Title: Non-Kähler Mirror Symmetry of the Iwasawa Manifold,
Abstract: We propose a new approach to the Mirror Symmetry Conjecture in a form
suitable to possibly non-Kähler compact complex manifolds whose canonical
bundle is trivial. We apply our methods by proving that the Iwasawa manifold
$X$, a well-known non-Kähler compact complex manifold of dimension $3$, is
its own mirror dual to the extent that its Gauduchon cone, replacing the
classical Kähler 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
Frölicher 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.