Toric models of LCYs

As stated last time, one of the motivations for learning the very classical story of elementary transformations was to better appreciate the role they play in GHK mirror symmetry. In turn, apart it from being a radical development in cluster theory and mirror symmetry itself that I want learn more about, I’d very much like to understand how the other versions of mirror symmetry that I’ve encountered (and some that I haven’t) can be reconciled with the GHK mentality.

It seems that much of the philosophy for mirror symmetry for Fano varieties or, roughly more generally, log Calabi-Yau (LCY) pairs is based on their proximity to toric varieties. In the Fano case as studied by Coates-Corti-Galkin-Golyshev-Kasprzyk-… the mirror is built from the toric degenerations of the Fano you started with. I’ve posted before about the conjecture that ‘smooth implies TG’. In the LCY case studied by GHK(K), it is toric models that play the part of mirror builders.

LCY is the log version of CY: these are pairs (Y,D) consisting of a smooth projective variety and a normal crossings divisor D\subset Y such that D\in|{-K_Y}|. Normal crossings is important for certain calculations to depend only on the interior U=Y\setminus D, which is also often referred to as an [open] LCY. There’s a good source of these: if Y is a toric variety and D is its toric boundary – that is, D=Y\setminus T removing the big torus, or it’s the divisor from the rays of the polytope defining Y – then (Y,D) is a LCY. A toric LCY is a ‘toric model’ of a LCY is it can be obtained by a series of suitable blowdowns.

The point is that for each toric model (\overline{Y},\overline{D}) of (Y,D) one gets an open torus inside U=Y\setminus D. Gathering all the tori from all the toric models of (Y,D) covers U up to codimension 2, which is good enough to study regular functions by Hartogs’ lemma (which is what one ends up doing). Of course, like collections of toric degenerations, the collection of toric models comes with much more structure than I’ve stated so far. One can ‘mutate’ between toric models via elementary transformations.

Here lies the problem with naively comparing GHK to CCGGK via comparing toric models and toric degenerations. I discussed the classical example last time of the elementary transformation from \mathbb{F}_0=\mathbb{P}^1\times\mathbb{P}^1 to $\mathbb{F}_1=\text{dP}_8$. These are then both toric models for each other. However, they are not deformation-equivalent! Hence, they get conflated in GHK and distinguished in CCGGK.

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