Mirror symmetry beyond Calabi-Yau and Fano

In a joint work with Mark Gross and Ludmil Katzarkov, we propose a mirror symmetry construction for varieties of any Kodaira dimension. In general, the mirror will be very singular and equipped with a sheaf of vanishing cycles. We associate Hodge numbers to the singular space with
this sheaf and show that these fulfil the mirror duality $h^{p,q}(X) = h^{d-p,q}(Y)$ for $X,Y$ a mirror pair of our construction. In my talk, I will introduce the construction, demonstrate it for the a genus two curve and draw connections to homological mirror symmetry if time
permits.