Riv. Mat. Univ. Parma, Vol. 9, No. 2, 2018

Dan Popovici [a]

Albanese map and self-duality of the Iwasawa manifold

Pages: 177-190
Accepted in revised form: 26 February 2018
Mathematics Subject Classification (2010): 32G05, 53C55, 14F43.
Keywords: Hodge theory, sGG manifolds, Albanese torus.
[a]: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France

Abstract: We prove that the three-dimensional Iwasawa manifold $$X$$, viewed as a locally holomorphically trivial fibration by elliptic curves over its two-dimensional Albanese torus, is self-dual in the sense that the base torus identifies canonically with its dual torus, the Jacobian torus of $$X$$, under a sesquilinear duality, while the fibre identifies with itself. To this end, we derive elements of Hodge theory for arbitrary sGG manifolds, introduced in earlier joint work of the author with L. Ugarte as those compact complex manifolds on which all the Gauduchon metrics are strongly Gauduchon, to construct in an explicit way the Albanese torus and map of any sGG manifold. These definitions coincide with the classical ones in the special Kähler and $$\partial\bar\partial$$ (i.e. satisfying the $$\partial\bar\partial$$-lemma) cases. The generalisation to the larger sGG class is made necessary by the Iwasawa manifold being an sGG, non-$$\partial\bar\partial$$, manifold. The main result of this paper can be seen as a complement from a different perspective to the author's very recent work where a non-Kähler mirror symmetry of the Iwasawa manifold was revealed. We also hope that it will suggest yet another approach to non-Kähler mirror symmetry for different classes of manifolds.

References
[Ang14]
D. Angella, Cohomological aspects in complex non-Kähler geometry, Lecture Notes in Math., 2095, Springer, Cham, 2014. MR3137419
[Bla58]
A. Blanchard, Sur les variétés analytiques complexes, Ann. Sci. Ecole Norm. Sup. (3) 73 (1956), 157-202. MR0087184
[Dem97]
J.-P. Demailly, Complex analytic and algebraic geometry, http://www-fourier.ujf-grenoble.fr/~demailly/books.html
[Pop13]
D. Popovici, Holomorphic deformations of balanced Calabi-Yau $$\partial\bar\partial$$-manifolds, Ann. Inst. Fourier (Grenoble), to appear.
[Pop14]
D. Popovici, Deformation openness and closedness of various classes of compact complex manifolds; examples, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), 255-305. MR3235516
[Pop15]
D. Popovici, Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds, Bull. Soc. Math. France 143 (2015), 763-800. MR3450501
[Pop17]
D. Popovici, Non-Kähler mirror symmetry of the Iwasawa manifold, arXiv:1706.06449, preprint, 2017. https://arxiv.org/abs/1706.06449
[PU14]
D. Popovici and L. Ugarte, Compact complex manifolds with small Gauduchon cone, Proc. London Math. Soc. (3) 116 (2018), 1161-1186. doi: 10.1112/plms.12110
[Sch07]
M. Schweitzer, Autour de la cohomologie de Bott-Chern, arXiv:0709.3528, preprint, 2007. https://arxiv.org/abs/0709.3528
[Uen75]
K. Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Math., 439, Springer-Verlag, Berlin-New York, 1975. MR0506253

Home Riv.Mat.Univ.Parma