# Lawrence Barrott, The Doran-Harder-Thompson conjecture via the Gross-Siebert program? (Thursday, July 21) :::{.remark} Goal: understand proportion of period integral via enumerative invariants on the mirror. For mirror symmetry, start with a family of CYs $X \mapsvia{f} \AA^1$ with $X$ smooth and $f\inv(0) = \Union D_i$ an SNC divisor. Locally this is the map $f^*: t\to \prod_{i=1}^n x_i$ for $t$ a coordinate on $\AA^1$, and we ask that $X$ is etale locally isomorphism to $\spec k[x_i, y_i]$ for $y_i$ some free variables. Want to encode fibers of this family via period integrals; use the dual intersection complex $\Sigma_X$, the subset of the cone \( \bigoplus \RR_{\geq 0} D_i \) where $\vector a\in \Sigma_X$ if the $D_i$ with $a_i > 0$ all intersect with nonempty intersection. ::: :::{.example title="?"} Take a quartic K3 degenerating to a toric boundary. Resolve 24 singularities via blowups to get a cone over a standard simplex. Sum coordinates to $\sum D_i$ to get a map $\Sigma_X \to \RR_{\geq 0}$, then $\Sigma_X(Z)$ gives a collection of integer points. ::: :::{.theorem title="Gross-Siebert"} There is a ring structure on formal monomials $k[\Sigma_X(Z)]$ extending the one induced by "conewise addition". ::: :::{.remark} This is a graded ring and the mirror is $\hat X = \Proj k[\Sigma_X(Z)]$ where $z^pz^q = \sum N_r z^r$ where $N_r$ is a count of punctured (open) GW invariants. This also works for pairs $(Y, D)$ of a variety and an anticanonical divisor $D\in \abs{-K_Y}$. Having a family with a map to $\AA^1$ yields a Landau-Ginzburg model. In this case, $\Proj \Sigma_{(Y, D)}(Z)$ has a canonical map to $\AA^1$, and superpotential $\hat{Y} \mapsvia{W} \AA^1$. ::: :::{.remark} Tyurin degenerations: a family $X \mapsvia{f} \AA^1$ CYs with $X$ smooth and $f\inv(0) = Y_1 \union_D Y_2$ glued along a anticanonical divisor $D \in \abs{-K_X}$. ::: ## DHT Conjecture :::{.remark} Let $A \mapsvia{f} \AA^2$ be a family of CYs and suppose - $f\inv(t, 0) = Y_1 \union_D Y_2$ - $f\inv(0, t)$ is an SNC with a maximality condition Here locally $t_1 = \prod x_i, t_2 = \prod y_i$, and $X$ is étale-isomorphic to $\spec k[x_i, y_i, z_i]$ where the $z_i$ are free variables. For some reason, $\Sigma_X$ looks like a Toblerone, and we now get *two* maps to $\AA^1$. ![](figures/2022-07-21_14-15-01.png) ::: :::{.remark} The DHT conjecture: $\hat{X}$ is a fibred Calabi-Yau and $\hat{X} \to \PP^1$ with fibers mirror to $D$. Topologically, the two maps to $\AA^1$ can be glued to get the map to $\PP^1$, and here $\hat{Y}_1 \mapsvia{W_1} \AA^1, \hat{Y}^2 \mapsvia{ W_2} \AA^1$ can be glued to get the fibration. Doran-Kostiuk-You construct examples as complete intersections on toric varieties of the form $\Pi\times \PP^1$ and related the periods of the fibres via Hadamard products of relative $I$ functions. Moral: can glue periods for $\hat{Y_1}, \hat{Y_2}$ to get periods for all of $X$. ::: :::{.remark} The GS program gives a scattering diagram (pairs of walls and functions) on the Toblerone which gives gluing instructions for the mirror family. Scattering diagrams have canonical differential forms, i.e. canonical sections of $\omega{\hat X}$, but $\ro{\omega_{\hat X}}{\hat D} \neq \omega_{\hat D}$ -- there are correction terms coming from other parts of the scattering diagram. Ruddat-Seibert give a way to compute period integrals associated to 1-dimensional balanced subsets of $\Sigma_X$ and form an $n\dash$cycle $\beta$ and integrating the canonical differential from from the scattering diagram, yielding $\int_\beta \omega_{\hat X}$. Since $\beta$ is contained in a fiber, one gets a relation \[ \int_\beta \omega_{\hat D} (1 + \sum N_k) = \int_\beta \ro{\omega_{\hat X}}{\hat D} ,\] and the $N_k$ count curves with $k+1$ marked points lying on the fiber, all tangent to $Y_1, Y_2$ (and possible negative or punctured). ::: :::{.remark} An ideal theorem: any purple curve (DZG: forgot to mark which was purple!) can be formed by gluing components on $\Sigma_{X_1, D}, \Sigma_{X_2, D}$: ![](figures/2022-07-21_14-27-38.png) :::