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We study SYZ mirror symmetry in the context of non-Kaehler Calabi-Yau manifolds. In particular, we study the six-dimensional Type II supersymmetric $SU(3)$ systems with Ramond-Ramond fluxes, and generalize them to higher dimensions. We show that Fourier-Mukai transform provides the mirror map between these Type IIA and Type IIB supersymmetric systems in the semi-flat setting. This is concretely exhibited by nilmanifolds.
Consider a Maslov zero Lagrangian submanifold diffeomorphic to a Lie group on which an anti-symplectic involution acts by the inverse map of the group. We show that the Fukaya $A_infty$ endomorphism algebra of such a Lagrangian is quasi-isomorphic to
We construct new smooth solutions to the Hull-Strominger system, showing that the Fu-Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. In particular, we prove that, for $13 leq k leq 22$ and $14leq
We construct new examples of solutions of the Hull-Strominger system on non-Kahler torus bundles over K3 surfaces, with the property that the connection $ abla$ on the tangent bundle is Hermite-Yang-Mills. With this ansatz for the connection $ abla$,
We show that every Kaehler algebraic curvature tensor is geometrically realizable by a Kaehler manifold of constant scalar curvature. We also show that every para-Kaehler algebraic curvature tensor is geometrically realizable by a para-Kaehler manifold of constant scalar curvature
Let $fcolon M^{2n}tomathbb{R}^{2n+ell}$, $n geq 5$, denote a conformal immersion into Euclidean space with codimension $ell$ of a Kaehler manifold of complex dimension $n$ and free of flat points. For codimensions $ell=1,2$ we show that such a subman