# Fenglong You, Degenerations, fibrations and higher rank Landau-Ginzburg models (Friday, July 22)

:::{.remark}
Mirror symmetry:

| A model | B model |
|---------|---------|
| Symplectic geometry | Complex geometry |
| GW invariants | Periods (related by Givental et al) |
| Degenerations to relative GW invariants | Relative periods (related by You et al) |

Today: how are periods related to relative periods? Fibration structures.
See Gross-Siebert program.
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:::{.remark}
Degenerations: let $X \mapsvia{\pi}  A$ with $\pi\inv(t) = X$ a Calabi-Yau and $\pi\inv(0) = X_1 \union_D X_2$ where $D\in \abs{-K_X}$ is smooth.
Example: a quintic degenerates to $\Bl \PP^3 \union_{\mathrm{K3}} Q_4$.
LG models: $X_i\dual \mapsvia{W_i}  \CC$ with generic fiber $D\dudal$?
Doran-Harder-Thompson conjecture: one can glue $W_1, W_2$ to get $X\dual \to \PP^1$ whose fiber $D\dual$ is mirror to $D$.
Thus the mirror of a degeneration is a fibration.
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:::{.example title="?"}
Examples of periods:

- $f_{Q_5} = \sum_{d\geq 0} q^d {(5d)! \over (d!)^5}$
- $f_{\mathrm{K3}} = \sum_{d\geq 0} q^d {(4d)! \over (d!)^4}$

How these are related: there is a rational function $A = {9\over y(y-1)^4}$?
One can check the following easily using the residue theorem:
\[
{1\over 2\pi i} \oint { f_{\mathrm{K3}} \over y(1-y)}\dy = \sum_{d\geq 0} q^d {(5d)! \over (d)^5}
.\]
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:::{.theorem title="?"}
Similar identities hold for relative periods (using the Hadamard product) for toric complete intersections and holds for a basis of solutions for DEs.
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:::{.remark}
Degenerations of Fanos, the quartic case: $Q_4\leadsto Q_3 \union_{D_0} \Bl \PP^3$ where $D_0$ is a cubic surface.
What is the mirror of $(Q_3, D_0)$ and $(\Bl\PP^3, D_0)$? 
These are not CY, so add an anticanonical divisor $D$ and degenerate not just the Fano but rather the pairs $(Q_4, D)$ where $D \leadsto D_1\union_{D_{12}} D_2$ for $D_i$ hypersurfaces in $Q_3$ and $\Bl \PP^3$ respectively.
This is degeneration of log CYs?
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:::{.remark}
Gluing conjecture: $(X, D) \leadsto (X_1, D_0 + D_1), (X_2, D_0 + D_2)$ which have mirrors (?) which can be glued along the first factors so obtain $(X\dual, W)$.
Evidence toward this: an Euler characteristic formula $\chi(X) = \cdots$, and an integral formula:
\[
{1\over 2\pi i}\oint f_{X_1, D_0 + D_1} \ast f_{X_2, D_0 + D_2} {\dy\over y} = f_{X, D} \ast f_{D_{0}, D_{12}}
.\]
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:::{.remark}
See degeneration to the normal cone, quantum Lefschetz principle.

> LG models are complex versions of Morse theory??

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