diff --git a/main.tex b/main.tex index 7e754d5a9181da8a43cfa82fbca67b9907a4a8be..88c3b8bd98474fe3845e49d17dcf412425bdc045 100644 --- a/main.tex +++ b/main.tex @@ -190,7 +190,13 @@ For the next subsections, we consider $q$ to be fixed with one of these values, and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail. -\subsection{$\Delta(E) + \Delta(G) \leq \Delta(F)$} +\subsection{ + \texorpdfstring{ + $\Delta(E) + \Delta(G) \leq \Delta(F)$ + }{ + Δ(E) + Δ(G) ≤ Δ(F) + } +} \label{subsect-d-bound-bgmlv1} This condition expressed in terms of $R,C,D,r,c,d$ looks as follows: @@ -300,7 +306,13 @@ In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have $\chern^{\beta}_2(F) = 0$, so some of these expressions simplify. -\subsection{$\Delta(E) \geq 0$} +\subsection{ + \texorpdfstring{ + $\Delta(E) \geq 0$ + }{ + Δ(E) ≥ 0 + } +} This condition expressed in terms of $R,C,D,r,c,d$ looks as follows: @@ -384,7 +396,13 @@ Notice that for $\beta = \beta_{-}$ (or $\beta_{+}$), that is when $\chern^{\beta}_2(F)=0$, the constant and linear terms match up with the ones for the bound found for $d$ in subsection \ref{subsect-d-bound-bgmlv1}. -\subsection{$\Delta(G) \geq 0$} +\subsection{ + \texorpdfstring{ + $\Delta(G) \geq 0$ + }{ + Δ(G) ≥ 0 + } +} \label{subsect-d-bound-bgmlv3} This condition expressed in terms of $R,C,D,r,c,d$ looks as follows: @@ -516,14 +534,14 @@ $\chern^{\beta}_2(F) = 0$, so some of these expressions simplify, and in particular, the constant and linear terms match those of the other bounds in the previous subsections. -\subsection{Bounds on $r$} +\subsection{Bounds on \texorpdfstring{$r$}{r}} Now, the inequalities from the last three subsections will be used to find, for each given $q=\chern^{\beta}_1(E)$, how large $r$ needs to be in order to leave no possible solutions for $d$. At that point, there are no Chern characters $(r,c,d)$ that satisfy all inequalities to give a pseudowall. -\subsubsection{All circular walls left of vertical wall} +\subsubsection{All circular pseudowalls left of vertical wall} Suppose we take $\beta = \beta_{-}$ in the previous subsections, to find all circular walls to the left of the vertical wall (TODO as discussed in ref).