Variation of gluing in homological mirror symmetry

This thesis is a collection of seven papers concerned with the relationship between variation of gluing spaces and categories in homological mirror symmetry(HMS). We divide it into three parts according to how we vary gluing what. The first part consists of three papers on algebraic deformations of Calabi–Yau 3-folds(CY3s), where we vary complex structures to glue locally trivial deformations. The second part consists of two papers on cut-and-reglue procedure for relative Jacobians of generic elliptic 3-folds, where we vary Brauer classes to glue smooth elliptic 3-folds with sections. The third part consists of two papers on local-to-global principle for wrapped Fukaya categories of very affine hypersurfaces(VAHs), where we vary Liouville structures to glue pairs of pants. Our main goal of the first two parts is to construct new Fourier– Mukai partners(FMPs), nonbirational derived-equivalent CY3s. While birational CY3s are derived-equivalent, FMPs give highly nontrivial multiple mirrors to the dual manifolds. Our main goal of the third part is to establish HMS for complete intersections of VAHs. Recently, Gammage–Shende established HMS for VAHs under some assumption essential to construct a global skeleton, which allows them to reduce gluing wrapped Fukaya categories to gluing local skeleta. For several reasons we need a different approach to remove their assumption. In the first paper, we prove that the derived equivalence of CY3s extends to their versal de formations over an affine complex variety. This is fundamental for our deformation methods to construct new examples of FMPs. Due to the main theorem of the second paper, the derived category of the generic fiber of a flat proper family can be described as a certain Verdier quo tient. As a consequence, the derived equivalence of the above versal deformations is inherited to their generic fibers. We analyze some good cases where also nonbirationality is inherited, establishing a deformation method to construct new FMPs from known examples. Conversely, the description enables us to prove specialization, i.e., the derived equivalence of the generic fibers extends to general fibers, completing all the relevant inductions of the derived equiva lence of CY3s through deformations. The main theorem of the third paper gives a rigorous explanation of these phenomena. Namely, deformations of a CY3 are equivalent to Morita deformations of its dg category of perfect complexes. We also prove that, analogous to isomor phisms of schemes, the derived equivalence is inherited from effectivizations to their enough close approximations. This is an improvement of the main theorem of the first paper, expected from the equivalence of the two deformation theories. In the fourth paper, we prove that any flat projective family must be what we call an almost coprime twisted power, whenever it is linear derived-equivalent over the base to a generic el liptic CY3. This should be the best possible reconstruction result for generic elliptic CY3s. Combining with the main theorem of the first paper, we obtain a family of pairs of coprime twisted powers whose closed fibers are nonbirational whenever they are nonisomorphic. Un winding our arguments, one sees that generic elliptic CY3s are linear derived-equivalent over the base if and only if their generic fibers are derived-equivalent. This is the key observation for the fifth paper where we give affirmative answers to two of the four conjectures raised by Knapp–Scheidegger–Schimannek. Namely, we prove that each of 12 pairs of elliptic CY3s constructed by them share the relative Jacobian and linear derived-equivalent over the base. Except one self-dual pair, the closed fibers of the family obtained by the above combination are nonisomorphic. Hence we obtain families of new FMPs, establishing another deformation method to construct FMPs. As far as we know, this is the first systematic construction of (fami lies of) FMPs. Moreover, it works for elliptic CY3s with higher multisections, whose examples some string theorists have been looking for. In the sixth paper, we establish HMS for complete intersections of VAHs. The main chal lenge is computing wrapped Fukaya categories of complete intersections. With the aid of equivariantization/de-equivariantization, we reduce it to unimodular case. Proving that locally complete intersections are products of lower dimensional pairs of pants, we reduce it further to hypersurface case without the assumption imposed on the previous result by Gammage– Shende. We extend it by inductive argument following Pascaleff–Sibilla which does not re quire any global skeleton. Besides the invariance of wrapped Fukaya categories under simple Liouville homotopies, one key is to find Weinstein structures on the initial exact symplectic manifold and the additional pair of pants which glue to yield that on the gluing, everytime we proceed the inductive argument. Another is to show that also their wrapped Fukaya categories glue to yield that of the gluing. Our method should work to compute wrapped Fukaya cat egories in other relevant settings. Finally, we glue HMS for pairs of pants along the global combinatorial duality over the tropical hypersurface. The geometry of VAHs is further studied in the seventh paper, where we complete the missing A-side of the SYZ picture over fanifolds. This can be regarded as a generalization of that over tropical hypersurfaces.