Title: SYZ transforms for immersed Lagrangian multi-sections,
Abstract: In this paper, we study the geometry of the SYZ transform on a semi-flat
Lagrangian torus fibration. Our starting point is an investigation on the
relation between Lagrangian surgery of a pair of straight lines in a symplectic
2-torus and extension of holomorphic vector bundles over the mirror elliptic
curve, via the SYZ transform for immersed Lagrangian multi-sections. This study
leads us to a new notion of equivalence between objects in the immersed Fukaya
category of a general compact symplectic manifold $(M, \omega)$, under which
the immersed Floer cohomology is invariant; in particular, this provides an
answer to a question of Akaho-Joyce. Furthermore, if $M$ admits a Lagrangian
torus fibration over an integral affine manifold, we prove, under some
additional assumptions, that this new equivalence is mirror to isomorphism
between holomorphic vector bundles over the dual torus fibration via the SYZ
transform.