## Geometry & Topology

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

#### Abstract

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 $ℤ ⊕ ℤ$, 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

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

Dates
Revised: 15 April 2017
Accepted: 15 June 2017
First available in Project Euclid: 13 April 2018

https://projecteuclid.org/euclid.gt/1523584818

Digital Object Identifier
doi:10.2140/gt.2018.22.2115

Mathematical Reviews number (MathSciNet)
MR3784517

Zentralblatt MATH identifier
06864333

#### Citation

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. https://projecteuclid.org/euclid.gt/1523584818

#### References

• 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)