanie 52(100) (2009) 355–375
  • S Papadima, A I Suciu, Bieri–Neumann–Strebel–Renz invariants and homology jumping loci, Proc. Lond. Math. Soc. 100 (2010) 795–834
  • J Park, Non-complex symplectic $4$–manifolds with $b_2^+=1$, Bull. London Math. Soc. 36 (2004) 231–240
  • I Smith, R P Thomas, S-T Yau, Symplectic conifold transitions, J. Differential Geom. 62 (2002) 209–242
  • W P Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976) 467–468
  • R Torres, J Yazinski, Geography of symplectic $4$– and $6$–manifolds, Topology Proc. 46 (2015) 87–115
  • A Tralle, J Oprea, Symplectic manifolds with no Kähler structure, Lecture Notes in Mathematics 1661, Springer (1997)
  • L-S Tseng, S-T Yau, Non-Kähler Calabi–Yau manifolds, from “String-Math 2011” (J Block, J Distler, R Donagi, E Sharpe, editors), Proc. Sympos. Pure Math. 85, Amer. Math. Soc., Providence, RI (2012) 241–254
  • C Voisin, Hodge structures on cohomology algebras and geometry, Math. Ann. 341 (2008) 39–69
  • B Wang, Torsion points on the cohomology jump loci of compact Kähler manifolds, Math. Res. Lett. 23 (2016) 545–563
  • G W Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics 61, Springer (1978)