Non-K\"ahler Mirror Symmetry of the Iwasawa Manifold

Abstract

We propose a new approach to the Mirror Symmetry Conjecture in a form suitable to possibly non-K\"ahler compact complex manifolds whose canonical bundle is trivial. We apply our methods by proving that the Iwasawa manifold $X$, a well-known non-K\"ahler compact complex manifold of dimension $3$, is its own mirror dual to the extent that its Gauduchon cone, replacing the classical K\"ahler cone that is empty in this case, corresponds to what we call the local universal family of essential deformations of $X$. These are obtained by removing from the Kuranishi family the two "superfluous" dimensions of complex parallelisable deformations that have a similar geometry to that of the Iwasawa manifold. The remaining four dimensions are shown to have a clear geometric meaning including in terms of the degeneration at $E_2$ of the Fr\"olicher spectral sequence. On the local moduli space of "essential" complex structures, we obtain a canonical Hodge decomposition of weight $3$ and a variation of Hodge structures, construct coordinates and Yukawa couplings while implicitly proving a local Torelli theorem. On the metric side of the mirror, we construct a variation of Hodge structures parametrised by a subset of the complexified Gauduchon cone of the Iwasawa manifold using the sGG property of all the small deformations of this manifold proved in earlier joint work of the author with L. Ugarte. Finally, we define a mirror map linking the two variations of Hodge structures and we highlight its properties.