3. More regularity [02DP]

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3. More regularity

3.1. Measures with density

We now turn to the study of the complex Monge-Ampère equation

(\omega+dd^{c}\varphi)^{n}=\mu,\text{ when }\mu=f\omega^{n}

is a measure with density $0\leq f\in L^{p}(\omega^{n})$ , $p>1$ .

Proposition 3.1.

Assume $\mu=f\omega^{n}$ is a probability measure with density $0\leq f\in L^{p}(X)$ , for some $p>1$ . Then for any $\alpha>0$ , there exists $A_{\alpha}>0$ such that $\mu$ satisfies ${\mathcal{H}}(\alpha,A_{\alpha},\omega)$ .

Proof.

It is enough to establish ${\mathcal{H}}(\alpha,A_{\alpha},\omega)$ for compact subsets, by regularity of $\mu$ and $Cap_{\omega}$ . Let $K$ be a compact subset of $X$ . It follows from Hölder’s inequality that

0\leq\mu(K)\leq||f||_{L^{p}(\omega^{n})}\left[\text{Vol}_{\omega}(K)\right]^{1/q},

where $1/p+1/q=1$ . Note that $||f||_{L^{p}(\omega^{n})}=1$ since we assume $\mu$ is a probability measure. We claim that

(4)

\text{Vol}_{\omega}(K)\leq C_{\omega}\exp\left[-(Cap_{\omega}(K))^{-\gamma_{\omega}/n}\right],

for some constants $C_{\omega},\gamma_{\omega}>0$ that only depend on $\omega$ . We will be done if we can prove (4) since we can then check by elementary computations that $\exp(-x^{-\delta})$ is dominated from above by $A_{\alpha}x^{\alpha}$ , for all $x\in[0,1]$ .

The set of functions ${\mathcal{F}}_{0}:=\{\varphi\in PSH(X,\omega)\,/\,\sup_{X}\varphi=0\}$ is compact in $L^{1}(X)$ (see proposition 2.7, [GZ 1]). These functions have Lelong numbers $\nu(\varphi,x)\leq\nu_{\omega}$ bounded from above by a uniform constant. It follows therefore from Skoda’s uniform integrability theorem [Z], that

\sup_{\varphi\in{\mathcal{F}}_{0}}\int\exp\left[-\frac{2\varphi}{\nu_{\omega}+1}\right]\omega^{n}\leq C_{2}<+\infty.

Set $\gamma_{\omega}:=2/(\nu_{\omega}+1)>0$ and let

V_{K,\omega}^{*}(x):=\left(\sup\{\varphi(x)\,/\,\varphi\in PSH(X,\omega),\,\varphi\leq 0\text{ on }K\}\right)^{*}

denote the Siciak extremal function of $K$ (see section 5.1 in [GZ 1]). Then

Vol_{\omega}(K)\leq\int_{X}\exp\left(-\gamma_{\omega}V_{K,\omega}^{*}\right)\omega^{n}\leq C_{2}T_{\omega}(K)^{\gamma_{\omega}},

where $T_{\omega}(K):=\exp(-\sup_{X}V_{K,\omega}^{*})$ denote the Alexander capacity of $K$ (see section 5.2 in [GZ 2]). It follows now from theorem 7.1 in [GZ 1] that

T_{\omega}(K)\leq e\exp\left[-Cap_{\omega}(K)^{-1/n}\right],

which yields (4). ∎

It follows therefore from theorem 2.1 that there exists a unique continuous function $\varphi\in PSH(X,\omega)$ such that

\mu=f\omega^{n}=(\omega+dd^{c}\varphi)^{n},\text{ with }\sup_{X}\varphi=-1,

when $0\leq f\in L^{p}(\omega^{n})$ , $p>1$ , with $\int_{X}f\omega^{n}=1$ .

Actually we will be interested in measures with $L^{p}$ -density with respect to a positive definite volume form $d\lambda$ , while the smooth measure $\omega^{n}$ may vanish along a divisor. This does not make much difference, as follows from Hölder’s inequality:

Lemma 3.2.

Let $V$ be a $n$ -dimensional compact normal Kähler space and $\Omega$ be a smooth Kähler form on $V$ . Let $\pi:X\to V$ a resolution, $\omega=\pi^{*}\Omega$ , and let $d\lambda$ be a positive definite smooth volume form on $X$ .

If $\mu=f_{1}d\lambda$ , with $f_{1}\in L^{p}(X,d\lambda)$ for some $p>1$ , then there exists $p^{\prime}>1$ such that $\mu=f\omega^{n}$ and $f\in L^{p^{\prime}}(X,\omega^{n})$ .

Proof.

Observe that $\omega^{n}=Ed\lambda$ for some smooth density $E\geq 0$ which vanishes along the exceptional divisor of $\pi$ , thus

\mu=f_{1}d\lambda=f\omega^{n},\text{ where }f=f_{1}/E.

Fix local coordinates $(z^{i})_{1\leq i\leq n}$ on a polydisk $\mathbb{D}\subset X$ and a local embedding $F:V\to\mathbb{C}^{m}$ . Note that $E$ is comparable to $|\frac{\partial F}{\partial z^{1}}\wedge\ldots\wedge\frac{\partial F}{\partial z^{1}}|^{2}\simeq\sum_{i=1}^{r}|f_{i}|^{2}$ , $f_{i}$ being holomorphic on $\mathbb{D}$ . Therefore $E\in L^{\infty}_{loc}(\mathbb{D})$ and $E^{-\alpha}\in L_{loc}^{1}(\mathbb{D},d\lambda)$ for some $0<\alpha<1$ .

Choose $0<\alpha^{\prime}<\alpha$ such that $\frac{1}{p}+\frac{1}{\alpha}=\frac{1}{\alpha^{\prime}}$ . Then $f^{\alpha^{\prime}}=f_{1}^{\alpha^{\prime}}E^{-\alpha^{\prime}}$ is the product of a function in $L^{p/\alpha^{\prime}}(d\lambda)$ and a function in $L^{\alpha/\alpha^{\prime}}(d\lambda)$ , hence it is in $L^{1}_{loc}(\mathbb{D},d\lambda)$ by Hölder’s inequality. A second application of Hölder’s inequality yields

\int_{{\mathbb{D}}}f^{1+\varepsilon}\omega^{n}=\int_{\mathbb{D}}f^{\epsilon}f_{1}d\lambda\leq\left(\int_{\mathbb{D}}f^{\epsilon q}d\lambda\right)^{1/q}\left(\int_{\mathbb{D}}f_{1}^{p}d\lambda\right)^{1/p}<+\infty,

where $q$ denotes the conjugate exponent to $p$ . This shows that $f\in L^{p^{\prime}}(\omega^{n})$ if $p^{\prime}=1+\varepsilon>1$ is chosen so small that $\varepsilon q<\alpha^{\prime}$ . ∎

3.2. Hölder continuity

Proposition 3.3.

Assume $\omega_{\varphi}^{n}=f\omega^{n}$ , $\omega_{\psi}=g\omega^{n}$ , where $\varphi,\psi\in PSH(X,\omega)$ are continuous and $f,g\in L^{p}(\omega^{n})$ , $p>1$ . Then for all $0<\gamma<2/(2+nq)$ ,

||\varphi-\psi||_{L^{\infty}(X)}\leq C||\varphi-\psi||_{L^{2}(\omega^{n})}^{\gamma},

where $q=p/(p-1)$ denotes the conjugate exponent to $p$ .

Proof.

Fix $\varepsilon>0$ and $\alpha>0$ to be chosen later. It follows from (2) and propositions 2.5, 3.1 that

||\varphi-\psi||_{L^{\infty}(X)}\leq\varepsilon+C_{1}\left[Cap_{\omega}(|\varphi-\psi|>\varepsilon)\right]^{\alpha/n}.

Applying the refined version of lemma 2.2 which involves the uniform bound on $||\varphi||_{L^{\infty}(X)},||\psi||_{L^{\infty}(X)}$ (see inequality (3)), we obtain

Cap_{\omega}(|\varphi-\psi|>\varepsilon)\leq\frac{C_{2}}{\varepsilon^{n+2/q}}\int_{X}|\varphi-\psi|^{2/q}(f+g)\omega^{n}.

It follows thus from Hölder’s inequality that

Cap_{\omega}(|\varphi-\psi|>\varepsilon)\leq\frac{C_{3}||f+g||_{L^{p}}}{\varepsilon^{n+2/q}}\left[||\varphi-\psi||_{L^{2}(\omega^{n})}\right]^{2/q}.

Choose now $\varepsilon:=||\varphi-\psi||_{L^{2}}^{\omega}$ where $0<\gamma<2/(2+nq)$ . Then

Cap_{\omega}(|\varphi-\psi|>\varepsilon)\leq C_{4}\left[||\varphi-\psi||_{L^{2}}\right]^{2/q-\gamma(n+2/q)}.

We infer

||\varphi-\psi||_{L^{\infty}(X)}\leq||\varphi-\psi||_{L^{2}}^{\gamma}+C_{5}||\varphi-\psi||_{L^{2}}^{\gamma^{\prime}},\;\text{ where }\gamma^{\prime}=\frac{\alpha}{n}\left[2/q-\gamma(n+2/q)\right].

We finally choose $\alpha>0$ so large that $\gamma\leq\gamma^{\prime}$ and adjust the value of the constant $C$ : this yields the desired estimate. ∎

Being able to control the $L^{\infty}$ -norm of $\varphi-\psi$ by its $L^{2}$ -norm is a powerful tool. If for instance $\psi=\varphi_{j}$ , $\varphi$ satisfy the assumptions of proposition 3.2 – with $\varphi_{j}$ being uniformly bounded –, and $\varphi_{j}\rightarrow\varphi$ in $L^{1}$ , then $\varphi_{j}\rightarrow\varphi$ in $L^{2}(\omega^{n})$ , hence $(\varphi_{j})$ actually uniformly converges towards $\varphi$ . This yields the continuity of the map

f\in L^{p}(\omega^{n})\mapsto\varphi\in{\mathcal{C}}^{0}(X),

where $\varphi$ is the unique $\omega$ -psh solution to $(\omega+dd^{c}\varphi)^{n}=f\omega^{n}$ , $\sup_{X}\varphi=-1$ . Thus Theorem A is proved.

We now give an application of this estimate, which is new even when the form $\omega$ is Kähler, but requires the manifold $X$ to be homogeneous, i.e. such that its group of holomorphic automorphisms acts transitively on it.⁴⁴ 4 In particular, the cohomology class of $\omega$ is Kähler and $\omega$ itself can be supposed to be Kähler without loss of generality..

Theorem 3.4.

Assume $X$ is a homogeneous manifold. If $\mu=f\omega^{n}$ is a probability measure with density $0\leq f\in L^{p}(\omega^{n})$ , $p>1$ , then the unique solution $\varphi\in PSH(X,\omega)\in{\mathcal{C}}^{0}(X)$ to the normalized Monge-Ampère equation

(\omega+dd^{c}\varphi)^{n}=\mu=f\omega^{n},\,\sup_{X}\varphi=-1,

is Hölder continuous of exponent $\gamma>0$ , for all $\gamma<2/(2+nq)$ , where $q=p/(p-1)$ is the conjugate exponent to $p$ .

Proof.

When $Aut(X)$ , the group of holomorphic automorphisms of $X$ , acts transitively on $X$ , one can regularize $\omega$ -psh functions by averaging over the Haar measure of the connected component of the identity of $Aut(X)$ . This is very similar to the way one regularizes psh functions in $\mathbb{C}^{n}$ by using convolutions with an approximation of the identity for the convolution product. We refer the reader to [Hu] and the Appendix of [G] for more details.

Let $\varphi_{h}$ be the $\omega$ -psh function which is the translate of $\varphi$ by an automorphism which is at distance $h$ from identity. We use the notation $\varphi_{h}$ by analogy with the $\mathbb{C}^{n}$ -situation, where $\varphi_{h}(x)=\varphi(x+h)$ . Since $\varphi$ is bounded, it has gradient in $L^{2}$ , hence

||\varphi_{h}-\varphi||_{L^{2}}\leq C|h|,

by using Cauchy-Schwarz inequality in a local chart. We can thus apply proposition 3.2 to obtain that

||\varphi_{h}-\varphi||_{L^{\infty}}\leq C^{\prime}|h|^{\gamma},

for all $\gamma<2/(2+nq)$ . Since $\varphi_{h}(x)\simeq\varphi(x+h)$ in a local chart, this precisely means that $\varphi$ is Hölder-continuous of exponent $\gamma$ . ∎

3.3. Regularity on the smooth locus

Theorem 3.5.

Let $X$ be projective algebraic complex manifold, $\omega_{0}$ a smooth semi Kähler form that is Kähler outside a complex subvariety $S\subset X$ , and fix $\Omega$ be a Kähler form on $X$ . Assume that $\omega_{o}^{n}=D\Omega^{n}$ , where $D^{-\varepsilon}$ is in $L^{1}(\Omega^{n})$ , and that $[\omega_{0}],[\Omega]\in NS_{\mathbb{R}}(X)$ .

Let $s_{1},...,s_{p}$ (resp. $t_{1},...,t_{q}$ ) be holomorphic sections of some line bundle $L$ (resp $L^{\prime}$ ) on $X$ . Fix $k\in\mathbb{R}_{\geq 0}$ , $l\in\mathbb{R}_{\geq 0}$ and $F\in{\mathcal{C}}^{\infty}(X,\mathbb{R})$ . Assume that

\int_{X}\frac{1}{|t_{1}|^{2l}+\ldots+|t_{q}|^{2l}}\Omega^{n}<\infty\text{ and }\int_{X}\frac{|s_{1}|^{2k}+\ldots+|s_{p}|^{2k}}{|t_{1}|^{2l}+\ldots+|t_{q}|^{2l}}e^{F}\Omega^{n}=\int_{X}\Omega^{n}.

Then the unique continuous function $\varphi\in PSH(X,\omega_{0})$ such that

(\omega_{0}+dd^{c}\varphi)^{n}=\frac{|s_{1}|^{2k}+\ldots+|s_{p}|^{2k}}{|t_{1}|^{2l}+\ldots+|t_{q}|^{2l}}e^{F}\Omega^{n}\,\text{ and }\sup_{X}\varphi=-1

is smooth outside $B=S\cup\cap_{i}\{s_{i}=0\}\cup\cap_{i}\{t_{i}=0\}$ .

Remark 3.6.

This result should be compared with [Y], Theorem 8. Yau’s result is stronger in many respects (there is no projectivity/rationality assumption and it gives a more precise regularity theory); on the other hand the conditions on the poles of the L.H.S. is less optimal than here.

We expect the projectivity/rationality assumptions to be superfluous. We also expect that a finer regularity theory might be developed for singular KE metrics depending on a finer analysis of the klt singularities involved.

The rest of this subsection will be devoted to the proof of Theorem 3.5. For the reader’s convenience, we will treat two special cases before tackling the general case ⁵⁵ 5 Notice that apart from the ${\mathcal{C}}^{0}$ -estimate with degenerate L.H.S., the methods used here are standard and in [Y], [Ts] and [Ko]. Higher regularity in [TZ] is treated along similar lines given the $L^{\infty}$ -estimate the authors announce.

Preliminary considerations

Thanks to Lemma 3.2 – here we use that $D^{-\varepsilon}\in L^{1}$ – and Theorem 2.1, for every $t\in[0,1]$ there is a unique continuous function $\varphi_{t}\in PSH(X,\omega_{0}+t\Omega)$ such that

(\omega_{o}+t\Omega+dd^{c}\varphi_{t})^{n}=C_{t}\frac{|s_{1}|^{2k}+\ldots+|s_{p}|^{2k}}{|t_{1}|^{2l}+\ldots+|t_{q}|^{2l}}e^{F}\Omega^{n}\text{ and }\sup_{X}\varphi_{t}=-1,

where $C_{t}>0$ is an adequate normalisation constant and $\|\varphi_{t}\|_{{\mathcal{C}}^{0}(X)}$ is uniformly bounded by a constant independant of $t\geq 0$ .

We cannot use right away [Y], Theorem 8 p. 403, to ensure that $(\varphi_{t})$ be smooth outside $B$ for $t>0$ , since our integral condition is stronger than his. However we can use [Y], Thm 3, p 365 to conclude that, in case $\cap_{i}\{t_{i}=0\}=\emptyset$ , $(\varphi_{t})$ is smooth outside $B$ and $dd^{c}\varphi_{t}$ is a form whose coefficients are globally bounded on $X$ , hence $\varphi_{t}\in{\mathcal{C}}^{1,1}(X)$ for $t>0$ . Since this does not imply ellipticity if $\cap_{i}\{s_{i}=0\}\not=\emptyset$ , this does not imply higher regularity on the whole of $X$ .

The required uniformity in $t>0$ is not proved in [Y]. To deal with this case, we use a nice trick due to H.Tsuji [Ts].

The simplest case

First, assume $\cap_{i}\{s_{i}=0\}\cup\cap_{i}\{t_{i}=0\}=\emptyset$ . Hence the family of equations under consideration can be rewritten as:

(\omega_{o}+t\Omega+dd^{c}\varphi_{t})^{n}=C_{t}e^{F}\Omega^{n},

$F$ being smooth.

Tsuji’s trick is as follows. By Kodaira’s lemma, there exists $E$ an effective Cartier divisor of $X$ such that $[\omega_{o}]=[\kappa_{\varepsilon}]+\varepsilon[E]$ where $[\kappa_{\varepsilon}]$ is ample, hence we may choose a representative $\kappa_{\varepsilon}$ which is a Kähler form for every $\varepsilon>0$ small enough. We may actually assume $E$ contains $B$ and use a family of $E$ such that $\cap Supp(E)=B$ , by Nakamaye’s theorem on base loci [Na].

Actually, despite the notation, it will NOT be necessary to let $\varepsilon$ decrease to $0$ ⁶⁶ 6 This technical device could be useful to study finer regularity results and we will fix once for all such an $\varepsilon>0$ .

Let $\sigma\in H^{0}(X,\mathcal{O}_{X}(E))$ be the canonical section vanishing on $E$ with the appropriate multiplicity. We can fix a smooth hermitian metric on this line bundle such that the Poincaré Lelong equation holds,

\omega_{o}=\kappa_{\varepsilon}+\varepsilon[E]-\varepsilon dd^{c}\log|\sigma|^{2}.

The function $\varphi_{t}:=\varphi_{t}-\varepsilon\log|\sigma|^{2}$ is smooth in $X\setminus E$ and is a classical solution to the PDE

(\kappa_{\varepsilon}+t\Omega+dd^{c}\varphi_{t})^{n}=e^{F_{\varepsilon,t}}(\kappa_{\varepsilon}+t\Omega)^{n},

where $(F_{\varepsilon,t})_{1\geq t>0}$ is uniformly bounded in the ${\mathcal{C}}^{\infty}(X)$ -topology of functions and $\kappa_{t}=\kappa_{\varepsilon}+t\Omega$ is uniformly bounded in the ${\mathcal{C}}^{\infty}$ -topology of Kähler forms on $X$ .

We can use the result of the calculation in [Y], section 2. The important formula is (2.22) p. 351 and in a subsidiary fashion (2.21). In these formulae, at each point $p\in X-E$ , an adequate system of normal coordinates for $\kappa_{t}$ is constructed and comparing the notations here and there, we substitute $n$ for $m$ , $\kappa_{t}$ for $g_{i\bar{j}}$ , $\kappa_{t}+dd^{c}\varphi_{t}$ for $g_{i\bar{j}}^{\prime}$ , $\varphi_{t}$ for $\varphi$ and $F_{\varepsilon,t}$ for $F$ . The operator $\Delta$ is the Laplace operator (with the analyst’s sign) of $\kappa_{t}$ and $\Delta^{\prime}$ the Laplace operator of $\kappa_{t}+dd^{c}\varphi_{t}$ . Also $R_{i\bar{i}l\bar{l}}=R^{t}_{i\bar{i}l\bar{l}}$ is the holomorphic bissectional curvature of $\kappa_{t}$ expressed in the above system of normal coordinates.

Since $\kappa_{t}$ is uniformly bounded in the ${\mathcal{C}}^{2}$ topology of Kähler forms then certainly there is constant $C=C_{\varepsilon}$ independent of $t$ such that (2.21) holds and $C^{\prime}$ also independent of $t$ such that $C^{\prime}>\inf R^{t}_{i\bar{i}l\bar{l}}$ .

After these substitutions are made, (2.22) p. 351 reads:

e^{C\varphi_{t}}\Delta^{\prime}(e^{-C\varphi_{t}}(n+\Delta\varphi_{t}))\geq\Delta(F_{\varepsilon,t})-n^{2}C^{\prime}-Cn(n+\Delta\varphi_{t})+e^{-\frac{F_{\varepsilon,t}}{n-1}}(n+\Delta\varphi_{t})^{\frac{n}{n-1}}

We can fix constants $C_{i}$ independent of $t$ such that

\Delta F_{\varepsilon,t}\geq C_{1}\;\;\;\;\text{ and }\;\;\;\;e^{-F_{\varepsilon,t}/n-1}\geq C_{3}>0.

Thus setting $y=n+\Delta\varphi_{t}$ yields

e^{C\varphi_{t}}\Delta^{\prime}(e^{-C\varphi_{t}}(n+\Delta\varphi_{t}))\geq C_{5}+C_{6}y+e^{C_{7}}y^{\frac{m}{m-1}}.

Now by definition

e^{-C\varphi_{t}}(n+\Delta\varphi_{t})=|\sigma|^{+C\varepsilon}e^{-C\varphi_{t}}(n+\Delta\varphi_{t}+\varepsilon\Delta\log|\sigma|^{2}).

For each $t>0$ the functions $\varphi_{t}$ , $\varepsilon\Delta\log|\sigma|^{2}$ and $\Delta\varphi_{t}$ are bounded on $X$ . Hence the positive function $e^{-C\varphi_{t}}(n+\Delta\varphi_{t})$ is continuous on $X$ , vanishes on $E$ and is smooth on $X-E$ . Its maximum is achieved at some point $p_{t}\not\in E$ . It follows from the maximum principle that

0\geq C_{5}+C_{6}y+e^{C_{7}}y^{\frac{n}{n-1}}\text{ at point }y=y(p_{t}).

Therefore $y\leq C_{8}$ with a constant independent of $t>0$ . Now $e^{-C\varphi_{t}(p_{t})}=|\sigma(p_{t})|^{+C\varepsilon}e^{-C\varphi_{t}(p_{t})}$ . Using the uniform ${\mathcal{C}}^{0}$ estimate for $\varphi_{t}$ , we get $0\leq(n+\Delta_{t}\varphi_{t})\leq C_{9}e^{+C\varphi_{t}}$ . Since $|\varphi_{t}|$ and $\varepsilon\Delta\log|\sigma|^{2}$ are uniformly bounded by a constant independent of $t>0$ , we infer

(n+\Delta_{t}\varphi_{t})\leq C_{10}|\sigma|^{-C\varepsilon}=C_{10}|\sigma|^{-C_{\varepsilon}\varepsilon}.

This yields a $t$ -independent ${\mathcal{C}}^{0}$ - estimate of $dd^{c}\varphi_{t}$ on the compact subsets of $X-E$ ⁷⁷ 7 Note that $C=C_{\varepsilon}$ and that $\varepsilon C_{\varepsilon}$ might blow up as $\varepsilon$ goes to $0$ . .

Standard arguments of the theory of complex Monge-Ampère equations give an interior estimate of $\varphi_{t}$ in ${\mathcal{C}}^{k,\alpha}_{loc}(X-E)$ for every $k\geq 2$ , $\alpha\in]0,1[$ which is independent of $t>0$ (see for instance Theorem 5.1, p. 15 in [Bl2]). Hence the family $(\varphi_{t})_{t>0}$ is precompact in every ${\mathcal{C}}^{k,\alpha}_{loc}(X-E)$ . Its cluster values are cluster values in ${\mathcal{C}}^{O}(X-E)$ hence they are all equal to $\varphi|_{X-E}$ . This implies $\varphi\in{\mathcal{C}}^{k,\alpha}_{loc}(X-E)$ , hence that $\varphi\in{\mathcal{C}}^{\infty}(X-E)$ .

Case where $\cap_{i}\{t_{i}=0\}=\emptyset$

⁸⁸ 8 It suffices to consider this case for constructing singular KE metrics on algebraic varieties with canonical singularities

We study here the equation

(\omega_{o}+t\Omega+dd^{c}\varphi_{t})^{n}=C_{t}(|s_{1}|^{2k}+\ldots+|s_{p}|^{2k})e^{F}\Omega^{n}

The first few steps of the preceding argument can be repeated without changes. Next we apply formula (2.22) in [Y] as earlier, except that we set $F=F_{\varepsilon,t}+\log||s||^{(2k)}$ , where $||s||^{(2k)}:=|s_{1}|^{2k}+\ldots+|s_{p}|^{2k}$ . This yields

	$\displaystyle e^{C\varphi_{t}}\Delta^{\prime}(e^{-C\varphi_{t}}(n+\Delta\varphi_{t}))$	$\displaystyle\geq$	$\displaystyle\Delta F_{\varepsilon,t}+\Delta\log\|\|s\|\|^{(2k)}-n^{2}C^{\prime}-Cn(n+\Delta\varphi_{t})$
			$\displaystyle+\left(\frac{e^{-F_{\varepsilon,t}}}{\|\|s\|\|^{(2k)}}\right)^{1/(n-1)}(n+\Delta\varphi_{t})^{\frac{n}{n-1}}$

We recall the two preceding inequalities and observe two new ones that are available:

	$\displaystyle\Delta F_{\varepsilon,t}\geq C_{1}\;$	and	$\displaystyle\;e^{-F_{\varepsilon,t}/n-1}\geq C_{3}>0;$
	$\displaystyle\Delta\log\|\|s\|\|^{(2k)}\geq C_{2}$	and	$\displaystyle C_{4}\geq\|\|s\|\|^{(2k)}.$

Setting as earlier $y=n+\Delta\varphi_{t}$ , we get

e^{C\varphi_{t}}\Delta^{\prime}(e^{-C\varphi_{t}}(n+\Delta\varphi_{t}))\geq C_{5}+C_{6}y+e^{C_{7}}y^{\frac{m}{m-1}}.

After this point, the proof is entirely the same as before.

Remark 3.7.

In order to carry out the second order a priori estimate, one needs information that only depend on $\sup_{X}F$ and $\inf_{X}\Delta F$ . This is pointed out in [Y], p. 351, and it is the basis for the proof of [Y], Thm 3.

3.4. Formal reduction of the general case to the second case using smooth orbifolds

In order to carry out the present argument, which in essence is just a change of variables $z\rightsquigarrow\zeta=z^{1/m}$ , we need to use analysis on certain smooth orbifolds. We will not give complete definitions since they are in the recent reference [BGK], section 2 pp. 560-564, see also [MO] and the references therein.

Let $(X,\Delta)$ be a smooth orbifold pair. By this we mean that we have the prime decomposition $\Delta=\sum_{i}(1-m_{i}^{-1})E_{i}$ where $m_{i}\in\mathbb{N}^{*}$ is an integer. We assume that $supp(\Delta)$ is a simple normal crossing divisor. Then, a classical construction surveyed in [BGK] enables to construct an orbifold $[X,\Delta]$ with a $1$ -morphism of orbifolds $c:[X,\Delta]\to X$ with the following properties:

•

$c$ is the reduction to the coarse moduli space of $[X,\Delta]$ .
•

$c_{X-Supp(\Delta)}:U:=[X,\Delta]\times_{X}\to X-Supp(\Delta)$ is an isomorphism. Hence $U$ is an open suborbifold of $[X,\Delta]$ which is an old-fashioned manifold).
•

For every open polydisk $\mathbb{D}\subset X$ with local coordinates $z_{1},\ldots,z_{n}$ such that $Supp(\Delta)=\{\prod_{j=1}^{p}z_{j}=0\}$ $[X,\Delta]\times_{X}\mathbb{D}=[\mathbb{D}^{\prime}/G_{loc}]$ .

In this formula, the local isotropy group is $G_{loc}=\prod_{j=1}^{p}\mathbb{Z}/m_{j}\mathbb{Z}$ , $m_{j}$ is the integer multiplicity of the divisor $E_{i_{j}}$ such that $E_{i_{j}}\cap\mathbb{D}=\{z_{j}=0\}$ , $G_{loc}$ acts on the polydisk $\mathbb{D}^{\prime}$ by $(\zeta_{1},...,\zeta_{p}).(z^{\prime}_{1},...,z^{\prime}_{n})=(\zeta_{1}z^{\prime}_{1},...,\zeta_{p}z^{\prime}_{p},z_{p+1}^{\prime},...)$ ⁹⁹ 9 The usual isomorphism of $\mathbb{Z}/m\mathbb{Z}$ with the group of $m$ -th root of unity is used..

The orbifold $1$ -morphism $[\mathbb{D}^{\prime}/G_{loc}]\to\mathbb{D}$ is induced by $\varkappa_{loc}:(z^{\prime}_{1},..,z^{\prime}_{n})\mapsto((z^{\prime}_{1})^{m_{1}},...)$ .
•

For sufficiently divisible $s$ , $c_{*}\mathcal{O}_{[X,\Delta]}(sK_{[X,\Delta]})=\mathcal{O}_{X}(s(K_{X}+\Delta))$ .

It is possible to define all the basic concepts of Kähler geometry on orbifolds such as smooth functions, Kähler metrics, etc… The principle is to think of $\varkappa_{loc}^{-1}$ as a (multivalued) smooth coordinate chart.

A continuous function on $[X,\Delta]$ is a continuous function on $X$ . A Radon measure on $[X,\Delta]$ is a Radon measure on $X$ .

A smooth function $f$ on $[X,\Delta]$ is a continuous function on $X$ such that for every local chart $\varkappa_{loc}^{*}f$ is smooth. In particular $f$ is Hölder continuous.

A Kähler metric $\Omega_{[X,\Delta]}$ on $[X,\Delta]$ is a Kähler metric $\Omega_{X-Supp(\Delta)}$ on $X-Supp(\Delta)$ with the property that $\varkappa_{loc}^{*}\Omega$ extends to a smooth Kähler metric on $\mathbb{D}^{\prime}$ . In particular, it also extends as a closed Kähler current on $X$ with Hölder potentials.

The pull back of a Kähler form on $X$ to $[X,\Delta]$ is a semi-Kähler form that is actually cohomologous to a Kähler class¹⁰¹⁰ 10 Here no reference can be given. But it is easy to extend the gluing methods for Kähler forms developed in [Dem 3] and [Pa] to orbifolds. Hence [DP] extends to Kähler orbifolds. .

Observe that $\varkappa^{*}_{loc}dz_{l}=m_{i_{l}}(z^{\prime}_{l})^{m_{i_{l}}-1}dz^{\prime}_{l}$ , hence a smooth volume form on $[X,\Delta]$ can be interpreted as a volume form $v$ on $X-Supp(\Delta)$ such that

v\text{ is comparable to }\frac{{\prod_{l=1}^{n}(\sqrt{-1}dz_{l}\wedge d\bar{z}_{l})}}{\prod_{l=1}^{p}|z_{l}|^{2(1-1/m_{i_{l}})}}.

In case the pair $(X,l^{-1}(t_{1}))$ is an orbifold pair, the equation

(5)

(\omega_{o}+dd^{c}\varphi_{t})^{n}=C_{t}\frac{|s_{1}|^{2k}+\ldots+|s_{p}|^{2k}}{|t_{1}|^{2l}}e^{F}\Omega_{X}^{n}

can be interpreted on $[X,\Delta]$ as an equation of the form

(c^{*}\omega_{o}+dd^{c}\varphi_{t})^{n}=C_{t}(|s_{1}|^{2k}+\ldots+|s_{p}|^{2k})e^{F}\Omega_{[X,\Delta]}^{n}.

The method used to analyze the case where $\cap_{i}\{t_{i}=0\}=\emptyset$ extends with almost no changes to the orbifold case. Hence the unique continuous solution of equation (5) is smooth outside its singular locus if $(X,\frac{1}{l}(t_{1}))$ is an orbifold pair.

Under the more general hypothesis that $\int_{X}|t_{1}|^{-2l}<\infty$ and $\frac{1}{l}(t_{1})$ is a divisor with simple normal crossings, then we can construct an orbifold pair $(X,\Delta)$ with $0\leq\frac{1}{l}(t_{1})\leq\Delta$ and we are back to the previous case.

For the most general case, consider the ideal $\mathcal{I}$ generated by the $t_{i}$ and fix $\mu:X^{\prime}\to X$ a log resolution of $(X,\mathcal{I})$ . Then we are back to the previous case, with an equation on $X^{\prime}$ . This ends the proof of Theorem 3.6.

In certain rare circumstances, there is a finite smooth covering $Y\to X$ such that $Y/G=X$ and $[Y/G]=[X,\Delta]$ and the argument we use here reduces to a $G$ -equivariant argument on $Y$ .

Remarks 3.8.

If we start with $\omega_{o}$ Kähler, and the log-resolution is non trivial, $\mu^{*}\omega_{o}$ is not Kähler anymore.

This method that dates back to [Ko] can be used to prove a variant of [Y], Theorem 7 p. 399 where the divisor of $s_{2}$ is a simple normal crossing divisor, under the sole assumption that $\int_{M}|s_{2}|^{-2k_{2}}<\infty$ .

Now, it could not have been used to prove Theorem 8 p. 403 in 1978 since log-resolutions force the use of Monge-Ampère equations with degenerate L.H.S, for which the ${\mathcal{C}}^{0}$ -estimate proved here was not available then.

3. More regularity [02DP]

3. More regularity

3.1. Measures with density

Proposition 3.1.

Proof.

Lemma 3.2.

Proof.

3.2. Hölder continuity

Proposition 3.3.

Proof.

Theorem 3.4.

Proof.

3.3. Regularity on the smooth locus

Theorem 3.5.

Remark 3.6.

Preliminary considerations

The simplest case

Case where ∩i{ti=0}=∅\cap_{i}\{t_{i}=0\}=\emptyset

Remark 3.7.

3.4. Formal reduction of the general case to the second case using smooth orbifolds

Remarks 3.8.

Case where $\cap_{i}\{t_{i}=0\}=\emptyset$