On the spectral sets of Inoue surfaces

The Inoue surfaces are certain non-Kaehler complex surfaces that have the structure of a $T^3$ bundle over the circle. We study the Inoue surfaces $S_M$ with the Tricerri metric and the canonical spin${}^c$ structure, and the corresponding chiral Dirac operators twisted by a flat $\mathbb C^$--connection\mathrm{.}Thetwistingconnectionisdeterminedby$z \in \mathbb C^$, and the points for which the twisted Dirac operators ${\mathcal{D}}_z^{\pm }$ are not invertible are called spectral points. We show that there are no spectral points inside the annulus ${\alpha }^{-1\mathrm{/}4}<\mathrm{\mid }z\mathrm{\mid }<{\alpha }^{1\mathrm{/}4}$, where $\alpha

1$istheonlyrealeigenvalueofthematrix$M$thatdetermines$S_M$, and find the spectral points on its boundary. Via Taubes’ theory of end-periodic operators, this implies that the corresponding Dirac operators are Fredholm on any end-periodic manifold whose end is modeled on $S_M$.