Non-calibrated framed processes, derived equivalence and Homological Mirror Symmetry
Michele Rossi
TL;DR
The paper develops a framework uniting framed duality ($f$-duality), Homological Mirror Symmetry (HMS), and derived/k-equivalence concepts to study multiple mirror models of projective complete intersections with nonnegative Kodaira dimension. It proves a calibrated/non-calibrated dichotomy, derives a Mirror Theorem in several calibrated/uncalibrated settings, and demonstrates that many distinct mirrors are in fact $K$-equivalent and often $D$-equivalent (notably in Calabi–Yau threefolds) via birational geometry and derived-category techniques. The work extends BB duality to generalized LT-mirrors, constructs large families of intermediate mirrors, and provides concrete evidence supporting the Bondal–Orlov–Kawamata conjectures within a mildly singular toric framework. These results solidify a web-like view of mirror symmetry, where multiple mirrors connect through uncalibrated $f$-processes while preserving deep categorical equivalences, with implications for the HMS program and the study of birational invariants in mirror pairs.
Abstract
The present paper is aimed to discussing three kinds of problems: (1) producing some ``mirror theorem'' for the recent mirror symmetric construction, called \emph{framed} duality ($f$-duality), described in \cite{R-fTV} and \cite{R-fpCI}: this is performed from the point of view proposed by Homological Mirror Symmetry (HMS), by studying \emph{derived equivalence} ($D$-equivalence) of multiple mirror models produced by means of a, so-called, \emph{uncalibrated $f$-process}; (2) proposing a general construction giving a big number of multiple mirror models to, in principle, any projective complete intersection of non-negative Kodaira dimension: these multiple mirrors turn out to be each other connected by means of uncalibrated $f$-processes and then, after (1), $D$-equivalent or $K$-equivalent, in the sense of Kawamata \cite{Kawamata}; (3) presenting a number of evidences for the Bondal-Orlov-Kawamata conjecture that $D$-equivalence is $K$-equivalence, and viceversa.