There is a set of ideas called mirror symmetry which, roughly speaking, relates symplectic and complex geometry of Calabi--Yau manifolds. There are also extensions to Fano and general type varieties but let's forget about them for now.

There are several more or less precise incarnations of mirror symmetry, including so-called SYZ and HMS programs. Each gives us a certain recipe to construct the manifold mirror to a given Calabi--Yau.

In SYZ, the expectation is that to construct the mirror manifold we need to put our Calabi--Yau in a family of Calabi--Yau's over the punctured disc so that the monodromy operator is maximally unipotent. Different maximally unipotent degenerations can lead to different mirror manifolds.

In HMS, let's assume that we have finally found the right definition of Fukaya category. After certain algebraic manipulations, we can cook up a triangulated category which should be equivalent to the derived category of coherent sheaves on the mirror manifold. If we pick a t-structure in that category, we can Gabriel-style reconstruct the mirror manifold itself from the heart of t-structure. However, if I understand correctly, there is no natural choice of t-structure so we may get several mirror manifolds.

The question is: are these two ambiguities (choosing a maximally unipotent degeneration and choosing a heart of t-structure) in some sort of bijective correspondence? Are there some references where this is studied?

P.S.: to avoid certain subtleties in hyperkaehler geometry, let's assume that the holonomy group of Calabi--Yau's is equal the special unitary group.

EDIT: on second thought, it feels like this should be wrong (Calabi--Yau's can be derived equivalent for reasons that have nothing to do with mirror symmetry). Is there some way to single out the mirrors arising from SYZ inside the set of mirrors arising from HMS?


Some relation between these two ambiguities is indeed part of the expected but still largely conjectural big picture.

Let $X$ be a compact Calabi-Yau manifold, viewed in a symplectic way. Let $M_X$ be the moduli space of complex structures on $X$. The various maximally unipotent degenerations are the various ways to go to infinity in $M_X$ in a maximally degenerated way. Some picture useful to have in mind for $M_X$ in some non-compact object, with various "cusps" corresponding to the various maximally unipotent degenerations.

Let $F(X)$ be the Fukaya category of $X$. It is expected that each choice of complex structure on $X$, i.e. of point of $M_X$, defines a stability condition on $F(X)$ in the sense of Bridgeland. A stability condition on $F(X)$ includes the data of a t-structure on $X$. The expectation is that this stability condition should be constructed in terms of the geometry of special Lagrangians in $X$ (the notion of special Lagrangian depends on the holomorphic volume on $X$ determined by the chosen complex structure on $X$), see for example https://arxiv.org/abs/1401.4949

For each choice $j$ of maximally unipotent degeneration, i.e. each "cusp" of $M_X$, we expect to have a SYZ fibration from which it is possible to construct a mirror $Y_j$, with some derived category of coherent sheaves $D(Y_j)$ equivalent to $F(X)$. The general expectation is that, if we consider the family of stability conditions defined by a family of complex structure going to $j$, then the corresponding $t$-structures have for limit the standard $t$-structure on $D(Y_j)$ whose heart is the abelian category of coherent sheaves on $Y_j$. More precisely, one should be able to identifiy a neighborhood of the cusp $j$ in $M_X$ with an open part of the Kähler cone of $Y_j$ and the limit "go to j" should be the "large volume limit" for $Y_j$. A semi-expository version of this story can be found in the book http://www.claymath.org/library/monographs/cmim04.pdf

Some more technical point: one should rather consider the universal cover of $M_X$, the fundamental group acting on $F(X)$ by autoequivalence, or equivalently stability conditions up to autoequivalences.

A remark about your "second thought": I think that all the known examples of derived equivalences between Calabi-Yau manifolds have something to do with mirror symmetry. There are no known a priori reasons why it should be the case, but I don't know a clear counterexample either.


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