greatestlower.tex

\chapter{$R(X)$ in complexity one} \label{chap:R(X)}
\chaptermark{The greatest lower bound on Ricci curvature}
Recall, as discussed in the introduction, that one approach to the existence of K\"ahler-Einstein metrics is the study of the continuity path, that is solutions \(\omega_t \in 2 \pi c_1(X) \) to the equation
\[
\Ric(\omega_t) = t\omega_t + (1-t) \omega.
\]
for \(t \in [0,1]\). By \cite{Yau1977} there is always a solution for \(t = 0\). However, Tian \cite{tian1992stability} showed that for some \(t\) sufficiently close to \(1\) there may not be a solution for certain Fano manifolds. It is natural to ask for the supremum of permissible \(t\), which turns out to be independent of the choice of \(\omega\).
\begin{definition}
Let \((X,\omega)\) be a K\"ahler manifold with \(\omega \in 2 \pi c_1(X)\). Define:
\[
R(X) := \sup ( t \in [0,1]   : \exists \ \omega_t \in 2 \pi c_1(X)  \ \Ric( \omega_t) = t \omega_t + (1-t) \omega ).
\]
\end{definition}
This invariant was first discussed, although not explicitly defined, by Tian in \cite{Tian87}. It was first explicitely defined by Rubenstein in \cite{rubinstein2008} and was further studied by Szekelyhidi in \cite{szekelyhidi2011}. It is sometimes referred to as the greatest lower bound on Ricci curvature.

In \cite{rubinstein2008} Rubenstein showed relation between \(R(X)\) and Tian's alpha invariant \(\alpha(X)\), and in \cite{rubinstein2009} conjectured that \(R(X)\) characterizes the \(K\)-semistability of \(X\). This conjecture was later verified by Li in \cite{li2017}.

In \cite{li2011} Li  determined a simple formula for \(R(X_\Delta)\), where \(X_\Delta\) is the polarized toric Fano manifold determined by a reflexive lattice polytope \(\Delta\). This result was later recovered in \cite{datar2016kahler}, by Datar and Sz\'ekelyhidi, using notions of \(G\)-equivariant \(K\)-stability. Previously \(R(X)\) has been calculated for group compactifications by Delcroix \cite{delcroix2017} and for homogeneous toric bundles by Yao \cite{yao2017}. Let us briefly recall the toric formula.
\begin{theorem}[{\cite[Theorem 1?]{li2009greatest}}]
Suppose \(X\) is a smooth Fano toric variety. Let \(P\) be the corresponding Fano polytope. If \(\bc(P) = 0\) then \(X\) is K\"ahler-Einstein and \(R(X) = 1\). Otherwise let \(q\) be the intersection of the ray generated by \(-\bc(P)\) with the boundary \(\partial P\). We then have:
\[
R(X) = \frac{|q|}{|q-\bc(P)|}.
\]
\end{theorem}

\begin{example}
Consider the toric variety \(X = \Bl_z \PP^1 \times \PP^1\) from Example~\ref{toricexample}. It is then easy to calculate \(R(X)\) from the polytope \(P\) given in Example~\ref{ex:toricpolytope}. We have \(\bc(P) = (-2/21,-2/21)\) and \(q = (1/2,1/2)\) so \(R(X) = 21/25\).
\begin{figure}[h]
\centering
  \begin{tikzpicture}[scale=1]
    (1,-1) -- (1,0) -- (0,1) -- (-1,1) -- (-1,-1) -- (1,-1);  
   	 %\draw[dotted,step=1,gray,] (-2,-2) grid (2,2);
   	 \draw[] (1,-1) -- (1,0) -- (0,1) -- (-1,1) -- (-1,-1) -- (1,-1);
   	 \draw (0,0) node {\textbullet};
   	 \draw (-2/21,-2/21) node[below] {\tiny{$\bc(P)$}};
  	 \draw (-4/21,-4/21) node {\tiny{\textbullet}};
   	 \draw (1/2,1/2) node {\tiny{\textbullet}};
  	 \draw (1/2,1/2) node[right] {\tiny{$q$}};
   	 \draw [->] (-4/21,-4/21) -- (1,1);
   	 \foreach \i in {-2,...,2}{
   	 	\foreach \j in {-2,...,2}{
   	 		\draw[gray] (\i,\j) node {\tiny{$\circ$}};
   	 	}
   	 }
	\end{tikzpicture}
	\caption{$R(X)$ calculation for a toric $X$}
\end{figure}
\end{example}

Using similar methods to \cite{datar2016kahler} we obtain an effective formula for manifolds with a torus action of complexity one, in terms its divisorial polytope. Let \(\Phi: \Box \to \wdiv_\QQ \PP^1\) be the Fano divisorial polytope corresponding to a smooth Fano complexity one \(T\)-variety \(X\). Let \(\{\Delta_y\}_{y \in \PP^1}\) be the finite set of degeneration polytopes corresponding to central fibres of the non-product test configurations of \(X\), as in Definition~\ref{def:configpoly}.

To state our result we must introduce a little more notation. Suppose we have \(\bc(\Delta_y) \neq 0\) for some i. Consider the halfspace \(H := N_\RR \times \RR^{\ge 0} \subset N'_\RR\). Let \(F_y\) be  the face of \(\Delta_y\) in which \(q_y\) lies, and let \(S\) be the set of points \(y\) for which \(\bc(\Delta_y) \neq 0\) and all outer normals to \(F_y\) lie in \(H\). Note, by definition of \(\Delta_y\), we have:
\[
S = \{ y \in \PP^1 | \bc(\Delta_y) \not\in H \}
\]
Recall the definition of the Duistermatt Heckman measure \(\nu\), and associated weighted barycenter of \(\Box\) given in Definition~\ref{def:divpoly2}. Suppose \(\bc_\nu(\Box) \neq 0\). Let \(q\) be the intersection of the ray generated by \(-\bc_\nu(\Box)\) with \(\partial \Box\). Let \(q_y\) be the point of intersection of \(\partial \Delta_y\) with the ray generated by \(-\bc(\Delta_y)\).

Note, by the equation for the Donaldson Futaki invariants (\ref{eq:futaki-character}) and Theorem~\ref{thm:DS}, we know that \(R(X) = 1\) if \(\bc(\Delta_y) \in \{0\} \times \RR^+\) for each \(y\). We may now state our result:

\begin{theorem}[{\cite[Theorem 1.1]{cable2019greatest}}] \label{thm:R(X)} Let \(X\) be a complexity one Fano \(T\)-variety as above. If \(\bc(\Delta_y) \in \{0\} \times \RR^+\). Otherwise:
\begin{equation} \label{eq:R(X)}
R(X) = \min \  \left\lbrace \frac{|q|}{|q-\bc_\nu(\Box)|}  \ \right\rbrace \cup \ \left\lbrace \frac{|q_y|}{|q_y - \bc(\Delta_y)|} \right\rbrace_{\bc(\Delta_y) \not\in H} .
\end{equation}
\end{theorem}
\begin{example}
To see that this formula is truly a generalization of Li's result, consider the situation of a toric downgrade. Suppose \(X\) is a Fano toric variety given by a polytope \(P\). Let \(\Phi = \Phi_0 \otimes \{0\} + \Phi \otimes \{\infty\}\) be the divisorial polytope obtained through the downgrade procedure in Example~\ref{ex:polytopedowngrade}. Under the associated surjection \(p: M' \to M\) we have \(p(\bc(\Delta_0)) = p(\bc(\Delta_\infty)) = \bc_\nu(\Box)\). It is easy to see \(\emptyset \neq S \subset \{0,\infty\}\). By convexity of \(P\) then for \(y \in S\) we have \(|q| \ge |p(q_y)|\), and moreover:
\[
\frac{|q|}{|q-\bc_\nu(\Box)|} \ge \frac{|p(q_y)|}{|p(q_y)-\bc_\nu(\Box)|} = \frac{|p(q_y)|}{|p(q_y-\bc(\Delta_y))|} = \frac{|q_y|}{|q_y-\bc(\Delta_y)|}
\]
Clearly then the minimum in \normalfont{(}\ref{eq:R(X)}\normalfont{)} is obtained at one of \(y \in \{0,\infty\}\)
\end{example}
\begin{example}
Consider the \((\CC^*)^2\)-threefold 2.30 from Example \ref{ex:sym}. There are \(3\) normal toric degenerations, given by the polytopes \(\Delta_{0},\Delta_1,\Delta_{\infty}\). It can easily be checked in this case that \(S = \emptyset\), see Figure~\ref{fig2:a} for example, or Appendix~\ref{appendix1}.
\begin{figure}[h]
	\subcaptionbox{\label{fig2:a}}{\degendiagram}\hfill%
	\subcaptionbox{\label{fig2:b}}{\momentdiagram}%
	\caption{Some of the calculation of $R(X)$ for threefold 2.30. (a) Degeneration polytope $\Delta_1$ with barycenter $\bc(\Delta_1)$ and $q_1,n_1$ shown, (b) Moment polytope $\Box$ with Duistermaat-Heckmann barycenter $\bc_\nu(\Box)$ and $q$ shown.}
\end{figure}
Therefore \(R(X)\) is given by the first term in the minimum. We calculate \(\bc_\nu(\Box)= (0,-6/23)\) and \(q = (0,1)\). Then:
\[
R(X) = \frac{1}{1+ 6/23} = \frac{23}{29}.
\]
\end{example}
\begin{corollary}[{\cite[Corollary 1.2]{cable2019greatest}}] \label{cor:R(X)}
In Table~\ref{table:name} below we give \(R(X)\) for \(X\) a Fano threefold  admitting a \(2\)-torus action appearing in the list of Mori and Mukai \cite{mori1981classification}. We include only those where \(R(X) <1\). Note all admit a K\"ahler-Ricci soliton by Theorem~\ref{thm:sol}.
\end{corollary}
\begin{table}[H] \centering
\begin{tabular}{l l}
\ & \ \\
\hline
X & R(X) \\
\hline
2.30 & \(23/29\) \\
2.31 & \(23/27\) \\
3.18 & \(48/55\) \\
3.21 & \(76/97\) \\
3.22 & \(40/49\) \\
3.23 & \(168/221\) \\
3.24 & \(21/25\) \\
4.5* & \(64/69\) \\
4.8 & \(76/89\) \\
\hline
\ & \ \\
\end{tabular}
\caption{Calculations for complexity \(1\) threefolds appearing in the list of Mori and Mukai for which \(R(X) <1\)} \label{table:name}
\end{table}
Let \(X\) be a \(T\)-variety of complexity one associated to a divisorial polytope \(\Psi: \Box \to \wdiv_\QQ(\PP^1)\), see Section~\ref{prelim:Tvar}.
It follows from Theorem~\ref{thm:DS} that: 
\begin{equation} \label{eq:R(X)inf}
R(X) = \inf_{(\X,\L)}( \sup(t | \DF_t(\X,\L,\cdot) \ge 0) ),
\end{equation}
where \((\X,\L)\) varies over all special test configurations for \((X,L)\). We have an explicit description of special test configurations and their Donaldson Futaki invariants, see Section~\ref{sec:eqKstab}. We will calculate \(R(X)\) by considering first the product configurations and then the non-product configurations. To calculate the values \(\sup(t | \DF_t(\X,\L) \ge 0)\) for a given configuration we need to first consider some elementary convex geometry.

\section{A short digression into convex geometry}
Let \(V\) be a real vector space and \(P \subset V\) be a full dimensional convex polytope containing the origin. Fix some point \(b \in \text{int}(P)\). Let \(q \in \partial P\) be the intersection of \(\partial P\) with the ray \(\tau = \RR^+ (-b)\). Suppose \(n \in V^\vee\) is an outer normal to a face containing \(q\). For \(a \in \partial P\) write \(\mathcal{N}(a) = \{w\in V^\vee \ | \langle a,w \rangle = \max_{x \in P} \langle x,c \rangle \}\). For \(w \in \mathcal{N}(a) \) let \(\Pi(a,w)\) be the affine hyperplane tangent to \(P\) at \(a\) with normal \(w\). For \(w \in \inte(\tau^\vee) \) there is a well-defined point of intersection of \(\Pi(a,w)\) and \(\tau\) which we denote \(r_w\). See Figure~\ref{schematic} for a schematic.
\begin{figure}[h] \centering
\diagram1
\caption{An Example in \(V \cong \RR^2\)}
\label{schematic}
\end{figure}
\begin{lemma} \label{R(X):Lemma3.1}
Fix \(w \in \inte(\tau^\vee) \backslash (\RR^+ n)\). For \(s \in [0,1]\) set \(w(s) := sn + (1-s)w\). As \(n \in \tau^\vee \) we may consider \(r(s) := r_{w(s)}\). For \(0 \le s' < s \le 1\) we then have:
\[
\frac{|r(s)|}{|r(s)-b|} < \frac{|r(s')|}{|r(s')-b|}.
\]
\end{lemma}
\begin{proof}
Without loss of generality we may assume \(s' = 0\). For \(s \in [0,1]\) the points \(r(s), q,b\) are collinear, so \(|r(s)|= |r(s)-q|+|q|\) and \(|r(s)-b| =|r(s)-q| +|q| + |b| \). Therefore:
\[
\frac{|r(s)|}{|r(s) - b|} = \frac{|r(s)-q|+|q|}{|r(s)-q| +|q| + |b|}.
\]
Hence it is enough for \(|r(s)-q| < |r(0)-q|\) for \(s >0\). Since \(q \neq 0\) is fixed this is equivalent to:
\[
\frac{|r(s) - q|}{|q|} < \frac{|r(0) - q|}{|q|}.
\]
For each \(s \in [0,1]\) choose \(a(s) \in \partial P\) such that \(w(s) \in \mathcal{N}(a(s))\). Write \(a = a(0)\) for convenience. We then have:
\[
\frac{|r(s)-q|}{|q|} = \frac{\langle a(s)-q, w \rangle }{\langle q,w \rangle}.
\]
Note \(n \in \mathcal{N}(q)\). Now \(\langle a(s)-q,n \rangle \le 0\)  and \( \langle a(s) - q,w \rangle \le \langle a - q,w \rangle\). Clearly we have \(\langle q , n \rangle > 0\). Then:
\begin{align*}
\frac{|r(s)-q|}{|q|}  &= \frac{\langle a(s) - q, w(s) \rangle }{\langle q, w(s) \rangle} \\ \\ &= \frac{s\langle a(s)-q, n \rangle + (1-s)\langle a(s)-q,w \rangle}{s \langle q , n \rangle + (1-s) \langle q, w \rangle} \\ \\ &\le \frac{(1-s)\langle a-q, w \rangle }{s \langle q , n \rangle + (1-s) \langle q, w \rangle} \\ \\ &< \frac{\langle a-q, w \rangle }{\langle q,w \rangle} \\ \\ &= \frac{|r(0) - q|}{|q|}.
\end{align*}
\end{proof}
Let \(V,P,b,q,\tau,n\) be as in the introduction to this section. Fix some open halfspace \(H \subset V^\vee\) given by \(u \ge 0\) for some \(u \in V \backslash \{0\}\). This defines a projection map \(p: V \to V/\langle u \rangle.\) Consider the function \(F_b: V^\vee \times  [0,1] \to \RR\) given by:
\[
F_b(w,t) := t \langle b,w \rangle+ (1-t) \max_{x \in P} \langle x, w \rangle
\]
\begin{corollary} \label{cor:convexgeomextra}
For any \(W \subseteq V^\vee\) containing \(n\) we have:
\begin{equation} \label{R(X):1}
\sup (t \in [0,1] \ | \ \forall_{w \in W} \  F_b(t,w) \ge 0) = \frac{|q|}{|q-b|}.
\end{equation}
If for some choice of \(n\) we have \(n \not\in H\) then:
\begin{equation} \label{R(X):2}
\sup (t \in [0,1] \ | \ \forall_{w \in H} \  F_b(t,w) \ge 0) = \frac{|\tilde{q}|}{|\tilde{q} - p(b)|},
\end{equation}
where \(\tilde{q}\) is the intersection of the ray \(p(\tau)\) with the boundary of \(p(P)\).
\end{corollary}
\begin{proof}
Note that:
\[
\sup (t \in [0,1] \ | \ \forall_{w \in W} \  F_b(t,w) \ge 0) = \inf_{w \in W} \sup (t \in [0,1] \ | \  F_b(t,w) \ge 0).
\]
Moreover \(\sup (t \in [0,1] \ | \  F_b(t,w) \ge 0) = 1 > F_b(t,n)\) for \(\langle b,w \rangle \ge 0\), so without loss of generality we may assume \(W \subseteq \inte(\tau^\vee)\). For \(w \in W\) then:
\begin{align*}
\sup (t \in [0,1] \ | \  F_b(t,w) \ge 0) &= \frac{\max_{x \in P} \langle x, w \rangle}{ \max_{x \in P} \langle x, w \rangle - \langle b, w \rangle } \\ &= \frac{ \langle a,w \rangle}{\langle a ,w \rangle - \langle b,w \rangle} \\ &= \frac{ \langle p_w,w \rangle}{\langle p_w ,w \rangle - \langle b,w \rangle} = \frac{|p_w|}{|p_w-b|}.
\end{align*}
Hence:
\[
\sup (t \in [0,1] \ | \ \forall_{w \in W} \  F_b(t,w) \ge 0) = \inf_{w \in W} \frac{|p_w|}{|p_w-b|}.
\]
Now for \(w \in W\) consider the continuity path \(w(s) = sn + (1-s)w\). By Lemma \ref{R(X):Lemma3.1} if \(n \in W\) then the above infimum is attained when \(s=1\) and we obtain \normalfont{(}\ref{R(X):1}\normalfont{)}. Otherwise the infimum is attained at some \(w \in \partial W\). For \normalfont{(}\ref{R(X):2}\normalfont{)} restricting \(F_b\) to \(\partial H \times [0,1]\) gives:
\begin{align*}
F_b(w,t) =  t \langle p(b) ,w \rangle+ (1-t) \max_{x \in p(P)} \langle x, w \rangle.
\end{align*}
Applying \normalfont{(}\ref{R(X):1}\normalfont{)} to the polytope \(p(P)\) in the vector space \(\partial H\) we obtain \normalfont{(}\ref{R(X):2}\normalfont{)}.
\end{proof}
\section{Proof of Theorem 4}
\subsection{Product Configurations}
Recall the formula \normalfont{(}\ref{eq:productDF}\normalfont{)} for the twisted Donaldson-Futaki invariant of a product configuration \(\X \cong X \times \A^1\).
Let \(q \in N_\RR\) be the point of intersection of the ray generated by \(-\bc(\Psi)\) with \(\partial \Box\). Applying \normalfont{(}\ref{R(X):1}\normalfont{)}, from Corollary~\ref{cor:convexgeomextra}, to \normalfont{(}\ref{eq:R(X)inf}\normalfont{)} we obtain:
\[
\sup(t | \DF_t(\X,\L,\cdot) \ge 0) = \frac{|q|}{|q-\bc_\nu(\Box)|}.
\]
\subsection{Non-Product Configurations}
Recall the description of special non-product test configurations of \(X\) from Section~\ref{sec:eqKstab}, and in particular the formula for the twisted Donaldson-Futaki in Lemma~\ref{lem:nonprodDF}. Set \(H := N_\QQ \times \RR^+\).
\begin{proposition}
For any non-product configuration \((\X,\L)\), with special fiber one of the \(\Delta_y\), let \(\sigma_y\) be the cone of outer normals to \(\Delta_y\) at the unique point of intersection of \(\partial \Delta_y\) with the ray generated by \(-\bc(\Delta_y)\). Denote this point of intersection by \(q_y\). Then:
\[
\sup(t | \DF_t(\X,\L,\cdot) \ge 0) =\begin{cases} 
     \frac{|q_y|}{|q_y - \bc(\Delta_y)|} & \sigma_y \cap H \neq \emptyset ;\\ \\
      \frac{|q|}{|q-\bc_\nu(\Box)|} & \sigma_y \cap H = \emptyset.
   \end{cases}
\]
\end{proposition}
\begin{proof}
Extend \(\DF_t(\X,\L,\cdot)\) linearly to the whole of \(N_\RR \times \RR\). In the case \(\sigma_y \cap H \neq \emptyset\) we may apply (1) from Corollary 2 with \(P = \Delta_y\) and \(b = \bc(\Delta_y)\). Otherwise we may apply \normalfont{(}\ref{R(X):2}\normalfont{)} from Corollary~\ref{cor:convexgeomextra}, noting that \(p(\Delta_y) = \Box\) and \(p(\bc(\Delta_y)) = \bc_\nu( \Box)\).
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:R(X)}]
With Remark 2 in mind, observe that a special test configuration must either be product or non-product. Any non-product configurations \(\Delta_y\) with \(\sigma_y \cap H \neq \emptyset\) have their contribution to the infimum already accounted for and we may exclude them. The result follows.
\end{proof}
\begin{proof}[of Corollary~\ref{cor:R(X)}]
We observe \(\bc(\Delta_y) \in H \) for every special test configuration polytope \(\Delta_y\) of each threefold in this list, see \ref{appendix1}.  We may then calculate \(R(X)\) using just the base polytope \(\Box\) and its Duistermaat-Heckman barycenter. The divisorial polytopes and Duistermaat-Heckman measures were originally given in \cite{suss2013fano}, and may be also be found in Appendix~\ref{appendix1}.
\end{proof}