Topics In Mirror Symmetry For Fano Varieties And Meromorphic Ddp Correspondence

This thesis consists of two parts, each of which can be read independently. The first part is about mirror symmetry of Fano varieties and related topics. We introduce the notion of a hybrid Landau-Ginzburg (LG) model, which is a mirror partner of a Fano variety with a chosen anti-canonical divisor. We formulate Kontsevich's homological mirror symmetry conjecture of such mirror pairs and show that it implies the mirror P=W conjecture, a refined Hodge number relation between associated mirror log Calabi-Yau varieties. Next, we discuss the deformation theory of hybrid LG models and related Hodge numbers. The second part is based on a joint work with Jia-choon Lee. We study the relation between Hitchin system and Calabi-Yau integrable system in the meromorphic setting of type A, motivated by the work of Diaconescu-Donagi-Pantev. We consider a symplectization of the meromorphic Hitchin integrable system, which is a semi-polarized integrable system in the sense of Kontsevich and Soibelman. On the Hitchin side, we show that the moduli space of unordered diagonally framed Higgs bundles forms an integrable system in this sense and recovers the meromorphic Hitchin system as the fiberwise compact quotient. Then we construct a new family of quasi-projective Calabi-Yau threefolds and show that its relative intermediate Jacobian fibration, as a semi-polarized integrable system, is isomorphic to the moduli space of unordered diagonally framed Higgs bundles.