Towards mirror symmetry for varieties of general type

Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds

“…Conjecture 38 ([PSh15a, Conjecture 1.1], see also [GKR12]). Let X be a smooth Fano variety of dimension n. Let f X be its toric Landau-Ginzburg model corresponding to the trivial divisor on X.…”

An Introduction to Hodge Structures

“…Using this it can be shown [15,Lemma 4.4] that there is a short exact sequence of cohomological mixed Hodge complexes…”

Mirror symmetry & supersymmetry on SU(4)-structure backgrounds

“…In this case, as it turns out, both the LTY 2A and 2B systems are related to supersymmetric solutions of type IIB, rather than 2A to IIA and 2B to IIB as was the case in d = 4. 11 The reason for this is that for SU(4), both e −iJ and Ω are even polyforms (or in the language of 10 Specifically, they describe the so called SU(3) case described by [5,6], with type (0,3) and type (3,0) generalized almost complex structures. 11 We shall nevertheless keep the "LTY 2A" and "LTY 2B" nomenclature for the respective systems, and hopefully this will not lead to confusion.…”

“…3 We shall also re-examine the Fourier-Mukai and T-duality transformations that relate the mirror backgrounds.…”

“…There is some progress in extending the SYZ approach to Kähler manifolds (see e.g. [2,3]) and generalized Calabi-Yau manifolds [4], and the immediate questions are which geometries admit mirrors and how to construct a mirror map. One immediate place to look for generalizations is in the supersymmetric geometries, i.e.…”