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\end{filecontents}
\addbibresource{references.bib}\begin{document}

\title{constructing mirrors}
 \maketitle
 \thispagestyle{firstpage}The first examples of mirror symmetry involved Calabi--Yau hypersurfaces in projective space. Let $H\subset \CP^n$ be a smooth hypersurface of degree $d$, so $H$ has complex dimension $n-1$. We can compute its first Chern class using the exact sequence
\[
0 \to T^{1,0}H\to T^{1,0}\CP^n|_H\to N_{H/\CP^n}\to 0.
\]
This tells us that
\[
c(T\CP^n)|_H=c(TH)\cdot c(N_{H/\CP^n}).
\]
Suppose that the hypersurface has degree $d$, and let $h=c_1(\mathcal O_{\CP^n}(1))|_H$. Then
\[
c(T\CP^n)|_H=(1+h)^{n+1},\qquad c(N_{H/\CP^n})=1+dh.
\]
Therefore,
\[
1+(n+1)h+\cdots=(1+c_1(TH)+\cdots)(1+dh),
\]
so
\[
c_1(TH)=(n+1-d)h.
\]
Equivalently, by adjunction,
\[
K_H\simeq \mathcal O_H(d-n-1).
\]
Thus $d=n+1$ is necessary and sufficient for $H$ to have trivial canonical bundle. If $H$ is smooth, a degree $n+1$ hypersurface in $\CP^n$ is a compact K\"ahler manifold with trivial canonical bundle, hence is Calabi--Yau in the sense of \cref{def:calabiYauManifold}. The quintic threefold in $\CP^4$ is the standard example.

\begin{definition}\cite{Mirror symmetry motivation notes}
\label{def:largeComplexStructureLimit}
Let
\[
\pi:\mathcal X^*\to \Delta^*
\]
be a one-parameter family of Calabi--Yau $n$-folds over a punctured disc. Parallel transport around the puncture gives a monodromy operator
\[
T:H^n(X_t;\QQ)\to H^n(X_t;\QQ).
\]
After replacing the punctured disc by a finite cover if necessary, assume that the monodromy is unipotent, and write $T$ for this unipotent monodromy. Then one can define
\[
N:=\log T=(T-I)-\frac{(T-I)^2}{2}+\frac{(T-I)^3}{3}-\cdots .
\]
The series is finite because $T-I$ is nilpotent. In this introductory convention, a boundary point $t=0$ is called a \emph{large complex-structure limit}, or a \emph{maximally unipotent monodromy point}, if
\[
N^n\neq 0
\]
on $H^n(X_t;\QQ)$. Equivalently, $N$ has nilpotency index $n+1$, the maximum possible for a weight-$n$ variation of Hodge structure. More refined definitions also impose conditions on the limiting mixed Hodge structure; the maximally unipotent monodromy criterion is the part used in this introductory discussion.
\end{definition}

For the degree $n+1$ hypersurface family, there is a particularly degenerate limit to the union of the $n+1$ coordinate hyperplanes. The central fiber is a singular toric normal-crossings variety, not a smooth toric manifold. A general strategy for constructing mirrors is to do mirror symmetry in families. Let $\mathcal X\to \Delta$ be a degeneration of Calabi--Yau manifolds whose central fiber $X_0$ is toric or toroidal and possibly singular. Then the strategy is:
\begin{enumerate}
\item find a toric or toroidal degeneration of the family to $X_0$;
\item construct a mirror degeneration $\check X_0$ using the dual toric or affine-combinatorial data;
\item deform or smooth $\check X_0$ to obtain the mirror family $\check{\mathcal X}$, and match the deformation parameters using the \underline{\href{https://jeffhicks.net/snippets/index.php?tag=def:mirrorMap}{ mirror map}}.
\end{enumerate}
There are several motivating reasons for taking this path:
\begin{itemize}
\item The SYZ proposal interprets mirror symmetry as fiberwise $T$-duality, which suggests looking at dual tori.
\item Homological mirror symmetry suggests that a Lagrangian torus fiber equipped with a flat unitary local system should correspond to a skyscraper sheaf at a point of the mirror. Thus torus fibrations are natural objects to examine.
\item The topological calculation below shows why dualizing torus fibers exchanges the two Hodge numbers.
\end{itemize}

\begin{definition}\cite{Mirror symmetry motivation notes}
\label{def:specialLagrangianTorusFibration}
Let $(X,\omega,\Omega)$ be a Calabi--Yau $n$-fold with its Ricci-flat K\"ahler metric, and assume that $\Omega$ is normalized with respect to this metric. For $\theta\in \RR/2\pi\ZZ$, an oriented real $n$-dimensional submanifold $L\subset X$ is \emph{special Lagrangian of phase $\theta$} if
\[
\omega|_L=0,\qquad \operatorname{Im}(e^{-i\theta}\Omega)|_L=0,
\]
and
\[
\operatorname{Re}(e^{-i\theta}\Omega)|_L=\operatorname{vol}_L
\]
as positive volume forms on $L$.

A \emph{special Lagrangian torus fibration} of phase $\theta$ is a continuous map
\[
f:X\to B
\]
to a real $n$-dimensional base such that, over a dense open subset $B_0\subset B$, the restriction
\[
f^{-1}(B_0)\to B_0
\]
is a smooth fiber bundle whose fibers are $n$-tori that are special Lagrangian submanifolds of phase $\theta$.

In the SYZ setting, nontrivial compact examples are expected to have singular fibers over a discriminant locus $\Delta=B\setminus B_0$.
\end{definition}

\begin{definition}\cite{Mirror symmetry motivation notes}
\label{def:dualTorusFibration}
Let
\[
f:X_0\to B_0
\]
be a smooth torus fibration with fibers $F_b\simeq V_b/\Lambda_b$, where $V_b$ is a real vector space and $\Lambda_b\subset V_b$ is a lattice. The \emph{dual torus fibration} is
\[
\check f:\check X_0\to B_0,
\qquad
\check F_b:=\check f^{-1}(b)=\operatorname{Hom}(\Lambda_b,U(1))
\simeq H^1(F_b;\RR)/H^1(F_b;\ZZ).
\]
Fiberwise,
\[
H_1(\check F_b;\ZZ)\simeq H^1(F_b;\ZZ),
\qquad
H^1(\check F_b;\ZZ)\simeq H_1(F_b;\ZZ).
\]
If the three-torus fibers are oriented, then Poincar\'e duality, with respect to the chosen orientations, gives the useful identifications
\[
H^1(\check F_b;\CC)\simeq H^2(F_b;\CC),
\qquad
H^2(\check F_b;\CC)\simeq H^1(F_b;\CC).
\]
These fiberwise identifications induce the corresponding dual local systems over $B_0$.
\end{definition}

Here is a motivating calculation from \cite{gross2001special}. It deliberately ignores many details about singular fibers, which are essential in the actual theory, but it captures the reason dual tori produce the mirror flip.

Suppose that we have a special Lagrangian torus fibration
\[
f:X\to B
\]
of a Calabi--Yau threefold. Assume, for this heuristic calculation, that the fibration is smooth, that the fibers are three-tori $F=f^{-1}(p)$, that $X$ has the Hodge diamond shape from \cref{art:hodgeDiamondAndDeformation}, and that the base $B$ is a closed oriented three-manifold with $H^1(B,\CC)=H^2(B,\CC)=0$. A globally smooth torus fibration with simply connected total space is not the actual situation for compact Calabi--Yau threefolds; singular fibers are necessary. The simplified model is still useful because it isolates the cohomological mechanism.

Let
\[
\mathcal H^q:=R^q f_*\CC
\]
denote the local system over $B$ whose fiber at $p$ is $H^q(F,\CC)$. The special Lagrangian condition gives an orientation of the fibers: after choosing the phase, $\operatorname{Re}\Omega$ restricts to a volume form on each fiber. Thus
\[
\mathcal H^0\simeq \CC,\qquad \mathcal H^3\simeq \CC.
\]
The Leray--Serre spectral sequence has
\[
E_2^{p,q}=H^p(B,\mathcal H^q)\Longrightarrow H^{p+q}(X,\CC).
\]
The $E_2$ page has the form
\[ \begin{tikzcd}[scale=.5]
H^0(B, \mathcal H^3) & H^1(B, \mathcal H^3) & H^2(B, \mathcal H^3) & H^3(B, \mathcal H^3)\\
H^0(B, \mathcal H^2) & H^1(B, \mathcal H^2) & H^2(B,  \mathcal H^2) & H^3(B,  \mathcal H^2) \\
H^0(B, \mathcal H^1) & H^1(B,  \mathcal H^1) & H^2(B, \mathcal H^1) & H^3(B, \mathcal H^1) \\
H^0(B, \mathcal H^0) & H^1(B, \mathcal H^0)& H^2(B,  \mathcal H^0) & H^3(B, \mathcal H^0).
\end{tikzcd}
\]Using $\mathcal H^0\simeq \mathcal H^3\simeq \CC$ and $H^1(B,\CC)=H^2(B,\CC)=0$, the top and bottom rows become

\[ \begin{tikzcd}[scale=.5]
\CC & 0& 0& \CC\\
H^0(B, \mathcal H^2) & H^1(B, \mathcal H^2) & H^2(B,  \mathcal H^2) & H^3(B,  \mathcal H^2) \\
H^0(B, \mathcal H^1) & H^1(B,  \mathcal H^1) & H^2(B, \mathcal H^1) & H^3(B, \mathcal H^1) \\
\CC & 0& 0  & \CC.
\end{tikzcd}
\]Assume further, as part of this simplified model, that the spectral sequence degenerates at $E_2$ and that the relevant invariant-cycle terms vanish:
\[
H^0(B,\mathcal H^1)=H^0(B,\mathcal H^2)=0.
\]
By duality, this also gives
\[
H^3(B,\mathcal H^1)=H^3(B,\mathcal H^2)=0.
\]
Then the relevant part of the spectral sequence is
\[ \begin{tikzcd}[scale=.5]
\CC & 0& 0& \CC\\
0 & H^1(B, \mathcal H^2) & H^2(B,  \mathcal H^2) & 0 \\
0 & H^1(B,  \mathcal H^1) & H^2(B, \mathcal H^1) & 0 \\
\CC & 0& 0  & \CC.
\end{tikzcd}
\]Consequently,
\[
H^2(X,\CC)\simeq H^1(B,\mathcal H^1),
\]
and
\[
H^3(X,\CC)\simeq \CC\oplus H^1(B,\mathcal H^2)\oplus H^2(B,\mathcal H^1)\oplus \CC.
\]
Since $X$ is a Calabi--Yau threefold with $h^{2,0}=0$, we have
\[
\dim H^2(X,\CC)=h^{1,1}(X),\qquad \dim H^3(X,\CC)=2h^{2,1}(X)+2.
\]
Moreover, Poincar\'e duality on the base with local coefficients, together with fiberwise duality, gives
\[
\dim H^1(B,\mathcal H^2)=\dim H^2(B,\mathcal H^1).
\]
Therefore,
\[
h^{1,1}(X)=\dim H^1(B,\mathcal H^1),\qquad
h^{2,1}(X)=\dim H^1(B,\mathcal H^2).
\]

Now look at the dual torus fibration
\[
\check f:\check X\to B.
\]
By \cref{def:dualTorusFibration}, fiberwise the dual torus satisfies
\[
H^1(\check F,\CC)\simeq H_1(F,\CC)\simeq H^2(F,\CC),
\]
and similarly
\[
H^2(\check F,\CC)\simeq H_2(F,\CC)\simeq H^1(F,\CC).
\]
Thus the local systems for the dual fibration are swapped:
\[
\check{\mathcal H}^1\simeq \mathcal H^2,
\qquad
\check{\mathcal H}^2\simeq \mathcal H^1.
\]
Applying the same calculation to $\check X$ gives
\[
h^{1,1}(\check X)=\dim H^1(B,\check{\mathcal H}^1)
=\dim H^1(B,\mathcal H^2)=h^{2,1}(X),
\]
and
\[
h^{2,1}(\check X)=\dim H^1(B,\check{\mathcal H}^2)
=\dim H^1(B,\mathcal H^1)=h^{1,1}(X).
\]
This is the topological shadow of the mirror flip of Hodge numbers.
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