Non-calibrated framed processes, derived equivalence and Homological Mirror Symmetry

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.