%
% Appendix : Tian's invariant and log canonical thresholds 
%
% Jean-Pierre Demailly,
% Universit\'e de Grenoble I, Institut Fourier
% 38402 Saint-Martin d'H\`eres, France
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\title{Appendix to I.\ Cheltsov and C.\ Shramov's article}
\title{``Log canonical thresholds of smooth Fano threefolds'' :}
\title{On Tian's invariant and log canonical thresholds}
\titlerunning{J.-P.\ Demailly,
Appendix: on Tian's invariant and log canonical thresholds}
\medskip

\noindent
{\twelvebf Jean-Pierre Demailly}\medskip

\noindent
{\twelverm Universit\'e de Grenoble I, Institut Fourier}
\bigskip\bigskip

The goal of this appendix is to relate log canonical thresholds with
the $\alpha$ invariant introduced by G.~Tian [Tia87] for the study of
the existence of K\"ahler-Einstein metrics. The approximation
technique of closed positive $(1,1)$-currents introduced in [Dem92] is
used to show that the $\alpha$ invariant actually coincides with the
log canonical threshold.

Algebraic geometers have been aware of this fact after [DK01]
appeared, and several papers have used it consistently in the latter
years (see e.g.\ [JK01], [BGK05]). However, it turns out that 
the required result is stated only in a local analytic form in [DK01],
in a language which may not be easily recognizable by algebraically
minded people. Therefore, we will repair here the lack of a proper
reference by stating and proving the statements required for the
applications to projective varieties, e.g.\ existence of
K\"ahler-Einstein metrics on Fano varieties and Fano orbifolds.

Usually, in these applications, only the case of the anticanonical
line bundle $L=-K_X$ is considered. Here we will consider more generally
the case of an arbitrary line bundle $L$ (or $\bQ$-line bundle $L$) on a 
complex  manifold $X$, with some additional restrictions which will
be stated later.

Assume that $L$ is equipped with a {\it singular hermitian metric} $h$
(see e.g.\ [Dem90]). Locally, $L$ admits trivializations
$\theta:L_{|U}\simeq U\times\bC$, and on $U$ the metric $h$ is given
by a weight function $\varphi$ such that
$$
\Vert\xi\Vert_h^2 =|\xi|^2e^{-2\varphi(z)},\qquad z\in U,~~\xi\in L_z
$$
when $\xi\in L_z$ is identified with a complex number. We are interested
in the case where $\varphi$ is (at the very least) a locally integrable
function for the Lebesgue measure, since it is then possible to compute the
curvature form
$$
\Theta_{L,h}={i\over \pi}\ddbar\varphi
$$
in the sense of distributions. We have $\Theta_{L,h}\ge 0$ as a $(1,1)$-current
if and only if the weights $\varphi$ are plurisubharmonic functions. In 
the sequel we will be interested only in that case. Let us first introduce
the concept of complex singularity exponent, following e.g.\
[Var82, 83], [ArGV85] and [DK01].

\claim (A.1) Definition|If $K$ is a compact subset of $X$, we define the
complex singularity exponent $c_K(h)$ of the metric $h$, written locally
as $h=e^{-2\varphi}$, to be the supremum of all
positive numbers $c$ such that  $h^c=e^{-2c\varphi}$ is
integrable in a neighborhood of every point $z_0\in K$, with respect to 
the Lebesgue measure in holomorphic coordinates centered at~$z_0$.
\endclaim

\noindent
Now, we introduce a generalized version of Tian's invariant $\alpha$, as
defined in [Tia87] (see also~[Siu88]).

\claim (A.2) Definition|Assume that $X$ is a compact manifold and that
$L$ is a pseudo-effective line bundle, i.e.\ $L$ admits a singular
hermitian metric $h_0$ with $\Theta_{L,h_0}\ge 0$. If $K$ is a compact 
subset of $X$, we put
$$
\alpha_K(L)=\inf_{\{h,\,\Theta_{L,h}\ge 0\}}c_K(h)
$$
where $h$ runs over all singular hermitian metrics on $L$ such that
$\Theta_{L,h}\ge 0$.
\endclaim

In algebraic geometry, it is more usual to look instead at linear systems
defined by a family of linearly independent sections 
$\sigma_0,\sigma_1,\ldots\sigma_N\in 
H^0(X,L^{\otimes m})$. We denote by $\Sigma$ the vector subspace generated by
these sections and by 
$$
|\Sigma|:=P(\Sigma)\subset|mL|:=P(H^0(X,L^{\otimes m}))
$$
the corresponding linear system. Such an $(N+1)$-tuple of sections 
$\sigma=(\sigma_j)_{0\le j\le N}$ defines a
singular hermitian metric $h$ on $L$ by putting in any trivialization
$$
\Vert \xi\Vert^2_h
=\frac{|\xi|^2}{\big(\sum_j|\sigma_j(z)|^2\big)^{1/m}}
=\frac{|\xi|^2}{|\sigma(z)|^{2/m}},\qquad \xi\in L_z,
$$
hence $h(z)=|\sigma(z)|^{-2/m}$ with
$\varphi(z)=\frac{1}{m}\log|\sigma(z)|=\frac{1}{2m}\log\sum_j|\sigma_j(z)|^2$
as the associated weight function. Therefore, we are interested in the
number $c_K(|\sigma|^{-2/m})$. In the case of a single section 
$\sigma_0$ (corresponding to a one-point linear system), this is the 
same as the log canonical threshold $\lct_K(X, \frac{1}{m}D)$ of the
associated divisor $D$, in the notation
of~Section~1 of [CS08]. We will also use the formal notation
$\lct_K(X, \frac{1}{m}|\Sigma|)$ in the case of a higher dimensional
linear system $|\Sigma|\subset|mL|$.

Now, recall that the line bundle $L$ is said to be
{\it big} if the Kodaira-Iitaka dimension $\kappa(L)$ equals $n=\dim_\bC X$. 
The main result of this appendix is~

\claim (A.3) Theorem|Let $L$ be a big line bundle on a compact complex
manifold $X$. Then for every compact set $K$ in $X$ we have
$$
\alpha_K(L)=\inf_{\{h,\,\Theta_{L,h}\ge 0\}} c_K(h) =\inf_{m\in\bZ_{>0}}\inf_{D\in|mL|}
\lct_K\Big(X,{1\over m}D\Big).
$$
\endclaim

\noindent
Observe that the inequality 
$$
\inf_{m\in\bZ_{>0}}\inf_{D\in|mL|}\lct_K\Big(X,{1\over m}D\Big)
\geq \inf_{\{h,\,\Theta_{L,h}\ge 0\}} c_K(h)
$$
is trivial, since any divisor $D\in|mL|$ gives rise to a singular
hermitian metric~$h$. The converse inequality will follow from the
approximation technique of [Dem92] and some elementary analysis.  The
proof is parallel to the proof of Theorem~4.2 of [DK01],
although the language used there was somewhat different. In any case,
we use again the crucial concept of multiplier ideal sheaves~: if
$h$ is a singular hermitian metric with local plurisubharmonic
weights $\varphi$, the multiplier ideal sheaf $\cI(h)\subset\cO_X$ 
(also denoted by $\cI(\varphi)$) is the ideal sheaf defined by
$$
\cI(h)_z=\big\{f\in\cO_{X,z}\,;\;\exists\;\hbox{a neighborhood $V\ni z$
such that~}\int_V|f(x)|^2e^{-2\varphi(x)}d\lambda(x)<+\infty\big\}
$$
where $\lambda$ is the Lebesgue measure. By Nadel [Nad89], this is a 
coherent analytic sheaf on $X$. Theorem (A.3)
has a more precise version which can be stated as follows.

\claim (A.4) Theorem|Let $L$ be a line bundle on a compact complex manifold
$X$ possessing a singular hermitian metric $h$ with $\Theta_{L,h}\ge 
\varepsilon\omega$ for some $\varepsilon>0$ and some smooth positive definite
hermitian $(1,1)$-form $\omega$ on~$X$. For every real number $m>0$, consider
the space $\cH_m=H^0(X,L^{\otimes m}\otimes\cI(h^m))$ of holomorphic sections 
$\sigma$ of $L^{\otimes m}$ on $X$ such that 
$$
\int_X|\sigma|_{h^m}^2dV_\omega
=\int_X|\sigma|^2e^{-2m\varphi}dV_\omega<+\infty,
$$
where $dV_\omega={1\over m!}\omega^m$ is the hermitian volume form.
Then for $m\gg 1$, $\cH_m$ is a non zero finite dimensional Hilbert space 
and we consider the closed positive $(1,1)$-current
$$
T_m={i\over 2\pi}\ddbar\Big({1\over 2m}\log\sum_k|g_{m,k}|^2\Big)
={i\over 2\pi}\ddbar\Big({1\over 2m}\log\sum_k|g_{m,k}|_h^2\Big)
+\Theta_{L,h}
$$
where $(g_{m,k})_{1\le k\le N(m)}$ is an orthonormal basis of~$\cH_m$. 
Then$\;:$
\smallskip
\item{\rm (i)} For every trivialization  $L_{|U}\simeq U\times\bC$
on a cordinate open set $U$ of $X$ and every compact set $K\subset U$,
there are constants $C_1,C_2>0$ independent of $m$ and $\varphi$ such that
$$
\varphi(z)-{C_1\over m}\le
\psi_m(z):={1\over 2m}\log\sum_k|g_{m,k}(z)|^2
\le\sup_{|x-z|<r}\varphi(x)+{1\over m}\log{C_2\over r^n}
$$
for every $z\in K$ and $r\le{1\over 2}d(K,\partial U)$. In particular,
$\psi_m$ converges to $\varphi$ pointwise and in $L^1_{\rm loc}$ topology
on~$\Omega$ when $m\to+\infty$, hence $T_m$ converges weakly 
to $T=\Theta_{L,h}$.
\smallskip
\item{\rm (ii)} The Lelong numbers $\nu(T,z)=\nu(\varphi,z)$ and 
$\nu(T_m,z)=\nu(\psi_m,z)$ are related by
$$
\nu(T,z)-{n\over m}\le\nu(T_m,z)\le
\nu(T,z)\quad\text{ for every $z\in X$.}
$$
\item{\rm (iii)} For every compact set $K\subset X$, the complex singularity
exponents of the metrics given locally by $h=e^{-2\varphi}$ and 
$h_m=e^{-2\psi_m}$ satisfy
$$
c_K(h)^{-1}-{1\over m}\le c_K(h_m)^{-1}\le c_K(h)^{-1}.
$$
\endclaim

\proof. The major part of the proof is a minor variation
of the arguments already explained in [Dem92] (see also [DK01] Theorem 4.2). 
We give them here in some detail for the convenience of the reader.\smallskip

\noindent
(i) We note that $\sum|g_{m,k}(z)|^2$ is the square of the norm of 
the evaluation linear form $\sigma\mapsto \sigma(z)$ on $\cH_m$, hence
$$
\psi_m(z)=\sup_{\sigma\in B(1)}{1\over m}\log|\sigma(z)|
$$
where $B(1)$ is the unit ball of $\cH_m$. For
$r\le {1\over 2}d(K,\partial\Omega)$, the mean value inequality 
applied to the plurisubharmonic function $|\sigma|^2$ implies
$$
\eqalign{
|\sigma(z)|^2&\le{1\over\pi^nr^{2n}/n!}\int_{|x-z|<r}
|\sigma(x)|^2d\lambda(x)\cr
&\le{1\over\pi^nr^{2n}/n!}\exp\Big(2m\sup_{|x-z|<r}\varphi(x)\Big)
\int_\Omega|\sigma|^2e^{-2m\varphi}d\lambda.\cr}
$$
If we take the supremum over all $\sigma\in B(1)$ we get
$$
\psi_m(z)\le\sup_{|x-z|<r}\varphi(x)+{1\over 2m}
\log{1\over\pi^nr^{2n}/n!}
$$
and the right hand inequality in (i) is proved. Conversely, the
Ohsawa-Takegoshi extension theorem [OhT87], [Ohs88] applied to 
the $0$-dimensional
subvariety $\{z\}\subset U$ shows that for any $a\in\bC$ there is
a holomorphic function $f$ on $U$ such that $f(z)=a$ and
$$
\int_U|f|^2e^{-2m\varphi}d\lambda\le C_3|a|^2e^{-2m\varphi(z)},
$$
where $C_3$ only depends on $n$ and~$\diam U$. Now, provided $a$ remains in
a compact set $K\subset U$, we can use a cut-off function $\theta$ with
support in $U$ and equal to $1$ in a neighborhood of $a$, and solve the
$\dbar$-equation $\dbar g=\dbar(\theta f)$ in the $L^2$ space associated
with the
weight $2m\varphi+2(n+1)|\log|z-a|$, that is, the singular hermitian
metric $h(z)^m|z-a|^{-2(n+1)}$ on~$L^{\otimes m}$. For this, we apply
the standard Andreotti-Vesentini-H\"ormander $L^2$ estimates (see e.g.\ [Dem82]
for the required version). This is possible for $m\ge m_0$ thanks to the 
hypothesis that $\Theta_{L,h}\ge \varepsilon\omega>0$, even if $X$ 
is non K\"ahler ($X$ is in any event a Moishezon variety from our
assumptions). The bound $m_0$ depends only on $\varepsilon$ and the geometry
of a finite covering of $X$ by compact sets $K_j\subset U_j$, where
$U_j$ are coordinate balls (say); it is independent of the point $a$ and 
even of the metric~$h$. It follows that $g(a)=0$ and therefore
$\sigma=\theta f-g$ is a holomorphic section of $L^{\otimes m}$ such that
$$
\int_X|\sigma|^2_{h^m}dV_\omega=
\int_X|\sigma|^2e^{-2m\varphi}dV_\omega\le C_4
\int_U|f|^2e^{-2m\varphi}dV_\omega\le C_5
|a|^2e^{-2m\varphi(z)},
$$
in particular $\sigma\in \cH_m=H^0(X,L^{\otimes m}\otimes\cI(h^m))$.
We fix $a$ such that the right hand side is~$1$. This gives
the inequality
$$
\psi_m(z)\ge{1\over m}\log|a|=\varphi(z)-{\log C_5\over 2m}
$$
which is the left hand part of statement (i).
\medskip

\noindent
(ii) The first inequality in (i) implies $\nu(\psi_m,z)\le\nu(\varphi,z)$. 
In the opposite direction, we find
$$
\sup_{|x-z|<r}\psi_m(x)\le\sup_{|x-z|<2r}\varphi(x)+
{1\over m}\log{C_2\over r^n}.
$$
Divide by $\log r<0$ and take the limit as $r$ tends to~$0$. The quotient
by $\log r$ of the supremum of a psh function over $B(x,r)$ tends to the
Lelong number at~$x$. Thus we obtain
$$
\nu(\psi_m,x)\ge\nu(\varphi,x)-{n\over m}.
$$
(iii) Again, the first inequality (in (i) immediately yields $h_m\le C_6 h$,
hence $c_K(h_m)\ge c_K(h)$. For the converse inequality, since we
have $c_{\,\cup K_j}(h)=\min_j c_{K_j}(h)$, we can assume without loss 
of generality that $K$ is contained in a trivializing open patch $U$ of~$L$.
Let us take $c<c_K(\psi_m)$. Then, by definition, if 
$V\subset X$ is a sufficiently small open neighborhood of $K$, 
the H\"older inequality
for the conjugate exponents $p=1+mc^{-1}$ and $q=1+m^{-1}c$
implies, thanks to equality ${1\over p}={c\over mq}$,
$$
\leqalignno{
\int_V & e^{-2(m/p)\varphi}dV_\omega
=\int_V\Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2e^{-2m\varphi}\Big)^{1/p}
  \Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2\Big)^{-c/mq}dV_\omega\cr
&\le
\left(\int_X\sum_{1\le k\le N(m)}|g_{m,k}|^2e^{-2m\varphi}dV_\omega\right)^{1/p}
\left(\int_V \Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2\Big)^{-c/m}dV_\omega
\right)^{1/q}\cr
&= N(m)^{1/p}
\left(\int_V \Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2\Big)^{-c/m}dV_\omega
\right)^{1/q}<+\infty.
\cr}
$$
From this we infer $c_K(h)\ge m/p$, i.e., $c_K(h)^{-1}\le p/m=1/m+c^{-1}$.
As $c<c_K(\psi_m)$ was arbitrary, we get $c_K(h)^{-1}\le 1/m+c_K(h_m)^{-1}$
and the inequalities of (iii) are proved.\qed
\medskip

\noindent
{\it Proof of Theorem (A.3).} Given a big line bundle $L$ on~$X$, there 
exists a modification
$\mu:\wt X\to X$ of $X$ such that $\wt X$ is projective and 
$\mu^*L=\cO(A+E)$ where $A$ is an ample divisor and $E$ an effective
divisor with rational coefficients. By pushing forward by $\mu$ a smooth 
metric $h_A$ with positive curvature on $A$, we get a singular hermitian 
metric $h_1$ on $L$ such that 
$\Theta_{L,h_1}\ge\mu_*\Theta_{A,h_A}\ge \varepsilon\omega$ on~$X$. Then
For any $\delta>0$ and any singular hermitian metric $h$ on $L$ with
$\Theta_{L,h}\ge 0$, the interpolated metric 
$h_\delta=h_1^\delta h^{1-\delta}$ satisfies
$\Theta_{L,h_\delta}\ge\delta\varepsilon\omega$. Since $h_1$ is bounded
away from $0$, it follows that $c_K(h)\ge (1-\delta)c_K(h_\delta)$
by monotonicity. By theorem (A.4, iii) applied to $h_\delta$, we infer
$$
c_K(h_\delta)=\lim_{m\to+\infty} c_K(h_{\delta,m}),
$$
and we also have
$$
c_K(h_{\delta,m})\ge \lct_K\Big({1\over m}D_{\delta,m}\Big)
$$
for any divisor $D_{\delta,m}$ associated with a section $\sigma\in
H^0(X,L^{\otimes m}\otimes\cI(h_\delta^m))$, since the metric $h_{\delta,m}$
is given by $h_{\delta,m}=(\sum_k|g_{m,k}|^2)^{-1/m}$ for an orthornormal basis
of such sections. This clearly implies
$$
c_K(h)\ge\liminf_{\delta\to 0}\;\liminf_{m\to+\infty}\;
\lct_K\Big({1\over m}D_{\delta,m}\Big)\ge
\inf_{m\in\bZ_{>0}}\inf_{D\in|mL|}
\lct_K\Big({1\over m}D\Big).\eqno\square
$$

In the applications, it is frequent to have a finite or compact group 
$G$ of automorphisms of $X$ and to look at $G$-invariant objects, namely
$G$-equivariant metrics on $G$-equivariant line bundles~$L\;$; in the case
of a reductive algebraic group $G$ we simply consider a compact real form
$G^\bR$ instead of $G$ itself.

One then gets an $\alpha$ invariant $\alpha_{G,K}(L)$ by
looking only at $G$-equivariant metrics in
Definition~A.2. All contructions made are then
$G$-equivariant, especially $\cH_m\subset |mL|$ is a $G$-invariant
linear system. For every $G$-invariant compact set $K$ in $X$, we thus
infer
$$
\alpha_{G,K}(L)
:=\inf_{\{h\;G\hbox{\sevenrm -equiv.},\,\Theta_{L,h}\ge 0\}}c_K(h)
=\inf_{m\in\bZ_{>0}}~~\inf_{|\Sigma|\subset|mL|,~\Sigma^G=\Sigma}~~
\lct_K\Big(\frac{1}{m}|\Sigma|\Big).
\leqno({\rm A}.5)
$$
When $G$ is a finite group, one can pick for $m$ large enough a
$G$-invariant divisor $D_{\delta,m}$ associated with a
$G$-invariant section $\sigma$, possibly after multiplying $m$ by
the order of $G$. One then gets the slightly simpler equality
$$
\alpha_{G,K}(L)
:=\inf_{\{h\;G\hbox{\sevenrm -equiv.},\,\Theta_{L,h}\ge 0\}}c_K(h)
=\inf_{m\in\bZ_{>0}}~~\inf_{D\in|mL|^G}~~\lct_K\Big(\frac{1}{m}D\Big).
\leqno({\rm A}.6)
$$
In a similar manner, one can work on an orbifold $X$ rather than on a non
singular variety. The $L^2$ techniques work in this setting with almost
no change ($L^2$ estimates are essentially insensitive  to
singularities,  since one can just use an orbifold metric on
the open set of regular points).

\bigskip\null
\noindent
{\fourteenbf References}
\bigskip

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}
\vskip30pt
\noindent
(June 12, 2008; printed on \today)

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