Peng Zhou

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====== Batyrev-Borisov Construction ======
This note is my attempt to understand how the mirror construction works, and how it compares with the monomial-divisor toric correspondence. 

===== Batyrev-Borisov construction =====
Let $V$ be a real vector space, $V^\vee$ its dual. For $x \in V, y \in V^\vee$, we use $(x,y)$ for their pairing. For $A, B \In V$, we use $A+B$ for their Minkowski sum, use $Conv(A \cup B)$ for convex hull. 

For any bounded subset $S \In V$, we define its "support function"
$$ h_S: V^\vee \to \R, \quad h_S(y) = \sup_{x \in S} (x,y). $$
it is clear that $h_S = h_{Conv(S)}$ and  $h_S$ is a convex function. Furthermore, we have
$$h_{S_1 + S_2} = h_{S_1} + h_{S_2}$$
$$h_{S_1 \cup S_2} = \max(h_{S_1}, h_{S_2})$$


**Definition** (nef partition)
Let $\Delta \In V$ be a (vertex simplicial) convex polytope with $0$ in its interior, with polar dual (facet simplicial)
$$ \Delta^\vee = \{y \in V^\vee \mid (x,y) \leq 1 \}. $$
Let $\Sigma[\Delta^\vee]$ be the face fan generated by cone of faces of $\Delta^\vee$. 
A nef partition is a decomposition
$$ Vert(\Delta^\vee) = E_1 \sqcup \cdots \sqcup E_r $$
such that for each $i=1,\cdots, r$, there exists piecewise linear convex function with linearity domain being the simplices
$$ \varphi_i: \Delta^\vee to \R, \varphi_i|_{E_i}=1, \quad \varphi_i|_{E_j}=0 \; \forall j \neq i. $$

We define 
$$ \nabla_i = Conv(0, E_i), \quad \nabla = \sum_i \nabla_i $$
and 
$$ \Delta_i = \{x \in V \mid (x, y) \leq \varphi_i(y) \forall y \in \Delta^\vee \} = Conv( d \varphi_i). $$

** Prop 1 **
$$ \Delta = \sum_i \Delta_i. $$
**proof**: it suffices to show
$$ h_{\Delta} = \sum_i h_{\Delta_i} $$
But $h_{\Delta}$ is the PL convex function on $T[\Delta^\vee]$ that is $1$ on all the vertices, and $h_{\Delta_i} = \varphi_i$ is the PL convex function that is $1$ on $E_i$ and $0$ on $E_{j \neq i}$. So we get $h_{\Delta} = \sum_i h_{\Delta_i}$ on $Vert(\Delta^\vee)$, hence on all $\Delta^\vee$. 

** Prop 2 **
$$ \nabla^\vee = Conv(\{\Delta_i\})$$
** proof **:  
$$ h_{\cup_i \Delta_i} = \max_i h_{\Delta_i} = \max_i \varphi_i =: \psi$$
We want to show that $\psi$ restricts to vertices of $\nabla = \sum_i \nabla_i$ equals $1$. Suppose $v \in \nabla$ is a vertex, then we claim there exists a face $F$ of $\Delta^\vee$, whose vertices $w_1, \cdots, w_k$ belongs to distinct $E_{i_1}, \cdots, E_{i_k}$,$v=w_1+\cdots+w_k$. Let barycenter of $F$ be $c_F = (w_1+ \cdots + w_k)/k$. Then $\psi(c_F) = 1/k$ and $\psi(v) = \psi(k c_F) = k \psi(c_F) = k*1/k = 1$. These claims needs to be justified. 

===== Skeleton of Complete intersection =====
We understood $\Delta^\vee$ in $M_A = N_B$, $\Sigma_B = \Sigma(\Delta^\vee)$ is the fan for $X_B$, $\Delta$ is in $N_A = M_B$. Given a nef partition of $\Sigma_B$'s rays, colored by $r$ colors, we will color the corresponding facet of $\Delta$ by $r$ colors. 

Let $Z_B(\Delta)$ be the union of codim-r faces that is adjacent to $r$ colors. 

The skeleton of the complete intersection is computed by Danil Kozevnikov, which surprisingly, is taking FLTZ skeleton, project to base, then restrict to some codim r sphere. that is really cool

===== BB construction and toric LG mirror =====
BB predicts that, complete intersection $Z_{\Delta_*} \In X_\Delta$ is mirror to $Z_{\nabla_*} \In X_{\nabla}$. 

Toric LG mirror says, compactification on one side is mirror to adding superpotential on the other side. 

The relation between the two is, passing to critical loci. 

If we realize $Z_{\Delta_*}$ as critical loci of some rank $r$ vector bundle $E_{\Delta_*}$ over $X_\Delta$, with some superpotential $W$, and do the same thing on the other side $E_{\nabla_*}$ with $W_{\nabla_*}$, then we can ask, do we have toric LG mirror symmetry
$$ B(E_\Delta, W_\Delta) \leftrightarrow A(E_\nabla, W_\nabla) $$
Is this just exchange monomials with divisors? We should count. 

On the $\Delta$-side, we count
  * divisors, we have $r$ divisors, together with number of vertices in $\Delta^\vee$
  * monomials, in $W_\Delta$, one group of term for each line bundle, each line bundle we have $\Delta_i$'s lattice points many monomials, so $ \sum_i |\Delta_i|_\Z $. 

On the $\nabla$-side, we count
  * divisors, we have $r$ divisors, together with number of nonzero vertices in $\nabla^\vee = Conv(\cup_i \Delta_i)$
  * monomials, in $W_\nabla$, one group of term for each line bundle, each line bundle we have $\nabla_i$'s lattice points many monomials, so $ \sum_i |\nabla_i|_\Z $. 

Ah, then we need to use some integral property of reflexive polytope now. We know $\Delta_i$ only intersects at $0$, so $ \sum_i |\Delta_i|_\Z $ equals $r$ plus non-zero vertices in $\nabla^\vee = Conv(\cup_i \Delta_i)$. Vice versa.