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We present a detailed study of a new mathematical object in$\mathrm{E}_{6(6)}\times \mathbb{R}^{+}$ generalised geometry called an`exceptional complex structure' (ECS). It is the extension of a conventionalcomplex structure to one that includes all the degrees of freedom of M-theoryor type IIB supergravity in six or five dimensions, and as such characterises,in part, the geometry of generic supersymmetric compactifications tofive-dimensional Minkowkski space. We define an ECS as an integrable$\mathrm{U}^{*}(6)\times \mathbb{R}^{+}$ structure and show it is equivalent toa particular form of involutive subbundle of the complexified generalisedtangent bundle $L_{1} \subset E_{\mathbb{C}}$. We also define a refinement, an$\mathrm{SU}^{*}(6)$ structure, and show that its integrability requires inaddition a vanishing moment map on the space of structures. We are able toclassify all possible ECSs, showing that they are characterised by two numbersdenoted `type' and `class'. We then use the deformation theory of ECS to findthe moduli of any $\mathrm{SU}^{*}(6)$ structure. We relate these structures tothe geometry of generic minimally supersymmetric flux backgrounds of M-theoryof the form $\mathbb{R}^{4,1}\times M$, where the $\mathrm{SU}^{*}(6)$ modulicorrespond to the hypermultiplet moduli in the lower-dimensional theory. Suchgeometries are of class zero or one. The former are equivalent to a choice of(non-metric-compatible) conventional $\mathrm{SL}(3,\mathbb{C})$ structure andstrikingly have the same space of hypermultiplet moduli as the fluxlessCalabi--Yau case.

Exceptional complex structures and the hypermultiplet moduli of 5d Minkowski compactifications of M-theory