# Ricci-flat manifolds of generalized ALG asymptotics

Yuanqi Wang University of Kansas, Lawrence, KS, USA<EMAIL_ADDRESS>
###### Abstract

In complex dimensions $\geq 3$, we provide a geometric existence for
generalized ALG complete non-compact Ricci flat Kähler manifolds with Schwartz
decay i.e. metric decay in any polynomial rate to an ALG model
$\mathbb{C}\times Y$ modulo finite cyclic group action, where $Y$ is Calabi-
Yau.

Consequently, for any $K3$ surface with a purely non-symplectic automorphism
$\sigma$ of finite order, a Kähler crepant resolution of the orbifold
$\frac{\mathbb{C}\times K3}{\langle\sigma\rangle}$ admits ALG Ricci-flat
Kähler metrics with Schwartz decay. It is known that Kähler crepant resolution
exists in our case. Hence there are $39$ integers, such that $2\pi$ divided by
each of them is the asymptotic angle of an ALG Ricci-flat Kähler $3-$fold with
Schwartz decay. We also exhibit a 1638 parameters family of ALG Ricci-flat
Kähler $3-$folds with asymptotic angle $\pi$ that realize $64$ distinct
triples of Betti numbers. They are iso-trivially fibred by $K3$ surface with a
non-symplectic Nikulin involution.

A simple version of local Kunneth formula for $H^{1,1}$/local
$i\partial\overline{\partial}-$lemma plays a role in both the Schwartz decay,
and the construction of ansatz that equals a Ricci flat ALG model outside a
compact set (isotrivial ansatz).

The proof of Schwartz decay relies on a non-concentration of the Newtonian
potential, and can not be immediately generalized to fibration with higher
dimensional base, due to existence of concentrating sequence of $L^{2}$
normalized eigen-functions on unit round spheres of (real) dimension $\geq 2$.

###### Contents

  1. 1 Introduction
    1. 1.1 Overview
    2. 1.2 Schwartz Decay
    3. 1.3 Representative examples
      1. 1.3.1 Asymptotic angle $\pi$ via branched double cover
      2. 1.3.2 Asymptotic angle $\pi$ via quartic
      3. 1.3.3 Asymptotic angle $\frac{2\pi}{3}$ via quartic
      4. 1.3.4 Asymptotic angle $\frac{2\pi}{44}$ via elliptic fibration
    4. 1.4 Sketch of the proof
  2. 2 Setup
    1. 2.1 For the main theorem
    2. 2.2 For Monge-Ampere equation
    3. 2.3 Other terms and conventions
  3. 3 Schwartz decay assuming an isotrivial ansatz
    1. 3.1 Preliminary and idea
    2. 3.2 Proof
    3. 3.3 Weighted Schauder estimate and Mazja-Plamenevskii technique
  4. 4 Local $i\partial\overline{\partial}-$lemma on the ALG end
  5. 5 Constraint, iso-trivial ansatz, and proof of main results
    1. 5.1 Hein and Haskins-Hein-Nordström technique
    2. 5.2 Hein decay estimate for Tian-Yau solution
      1. 5.2.1 Finishing the proof of main results
  6. 6 Bergman metric
    1. 6.1  $A_{1}-$singularity and the crepant resolution
    2. 6.2 Nikulin poly-cylinder
    3. 6.3 Resolution along fixed curves and Nikulin-Reid poly-cylinders
    4. 6.4 Projectivity
  7. 7 Topology and table of examples
    1. 7.1 Fundamental group
    2. 7.2 Betti numbers
    3. 7.3 Table of examples and proof of Corollary 1.5
  8. 8 Appendix
    1. 8.1 Poincare inequality under model radius exhaustion
    2. 8.2 Hein energy decay estimate for model radius function
    3. 8.3 Tian-Yau quasi atlas and Schauder norms
      1. 8.3.1 Review
      2. 8.3.2 Poly-Cylinder
      3. 8.3.3 Weighted Schauder norm
    4. 8.4 Gluing for ansatz
    5. 8.5 “Un-conditional” solvability of Laplace/$i\partial\overline{\partial}$ equation in dimension $1$.

##  1 Introduction

###  1.1 Overview

ALG gravitational instantons exist in abundance and are intensively studied.
For example, see the construction by Cherkis-Kapustin [22] via moduli of
periodic monopoles, Biquard-Minerbe [11] by Kummer gluing constructions, and
Hein [52] via Tian-Yau type [78] Monge-Amperè method and rational elliptic
surfaces with a singular fiber removed. Chen-Chen [19] classified the removed
fibers.

Along another thread, gluing construction for manifolds with holonomy $G_{2}$
requires quasi asymptotic locally Euclidean and asymptotic cylindrical Ricci
flat Kähler manifolds as building blocks [55, 56, 59, 60, 34, 35]. These
Kähler metrics generalize ALE and ALH gravitational instantons respectively.
Moreover, ALF gravitational instantons is the local model of co-dimension $1$
collapsing of certain hyper-Kähler metrics on $K3$ surfaces [41].

Motivated by the above, we provide examples of generalized ALG Ricci flat
Kähler $3-$folds on iso-trivial $K3-$fibrations. We apply Tian-Yau Monge-
Ampere solvability [78] and Hein’s version [51, 52]. These metrics are
asymptotic to a group quotient of $\mathbb{C}\times K3$, where the $K3$ fiber
has a purely non-symplectic automorphism of finite order.

######  Definition 1.1.

Let $(Y,\sigma)$ be a compact Calabi-Yau manifold with a purely non-symplectic
automorphism of finite order such that
$\sigma^{\star}\Omega_{Y}=\zeta_{n}\Omega_{Y}$, where $\zeta_{n}$ is a
primitive $n-$th unit root, and $\Omega_{Y}$ is the holomorphic $(d-1,0)-$form
on $Y$ (complex dimension of $Y$ is assumed to be $d-1$). Let $\sigma$ act on
$\mathbb{C}$ by multiplication of $\frac{1}{\zeta_{n}}$.

Our main geometric existence of ALG Ricci flat metric on crepant resolutions
works in general dimensions.

######  Theorem 1.2.

Under Definition 1.1 and 2.1 below, suppose $\widetilde{\frac{\mathbb{C}\times
Y}{\langle\sigma\rangle}}$ is a Kähler crepant resolution of the orbifold
$\frac{\mathbb{C}\times Y}{\langle\sigma\rangle}$ and $b^{1}(Y)=0$. For any
Kähler metric $\omega$ on $\widetilde{\frac{\mathbb{C}\times
Y}{\langle\sigma\rangle}}$, there is a smooth function $\phi_{RFALG}$ such
that

$\omega_{RFALG}\triangleq\omega+i\partial\overline{\partial}\phi_{RFALG}$ (1)

is an $ALG_{\infty}(\frac{2\pi}{n})$ Ricci flat metric (with Schwartz decay).

Suppose $\omega^{\prime}=\omega_{RFALG}+i\partial\overline{\partial}\phi$ for
some bounded $C^{2}$ real valued function $\phi$. Suppose $\omega^{\prime}$ is
Kähler, quasi-isometric to $\omega_{RFALG}$, and has the same volume form as
$\omega_{RFALG}$ (therefore $\omega^{\prime}$ is still Ricci flat). Then it
must equal $\omega_{RFALG}$, and $\phi$ must be a constant.

######  Remark 1.3.

A Kähler crepant resolution simply means a crepant resolution that admits a
smooth Kähler metric. Neither this Kähler metric nor the $\omega$ in the
theorem is required to be complete, or has any other property beyond being
Kähler. The reason is that Kähler condition suffices for the local
$i\partial\overline{\partial}-$lemma 4.1 below, and that the gluing region for
ansatz is compact (Lemma 8.2 below). Thus any smooth function has bounded
$C^{k}-$norms thereon, for any $k$.

The theorem assumes existence of a crepant resolution, which is true for
$3-$folds but can possibly fail for higher dimensional orbifolds.

$K3-$surfaces with purely non-symplectic automorphism of finite order
(including $2,\ 3$ and the larger ones) are intensively studied and classified
to various generality/extent. For examples, see work conducted by Nikulin [63,
64, 65], Artebani-Sarti [3, 4], Artebani-Comparin-Valdés [2], Schütt [71],
Artebani-Sarti-Taki [5], Brandhorst [16], Brandhorst-Hoffman [17], Machida-
Oguiso [62], Kondo [58], Dillies [37], Sarti-Al Tabbaa-Taki [70] etc. These
yield our examples as $3-$dimensional analogies of ALG gravitational
instantons in complex dimension $2$. Their volume growth is still quadratic.

######  Corollary 1.4.

For any $K3$ surface with a purely non-symplectic automorphism $\sigma$ of
finite order, any Kähler crepant resolution $\widetilde{\frac{\mathbb{C}\times
K3}{\langle\sigma\rangle}}$ admits an $ALG_{\infty}(\frac{2\pi}{n})$ Ricci
flat metric $\omega_{RFALG}$ with Schwartz decay. Because Kähler crepant
resolution exists for the $3-$orbifold $\frac{\mathbb{C}\times
K3}{\langle\sigma\rangle}$, for any integer $n$ among the following $8$ prime
numbers and $31$ composite numbers, there is $ALG_{\infty}(\frac{2\pi}{n})$
Ricci flat Kähler $3-$fold with Schwartz decay.

$\displaystyle 2,\ 3,\ 5,\ 7,\ 11,\ 13,\ 17,\ 19$ $\displaystyle 4,\ 6,\ 8,\
9,\ 10.$ $\displaystyle 12,\ 14,\ 15,\ 16,\ 18,\ 20$ $\displaystyle 21,\ 22,\
24,\ 25,\ 26,\ 27,\ 28,\ 30$ $\displaystyle 32,\ 33,\ 34,\ 36,\ 38,$
$\displaystyle 40,\ 42,\ 44,\ 48,\ $ $\displaystyle 50,\ 54,\ $ $\displaystyle
66.$ (2)

When the order of the purely non-symplectic automorphism is $2$, topological
data and number of parameters for examples can be calculated fairly easily.

######  Corollary 1.5.

There is a 1638 parameters family of weakly distinct simply-connected
$ALG_{\infty}(\pi)$ Ricci flat Kähler $3-$folds (with Schwartz decay)
realizing $64$ distinct triples of Betti numbers $(b^{2},b^{3},b^{4})$. They
live on iso-trivial $K3-$fibrations of which the generic fiber admits a non-
symplectic Nikulin involution.

The examples have Betti numbers $b^{i}=0$ when $i\geq 5$. The pullback of each
onto the blowup $Bl_{\mathcal{C}}(\mathbb{C}\times K3)$ along fixed locus of
$\sigma$ is a Ricci flat Kähler metric with conic angle $4\pi$ along the
exceptional divisor, and also with Schwartz decay to the ALG model on
$\mathbb{C}\times K3$ (not the quotient).

ALE/Asymptotically conical Ricci flat metrics with additional conic
singularity along a compact divisor with angle $\in(0,2\pi)$ are constructed
by de Borbon [13] and de Borbon-Spotti [14] etc. See Remark 6.1 below.

$K3$ surfaces with a non-symplectic Nikulin involution [63, 64, 65] also
appear in the gluing construction for manifolds with $G_{2}-$holonomy by
Kovalev-Lee [60] and Joyce-Karigiannis [56].

Two complete Ricci flat manifolds in Corollary 1.5 are said to be weakly
distinct if one of the following holds.

  * •

The generic $K3$ fibers are not isomorphic as $\rho-$polarized $K3-$surfaces,
in the sense of Dolgachev-Kondo [38].

  * •

The generic $K3$ fibers are isomorphic as $\rho-$polarized $K3-$surfaces, but
the Kähler classes of the Ricci flat metrics on the $K3-$fiber component in
ALG model (6) below are not identified by the isomorphism (as
$\sigma$-invariant effective Hodge isometry

$H^{2}(K3,\mathbb{Z})\longrightarrow H^{2}(K3^{\prime},\mathbb{Z}),\
H^{2}(K3,\mathbb{C})\longrightarrow H^{2}(K3^{\prime},\mathbb{C})).$

  * •

The generic $K3$ fibers are isomorphic as $\rho-$polarized $K3-$surfaces, and
the Kähler classes of the Ricci flat metrics on the $K3-$fiber component in
ALG model (6) below) are identified by the isomorphism, but the $\lambda$
factors in (6) are different.

We take all $\sigma-$invariant Kähler classes on the fixed $K3$ into account,
but do not study which Kähler classes on the $K3$ or the ALG manifold are
identified by a bi-holomorphism thereon.

By $ALG$, we always mean a positive rate of decay to the $ALG$ model (see (5)
below and compare it to the concept of weak ALG). We need purely non-
symplectic automorphism because if it is not, the fixed points of the action
on $\mathbb{C}\times Y$ does not entirely lie in the fibre $O\times Y$.
Namely, $\sigma^{n}$ is a none trivial symplectic automorphism which could fix
points on every $Y-$fibre. When asymptotic angle is $\pi$, the compactifying
divisors of the examples in Corollary 1.5 are smooth rational surfaces.

In conjunction with the gluing construction of gravitational instantons by
Biquard-Minerbe [11] in complex dimension $2$, it is natural to ask whether it
is possible to construct these Ricci-flat metrics by gluing. The difficulty is
that the fixed locus consists of curves, and we do not known whether a
singularity model/building block for metric resolution exists, though we
expect such a perspective metric building block to live on the rank $2$
orbifold normal bundle of a (irreducible/connected component of point-wisely)
fixed curve $\mathcal{C}$. We do not known whether a holomorphic orbifold
tubular neighborhood exists for fixed curves, and the expectation is non-
existence in most of the cases. To construct exotic Ricci-flat metrics with
maximal volume growth on $\mathbb{C}^{n},\ n\geq 3$, in the fundamental
$\mathbb{C}\times A_{1}$ case for [61, 32, 77, 23], schematically speaking,
metric resolution along the $1-$dimensional singularity $\mathbb{C}\times O$
with trivial orbifold normal bundle is obtained. In our case, when the area of
the exceptional divisor tends to $0$, it is tempting to expect the ALG Ricci
flat metrics to sub-converge in Gromov-Hausdorff topology to the product
orbifold metric (6).

It is tempting to study whether Corollary 1.5, 1.4 can possibly yield more
examples of generalized Kummer $G_{2}-$manifolds similar to those in [55], and
whether the examples therein can arise as model of singularity formation of
metrics with Ricci curvature bounded from below at some scale.

Asymptotically (real) cylindrical Ricci flat Kähler $3-$folds are building
blocks for twisted connected sum in [59, 35, 34]. Though far from reach, it is
tempting ask if there is a certain kind of connected sum construction for
$G_{2}-$manifolds using asymptotically complex cylindrical Ricci flat Kähler
$3-$folds in Corollary 1.4, 1.5 as building blocks.

Historically, in general dimensions, Tian-Yau [78, 79] utilize Monge-Amperè
methods to construct complete non-compact Ricci flat Kähler metrics. In [78],
they established analytic existence and examples with sub-quadratic volume
growth. Also see work of Bando-Kobayashi [7, 8]. More recently, Hein [51, 52]
and Haskins-Hein-Nordström [49] refined Tian-Yau existence theory. Kovalev
[59], Haskins-Hein-Nordström [49], and Corti-Haskins-Nordström-Pacini [34]
obtained asymptotic cylindrical examples. By similar method, asymptotic
conical and other examples with faster than quadratic volume growth are
constructed by Hein [51], Joyce [55], Conlon-Hein [28], van Coevering [26],
Goto [43], and Santoro [68]. Exotic asymptotic conical Ricci flat metrics on
$\mathbb{C}^{n},\ n\geq 3$ constructed by Li [61], Szekelyhidi [77], Conlon-
Rochon [32], Chiu [23], etc.

On non-compact manifolds, uniqueness of solutions to Monge-Ampere equations
are considered in most of the existence results mentioned above. Our
uniqueness in Theorem 1.2 is not expected to be even nearly optimal. Namely,
classifications of the ends, Torelli-type results, moduli theory, and metric
uniqueness under essentially weaker conditions are also intensively studied in
the setting of asymptotic cylindrical, asymptotically conical, and
gravitational instantons, by [51, 52, 53, 76, 33, 20, 21, 77, 24, 23, 29, 30]
etc. The strong uniqueness in [24] allows slower than quadratic growth on
Kähler manifolds with maximal volume growth.

Part of our work is related to a special case considered in [68]. Namely,
Santoro [68] constructed $ALF-$Ricci flat metrics on crepant
resolutions/isotrivial torus fibration. We focus on ALG case here with
quadratic volume growth, and provide examples with non-flat $K3-$fibers.

###  1.2 Schwartz Decay

######  Proposition 1.6.

Let

  * •

$(X^{d},\omega_{0})$ be an iso-trivial ALG Kähler manifold,

  * •

$k_{0}$ be the smallest positive eigenvalue of $-\Delta_{Y}$, where
$\Delta_{Y}$ is the Laplace Beltrami of the fiber $Y-$component in the ALG
model (6).

Then any smooth solution $\phi$ to the target Monge-Ampere equation (10) such
that

  * •

$\omega_{0}+i\partial\overline{\partial}\phi$ is Kähler,

  * •

$\phi=O(e^{\tau_{0}r})$ for some $\tau_{0}<k_{0}$,

  * •

$i\partial\overline{\partial}\phi=O(\frac{1}{r^{\delta_{0}}})$ for some
$\delta_{0}>0$

has Schwartz Decay i.e. for any positive real number $\beta$ and integer $k$,

$|i\partial\overline{\partial}\phi|_{\omega_{0}}=O(\frac{1}{r^{\beta}})\
\textrm{and}\ i\partial\overline{\partial}\phi\in C^{k}_{\beta}(X).$

Consequently, in complex dimension $2$, the ALG gravitational instantons in
Biquard-Minerbe gluing construction [11, Theorem 2.3] has Schwartz decay.

We allow mild exponential growth of $\phi$. This holds particularly when
$\phi$ is bounded. The decay is on the Kähler form difference
$i\partial\overline{\partial}\phi$, not necessarily on the potential function
$\phi$. It does not explicitly require the constraint. We are not sure whether
Hein’s ALG reference metric [52, Proposition 3.7] on surfaces are iso-trivial.
If they are, among the examples provided by [52, Theorem 1.5 (ii)], the ALG
gravitational instantons on iso-trivial elliptic fibrations have Schwartz
decay.

###  1.3 Representative examples

####  1.3.1 Asymptotic angle $\pi$ via branched double cover

Let $\mathcal{C}(10)$ be a (smooth) sextic in $\mathbb{P}^{2}$ (of genus
$10$). By the construction [9], there is a smooth K3 surface denoted by
$K3_{\mathcal{C}(10)}$ with a non-symplectic involution $\sigma$, such that
there is a covering $K3_{\mathcal{C}(10)}\rightarrow\mathbb{P}^{2}$ branched
along $\mathcal{C}(10)$, and $\sigma$ is the covering transformation. As a
particular case of Corollary 1.5, Corollary 1.4 implies there is an
$ALG_{\infty}(\pi)$ Ricci flat Kähler metric on
$\widetilde{\frac{\mathbb{C}\times
K3_{\mathcal{C}(10)}}{\langle\sigma\rangle}}$ with Schwartz decay. The
exceptional divisor is the ruled surface $\mathbb{P}[O_{\mathcal{C}(10)}\oplus
T^{1,0}_{\mathcal{C}(10)}]$. For example, let $\mathcal{C}(10)$ be Fermat
sextic

$\\{z^{6}_{0}+az^{6}_{1}+bz^{6}_{2}=0\\},\ a\neq 0,\ b\neq 0.$

The compactifying divisor is simply $\mathbb{P}^{2}$. The Betti numbers
$(b^{2},b^{3},b^{4})$ of $\widetilde{\frac{\mathbb{C}\times
K3_{\mathcal{C}(10)}}{\langle\sigma\rangle}}$ equal $(2,20,2)$ respectively.

####  1.3.2 Asymptotic angle $\pi$ via quartic

As studied by Sarti [69], consider the Fermat quartic:

$\\{X^{4}_{0}+aX^{4}_{1}+bX^{4}_{2}+cX^{4}_{3}=0\\},\ a\neq 0,\ b\neq 0,\
c\neq 0$

with the non-symplectic involution $\sigma$ defined by

$\sigma\cdot[X_{0},X_{1},X_{2},X_{3}]=[-X_{0},X_{1},X_{2},X_{3}].$

The fixed degree $4$ curve
$\\{aX^{4}_{1}+bX^{4}_{2}+cX^{4}_{3}=0\\}\triangleq\mathcal{C}(3)$ lies in the
plane $\\{X_{0}=0\\}$ and has genus $3$. The exceptional divisor is the ruled
surface $\mathbb{P}[O_{\mathcal{C}(3)}\oplus T^{1,0}_{\mathcal{C}(3)}].$
Theorem 1.2 implies there is an $ALG_{\infty}(\pi)$ Ricci flat Kähler metric
on $\widetilde{\frac{\mathbb{C}\times K3_{a,b,c}}{\langle\sigma\rangle}}$ with
Schwartz decay. The Betti numbers $(b^{2},b^{3},b^{4})$ of
$\widetilde{\frac{\mathbb{C}\times K3_{a,b,c}}{\langle\sigma\rangle}}$ equal
$(9,6,2)$ respectively. Via the projection

$[X_{0},X_{1},X_{2},X_{3}]\longrightarrow[0,X_{1},X_{2},X_{3}],$

the compactifying divisor is the obvious double cover of the plane

$\mathbb{P}^{2}=\\{X_{0}=0\\}$

branched along $\mathcal{C}(3).$ There is another involution

$\sigma\cdot[X_{0},X_{1},X_{2},X_{3}]=[-X_{0},X_{1},-X_{2},X_{3}]$

which has one more negative sign. It is symplectic because it preserves the
form $\frac{dx_{0}dx_{1}}{x^{3}_{2}}$ when $X_{2},\ X_{3}\neq 0$. Symplectic
involutions are not applied to ALG Ricci flat manifolds here i.e. we do not
consider asymptotic angle $2\pi$.

####  1.3.3 Asymptotic angle $\frac{2\pi}{3}$ via quartic

As studied by Artebani-Sarti [3], consider the smooth quartic
$K3_{quartic,3}\in\mathbb{P}^{3}$ :

$\\{X^{4}_{0}+X^{4}_{1}+X^{4}_{2}+X_{2}X^{3}_{3}=0\\}$

with the non-symplectic automorphism
$\sigma\cdot[X_{0},X_{1},X_{2},X_{3}]=[X_{0},X_{1},X_{2},e^{\frac{2\pi
i}{3}}X_{3}].$ Theorem 1.2 implies there is an $ALG_{\infty}(\pi)$ Ricci flat
Kähler metric on $\widetilde{\frac{\mathbb{C}\times
K3_{quartic,3}}{\langle\sigma\rangle}}$ with Schwartz decay.

####  1.3.4 Asymptotic angle $\frac{2\pi}{44}$ via elliptic fibration

As studied by Kondo [58], consider the smooth quartic $K3_{44}$ that is the
Kodaira-Néron model of the Weierstrass fibration:

$y^{2}=x^{3}+x+t^{11}$

with the non-symplectic automorphism
$\sigma\cdot(x,y,t)\triangleq(\zeta^{22}x,\zeta^{11}y,\zeta^{2}t)$, where
$\zeta=e^{\frac{2\pi i}{44}}$. Then $\sigma^{\star}\Omega=\zeta^{13}\Omega$
which implies $\sigma$ is purely non-symplectic. Let $\sigma$ act on
$\mathbb{C}$ by multiplication of $\zeta^{31}$ i.e. counter-clockwise rotation
in angle $\frac{31\pi}{22}$. Theorem 1.2 implies there is an
$ALG_{\infty}(\pi)$ Ricci flat Kähler metric on
$\widetilde{\frac{\mathbb{C}\times K3_{44}}{\langle\sigma\rangle}}$ with
Schwartz decay.

###  1.4 Sketch of the proof

Step 1: The geometric existence is via Tian-Yau existence and Hein decay. The
production of the ansatz can be described by the following diagram. The
horizontal arrow is by local Kunneth formula/local
$i\partial\overline{\partial}-$solvability (Section 4), and gluing Lemma 8.2.
The vertical arrow is by a version of Hein and Haskins-Hein-Nordström
technique (Lemma 5.1).

Isotrivial ALG metricKähler metricIsotrivial ansatz satisfying constraint

Also see (54) and (55) below about the ALG Ricci flat potential (1) relative
to the initial Kähler metric in Theorem 1.2.

The condition $b^{1}(Y)=0$ is only applied to the equalities (34) for the
mixed terms. This is crucial for the local
$i\partial\overline{\partial}-$Lemma 4.1 that yields ALG metric that
(strictly) equals the model outside a compact set. Once such an iso-trivial
ansatz is constructed, the Schwartz decay does not explicitly require
$b^{1}(Y)=0$.

Step 2: The Schwartz decay is via the weighted $L^{2}-$estimates [82] adapted
to our ALG manifolds here. To turn it into (pointwise) weighted Schauder
estimate, we need no concentration of $L^{2}-$norm of the Newtonian potential.
Namely, because when $R$ is large, an annuli $\\{R<r<R+2\\}$ on $\mathbb{C}$
has area $O(R)$, that a function $Q_{0}=O(\frac{1}{r^{\beta}})$ implies the
$L^{2}-$norm of $Q_{0}$ on the fibred annuli $A(R,R+2)$ is
$O(\frac{1}{R^{\beta-\frac{1}{2}}})$. Naive application of the weighted
$L^{2}-$estimate [82] on fibered annuli only says that the Newtonian potential
$G(Q_{0})$ (21) is $O(\frac{1}{r^{\beta-\frac{1}{2}}})$ pointwisely. In other
words, this causes “serious” weight loss which fails the iteration for
Schwartz decay. To overcome, as opposed to doing Schauder estimate on the
fibred annuli, we do it on the fibred sector of angle width comparable to
$\frac{1}{R}$ (see Figure 2). This is a “small” part of the fibred annuli that
has definite size. The $L^{2}-$norm of the Newtonian potential $G(Q_{0})$ in
the fibred sector is bounded by constant comparable to
$\frac{1}{R^{\frac{1}{2}}}$ times the $L^{2}-$norm of $Q_{0}$ on the fibred
annuli. The no concentration is because the base of the fibration is
$1-$dimensional. This method is expected to fail if the base is of complex dim
$2$ or higher. Eguchi-Hansen metric on the crepant resolution of
$A_{1}-$singularity $\frac{\mathbb{C}^{2}}{\pm Id}$ decays to the flat ALE
model in the rate of $O(\frac{1}{r^{4}})$, but not faster. The advantage of
$\mathbb{S}^{1}$ as the link of the simplest $1-$dimensional cone
$\mathbb{C}\setminus O$ is that the eigen functions $e^{ik\theta}$ has
constant norm $1$, therefore has no concentration. This is applied in line 2–3
of (22) below to obtain the factor of $\frac{2}{R}$ which guarantees no loss
of weight. However, on spheres of dimension $\geq 2$, a sequence of
$L^{2}-$normalized eigen-functions indexed by increasing eigenvalues, called
Zonal functions, concentrates at a point [74].

The above issue does not happen in the asymptotic cylindrical setting. The
reason is in Figure 1 omitting fiber direction. Also see the more
comprehensive Figure 2.

vs$R_{1}$$R_{1}+2$$R_{2}$$R_{2}+2$ Figure 1: Two annuli of the same radius. In
the complex plane $\mathbb{C}$, the shape/area depends on their inner radius.
But in the standard cylinder $\mathbb{R}\times\mathbb{S}^{1}$, they look the
same.

Organization: Most definitions and background can be found in Section 2 and
the introduction. In Section 3, assuming an ansatz that equals a Ricci flat
ALG model outside a compact set (of which the construction will be deferred to
Section 5), we utilize the weighted $L^{2}-$estimates in [82] to prove
Schwartz decay. In Section 4, on a Kähler $Y-$fibration over the punched
$1-$dimensional disk such that $b^{1}(Y)=0$, we show that a closed
$(1,1)-$form is $i\partial\overline{\partial}-$cohomologous to a fixed
$(1,1)-$form on $Y$ (independent of the fiber). With this “local Kunneth for
$H^{1,1}$”, in Section 5, we construct ansatz that equals the (Ricci flat) ALG
model on the end and satisfies the constraint in Tian-Yau solvability [78,
Theorem 1.1] and Hein’s version [51, Proposition 4.1]. The desired Ricci flat
ALG metric is therefore constructed, and we review Hein’s polynomial decay
estimate. In Section 6, as fairly explicit examples of initial Kähler metrics
required by Theorem 1.2, we describe the crepant resolution of the orbifold
and the Bergman metric as a consequence of Kodaira-Baily embedding [6]. In
Section 7, we calculate the fundamental group and Betti numbers of the
examples, and count the parameters of the family.

Acknowledgement: The author is grateful to Simon Donaldson, Song Sun, Yunfeng
Jiang, Yongqiang Liu, Craig van Coevering, and Simon Brandhorst for valuable
discussions.

##  2 Setup

###  2.1 For the main theorem

Manifolds are assumed connected throughout unless otherwise indicated. For
example, the exceptional divisor of the resolution and fixed locus of the
automorphism $\sigma$ could be reducible.

######  Definition 2.1.

  1. 1.

An admissible Kähler configuration means a closed Kähler manifold $Y$ of
complex dimension $d-1$, with an automorphism $\sigma$ of finite order $n$.

  2. 2.

A non-compact complete Kähler $d-$fold $(X^{d},\omega)$ is said to be weak
$\beta-$ALG if there is an admissible Kähler configuration
$(Y,\omega_{Y},\sigma)$, compact sets $K\subset X$,
$K^{\prime}\subset\frac{\mathbb{C}\times Y}{\langle\sigma\rangle}$, and a
diffeomorphism

$\Phi:\ \frac{\mathbb{C}\times Y}{\langle\sigma\rangle}\setminus
K^{\prime}\rightarrow X\setminus K$ (3)

such that $\Phi^{-1}$ composed with (restriction of) the projection

$\rho_{1}:\ \ \frac{\mathbb{C}\times
Y}{\langle\sigma\rangle}\longrightarrow\frac{\mathbb{C}}{\langle\sigma\rangle}$

is holomorphic from $X\setminus K\longrightarrow
Range(\rho_{1}\circ\Phi^{-1})\subset\frac{\mathbb{C}}{\langle\sigma\rangle}$,
and

$\frac{\frac{i}{2}\lambda^{2}|z|^{2\beta-2}dzd\overline{z}+\omega_{Y}}{C_{X,g}}\leq\Phi^{\star}(\omega)\leq
C_{X,g}(\frac{i}{2}\lambda^{2}|z|^{2\beta-2}dzd\overline{z}+\omega_{Y}),$ (4)

for some positive number $C_{X,g}$. $\omega_{Y}$ is a $\sigma-$invariant
Kähler form on $Y$, and $\lambda^{2}$ is a positive real number called the
radial scaling factor. The action of $\sigma$ on $\mathbb{C}$ is
multiplication by a primitive $n-$th unit root. When $\beta=1$, we call it
weak ALG. A weak ALG manifold is said to be $ALG_{\delta}(\frac{2\pi}{n})$ for
some $\delta>0$, or simply ALG (sometimes $\delta$ and/or $\frac{2\pi}{n}$ are
suppressed) if for any natural number $k$,

$|\nabla^{k}[\Phi^{\star}(\omega)-(\frac{i}{2}\lambda^{2}dzd\overline{z}+\omega_{Y})]|\leq
C_{k,\delta}r^{-\delta},$ (5)

for some $C_{k,\delta}$ independent of $r\triangleq|z|$. We call

$\omega_{model}\triangleq\frac{i}{2}\lambda^{2}dzd\overline{z}+\omega_{Y}\ \
\textrm{the ALG model},$ (6)

under which the norms/covariant derivatives in (5) are defined. We say such a
manifold has Schwartz decay if it is $ALG_{\delta}(\frac{2\pi}{n})$ for any
$\delta>0$, in which case we call it $ALG_{\infty}(\frac{2\pi}{n})$ as well.
The compact sets $K$ and $K^{\prime}$ are not necessarily required to be
uniform in $\delta$.

Unlike [51, 19], we do not require faster decay for higher derivatives
formally in (5). But there is no essential difference because we prove
Schwartz decay anyway.

  3. 3.

A weak ALG manifold (without specifying metric) is called an isotrivial ALG
manifold if the $\Phi$ in (3) is a biholomorphism. An ALG Kähler metric on an
iso-trivial ALG manifold is called iso-trivial if it (strictly) equals a model
(6) away from a compact set in $X$ under the holomorphic coordinate $\Phi$
near $\infty$.

The following implication on metrics holds.

$Isotrivial\ ALG\Longrightarrow ALG_{\infty}\Longrightarrow
ALG\Longrightarrow\textrm{weak}\ ALG.$

  4. 4.

Continuing Definition 1.1 and Theorem 1.2, we usually denote the Kähler
crepant resolution $\widetilde{\frac{\mathbb{C}\times
Y}{\langle\sigma\rangle}}$ by $X$.

The pullback $\Phi^{-1,\star}(dz\wedge\Omega_{Y})$ of the invariant
holomorphic volume form extends to a now-where vanishing holomorphic
$(d,0)-$form on $X$, since the discrepancy is $0$ (see [67, Lemma 6, extension
in the proof] and [34, Example 1.9, the application of Theorem D]).

The compactification $\overline{X}$ is

$\widetilde{\frac{\mathbb{P}^{1}\times Y}{\langle\sigma\rangle}},$

where $\sigma$ still acts by $\frac{1}{\zeta_{n}}$ on
$\mathbb{P}^{1}=\mathbb{C}\cup\\{\infty\\}$, and the resolution is only along
$\\{0\\}\times Fix(\sigma)$, but not along $\\{\infty\\}\times Fix(\sigma)$.
$u=\frac{1}{z}$ defines the coordinate $\Phi$ near $\infty$, and the
compactifying divisor is $\frac{\\{0_{u}\\}\times Y}{\langle\sigma\rangle}$
(also denoted by $\frac{\\{\infty\\}\times Y}{\langle\sigma\rangle}$ in terms
of $z$).

For any $\infty\geq b>a>0$, let $A(a,b)$ denote the (open) annulus

$(a,b)\times\mathbb{S}^{1}\times Y\subseteq\mathbb{C}^{\star}\times Y$

When $b=\infty$, we also call it “the end”. Let $\psi_{\mu}$ denote the
biholomorphism from $A(r_{0},\infty)$ to $A(\mu r_{0},\infty)$ defined by the
radial scaling

$\psi_{\mu}(z,y)=(\mu z,y).$

When $r_{0}$ is large enough, the model metric (4) becomes the normalized

$\frac{i}{2}|z|^{2\beta-2}dzd\overline{z}+\omega_{Y}\ \textrm{under the
coordinate}\ \psi_{\lambda^{-\frac{1}{\beta}}}\circ\Phi.$ (7)

The proof of Theorem 1.2 (including the lemmas, propositions etc it requires)
are under the normalizing chart $\psi_{\lambda^{-\frac{1}{\beta}}}\circ\Phi$
unless otherwise specified. Under the fixed chart $\Phi$ of the crepant
resolution $X$, the $\lambda$ in the ALG model (6) (for $\beta=1$) adds one
more parameter to the family of examples.

###  2.2 For Monge-Ampere equation

The identity

$\omega_{\mathbb{C}^{d}}^{d}=\nu_{d}\Omega_{\mathbb{C}^{d}}\wedge\overline{\Omega}_{\mathbb{C}^{d}},\
\nu_{d}\triangleq\left\\{\begin{array}[]{c}\frac{d!}{2^{d}}\ \textrm{when}\ d\
\textrm{is even},\\\ \frac{d!}{2^{d}}\cdot i\ \textrm{when}\ d\ \textrm{is
odd}\end{array}\right.$ (8)

holds for the standard Kähler metric and holomorphic volume forms on the
Euclidean space:

$\omega_{\mathbb{C}^{d}}=\frac{i}{2}(dz_{1}d\overline{z}_{1}+...+dz_{d}d\overline{z}_{d}),\
\ \Omega_{\mathbb{C}^{d}}=dz_{1}dz_{2}\wedge...\wedge dz_{d}.$

On an iso-trivial Calabi-Yau fibration $\widetilde{\frac{\mathbb{C}\times
Y}{\langle\sigma\rangle}}$ as Definition 2.1, define the Ricci
potential/volume density

$f\triangleq\log(\frac{\nu_{d}\Omega\wedge\bar{\Omega}}{\omega^{d}_{0}}).$ (9)

It suffices to solve the target (elliptic) Monge-Ampere equation

$(\omega^{d}_{0}+i\partial\overline{\partial}\phi)^{d}=e^{f}\omega^{d}_{0},$
(10)

which is equivalent to the volume form equation:

$\omega_{\phi}^{d}=\nu_{d}\Omega\wedge\bar{\Omega},\ \textrm{where}\
\omega_{\phi}\triangleq\omega_{0}+i\partial\overline{\partial}\phi.$ (11)

Tian-Yau existence [78, Theorem 1.1] and Hein’s version [51, Proposition 4.1]
require an ansatz that obeys the constraint

$\int_{X}(\omega_{0}^{d}-\nu_{d}\Omega\wedge\overline{\Omega})=0.$ (12)

As a premise we need the top form
$\omega_{0}^{d}-\nu_{d}\Omega\wedge\overline{\Omega}$ to be integrable over
$X$.

###  2.3 Other terms and conventions

Terms:

  * •

Let $orb$ denote $\frac{\mathbb{C}\times Y}{\langle\sigma\rangle}$ or the
special case $\frac{\mathbb{C}\times K3}{\langle\sigma\rangle}$ of interest,
and call it “the orbifold”.

  * •

Sometimes or always, by “Kähler metric”, we mean the Kähler form.

  * •

Let $R_{0}>>1$ be large enough so all the subsequent desired operations are
valid. We extend the model radius $r$, from $A(R_{0},\infty)$ to be valued in
$[0,1]$ on the whole $X$ and $\equiv 0$ in $X\setminus A(R_{0}-1000,\infty)$.

  * •

Let $\Delta^{\star}$ or $\Delta^{\star}_{u}$ be the punched disk
$\\{u\in\mathbb{C}|\ 0<|u|<\rho_{0}\\}$, $u\triangleq\frac{1}{z}$ is the base
coordinate near $\infty$. The radius $\rho_{0}$ is suppressed and is usually
taken large enough depending on context. See for example section 4.

Convention for constants: Constants $C$ (might be different at different
places) and radius $R_{i}$ are positive and depend on the underlying
manifold/orbifold which is fixed and clear from context. They might depend on
existing/preceding reference metrics, but never on the reference metric we are
constructing, or the Ricci flat metric/Kähler potential we solve for in the
current step. Sometimes we add subscript to indicate dependence, and/or use
other symbols.

##  3 Schwartz decay assuming an isotrivial ansatz

###  3.1 Preliminary and idea

On the end of the resolution $X=\widetilde{\frac{\mathbb{C}\times
Y}{\langle\sigma\rangle}}$, we study the metrics and functions in the orbifold
chart upstairs i.e. we view them as being $\sigma-$invariant on the fibred
annuli $A(\rho,\infty)=[\mathbb{C}\setminus D(\rho)]\times Y$ where
$D(\rho)\subset\mathbb{C}$ is a disk of radius $\rho$ centered at the origin.
Iso-triviality of the ansatz $\omega_{0}$ constructed at the end of Section 4
implies the Ricci potential/volume density $f$ (9) is compactly supported.
Away from the support of $f$, the target equation (10) reads

$\omega_{\phi}^{d}=\omega_{model}^{d},\ \textrm{where}\
\omega_{\phi}\triangleq\omega_{model}+i\partial\overline{\partial}\phi.$ (13)

The equality (48) below becomes

$\frac{i\partial\overline{\partial}\phi\wedge[\omega_{\phi}^{n-1}+...+\omega_{model}^{j}\wedge\omega_{\phi}^{n-1-j}+...+\omega_{model}^{n-1}]}{\omega_{model}^{d}}=0.$
(14)

We foil the left side and observe that the lowest degree term in
$i\partial\overline{\partial}\phi$ is $\underline{\omega}^{n-1}$ up to a
positive real constant. Then dividing by another constant if necessary, the
target equation on the ALG end becomes

$\underline{\Delta}\phi=Q(i\partial\overline{\partial}\phi),$ (15)

where $Q$ is quadratic in $i\partial\overline{\partial}\phi$, and
$\underline{\Delta}$ is the Laplace Beltrami of the product orbifold metric
$\underline{\omega}$. Then we bootstrap the decay rate using the main theorem
in [82]

$i\partial\overline{\partial}\phi=O(\frac{1}{r^{\beta}})\Longrightarrow
Q(i\partial\overline{\partial}\phi)=O(\frac{1}{r^{2\beta}})\Longrightarrow
i\partial\overline{\partial}\phi=O(\frac{1}{r^{2\beta}}).$

Successive application of the above yields that
$|i\partial\overline{\partial}\phi|$ has faster than polynomial decay i.e.
$|i\partial\overline{\partial}\phi|=o(\frac{1}{r^{p}})$ for any $p>0$.

###  3.2 Proof

We say a continuous function $f$ on a cone end $A(\varrho,\infty)$ has
vanishing fiber-wise integrals if for any $z$ with $|z|>\varrho$ (where
$\varrho>0$ is fixed), the integral of $f$ over $z\times Y$ vanishes. We
divide equation (15) into the $Ker\Delta_{Y}-$component and the component on
the orthogonal complement i.e. we decompose

$\phi=\phi_{0}+\underline{\phi},\ Q=Q_{0}+\underline{Q}$ (16)

where $\underline{\phi},\ \underline{Q}$ are pullback functions from
$\mathbb{C}$, and $\phi_{0},\ Q_{0}$ have vanishing fiber-wise integrals along
$z\times Y$, for any $|z|>R_{0}$. We do not decompose the
$i\partial\overline{\partial}\phi$ in $Q$, but directly decompose $Q$. Because
$\Delta_{Y}\underline{\phi}=0$, (15) is decomposed into the following two
equations

$\Delta\phi_{0}=Q_{0},\ \Delta_{\mathbb{C}}\underline{\phi}=\underline{Q}.$
(17)

The point is that $\Delta_{\mathbb{C}}=4\frac{\partial^{2}}{\partial
z\partial\overline{z}}$. Hence the second equation in (17) says

$i\partial\overline{\partial}\underline{\phi}=\frac{\underline{Q}}{4}idzd\overline{z}\
\ \ \textrm{automatically decays because}\ \underline{Q}\ \textrm{does}.$ (18)

The main theorem in [82] gives a Green’s function for the second equation in
(17) with certain bounds. This yields the following weighted Schauder
(pointwise) estimate.

######  Lemma 3.1.

On the cone end, for any real number $\beta$, natural number
$k\geq 1000(1000^{d}+1)$, and non-negative real number $\tau_{0}$ as
Proposition 1.6, there exists a constant $C$ such that for any $R>1000$, the
following estimate holds whenever $\phi$ have vanishing fiber-wise integral,
and is $C^{k+2,\frac{1}{2}}$ in $A(R-1,\infty)$.

$|i\partial\overline{\partial}\phi_{0}|_{C_{\beta}^{k,\frac{1}{2}}[A(R-1,\infty)]}\leq
C\\{|e^{-\tau_{0}r}\phi_{0}|_{C^{0}[A(R-2,\infty)]}+|\Delta\phi_{0}|_{C_{\beta}^{k,\frac{1}{2}}[A(R-2,\infty)]}\\}.$
(19)

By smoothness of Tian-Yau solution [78, Theorem 1.1, last sentence], we only
need the a priori estimate. Combining the trivial identification (18), we find
for any large $R_{1}$ that

$|i\partial\overline{\partial}\phi|_{C_{\beta}^{k,\frac{1}{2}}[A(R_{1}+1,\infty)]}\leq
C_{\beta}\\{|\phi|_{C^{0}[A(R_{1},\infty)]}+|Q|_{C_{\beta}^{k,\frac{1}{2}}[A(R_{1},\infty)]}\\}.$
(20)

Starting from the initial decay (5.2), induction yields

$\displaystyle|i\partial\overline{\partial}\phi|_{C_{2^{j}\beta_{0}}^{k,\frac{1}{2}}[A(R_{1}+j,\infty)]}$
$\displaystyle\leq$ $\displaystyle C_{2^{j}\beta_{0}}\cdot
C_{2^{j-1}\beta_{0}}\cdot...C_{2\beta_{0}}(|\phi|_{C^{0}[A(R_{1},\infty)]}+|i\partial\overline{\partial}\phi|_{C_{\beta_{0}}^{k,\frac{1}{2}}[A(R_{1},\infty)]})$

for any non-negative integer $j$. The $k$ is fixed, large, and arbitrary
during the induction. The proof for Schwartz decay is complete.

######  Remark 3.2.

Though the Schwartz decay is not affected even if we allow the constant in
(19) to depend on $R$, it can be taken independent of $R$.

###  3.3 Weighted Schauder estimate and Mazja-Plamenevskii technique

###### Proof of Lemma 3.1:.

Step 1: Preliminary.

Let $Q_{0}$ denote $\Delta\phi_{0}$, and $G$ denote the Green’s function
constructed in [82]. Here we extend $Q_{0}$ to be $0$ outside the end
$A(R,\infty)$. This does not affect $L^{2}-$integrability. Then we apply the
$G$ in [82] and restrict $G\circ Q_{0}$ on the end. We find the decomposition

$\phi_{0}=G\circ Q_{0}+h\ \textrm{where}\ \ h\ \ \textrm{is harmonic}.$ (21)

We estimate them separately.

Let $p=(R,\theta_{0},x)\in(0,\infty)\times\mathbb{S}^{1}\times Y$ be an
arbitrary point such that $R>1000$. The regularity we assumed ensures that the
summation of Fourier-Series equals the function itself i.e.

$Q_{0}=\Sigma_{k\in\mathbb{Z},\mu\in
Spec_{mul}\Delta_{Y}}(Q_{0})_{k,\mu}e^{ik\theta}\psi_{\mu},$

where $Spec_{mul}\Delta_{Y}$ is the spectrum (counted with multiplicity) of
the Laplace Beltrami on $Y$ with respect to the invariant Kähler metric
$\omega_{Y}$. The $\psi_{\mu}$ is the corresponding eigen-function. The
Green’s function constructed in [82, Theorem 1.2] respects the Fourier
decomposition i.e. for any $k,\mu$ as indicated, there exists a Green’s
function $G_{k,\mu}$ on functions in $r$ only, such that

$G(Q_{0})=\Sigma_{k\in\mathbb{Z},\mu\in
Spec_{mul}}G_{k,\mu}[(Q_{0})_{k,\mu}]e^{ik\theta}\psi_{\mu}.$

Step 2: Estimate for the Newtonian potential $G(Q_{0})$.

The estimates in [82] are all about $L^{2}$/Sobolev norms. Here we translate
them to pointwise estimates.

We apply the usual interior Schauder estimate in a small scale with respect to
the convex, harmonic radius of $Y$, and Tian-Yau “radius”. The idea is to
integrate the Mazja-Plamenevsky estimate [82, Theorem 1.2.(4)] on a sector
over the “small” portion $(\theta_{0}-\frac{1}{R},\theta_{0}+\frac{1}{R})$ of
the circle (Figure 2).

$\theta_{0}+\frac{1}{R}$$\theta_{0}-\frac{1}{R}$$\ \times\ Y\ \ \ \ \ (R$ is
large regarding the manifold)$R-1$$R+1$ Figure 2: gray region$\times Y$ is a
sector-shaped annulus

The weighted $L^{2}-$norm on the sector-shaped annulus is estimated as
follows, where the measure for the integrals is the standard $rdrd\theta dY$.

$\displaystyle|r^{\beta}G(Q_{0})|^{2}_{L^{2}[(R-1,R+1)\times(\theta_{0}-\frac{1}{R},\theta_{0}+\frac{1}{R})\times
Y]}$ (22) $\displaystyle=$ $\displaystyle\Sigma_{k\neq
0,\mu}\int_{Y}\int^{\theta_{0}+\frac{1}{R}}_{\theta_{0}-\frac{1}{R}}\int_{R-1}^{R+1}|G_{k,\mu}[(Q_{0})_{k,\mu}]|^{2}|e^{ik\theta}|^{2}\cdot|\psi_{\mu}|^{2}r^{2\beta}rdrd\theta
dY$ $\displaystyle=$ $\displaystyle\frac{2}{R}\Sigma_{k\neq
0,\mu}\int_{R-1}^{R+1}|G_{k,\mu}[(Q_{0})_{k,\mu}]|^{2}r^{2\beta}rdr=\frac{1}{\pi
R}|r^{\beta}G(Q_{0})|^{2}_{L^{2}[A(R-1,R+1)]}$ $\displaystyle\leq$
$\displaystyle C\sup_{\varrho\geq
R}\frac{1}{\varrho}|r^{\beta}G(Q_{0})|^{2}_{L^{2}[A(\varrho-1,\varrho+1)]}\leq
C\sup_{\varrho\geq
R}\varrho^{2\beta}|Q_{0}|^{2}_{C^{0}[A(\varrho-1,R\varrho+1)]}.$

The reason we lose the $\varrho-$factor in line $4$ is that the area of the
fibred annuli is comparable to $R$. In line 3, we must sum up before applying
Mazja-Plamenevski estimate [82, Theorem 1.2, (4)] to obtain line 4, where the
$L^{2}-$norms are on the full fibred annuli, not only the sector.

Write $S^{a,b}_{\theta_{0},\theta_{1}}$ for the sector
$(a,b)\times(\theta_{0},\theta_{1})\times Y$. The ball $B_{p}(\frac{1}{10})$
(deeper gray ball in Figure 2) is contained in the interior of the sector
$S^{R-1,R+1}_{\theta_{0}-\frac{1}{R},\theta_{0}+\frac{1}{R}}$, using the
$L^{2}$ norm as the lowest order term in the Schauder interior estimate, we
find

$\displaystyle R^{\beta}|G(Q_{0})|_{C^{k+2,\frac{1}{2}}[B_{p}(\frac{1}{10})]}$
$\displaystyle\leq$ $\displaystyle
CR^{\beta}\\{|G(Q_{0})|_{L^{2}[S^{R-1,R+1}_{\theta_{0}-\frac{1}{R},\theta_{0}+\frac{1}{R}}]}+|Q_{0}|_{C^{k,\frac{1}{2}}[S^{R-1,R+1}_{\theta_{0}-\frac{1}{R},\theta_{0}+\frac{1}{R}}]}\\}$
$\displaystyle\leq$ $\displaystyle C\cdot\sup_{\varrho\geq
R}R^{\beta}|Q_{0}|_{C^{k,\frac{1}{2}}[A(\varrho-1,\varrho+1)]}\ \ \
(\textrm{we applied}\ \eqref{equ 1 Green func est})$ $\displaystyle\leq$
$\displaystyle C|Q_{0}|_{C_{\beta}^{k,\frac{1}{2}}[A(\varrho-1,\varrho+1)]}.$

Step 3: Estimate for the harmonic part. Using the spectrum gap, let
$\tau\triangleq\frac{\tau_{0}+k_{0}}{2}\in(\tau_{0},k_{0})$ in Proposition
1.6, we find

$\displaystyle R^{\beta}|h|_{C^{k+2,\frac{1}{2}}[B_{p}(\frac{1}{10})]}\leq
C|r^{\beta}h|_{L^{2}[B_{p}(\frac{1}{10})]}\leq C|r^{\beta}e^{-\tau
r}h|_{L^{2}[A(R-1,\infty)]}$ (24) $\displaystyle\leq$ $\displaystyle
C|e^{-\tau_{0}r}h|_{C^{0}[A(R-1,\infty)]}\leq
C[|e^{-\tau_{0}r}\phi_{0}|_{C^{0}[A(R-1,\infty)]}+|e^{-\tau_{0}r}G\circ
Q_{0}|_{C^{0}[A(R-1,\infty)]}]$ $\displaystyle\leq$ $\displaystyle
C[|e^{-\tau_{0}r}\phi_{0}|_{C^{0}[A(R-2,\infty)]}+|Q_{0}|_{C_{\beta}^{\frac{1}{2}}[A(R-2,\infty)]}].$

Step 4: Summing (3.3) and (24) up, we find

$\displaystyle|i\partial\overline{\partial}\phi_{0}|_{C_{\beta}^{k,\frac{1}{2}}[A(R-1,\infty)]}\leq|i\partial\overline{\partial}(G\circ
Q_{0})|_{C_{\beta}^{k,\frac{1}{2}}[A(R-1,\infty)]}+|i\partial\overline{\partial}h|_{C_{\beta}^{k,\frac{1}{2}}[A(R-1,\infty)]}$
$\displaystyle\leq$ $\displaystyle
C\\{|e^{-\tau_{0}r}\phi_{0}|_{C^{0}[A(R-2,\infty)]}+|Q_{0}|_{C_{\beta}^{k,\frac{1}{2}}[A(R-2,\infty)]}\\}.$

The proof is complete. ∎

##  4 Local $i\partial\overline{\partial}-$lemma on the ALG end

######  Lemma 4.1.

Let $Y$ be a compact Kähler manifold, and $[\ \cdot\ ]$ denote the cohomology
class of a $(1,1)-$form in $H^{1,1}[Y,\mathbb{C}]$. Then the restricted
$(1,1)-$cohomology class $[\omega(u)]$ on $Y$ is independent of $u$. Suppose
$b^{1}(Y)=0$ additionally. Then for any $u_{0}\in\Delta^{\star}$, there is a
smooth function $\phi_{0}$ on $\Delta^{\star}\times Y$ such that

$\omega=i\partial\overline{\partial}\phi_{0}+\omega_{Y,0},$ (25)

where $\omega_{Y,0}\triangleq\omega_{Y}(u_{0})$ is the restriction of $\omega$
onto the fiber $u_{0}\times Y$, viewed as a $\Delta^{\star}-$independent form.

If $\sigma$ is a non-id automorphism of $Y$ of finite order $ord_{\sigma}$,
let $\sigma$ act on $\Delta^{\star}$ via multiplication by a primitive
$ord_{\sigma}-$th unit root. If $\omega$ is real and/or $\sigma-$invariant,
then $\phi_{0}$ can be taken real and/or $\sigma-$invariant as well.

Related solvability for elliptic fibrations are obtained in [51, 52, 21, 47]
etc. Standard Hodge decomposition obviously “fails” in this local “toy” case
i.e. $b^{1}[\Delta^{\star}\times Y]=1$, but our point is that the Kunneth
formula for $H^{1,1}$ is still true i.e.

$H^{1,1}[\Delta^{\star}\times Y,\mathbb{C}]=H^{1,1}[Y,\mathbb{C}],\
H^{1,1}[\Delta^{\star}\times Y,\mathbb{R}]=H^{1,1}[Y,\mathbb{R}].$

We recall the pedestrian language that $H^{1,1}$ means the vector space of all
$(1,1)-$classes i.e. cohomology classes in $H^{2}$ represented by
$(1,1)-$forms (either complex or real, depending on the co-efficient symbol).

###### Proof.

Let $\rho_{1},\ \rho_{2}$ be projections from $\Delta^{\star}\times Y$ to the
$\Delta^{\star}$ and $Y$ components, respectively. Under the decomposition

$\displaystyle\wedge^{1,1}[\Delta^{\star}\times Y]$ $\displaystyle=$
$\displaystyle\rho_{1}^{\star}\wedge^{1,1}[\Delta^{\star}]\oplus(\rho_{1}^{\star}\wedge^{1,0}[\Delta^{\star}]\wedge\rho_{2}^{\star}\wedge^{0,1}[Y])\oplus(\rho_{1}^{\star}\wedge^{0,1}[\Delta^{\star}]\wedge\rho_{2}^{\star}\wedge^{1,0}[Y])$
$\displaystyle\oplus\rho_{2}^{\star}\wedge^{1,1}[Y],$

we write

$\omega=aidud\overline{u}+du\wedge F^{0,1}+d\overline{u}\wedge
F^{1,0}+\omega_{Y}(u)$ (26)

according to the order of the $4-$components where $a$ is a function, and
$\ F^{1,0},F^{0,1},\ \omega_{Y}(u)$ are sections of
$\rho_{2}^{\star}\Omega^{1,0}[Y],\ \rho_{2}^{\star}\Omega^{0,1}[Y],\
\rho_{2}^{\star}\Omega^{1,1}[Y]$ respectively.

Step 1: Independence of fiber.

Since the restriction of $\omega$ on each fiber is closed under the fiberwise
exterior differential $d_{Y}$, and the exterior differential in the
$\Delta^{\star}\subset\mathbb{C}-$component can be decomposed as the following

$d_{\mathbb{C}}=du\wedge\frac{\partial}{\partial
u}+d\overline{u}\wedge\frac{\partial}{\partial\overline{u}}\ (\textrm{and}\
d=d_{\mathbb{C}}+d_{Y}),$

we calculate

$\displaystyle 0$ $\displaystyle=$ $\displaystyle d\omega$ $\displaystyle=$
$\displaystyle idud\overline{u}[d_{Y}a+i\frac{\partial
F^{0,1}}{\partial\overline{u}}-i\frac{\partial F^{1,0}}{\partial
u}]+du[\frac{\partial\omega_{Y}}{\partial
u}-d_{Y}F^{0,1}]+d\overline{u}[\frac{\partial\omega_{Y}}{\partial\overline{u}}-d_{Y}F^{1,0}].$

Then

$\frac{\partial\omega_{Y}}{\partial u}=d_{Y}F^{0,1},\ \
\frac{\partial\omega_{Y}}{\partial\overline{u}}=d_{Y}F^{1,0}.$ (28)

Take cohomology class $[\ \cdot\ ]$ in $H^{1,1}[Y,\mathbb{C}]$, we find

$\frac{\partial[\omega_{Y}]}{\partial
u}=\frac{\partial[\omega_{Y}]}{\partial\overline{u}}=0.$ (29)

This implies $[\omega_{Y}(u)]\in H^{1,1}[Y,\mathbb{C}]$ is constant in $u$.

Step 2: There is an unique smooth function $\phi$ on $\Delta^{\star}\times Y$
with fiber-wise $0-$average such that

$\omega_{Y}(u)=\omega_{Y,0}+i\partial_{Y}\overline{\partial}_{Y}\phi.$ (30)

This is because $\omega_{Y}(u)$ and $\omega_{Y,0}$ are cohomologous. Fiber-
wise $i\partial_{Y}\overline{\partial}_{Y}-$solutions with $0-$average define
a function on the fibration $\Delta^{\star}\times Y$ that is smooth by
implicit function theorem. Please see [80, 2.1] about related discussion for
semi Ricci flat metrics introduced by [45].

Step 3: If $b^{1}(Y)=0$, the solution $\phi$ in (30) actually solves the
equation (25) on the fibration modulo pullback of a $(1,1)-$form on
$\Delta^{\star}$ i.e. $i\partial\overline{\partial}$ of a pullback function
from $\Delta^{\star}$ (independent of $Y$).

Namely, the decomposition (30) and the vanishing of each component in (4) says

$\displaystyle d_{Y}a+i\frac{\partial
F^{0,1}}{\partial\overline{u}}-i\frac{\partial F^{1,0}}{\partial u}=0,$ (31)
$\displaystyle
i\partial_{Y}\overline{\partial}_{Y}\frac{\partial\phi}{\partial
u}-d_{Y}F^{0,1}=0,$ (32) $\displaystyle
i\partial_{Y}\overline{\partial}_{Y}\frac{\partial\phi}{\partial\overline{u}}-d_{Y}F^{1,0}=0.$
(33)

The $(1,1)-$part of (33) must vanish, which means the $(1,0)-$form

$i\partial_{Y}\frac{\partial\phi}{\partial\overline{u}}+F^{1,0}\ \
\textrm{is}\ \ \overline{\partial}_{Y}-\textrm{closed. Hence it is a
holomorphic}\ (1,0)-\textrm{form on}\ Y.$

The assumption $b^{1}(Y)=0$ and Hodge decomposition implies vanishing of
$H^{1}(Y,O_{Y})=H^{0}(Y,\Omega^{1}_{Y})$. Therefore the above form must be $0$
i.e.

$F^{1,0}=-i\partial_{Y}\frac{\partial\phi}{\partial\overline{u}}.\ \ \
\textrm{Similarly by}\ \eqref{equ 1 iddar},\
F^{0,1}=i\overline{\partial}_{Y}\frac{\partial\phi}{\partial u}.$ (34)

Plugging (34) into (31), we find

$d_{Y}a-\partial_{Y}(\frac{\partial^{2}\phi}{\partial
u\partial\overline{u}})-\overline{\partial}_{Y}(\frac{\partial^{2}\phi}{\partial
u\partial\overline{u}})=0.$

This means $d_{Y}(a-\frac{\partial^{2}\phi}{\partial
u\partial\overline{u}})=0$. Therefore

$a=\frac{\partial^{2}\phi}{\partial u\partial\overline{u}}+f(u)$ (35)

where $f$ only depends on $u$, not $Y$. Plugging (30), (34), (35) into the
initial decomposition (26), we find

$\omega=f(u)idud\overline{u}+i\partial\overline{\partial}\phi+\omega_{Y,0}.$
(36)

Apply the “un-conditional” solvability Claim 8.3, we find a function $h$ of
only $u$ such that

$i\partial\overline{\partial}h=f(u)idud\overline{u}.$

Denote $\phi+h$ by $\phi_{0}$, the proof of (25) is complete.

If $\omega$ is real, simply take the real part of $\phi_{0}$ so that (25)
still holds. In the $\sigma-$invariant case, because $\sigma$ is an bi-
holomorphism (on both the fibration and the fiber), and the restriction
$\omega_{Y,0}$ is still $\sigma-$invariant on $Y$, we simply take the average

$\underline{\phi}_{0}\triangleq\frac{\phi_{0}+\sigma^{\star}\phi_{0}+...+(\sigma^{ord_{\sigma}-1})^{\star}\phi_{0}}{ord_{\sigma}}$
(37)

which is $\sigma-$invariant and still solves (25). ∎

##  5 Constraint, iso-trivial ansatz, and proof of main results

###  5.1 Hein and Haskins-Hein-Nordström technique

The version of H-HHN technique here does not immediately work in the
cylindrical case c.f. [52, 51, 49]. It requires the volume growth of the model
metric to be strictly faster than linear (which we do not expect to be
sufficient). See how $\beta>0$ in (44) implies the modified real $(1,1)-$form
$\omega+\eta$ is positive definite when error $<0$.

######  Lemma 5.1.

Let $(X,\omega)$ be a weak $\beta-$ALG Kähler manifold, $R_{3}>0$ be large
enough, and $\Delta^{\star}=\\{|z|>R_{3}\\}=\\{|u|<\frac{1}{R_{3}}\\}$.
Suppose $\Upsilon$ is a (real) volume form such that
$\frac{\Upsilon}{\omega^{d}}>0$ everywhere and $\frac{\Upsilon}{\omega^{d}}-1$
is integrable with respect to $\omega^{d}$. Then there is a positive constant
$t_{0}$ (see (45) below) with the following property. Consider the coordinate
$u=\frac{1}{z}$ near $\infty$ (denote $u=\rho e^{i\theta}$) and let

$\eta\triangleq s\chi(\rho t)\rho d\rho\wedge d\theta=s\chi(\rho
t)\frac{i}{2}du\wedge d\overline{u},$ (38)

where $\chi(y)$ is a standard cutoff function that $\equiv 1$ for $2\leq y\leq
3$, but vanishes if $y\leq 1$ or $y\geq 4$. For any $t>t_{0}$, there is $s$
such that $\omega_{0}\triangleq\omega+\eta$ is Kähler (positive definite) that
satisfies constraint $\int_{X}(\omega^{d}_{0}-\Upsilon)=0$.

Since $\eta$ is closed, $\omega+\eta$ is closed. The parameter $t$ controls
the distance from supp$\eta$ to the origin. However, $s$ is not necessarily
positive. Therefore $\eta$ is not necessarily positive on $\Delta^{\star}$.

###### Proof.

Because $\eta\wedge\eta=0$, constraint (12) for
$\omega_{0}\triangleq\omega+\eta$ is equivalent to the following integral
condition:

$d\int_{X}\eta\wedge\omega^{d-1}=-\int_{X}(\omega^{d}-\Upsilon).\ \ \ $ (39)

Denote the right side of (39) by the real number $Err$. Let the sign of $s$ be
that of $Err$, then $\eta$ descends to the quotient
$\Delta^{\star}/\langle\sigma\rangle$ as it only depends on $\rho$ thus is
invariant under the action of $\sigma$. Let $K\geq 10$ be a quasi-isometry
constant such that

$\frac{(\frac{i}{2}\cdot\frac{dud\overline{u}}{|u|^{2+2\beta}}+\omega_{Y})}{K}\leq\omega\leq
K(\frac{i}{2}\cdot\frac{dud\overline{u}}{|u|^{2+2\beta}}+\omega_{Y}).$ (40)

If $Err=0$, we are done. The point is to find $s$ when $Err\neq 0$.

Case I: Suppose $Err>0$. Integrating the left side of (39) using the explicit
$\eta$ (38) and (40), we find

$\displaystyle\frac{2\pi\cdot
d!sVol(Y)}{ord_{\sigma}K^{d-1}}\int^{\infty}_{0}\chi(\rho t)\cdot\rho d\rho$
(41) $\displaystyle\leq$ $\displaystyle d\int_{X}\eta\wedge\omega^{d-1}=Err$
$\displaystyle\leq$ $\displaystyle\frac{2\pi\cdot
K^{d-1}(d!)sVol(Y)}{ord_{\sigma}}\int^{\infty}_{0}\chi(\rho t)\cdot\rho d\rho$

When $Err$ is negative, the reverse inequality holds. The $d!$ is the product
of the $d$ in the middle line and the denominator of
$\frac{\underline{\omega}^{d-1}_{Y}}{(d-1)!}$ in the volume. The order of the
orbifold group appears because the integral on the orbifold is
$\frac{1}{ord_{\sigma}}$ times the integral on the cover.

Let $t>\frac{10}{R_{3}}$ be large enough such that supp$\chi(\rho
t)\subset\Delta^{\star}$. Using that

$\int^{\infty}_{0}\chi(\rho t)\cdot\rho d\rho=\frac{\Gamma_{\chi}}{t^{2}}$

where $\Gamma_{\chi}=\int^{\infty}_{0}\chi(y)ydy$ only depends on the model
cutoff function, the target integral $d\int_{X}\eta\wedge\omega^{d-1}_{0}$ is
bounded from above and below respectively by

$\frac{2\pi\cdot K^{d-1}(d!)sVol(Y)\Gamma_{\chi}}{ord_{\sigma}\cdot t^{2}}\ \
\textrm{and}\ \ \frac{2\pi\cdot(d!)sVol(Y)\Gamma_{\chi}}{K^{d-1}\cdot
ord_{\sigma}t^{2}}.$

If $s>\frac{K^{d-1}ord_{\sigma}\cdot Err\cdot
t^{2}}{2\pi\cdot(d!)Vol(Y)\Gamma_{\chi}}$, or $s<\frac{ord_{\sigma}\cdot
Err\cdot t^{2}}{K^{d-1}\cdot 2\pi\cdot(d!)Vol(Y)\Gamma_{\chi}}$, the target
integral is either $>Err$ or $<Err$ respectively. Since it is a linear
function in $s$, there exists a unique $s_{0}$ in the indicated range

$(\frac{ord_{\sigma}\cdot Err\cdot t^{2}}{K^{d-1}\cdot
2\pi\cdot(d!)Vol(Y)\Gamma_{\chi}},\ \frac{K^{d-1}ord_{\sigma}\cdot Err\cdot
t^{2}}{2\pi\cdot(d!)Vol(Y)\Gamma_{\chi}}),$ (42)

such that the target integral equals $Err$. $Err>0$ implies positivity of $s$
and non-negativity of $\eta$, therefore also the positivity of $\omega+\eta$.
The point is when $Err<0$, we can still make it Kähler.

Case II. Suppose $Err<0$. With sign changes, similar argument as from (41) to
(42) still produces a $s_{0}$ in the interval

$(-\frac{K^{d-1}ord_{\sigma}\cdot|Err|\cdot
t^{2}}{2\pi\cdot(d!)Vol(Y)\Gamma_{\chi}},\ -\frac{ord_{\sigma}\cdot|Err|\cdot
t^{2}}{K^{d-1}\cdot 2\pi\cdot(d!)Vol(Y)\Gamma_{\chi}}).$ (43)

such that (39) holds for $\eta$ with $s=s_{0}$. On the support of $\chi(\rho
t)$ that lies in the annuli $\\{\rho\in(\frac{1}{t},\frac{4}{t})\\}$ (we
assume $t>\frac{10}{R_{0}}$ in order for $supp\eta\subset\Delta^{\star}$), the
metric $\omega$ satisfies

$\omega\geq\frac{(\frac{i}{2}\frac{dud\overline{u}}{|u|^{2+2\beta}}+\omega_{Y})}{K}\geq\frac{(\frac{i}{4}\frac{dud\overline{u}}{|u|^{2+2\beta}}+\frac{i}{4}dud\overline{u}\cdot\frac{1}{32^{2+2\beta}}t^{2\beta+2}+\omega_{Y})}{K}.$
(44)

We relaxed $\frac{1}{|u|^{2\beta+2}}$ to its lower bound
$\frac{1}{32^{2+2\beta}}t^{2\beta+2}$ on $supp\chi$. That $s_{0}=O(t^{2})$
implies

$\eta=O(t^{2})\cdot\frac{i}{2}du\wedge d\overline{u}$

as the cutoff function has magnitude $\leq 1$. But $\beta>0$ and the factor of
$t^{2\beta+2}$ implies the growth of $\omega$ in $t$ is strictly faster than
$O(t^{2})$. Namely, using that $s_{0}$ lies in the interval (43), when

$t^{2\beta}>\frac{1000\cdot
32^{2+2\beta}K^{d}ord_{\sigma}\cdot|Err|}{2\pi\cdot(d!)Vol(Y)\Gamma_{\chi}},$

we have

$\frac{i}{4}dud\overline{u}\cdot\frac{1}{32^{2+2\beta}}t^{2\beta+2}>\eta.$

Therefore $\omega+\eta$ is still a Kähler form and

$\omega\geq\frac{(\frac{i}{2}\cdot\frac{dud\overline{u}}{|u|^{2+2\beta}}+\omega_{Y})}{2K}$

on $supp\eta$. In summary, no matter what the sign of $Err$ is, let

$t>t_{0}\triangleq\max\\{\frac{10}{R_{0}},[\frac{1000\cdot
32^{2+2\beta}K^{d}ord_{\sigma}\cdot|Err|}{2\pi\cdot(d!)Vol(Y)\Gamma_{\chi}}]^{\frac{1}{2\beta}}\\},$
(45)

$\omega+\eta$ is Kähler. ∎

###  5.2 Hein decay estimate for Tian-Yau solution

An $ALG(\frac{2\pi}{n})-$metric has $(K,2,0)-$polynomial growth [78, Section
1] and positive injective radius due to curvature bound and volume ratio lower
bound for all balls with radius $<1$ [44, 18] etc. The bound on covariant
derivatives of curvatures also implies the quasi-finite geometry of order
$2+\frac{1}{2}$ [78, Proposition 1.2]. The general existence [78, Theorem 1.1]
provides a smooth solution $\phi$ to the target Monge-Ampere equation (10)
such that $\omega_{\phi}$ is quasi isometric to the reference metric
$\omega_{0}$ i.e. there is a positive constant $C$ such that

$\frac{\omega_{0}}{C}\leq\omega_{0}+i\partial\overline{\partial}\phi\triangleq\omega_{\phi}\leq
C\omega_{0}.$ (46)

The target equation (10) can be written as

$\frac{(\omega_{0}+i\partial\overline{\partial}\phi)^{n}-\omega_{0}^{n}}{\omega_{0}^{n}}=e^{f}-1.$
(47)

We further write it as a second order elliptic equation with co-efficients
depending on the unknown, as [78, 79, 49, 59] etc.

$\frac{i\partial\overline{\partial}\phi\wedge[\omega^{n-1}_{\phi}+...+\omega_{0}^{j}\wedge\omega_{\phi}^{n-1-j}+...+\omega_{0}^{n-1}]}{\omega_{0}^{n}}=e^{f}-1.$
(48)

See related techniques in [78, Theorem 1.1], [51], [59], and [49].

In the energy decay (80), abusing notation, we still denote
$\min\\{\beta_{0},\frac{1}{1000}\\}$ by $\beta_{0}$. Under the poly-cylinder
that utilizes the Tian-Yau atlas on the fiber direction $Y$, the
$i\partial\overline{\partial}\phi$ below is viewed as a matrix under the
chart. This norm is equivalent to the one viewing it as a tensor with
derivative being co-variant derivatives, by the bounds on the pullback metric
under Tian-Yau coordinates. See Appendix 8.3 for a more comprehensive review
and the standard norms.

######  Lemma 5.2.

(Hein [51])
$|i\partial\overline{\partial}\phi|_{C^{k}[A(r,\infty)]}\leq\frac{C_{\beta_{0},k}}{r^{\frac{\beta_{0}}{2}}}$.

###### Proof.

This is direct from Hein Dirichlet energy decay (80).

Neumann Poincare inequality on $B_{euc,x}(\mu_{2})$ implies

$\int_{B_{euc,x}(\frac{\mu_{2}}{2})}|\phi-c_{\phi}|^{2}\leq
C\int_{B_{euc,x}(\mu_{2})}|\nabla\phi|^{2}\leq C_{\beta_{0}}r^{-\beta_{0}},$
(49)

where $c_{\phi}$ is a constant depending on $\phi$. Tian-Yau higher-order
bound [78, Theorem 1.1, last sentence] implies the co-efficients of (48), as a
second order elliptic equation in $\phi$, are bounded in
$C^{1}[B_{euc,x}(\frac{\mu_{2}}{4})]$ in terms of the Ricci potential/volume
density $f$, and the ansatz $\omega_{0}$. Then interior Schauder estimate for
linear equation and asymptotics of $e^{f}-1$ says

$|\phi-c_{\phi}|_{C^{2,\frac{1}{2}}[B_{euc,x}(\frac{\mu_{2}}{8})]}\leq
C|e^{f}-1|_{C^{\frac{1}{2}}[B_{euc,x}(\frac{\mu_{2}}{4})]}+C|\phi-
c_{\phi}|_{L^{2}[B_{euc,x}(\frac{\mu_{2}}{4})]}\leq\frac{C_{\beta_{0}}}{r_{x}^{\frac{\beta_{0}}{2}}}.$

Lemma 5.2 is proved when $k=2$. By the Schauder bounds on the metric under
quasi-atlas, differentiating the equation, we find higher order bounds.

$|\phi-
c_{\phi}|_{C^{k+2,\frac{1}{2}}[B_{euc,x}(\frac{\mu_{2}}{10})]}\leq\frac{C_{\beta_{0},k}}{r_{x}^{\frac{\beta_{0}}{2}}}$
(50)

This particularly implies
$|i\partial\overline{\partial}\phi|_{C^{k}[A(r,\infty)]}\leq\frac{C_{\beta_{0},k}}{r^{\frac{\beta_{0}}{2}}}$.
∎

####  5.2.1 Finishing the proof of main results

###### Proof of Theorem 1.2, Corollary 1.4, and existence part of Corollary
1.5:.

.

Step 1: ALG metric that is equal to a (Ricci flat) ALG model outside a compact
set.

The assumed Kähler metric yields an iso-trivial ALG metric $\omega_{isotr}$.
Namely, the difference of the expression (25) for $\omega$ and the ALG model
(6) yields

$\displaystyle\omega-\frac{i}{2}dzd\overline{z}-\omega_{RF,Y}=i\partial\overline{\partial}\phi_{0}-\frac{i}{2}dzd\overline{z}+\omega_{Y}-\omega_{RF,Y}$
(51) $\displaystyle=$ $\displaystyle
i\partial\overline{\partial}(\phi_{0}-\phi_{RF,Y}-\frac{|z|^{2}}{2}).$

where $\omega_{RF,Y}$ is the unique Ricci flat Kähler metric cohomologous to
$\omega_{Y}$, and $\phi_{RF,Y}$ is the Ricci flat potential:

$\omega_{RF,Y}-\omega_{Y}=i\partial\overline{\partial}\phi_{RF,Y}.$ (52)

Here we applied Lemma 4.1 with the coordinate change $\rho=\frac{1}{r},\
u=\frac{1}{z}$ near $z=\infty$.

Gluing Lemma 8.2 says that when $t$ is large enough,

$\omega_{isotr}\triangleq\omega+i\partial\overline{\partial}\chi(-\phi_{0}+\phi_{RF,Y}+\frac{|z|^{2}}{2})+tb_{0}.$

is the desired iso-trivial Kähler metric, where $\chi$ is the cutoff function
supported on the end and vanishes in a compact set containing the exceptional
divisor.

Step 2: Ansatz with constraint.

The desired ansatz $\omega_{0}$ is obtained applying Lemma 5.1 to
$\omega_{isotr}$ i.e.

$\omega_{0}=\omega_{isotr}+\eta=\omega+i\partial\overline{\partial}\chi(\phi_{0}-\phi_{RF,Y}+\frac{|z|^{2}}{2})+tb_{0}+\eta.$
(53)

Since the difference between $\omega_{0}$ and $\omega_{isotr}$ is compactly
supported, $\omega_{0}$ is iso-trivial as well. Since $f$ is compactly
supported, it meets the decay condition required in Tian-Yau solvability [78,
Thm 1.1].

Step 3: Tian-Yau solution, Hein decay, and Schwartz decay.

Apply Tian-Yau solvability to the isotrivial ansatz $\omega_{0}$ with
constraint, we find the solution. We can write down the difference between the
ALG Ricci flat metric $\omega_{RFALG}$ with the initial Kähler metric $\omega$
more explicitly. Applying the radial solvability in Claim 8.3 to the two
radial compact supported bump forms in (1), we find smooth radial functions
$\phi_{rb,g}$ and $\phi_{rb,c}$ (potentials of radial bump forms for gluing
and constraint respectively) such that

$tb_{0}=i\partial\overline{\partial}\phi_{rb,g},\ \
\eta=i\partial\overline{\partial}\phi_{rb,c}.$

Then

$\omega_{RFALG}-\omega=i\partial\overline{\partial}[\chi(\phi_{0}-\phi_{RF,Y}+\frac{|z|^{2}}{2})+\phi_{rb,g}+\phi_{rb,c}+\phi_{TY}],$
(54)

and the ALG Ricci flat potential $\phi_{RFALG}$ (1) equals

$\chi(\phi_{0}-\phi_{RF,Y}+\frac{|z|^{2}}{2})+\phi_{rb,g}+\phi_{rb,c}+\phi_{TY}.$
(55)

Except the Tian-Yau potential $\phi_{TY}$ [78, Theorem 1.1] relative to
$\omega_{0}$, the other potential functions (55) relative to the initial
Kähler metric $\omega$ in Theorem 1.2 are supported on the end i.e. they
vanish in a compact set containing the exceptional divisor. Particularly, they
are smooth functions on the crepant resolution $X$. Since the resolution map
is not necessarily a smooth orbifold map, the pullback of a smooth function
from the orbifold to the resolution is not known by us to still be smooth, if
it does not vanish near the orbifold locus $O\times Fix(\sigma)$.

Hein decay (Lemma 5.2) fulfills the condition in Schwartz decay Proposition
1.6. Therefore the metric has Schwartz decay toward the ALG model. Proof for
existence is complete.

Step 4: Monge-Ampere partial uniqueness.

By metric equivalence between the ansatz $\omega_{0}$ and $\omega_{RFALG}$,
the Poincare inequality Lemma 8.1 for ALG model yields the weak Poincaré
inequality for $\omega_{RFALG}$. This fulfills the requirement for uniqueness
of bounded $C^{1,1}-$solutions [83] that implies the desired uniqueness here.

Step 5: Examples of $ALG_{\infty}$ Ricci flat Kähler $3-$folds via $K3$ with
purely non-symplectic automorphism of finite order.

Because all $K3^{\prime}$s with a non-symplectic automorphism $\sigma$ of
finite order is polarized i.e. admits positive line bundle $L_{K3}$ (for
example, see [5]). The “average”

$L_{0}\triangleq
L_{K3}\otimes\sigma^{\star}L_{K3}\otimes...\otimes(\sigma^{ord_{\sigma}-1})^{\star}L_{K3}$

is a $\sigma-$invariant positive line bundle on the $K3$. Denote the
$\sigma-$invariant positive curvature form of $L_{0}$ by $\omega_{K3,\sigma}$.
The ALG model

$\frac{i}{2}\lambda^{2}dzd\overline{z}+\omega_{K3,\sigma}$

is a Kähler orbifold metric on the $3-$orbifold

$\frac{\mathbb{C}\times K3}{\langle\sigma\rangle}$

whose orbifold groups are obviously in $SL(3,\mathbb{C})$ by our construction.
Then discussions [55, Page 137 above 6.6.1] say there is a Kähler crepant
resolution. The point (of Lemma 4.1) is that we only need a Kähler metric at
this step, and no completeness is required.

For the examples in Corollary 1.5, the Kähler metric on $X$ can be taken as
the Bergman metric (68) below.

By [62, Main Theorem 3, Corollary 5, Table 1], an integer among the list (1.4)
is the order of an purely non-symplectic automorphism on a $K3$. Corollary 1.4
and existence part of Corollary 1.5 are proved. ∎

##  6 Bergman metric

###  6.1  $A_{1}-$singularity and the crepant resolution

The action of $-Id$ on $\mathbb{C}^{2}$ defines the $A_{1}-$singularity
$\frac{\mathbb{C}^{2}}{\langle\pm Id\rangle}$. This orbifold is isomorphic to
the affine variety

$V=\\{(u_{0},u_{1},u_{2})|u_{0}u_{2}=u^{2}_{1}\subset\mathbb{C}^{3}\\}$

via the map

$u_{0}=x^{2},\ u_{1}=xy,\ u_{2}=y^{2}.$ (56)

We define two copies of $\mathbb{C}^{2}$ and denote them by
$\mathbb{C}_{0}^{2}$ and $\mathbb{C}_{1}^{2}$, with coordinates
$(\lambda_{0},\mu_{2})$ and $(\lambda_{1},\mu_{1})$ respectively. The crepant
resolution of $A_{1}-$singularity is the gluing of them by the map

$\lambda_{1}=\frac{1}{\mu_{0}},\ \mu_{1}=\lambda_{0}\mu^{2}_{0}.$ (57)

  * •

The map

$u_{0}=\lambda_{0},\ u_{1}=\lambda_{0}\mu_{0},\ u_{2}=\lambda_{0}\mu^{2}_{0}.$

is an isomorphism
$\mathbb{C}_{0}^{2}\setminus\\{\lambda_{0}=0\\}\longrightarrow
V\setminus\\{x=0\\}$. It maps the whole $\mu_{0}-$axis to the origin. The
inverse when $x\neq 0$ is

$\lambda_{0}=x^{2},\ \ \mu_{0}=\frac{y}{x}.$ (58)

  * •

The map

$u_{0}=\lambda^{2}_{1}\mu_{1},\ u_{1}=\lambda_{1}\mu_{1},\ u_{2}=\mu_{1}.$

is an isomorphism $\mathbb{C}_{0}^{2}\setminus\\{\mu_{1}=0\\}\longrightarrow
V\setminus\\{y=0\\}$. It maps the whole $\lambda_{1}-$axis to the origin. The
inverse when $y\neq 0$ is

$\lambda_{1}=\frac{x}{y},\ \ \mu_{1}=y^{2}.$ (59)

The $\mu_{0}-$axis in $\mathbb{C}_{0}^{2}$ is glued to the $\lambda_{1}-$axis
in $\mathbb{C}_{1}^{2}$ along the non-zero part via the transition map
$\lambda_{1}=\frac{1}{\mu_{0}}$, in conjunction with (57). This is the
exceptional (smooth) $\mathbb{P}^{1}$ which is a $(-2)-$curve.

$\lambda_{0}$$\mu_{0}$$\lambda_{1}$$\mu_{1}$ Figure 3: Manifold structure for
crepant resolution of $A_{1}-$singularity

The isomorphism from the resolution
$\widetilde{\frac{\mathbb{C}^{2}}{\langle\sigma\rangle}}=\mathbb{C}_{0}^{2}\underline{\cup}\mathbb{C}_{1}^{2}$
to the total space $O(-2)\rightarrow\mathbb{P}^{1}$ is :

$\left\\{\begin{array}[]{c}(\lambda_{0},\mu_{0})\rightarrow\lambda_{0}(1,\mu_{0})^{\otimes
2}\\\ (\lambda_{1},\mu_{1})\rightarrow\mu_{1}(\lambda_{1},1)^{\otimes
2}.\end{array}\right.$

We define the $0$ in $\lambda_{1}-$axis in $\mathbb{C}_{1}^{2}$ as the “$0$”
in the exceptional divisor, and the $0$ in $\mu_{0}-$axis as the “$\infty$”.
See [66] for more explicit information on Du Val singularities.

###  6.2 Nikulin poly-cylinder

At a fixed point $p$ of the Nikulin involution on a $K3$, we call a coordinate
neighborhood $\\{U_{p},(y,v)\\}$ a Nikulin coordinate at $p$, if $p$ maps to
$(0,0)$ in the chart, and the Nikulin-involution $\sigma$ is diagonal therein:

$\sigma\cdot(y,v)=(y,v)\cdot\left|\begin{array}[]{cc}-1&\\\ &1\\\
\end{array}\right|.$ (60)

Such a coordinate always exists [63]. On the other hand, the fixed points of
$\sigma$ form a disjoint union of holomorphic curves. Let $\mathcal{C}_{i}$
denote a component of the fixed locus $\mathcal{C}$ of $\sigma$ i.e. a
(connected) curve point-wisely fixed by $\sigma$. In a Nikulin coordinate at a
$p\in\mathcal{C}$, the curve is defined by $y=0$. Consider a poly-cylinder in
$y$ and $v$ of equal radius (contained in the coordinate chart) with closure
in a Nikulin coordinate neighborhood:

$PC_{p}(r)\triangleq D_{y}(r)\times D_{v}(r),$

where $D_{y}(r)$ is open disk of radius $r$ in the complex plane with
coordinate $y$ centered at $y=0$, and $D_{v}(r)$ is likewise. We call it a
Nikulin poly-cylinder. The transition function between two Nikulin poly-
cylinders at a point $p$ has a Taylor expansion near the origin (that
corresponds to $p$):

$\widehat{y}=a_{1}(v)y+a_{3}(v)y^{3}+a_{5}(v)y^{5}...$ (61)

The reason why only odd degree terms in $y$ appear is the invariance

$\widehat{y}(-y,v)=-\widehat{y}(y,v).$

Non-vanishing of Jacobian of the transition function says

$a_{1}(v)=\lim_{y\rightarrow
0}\frac{\widehat{y}}{y}(y,v)=\frac{\partial\widehat{y}}{\partial y}(0,v)\neq
0$

for any $v$ on the overlap on $\mathcal{C}_{i}$. It is the transition function
for the the co-normal bundle $N^{\star}_{\mathcal{C}_{i}}$ of
$\mathcal{C}_{i}$. Let $Exc$ denote the dis-joint union $\cup_{i}Exc_{i}$,
where $Exc_{i}\triangleq\pi^{-1}C_{i}$.

###  6.3 Resolution along fixed curves and Nikulin-Reid poly-cylinders

We simply do product resolution which is independent of coordinate. $x$ still
parametrizes $\mathbb{C}$. Abusing terms, we also call

$D_{x}(r)\times D_{y}(r)\times D_{v}(r)$ (62)

and the quotient

$\widetilde{\frac{D_{x}(r)\times D_{y}(r)}{\langle\sigma\rangle}}\times
D_{v}(r)\triangleq PC_{resol}$

Nikulin poly-cylinders. According to the inverse resolution morphisms (58),
(59) and the transition, Corresponding to the manifold structure of crepant
resolution of $\frac{\mathbb{C}^{2}}{\\{\pm Id\\}}$, we have the decomposition

$PC_{resol}=PC_{resol,0}\cup PC_{resol,1}.$

where $PC_{resol,i}$ are parametrized by $\lambda_{i},\mu_{i}$ and correspond
to the two open sets in (6.3), (59) are called Nikulin-Reid poly-cylinders.
Unlike (62), they are not of equal radius.

Let $(x,y,v)$ and $(x,\widehat{y},\widehat{v})$ be two Nikulin poly-cylinders.
On the overlap, the transition map lifts to the resolution because
$\frac{\widehat{y}}{y}$ extends holomorphically in
$y^{2}=\lambda_{1}\lambda_{0}=\mu_{0}^{2}$ and $v$. Moreover,

$\widehat{\lambda}_{0}=\lambda_{0}=x^{2},\ \ \ \
\widehat{\mu}_{0}=\mu_{0}\cdot\frac{\widehat{y}}{y},$
$\widehat{\lambda}_{1}=\lambda_{1}(\frac{\widehat{y}}{y})^{-1},\ \ \ \
\widehat{\mu}_{1}=\mu_{1}\cdot(\frac{\widehat{y}}{y})^{2}.$

When $y^{2}=0$, the transition map between Nikulin-Reid poly-cylinders becomes
the fiberwise linear map:

$\widehat{\lambda}_{0}=\lambda_{0},\ \ \ \ \widehat{\mu}_{0}=\mu_{0}\cdot
a_{1},\ \ \ \widehat{\lambda}_{1}=\frac{\lambda_{1}}{a_{1}(v)},\ \ \ \
\widehat{\mu}_{1}=\mu_{1}\cdot a^{2}_{1}.$

Therefore, the map

$\lambda_{1}\longrightarrow[1\oplus\lambda_{1}dy],\ \ \
\mu_{0}\longrightarrow[\mu_{0}\oplus dy]$

is independent of coordinate, and yields an isomorphism

$Exc_{i}\longrightarrow\mathbb{P}[O_{\mathcal{C}_{i}}\oplus
N^{\star}_{\mathcal{C}_{i}}]=\mathbb{P}[N_{\mathcal{C}_{i}}\oplus
O_{\mathcal{C}_{i}}]=\mathbb{P}[\textrm{Normal Bundle of}\ \mathcal{C}_{i}].$

Adjunction says

$N_{\mathcal{C}_{i}}=[\mathcal{C}_{i}]|_{\mathcal{C}_{i}}=K_{\mathcal{C}_{i}},\
\ \
N^{\star}_{\mathcal{C}_{i}}=K^{\star}_{\mathcal{C}_{i}}=T^{1,0}_{\mathcal{C}_{i}}\
(\textrm{holomorphic tangent bundle of}\ \mathcal{C}_{i}).$

###  6.4 Projectivity

Let $Bl^{O\times Fix(\sigma)}(\mathbb{P}^{1}\times K3)$ denote the blow up of
$\mathbb{P}^{1}\times K3$ along the curve $O\times Fix(\sigma)$. It is still a
smooth projective variety. See [73, Remark 1, 6.2.2], [48, II. Proposition
7.16] for related discussions. Then there is a positive line bundle
$L_{0}\rightarrow Bl^{O\times Fix(\sigma)}(\mathbb{P}^{1}\times K3).$ By the
very ampleness discussions in [48, II Propsition 7.10 (b), Remark 5.16.1], and
that the invertible sheaf “$O_{P}(1)$” in [48, II Propsition 7.10 (b)] is of
the divisor $-Exc$ (see [81, 4]), we can simply take $L_{0}=[-Exc]\otimes
m\pi^{\star}L_{K3}$ for sufficiently large positive $m$. Hence
$L\triangleq\sigma^{\star}L_{0}\otimes L_{0}$ is a $\sigma-$invariant positive
line bundle. The quotient

$\frac{Bl^{O\times Fix(\sigma)}(\mathbb{P}^{1}\times K3)}{\langle\pm
Id\rangle}\triangleq\overline{X}$ (63)

is a complex orbifold, and is also a projective variety by Kodaira-Baily
embedding theorem [6]. By coordinates for blow ups of a smooth complex
manifold [46, 4.6], and of the orbifold with $A_{1}-$singularities in Section
6.1, it is routine to check

$\frac{Bl^{O\times Fix(\sigma)}(\mathbb{C}\times K3)}{\langle\pm
Id\rangle}=\widetilde{\frac{\mathbb{C}\times K3}{\langle\pm Id\rangle}}=X.$
(64)

i.e. the crepant resolution of the $Z_{2}-$quotient is isomorphic to the
$Z_{2}-$quotient of the blow up. Namely, similar to (58) its subsequent
material, let $x,\ y,\ v$ be Nikulin coordinate near a point on $O\times
Fix(\sigma)$, we have the following coordinates $z_{i},w_{i}$ near the
exceptional divisor of the blow up of $\mathbb{P}^{1}\times K3$:

$z_{0}=x,\ w_{0}=\frac{y}{x}\ \textrm{when}\ x\neq 0,\ \textrm{and}\
z_{1}=\frac{x}{y},\ w_{1}=y,\ \textrm{when}\ y\neq 0.$

Moreover, we have the transition on the overlap of Nikulin- coordinates:

$\widehat{z}_{0}=z_{0}=x,\ \ \ \
\widehat{w}_{0}=w_{0}\cdot\frac{\widehat{y}}{y}\ ;\
\widehat{z}_{1}=z_{1}(\frac{\widehat{y}}{y})^{-1},\ \
\widehat{w}_{1}=w_{1}\cdot\frac{\widehat{y}}{y}\ .$ (65)

The action of $\sigma$ on $\mathbb{P}^{1}\times K3$, locally near a point on
$O\times Fix(\sigma)$ given by

$\sigma(x,y,v)=(-x,-y,v),$

obviously lift to the blow-up by

$\sigma(z_{0},w_{0},v)=(-z_{0},w_{0},v),\
\sigma(z_{1},w_{1},v)=(z_{1},-w_{1},v)$ (66)

and is independent of the coordinate chosen, as when we replace $x,\ y,\
\widehat{x},\ \widehat{y}$ by $-1$ of themselves in (65) and (66), the
negative signs cancel out. The morphism

$Bl^{O\times Fix(\sigma)}(\mathbb{P}^{1}\times
K3):\longrightarrow\widetilde{\frac{\mathbb{P}^{1}\times K3}{\langle\pm
Id\rangle}},$ (67)

locally near the exceptional divisor, is defined by

$\lambda_{0}=z^{2}_{0},\ \mu_{0}=w_{0},\ \lambda_{1}=z_{1},\
\mu_{1}=w^{2}_{1}.$

######  Remark 6.1.

Fu-Namikawa [40] showed existence of lifting for surface singularities. By
$d\lambda_{0}=z_{0}dz_{0}$, we find $d\lambda_{0}\wedge
d\overline{\lambda}_{0}=|z_{0}|^{2}dz_{0}\wedge d\overline{z}_{0}$. Since
$\lambda_{0}$ is the defining holomorphic/regular function of the exceptional
divisor in Nikulin coordinate, the pullback of the ALG Ricci flat metric from
$X$ to $Bl^{O\times Fix(\sigma)}(\mathbb{C}\times K3)$ is still a Ricci flat
Kähler metric with an ALG end and conic singularity along the exceptional
divisor of angle $4\pi$.

The Bergman metric is written as

$\omega_{Bergman}=i\partial\overline{\partial}\log(|s_{0}|^{2}+...+|s_{n}|^{2}).$
(68)

The biregular map $\Phi$ produced by Baily [6, Global imbedding theorem]
locally generates the whole stalk of invariant holomorphic functions [6, below
local imbedding theorem]. Restricted to $X$, it is a genuine complex manifold
embedding.

##  7 Topology and table of examples

###  7.1 Fundamental group

The global quotient $\frac{\mathbb{\mathbb{C}\times
Y}}{\langle\sigma\rangle}$, denoted by $Orb$, is normal. The result of Kollar
[57, Theorem 7.8] says the induced homorphism

$p_{\star}:\ \pi_{1}[X]\rightarrow\pi_{1}[Orb].$

between the fundamental groups is an isomorphism. The radial retraction of the
$\mathbb{C}-$component to the origin is invariant under the group action, thus
it induces an isomorphism

$\pi_{1}[X]\rightarrow\pi_{1}[Y/\langle\sigma\rangle].$

######  Lemma 7.1.

Let $Y$ be a smooth compact path connected manifold. Let $\sigma$ be
diffeomorphism of $Y$ with prime order i.e. there is a prime number $p>1$ such
that $\sigma^{p}=id$, but $\sigma^{k}\neq id$ when $1<k<p$. Suppose $\sigma$
has at least one fixed point $x_{0}$. Then the induced homorphism
$\pi_{1}(Y)\longrightarrow\pi_{1}(\frac{Y}{\langle\sigma\rangle})$ is
surjective.

Because $Y$ is simply connected and compact, Lemma 7.1 implies both the
orbifold and the resolution $X$ are simply-connected.

###### Proof of Lemma 7.1:.

The prime order condition implies fixed locus of a non-zero power $\sigma^{k}$
is the same as fixed locus of $\sigma$. This is straight-forward: there are
integers $a,b$ such that $ak-bp=1$, therefore

$\sigma^{k}(x)=x\ \textrm{if and only if}\ \sigma(x)=\sigma^{ak-bp}(x)=x.$

Thus

$Fix(\sigma)=Fix(\langle\sigma\rangle)\triangleq\cup_{1\leq k\leq
ord_{\sigma}-1}Fix(\sigma^{k})=Fix(\sigma^{j})\ \textrm{for any}\ 1\leq j\leq
ord_{\sigma}-1,$ (69)

Henceforth we denote it by the first.

The observation is that a loop

$\gamma:\ [0,1]\longrightarrow\frac{Y}{\langle\sigma\rangle}$

based at the image

$[x_{0}]\in\frac{Y}{\langle\sigma\rangle}$

of the fixed point can be lifted to a loop in $Y$ based at $x_{0}$. Namely,
$x_{0}$ is the only pre-image of $[x_{0}]\in\frac{Y}{\langle\sigma\rangle}$
under the quotient $\rho:\ Y\longrightarrow\frac{Y}{\langle\sigma\rangle}$. As
the group $\langle\sigma\rangle$ is cyclic and finite, the restricted map away
from the fixed locus:
$Y-Fix\sigma\longrightarrow\frac{Y-Fix(\sigma)}{\langle\sigma\rangle}$ is a
covering. There is a closed subset $S\subset[0,1]$ such that

  * •

$\gamma|_{S}$ lies on $\rho[Fix(\sigma)]$,

  * •

$\gamma|_{[0,1]\setminus S}$ lies on
$\frac{Y-Fix(\sigma)}{\langle\sigma\rangle}$.

Any open (closed) subset in $[0,1]\subset\mathbb{R}$ is a countable union of
bounded open (closed) intervals in $[0,1]$. Since
$\gamma(0)=\gamma(1)=x_{0}\in S$, we denote

$[0,1]\setminus S=\cup_{k=1}^{\infty}(a_{k},b_{k}).$

By (69), $\gamma|_{S}$ automatically lifts by the identity homeomorphism

$\pi:Fix(\sigma)\longrightarrow Fix(\langle\sigma\rangle)$

to $Y$. For each $k$, denote a lift (by property of covering space) of

$\gamma:\ (a_{k},b_{k})\longrightarrow\frac{Y}{\langle\sigma\rangle}$

by $\gamma_{Y}(a_{k},b_{k})$. The point is to show it “matches” the trivial
lift $\gamma|_{S}$ at the end points.

Equip $Y$ with a $\sigma-$invariant Riemannian metric, by the average
$\frac{1}{p}\Sigma_{i=0}^{p-1}(\sigma^{i})^{\star}g_{0}$, where $g_{0}$ is a
smooth metric on $Y$, and $m$ is the order of $\sigma$. Because $Y$ is compact
and $\sigma$ is continuous, for any sequence $t_{j}\rightarrow a^{+}_{k}$,
$\gamma_{Y}(t_{j})$ sub-converges to point $x_{k,left}$ whose image in
$\frac{Y}{\langle\sigma\rangle}$ is $\gamma_{a_{k}}$. Thus
$x_{k,left}=\gamma_{a_{k}}$ as fixed point in $Y$. Similarly, for any sequence
$t_{j}\rightarrow b^{-}_{k}$, $\gamma_{Y}(t_{j})$ sub-converges to point
$x_{k,right}$ whose image in $\frac{Y}{\langle\sigma\rangle}$ is
$\gamma_{b_{k}}$. Thus $x_{k,right}=\gamma_{b_{k}}$ as fixed point in $Y$.
Hence the limits exist and is correct. This means the lift $\gamma_{Y}(t)$ is
continuous at the end points. ∎

###  7.2 Betti numbers

######  Proposition 7.2.

As in Corollary 1.5, let $(K3,\sigma)$ be a $K3-$surface with a non-symplectic
Nikulin involution $\sigma$. Then the crepant resolution
$X\triangleq\widetilde{\frac{\mathbb{C}\times K3}{\langle\sigma\rangle}}$
(which is simply-connected) has Betti number

$b^{\star}(X)=b^{\star}(\frac{\mathbb{C}\times
K3}{\langle\sigma\rangle})+b^{\star}(Exc)-b^{\star}(\mathcal{C}).$ (70)

Consequently,

$\displaystyle b^{2}=rkNS(\sigma)+\textrm{Number of all fixed curves},$ (71)
$\displaystyle b^{3}=2\Sigma_{\textrm{all fixed curves}}g_{\mathcal{C}},$ (72)
$\displaystyle b^{4}=1+\textrm{Number of all fixed curves}.$ (73)

Moreover, $b^{i}=0$ when $i\geq 5$.

###### Proof.

By [46, 4.6 blow up sub-manifolds] and [39, 4.1], we find the following
decomposition for $\mathbb{R}-$co-efficient singular cohomology

$H^{i}[Bl_{\mathcal{C}}(\mathbb{C}\times K3)]=H^{i}[\mathbb{C}\times
K3]\oplus\frac{H^{i}[Exc]}{I\cdot H^{i}[\mathcal{C}]},$ (74)

where $I$ is an injection. De Rham isomorphism says (74) is also true for De
Rham cohomology. On the $Z_{2}-$quotient $X$, by Satake isomorphism between
invariant De Rham cohomology and singular cohomology i.e.

$H_{DR_{\sigma}}^{\star}(\frac{\mathbb{C}\times
K3}{\langle\sigma\rangle})=H_{Sing}^{\star}(\frac{\mathbb{C}\times
K3}{\langle\sigma\rangle})\ \ (\textrm{see}\
\cite[cite]{[\@@bibref{}{Satake,Clader,ALR}{}{}]}).$

Because $Exc$ and $\mathcal{C}$ are point-wisely fixed by $\sigma$, the De
Rham cohomologies of $H^{i}[X]$ and $H^{i}[\frac{\mathbb{C}\times
K3}{\langle\sigma\rangle}]$ are $\sigma-$invariant parts of
$H^{i}[Bl_{\mathcal{C}}(\mathbb{C}\times K3)]$ and $H^{i}[\mathbb{C}\times
K3]$, respectively. We find

$H^{i}[X]=H^{i}[\frac{\mathbb{C}\times
K3}{\langle\sigma\rangle}]\oplus\frac{H^{i}[Exc]}{I\cdot H^{i}[\mathcal{C}]}.$
(75)

It suffices to calculate $H^{\star}[\frac{\mathbb{C}\times
K3}{\langle\sigma\rangle}]$ and $H^{\star}(Exc)$.

  * •

For $H^{\star}[\frac{\mathbb{C}\times K3}{\langle\sigma\rangle}]$, the obvious
radial retraction of $\mathbb{C}-$component says

$H^{i}[\frac{\mathbb{C}\times
K3}{\langle\sigma\rangle}]=H^{i}[\frac{K3}{\langle\sigma\rangle}].$

$H^{2}[\frac{K3}{\langle\sigma\rangle}]$ is simply the rank of
$\sigma-$invariant Néron-Severi lattice [5, Theorem 2.1] i.e. an integer
cohomology class is $\sigma-$invariant only if it in Picard lattice/Néron-
Severi lattice. Satake isomorphism again says that except
$H^{0}[\frac{K3}{\langle\sigma\rangle}]$ and
$H^{4}[\frac{K3}{\langle\sigma\rangle}]$ which are both $1-$dimensional, the
other cohomologies vanish.

  * •

For $H^{\star}(Exc)$, recall that Leray-Hirsch theorem [15] says

$H^{\star}(Exc)=H^{\star}(\mathcal{C})\otimes H^{\star}(\mathbb{P}^{1})$

as in Kunneth-formula for trivial fibration, because the Euler class of the
universal bundle generates $H^{2}$ of each $\mathbb{P}^{1}-$fiber.

With (75), the above calculation for
$H^{\star}[\frac{K3}{\langle\sigma\rangle}]$ and $H^{\star}(Exc)$ proves
formulas (71)–(73). The reason why there is factor of “$2$” in (72) is that
$b^{1}$ of a curve is twice of the genus. ∎

###  7.3 Table of examples and proof of Corollary 1.5

By Proposition 7.2 and lattice data of $K3-$surfaces with a non-symplectic
Nikulin involution [64, 63, 65], we calculate the Betti numbers, and count the
number of parameters for $ALG_{\infty}(\pi)$ Ricci flat manifolds.

In Table 1 – 3 below, the column “$rkNS(\sigma)$” is the rank of the
$\sigma-$invariant sub-lattice in Néron-Severi lattice. The column “largest
genus $g$” is the largest genus of a fixed curve. The column “number of
rational curves” is the number of fixed curves counted without the largest
genus curve, which must be rational (this is not the case for the exceptional
examples from Table 3).

In conjunction with Betti number Proposition 7.2, the column “$b^{2}_{local}$”
means the local contribution from the resolution to the $(1,1)-$Hodge number
which is the same as the $b^{2}$ of the resolution. It equals number of all
fixed curves (connected components of the fixed locus). The $b^{2},\ b^{3},\
b^{4}$ mean the Betti numbers $\widetilde{\frac{\mathbb{C}\times
K3}{\langle\sigma\rangle}}$, abbreviated to $X$. The column “Dim
$\mathcal{M}_{\sigma}$” means dimension of moduli of $K3$ surfaces that with
the data in the underlying row. It equals $20-rkNS(\sigma)$. The column “ALGs”
is the number of parameters of the ALG Ricci flat metrics family obtained,
which equals $b^{2}+Dim\mathcal{M}_{\sigma}$+1. The “1” means the scaling
factor $\lambda$ (6).

  * •

The component of the exceptional divisor, corresponding to the fixed
(connected) curve of genus $g$ denoted by $\mathcal{C}(g)$ in the first
column, is the projective bundle $\mathbb{P}[O_{\mathcal{C}(g)}\oplus
T^{1,0}_{\mathcal{C}(g)}]$.

  * •

The component of the exceptional divisor, corresponding to the rational curves
in column 2, are disjoint unions of Hirzebruch surfaces $\Sigma_{2}$.

  * •

For the exceptional examples in Table 3 below, the exceptional divisor is the
disjoint union of the trivial $\mathbb{P}^{1}-$fibrations
$E_{1}\times\mathbb{P}^{1}$ and $E_{2}\times\mathbb{P}^{1}$, where $E_{i}$ are
the fixed elliptic curves.

  * •

In all 3 tables, the compactifying divisor at $\infty$ is $\frac{\textrm{the
K3}}{\langle\sigma\rangle}$, which is always a smooth rational surface.

The calculations are computer aided. The sum of the last columns of all tables
below is 1638. All triples of Betti numbers realized by Table 1 and 2 are
distinct. There are 64 rows in total in Table 1 (36 rows) and 2 (28 rows).
Table 3 does not give new triple of Betti number as the Betti numbers coincide
with one row in Table 1. The counting for Corollary 1.2 is complete.

g |  Number of rational Curves |  $rkNS(\sigma)$ |  $b^{2}_{local}$ |  $b^{2}$ |  $b^{3}$ |  $b^{4}$ |  Dim $\mathcal{M}_{\sigma}$ |  Dim ALG family
---|---|---|---|---|---|---|---|---
10 |  0 |  1 |  1 |  2 |  20 |  2 |  19 |  22
10 |  1 |  2 |  2 |  4 |  20 |  3 |  18 |  23
9 |  0 |  2 |  1 |  3 |  18 |  2 |  18 |  22
8 |  0 |  3 |  1 |  4 |  16 |  2 |  17 |  22
9 |  1 |  3 |  2 |  5 |  18 |  3 |  17 |  23
7 |  0 |  4 |  1 |  5 |  14 |  2 |  16 |  22
8 |  1 |  4 |  2 |  6 |  16 |  3 |  16 |  23
6 |  0 |  5 |  1 |  6 |  12 |  2 |  15 |  22
7 |  1 |  5 |  2 |  7 |  14 |  3 |  15 |  23
5 |  0 |  6 |  1 |  7 |  10 |  2 |  14 |  22
6 |  1 |  6 |  2 |  8 |  12 |  3 |  14 |  23
7 |  2 |  6 |  3 |  9 |  14 |  4 |  14 |  24
4 |  0 |  7 |  1 |  8 |  8 |  2 |  13 |  22
5 |  1 |  7 |  2 |  9 |  10 |  3 |  13 |  23
6 |  2 |  7 |  3 |  10 |  12 |  4 |  13 |  24
3 |  0 |  8 |  1 |  9 |  6 |  2 |  12 |  22
4 |  1 |  8 |  2 |  10 |  4 |  3 |  12 |  23
5 |  2 |  8 |  3 |  11 |  10 |  4 |  12 |  24
6 |  3 |  8 |  4 |  12 |  12 |  5 |  12 |  25
2 |  0 |  9 |  1 |  10 |  4 |  2 |  11 |  22
3 |  1 |  9 |  2 |  11 |  6 |  3 |  11 |  23
4 |  2 |  9 |  3 |  12 |  8 |  4 |  11 |  24
5 |  3 |  9 |  4 |  13 |  10 |  5 |  11 |  25
6 |  4 |  9 |  5 |  14 |  12 |  6 |  11 |  26
1 |  0 |  10 |  1 |  11 |  2 |  2 |  10 |  22
2 |  1 |  10 |  2 |  12 |  4 |  3 |  10 |  23
3 |  2 |  10 |  3 |  13 |  6 |  4 |  10 |  24
4 |  3 |  10 |  4 |  14 |  8 |  5 |  10 |  25
5 |  4 |  10 |  5 |  15 |  10 |  6 |  10 |  26
6 |  5 |  10 |  6 |  16 |  12 |  7 |  10 |  27
0 |  0 |  11 |  1 |  12 |  0 |  2 |  9 |  22
1 |  1 |  11 |  2 |  13 |  2 |  3 |  9 |  23
2 |  2 |  11 |  3 |  14 |  4 |  4 |  9 |  24
3 |  3 |  11 |  4 |  15 |  6 |  5 |  9 |  25
4 |  4 |  11 |  5 |  16 |  8 |  6 |  9 |  26
5 |  5 |  11 |  6 |  17 |  10 |  7 |  9 |  27

Table 1: Non-exceptional examples Part I . Number of parameters obtained is
$848$. 36 Distinct triples of Betti numbers $(b^{2},b^{3},b^{4})$ are
realized.

g |  Number of rational Curves |  $rkNS(\sigma)$ |  $b^{2}_{local}$ |  $b^{2}$ |  $b^{3}$ |  $b^{4}$ |  Dim $\mathcal{M}_{\sigma}$ |  Dim ALG family
---|---|---|---|---|---|---|---|---
0 |  1 |  12 |  2 |  14 |  0 |  3 |  8 |  23
1 |  2 |  12 |  3 |  15 |  2 |  4 |  8 |  24
2 |  3 |  12 |  4 |  16 |  4 |  5 |  8 |  25
3 |  4 |  12 |  5 |  17 |  6 |  6 |  8 |  26
4 |  5 |  12 |  6 |  18 |  8 |  7 |  8 |  27
0 |  2 |  13 |  3 |  16 |  0 |  4 |  7 |  24
1 |  3 |  13 |  4 |  17 |  2 |  5 |  7 |  25
2 |  4 |  13 |  5 |  18 |  4 |  6 |  7 |  26
3 |  5 |  13 |  6 |  19 |  6 |  7 |  7 |  27
0 |  3 |  14 |  4 |  18 |  0 |  5 |  6 |  25
1 |  4 |  14 |  5 |  19 |  2 |  6 |  6 |  26
2 |  5 |  14 |  6 |  20 |  4 |  7 |  6 |  27
3 |  6 |  14 |  7 |  21 |  6 |  8 |  6 |  28
0 |  4 |  15 |  5 |  20 |  0 |  6 |  5 |  26
1 |  5 |  15 |  6 |  21 |  2 |  7 |  5 |  27
2 |  6 |  15 |  7 |  22 |  4 |  8 |  5 |  28
0 |  5 |  16 |  6 |  22 |  0 |  7 |  4 |  27
1 |  6 |  16 |  7 |  23 |  2 |  8 |  4 |  28
2 |  7 |  16 |  8 |  24 |  4 |  9 |  4 |  29
0 |  6 |  17 |  7 |  24 |  0 |  8 |  3 |  28
1 |  7 |  17 |  8 |  25 |  2 |  9 |  3 |  29
2 |  8 |  17 |  9 |  26 |  4 |  10 |  3 |  30
0 |  7 |  18 |  8 |  26 |  0 |  9 |  2 |  29
1 |  8 |  18 |  9 |  27 |  2 |  10 |  2 |  30
2 |  9 |  18 |  10 |  28 |  4 |  11 |  2 |  31
0 |  8 |  19 |  9 |  28 |  0 |  10 |  1 |  30
1 |  9 |  19 |  10 |  29 |  2 |  11 |  1 |  31
0 |  9 |  20 |  10 |  30 |  0 |  11 |  0 |  31

Table 2: Non-exceptional examples Part II. Number of parameters obtained is
767. 28 distinct triples of Betti numbers $(b^{2},b^{3},b^{4})$ is realized
(beyond Table 1).

$rkNS(\sigma)$ |  $b^{2}_{local}$ |  $b^{2}$ |  $b^{3}$ |  $b^{4}$ |  Dim $\mathcal{M}_{\sigma}$ |  Dim ALG family
---|---|---|---|---|---|---
10 |  2 |  12 |  4 |  3 |  10 |  23

Table 3: Exceptional examples. Fixed locus consists of two disjoint elliptic
curves. Triple of Betti numbers $(b^{2},b^{3},b^{4})=(12,4,3)$ coincides with
a row in Table 1.

##  8 Appendix

###  8.1 Poincare inequality under model radius exhaustion

Instead of distance function of the whole manifold, the argument in [51, Prop
4.8 i, ia, ib] for the initial decay works verbatim for the model distance $r$
on the end. The reason why we need this is that our end is a twisted product.
It seems harder for us to show the annulus are connected, than to work with
the model distance $r$. For the Poincare inequality and initial decay, we
essentially only need the reference metric to be equivalent to the model
orbifold metric on the end.

######  Lemma 8.1.

Let $A=\\{\rho\leq r\leq\rho^{\prime}\\}$, $\rho^{\prime}>\rho>0$ be a non-
trivial annulus in the cone with (smooth) closed link $\mathbb{S}$:

$\mathbb{S}\times\mathbb{R}^{+},\ \ \ dr^{2}+r^{2}(d\mathbb{S})^{2}.$

Let $Y$ be a (smooth) closed manifold, and $P_{A},\ P_{Y}$ be the Neumann
Poincare constant on the annulus and $Y$ respectively. The following holds for
any function $u\in C^{2}[A\times Y]$.

$\int_{A}\int_{Y}u^{2}\leq(P_{A}+P_{Y})\int_{A}\int_{Y}|\nabla u|^{2}.$

###### Proof.

Write $u_{Ave,Y}$ for $\frac{\int_{Y}u}{Vol(Y)}$ and $v$ for $u-u_{Ave,Y}$.
Then $u_{Ave,Y}$ is a function in $A$ only, and is constant along each fiber.
$v(a,\cdot)$ has $0$ integral/average on any fiber $Y$, for any $a\in A$.
Obviously, $u_{Ave,Y}$ and $v$ are perpendicular under $L^{2}[Y]-$inner
product, and

$u=u_{Ave,Y}+v.$

We calculate

$\int_{A}\int_{Y}u^{2}=\int_{A}\int_{Y}u^{2}_{Ave,Y}+\int_{A}\int_{Y}v^{2}\leq
P_{A}\int_{Y}\int_{A}|\nabla_{A}u_{Ave,Y}|^{2}+P_{Y}\int_{A}\int_{Y}|\nabla_{Y}v|^{2}.$
(76)

We have the following orthogonality because $\nabla_{A}$ obviously commutes
with integration in $Y$, and $u_{Ave,Y}$ is a constant along each fiber
$\\{a\\}\times Y$.

$\int_{Y}\nabla_{A}u_{Ave,Y}\cdot\nabla_{A}v=0.$ (77)

Hence (77) implies

$\int_{A}\int_{Y}|\nabla_{A}u|^{2}=\int_{A}\int_{Y}|\nabla_{A}u_{Ave,Y}|^{2}+\int_{A}\int_{Y}|\nabla_{A}v|^{2}.$
(78)

Plugging the above into (76) and throwing away the “good” term

$P_{A}\int_{A}\int_{Y}|\nabla_{A}(u-u_{Ave,Y})|^{2},$

the proof is complete.

For the reader’s convenience, we provide the routine proof for (77). Let $a\in
A$ and $a^{i},\ i=1...dimA$ be a normal coordinate at/near $a$ under the cone
metric. Then $\frac{\partial u_{Ave,Y}}{\partial a^{i}}$ is still a constant
along the fiber $\\{a\\}\times Y$. On the other hand, because

$\int_{Y}v\equiv 0\ \textrm{for any}\ b\in A,$

we find

$0=\frac{\partial}{\partial a^{i}}\int_{Y}v=\int_{Y}\frac{\partial v}{\partial
a^{i}}.$

Finally, we multiply them to get

$\int_{Y}\frac{\partial u_{Ave,Y}}{\partial a^{i}}\cdot\frac{\partial
v}{\partial a^{i}}=\frac{\partial u_{Ave,Y}}{\partial
a^{i}}\cdot\int_{Y}\frac{\partial v}{\partial a^{i}}=0.$

This is similar to the integration along fiber in [15]. ∎

###  8.2 Hein energy decay estimate for model radius function

Define $E(\rho)=\int_{\\{r\geq\rho\\}}|\nabla\phi|^{2}$. The argument in [51,
Prop 4.8 i, ia, ib] says there exists $0\leq\mu<1$ such that

$E(2^{n}r_{0})\leq\mu^{n}E(r_{0}).$ (79)

Standard iterative argument (see [42, Proof of Lemma 8.23] for a sophisticated
version) implies polynomial energy decay i.e.

$E(r)\leq C_{\beta_{0}}r^{-\beta_{0}},$ (80)

where $C_{\beta_{0}}=E(r_{0})\cdot r_{0}^{-\frac{\log\mu}{\log 2}},\
\beta_{0}=-\frac{\log\mu}{\log 2}$.

###### Hein’s proof of (79) under the model radius function:.

.

The energy inequality [51, (4.14)] gives

$\int_{\\{r\geq
2^{i+1}r_{0}\\}}|\nabla\phi|^{2}\leq\frac{C}{2^{i}r_{0}}|\phi-\phi_{Ave}|_{L^{2}[2^{i}r_{0}\leq
r\leq 2^{i+1}r_{0}]}|\nabla\phi|_{L^{2}[2^{i}r_{0}\leq r\leq 2^{i+1}r_{0}]}.$
(81)

The radius $r_{0}$ is chosen large enough such that $f=0$ on
$\\{r\geq\frac{r_{0}}{2}\\}$. Therefore the $f-$term in [51, (4.14)] vanishes.
The annulus $2^{i}r_{0}\leq r\leq 2^{i+1}r_{0}$ are scaling equivalent to the
standard $\\{1\leq r\leq 2\\}$. By Lemma (8.1) and the Poincare constant bound

$P_{A}+P_{Y}\leq C(2^{i}r_{0})^{2},$

we find

$\int_{\\{r\geq 2^{i+1}r_{0}\\}}|\nabla\phi|^{2}\leq C\int_{\\{2^{i}r_{0}\leq
r\leq 2^{i+1}r_{0}\\}}|\nabla\phi|^{2}=C(\int_{\\{r\geq
2^{i}r_{0}\\}}|\nabla\phi|^{2}-\int_{\\{r\geq
2^{i+1}r_{0}\\}}|\nabla\phi|^{2}).$

This means

$\int_{\\{r\geq
2^{i+1}r_{0}\\}}|\nabla\phi|^{2}\leq\frac{C}{C+1}\int_{\\{r_{0}\geq
2^{i}r_{0}\\}}|\nabla\phi|^{2}.$

This means

$E(2^{i+1}r_{0})\leq\mu\cdot E(2^{i}r_{0})\ \textrm{for any}\ i\geq 0,$ (82)

if we take $\mu=\frac{C}{C+1}<1$. Then (79) follows from (82). ∎

###  8.3 Tian-Yau quasi atlas and Schauder norms

####  8.3.1 Review

Unlike injectivity radius and harmonic radius, the concept of “holomorphic
radius ” does not seem immediate and obvious. We apply the quasi atlas [78,
51] to the compact Kähler manifold $Y$, even though it is mainly applied to
non-compact Kähler manifolds. By Hein’s version of Tian-Yau quasi atlas [51,
Lemma 4.3], there is a $C^{1,\alpha}$ quasi atlas which is also a
$C^{k,\alpha}-$quasi atlas, for any $k>0$ and $\alpha\in(0,1)$. Henceforth we
refer to it as “the quasi atlas”. In Tian-Yau coordinate, the crucial
properties we rely on here is the quasi-isometry between the pullback
$\Phi_{\underline{x}}^{\star}\omega_{0}$ and the Euclidean metric, and the
uniform $C^{k,\alpha}$ bound on $\Phi_{\underline{x}}^{\star}\omega_{0}$.

Namely, there is a $\mu_{0}>0$ such that for any $x\in Y$, there is a
$\underline{x}\in A\subset Y$ such that $x\in\Phi_{\underline{x}}(B_{euc})$
and

  * •

$B_{euc,\Phi^{-1}_{\underline{x}}(x)}(\mu_{0})\subset B_{euc,O}(1)$,

  * •

$dist[B_{euc,x}(\mu_{0}),\partial B_{euc,O}(1)]>\mu_{0}$.

$A$ is the subset of “centers” of open sets in the quasi-atlas,
$\Phi_{\underline{x}}$ is the holomorphic coordinate map, $B_{euc,x}(r)$ is
the unit ball in Euclidean space $\mathbb{C}^{d-1}$ with radius $r$ centered
at $x$, and $\Phi_{\underline{x}}$ identifies the origin to $\underline{x}$ on
the manifold. We interpret the conditions in [51] to coordinate chart. A
schematic description of $x$, $\underline{x}$, and a point $y$ with
$dist(x,y)<\mu_{0}$ under the coordinate in quasi-atlas is as Figure 4.

$\underline{x}$$x$$y$ Figure 4: Positions of $x,\ y$ and $\underline{x}$

####  8.3.2 Poly-Cylinder

Let

$\mu_{1}<<\frac{\min\\{\mu_{0},h_{0,Y},inj_{0,Y},conv_{0,Y}\\}}{c_{TY,ALG}}\subseteq\mathbb{C}\times
Y$

be a positive radius where $c_{TY,ALG}$ is a large enough constant depending
on ALG model (6) and Tian-Yau atlas, where $h_{0,Y},\ inj_{0,Y},\ conv_{0,Y}$
are harmonic, injectivity, and convex radius of $Y$ under the metric
$\omega_{Y}$ respectively. Let the Schauder norm on the fibred annuli
$A(R_{1},R_{2})$ be defined in the standard way (for example, see [55,
1.2.2]): supremum of norms of co-variant derivatives under ALG model, together
with $\alpha-$Hölder semi norms by the distance on $\mathbb{C}\times Y$. To
estimate the Lipschitz quotient $\frac{|v(x)-v(y)|}{d^{\alpha}(x,y)}$, where
$v$ is either a function or a matrix valued function representing a
$(1,1)-$form under coordinate chart, it is well known that it suffices to
assume $d(x,y)<\mu$ for some uniform number $\mu$ independent of $v$, because
otherwise the Lipschitz quotient is bounded by the $C^{0}-$norm of $v$.
Therefore, it suffices to estimate in poly-cylinders

$PC_{z,\underline{x},y}(\mu_{1})\triangleq
D_{z,\mathbb{C}}(\mu_{1})\times[B_{euc,y}(\mu_{1})]$

in Tian-Yau chart. The symbol $D_{z,\mathbb{C}}(\mu_{1})$ denotes disk/ball of
radius $\mu_{1}$ centered at $z$ in the $\mathbb{C}-$component. When
$\mu_{2}<<\mu_{1}$, the above poly-cylinder contains the Euclidean ball
$B_{euc,(z,y)}(\mu_{2})$ where the sub-scripts are usually suppressed. In the
energy decay estimate (49), Neumann-Poincaré inequality is applied on such a
ball.

####  8.3.3 Weighted Schauder norm

We define weighted Schauder norms on the end by supremum of naively weighted
norms on fibred annuli:

$|v|_{C^{k,\frac{1}{2}}_{\beta}[A(R,\infty)]}\triangleq\sup_{l\geq
R}l^{\beta}|v|_{C^{k,\frac{1}{2}}[A(l,l+2)]}.$ (83)

On one hand,

$|v|_{C^{k,\frac{1}{2}}[A(l,l+2)]}\leq
C\sup_{{PC_{z,\underline{x},y}(\mu_{1})}\cap[A(l,l+2)]\neq\emptyset}|v|_{C^{k,\frac{1}{2}}[PC_{z,\underline{x},y}(\mu_{1})]}.$

The width $2$ of the annuli is not serious. Any universal positive constant
will do. On the other hand, when
$\Phi_{\underline{x}}[PC_{z,\underline{x},y}(\mu_{1})]\subset A(l,l+2)$ (the
map in $\mathbb{C}-$component is identity), we have

$|v|_{C^{k,\frac{1}{2}}[A(l,l+2)]}\geq
C|v|_{C^{k,\frac{1}{2}}[PC_{z,\underline{x},y}(\mu_{1})]}.$

The norm on the right means the (usual) norm under Tian-Yau coordinate (and
Euclidean metric therein), but the norm on the left employs geodesic distance
as in [55].

###  8.4 Gluing for ansatz

######  Lemma 8.2.

(Gluing for ansatz [52, 51, 49]) Under Theorem 1.2, on an iso-trivial ALG
Calabi-Yau fibration $\widetilde{\frac{\mathbb{C}\times
Y}{\langle\sigma\rangle}}$, let $\chi(r)$ be a cutoff function $\equiv 1$ when
$r\geq R+1$ but $\equiv 0$ when $r\leq R$, for some $R>>1$. Let $\varphi$ be
another cutoff function that $\equiv 1$ when $r\geq R-1$ but $\equiv 0$ when
$r\leq R-2$. Let $b_{0}\triangleq\varphi(r)\frac{i}{2}dzd\overline{z}$.
Suppose $\widetilde{\omega}$ is a real $(1,1)-$form on $X$ that is
positive/Kähler on $\\{r\leq R+1\\}$, and $\phi$ is a real-valued function on
$\\{r>R-3\\}$ such that $\widetilde{\omega}+i\partial\overline{\partial}\phi$
is positive/Kähler on $\\{r\geq R\\}$. Then when $t$ is large enough, the real
closed $(1,1)-$form

$\underline{\omega}\triangleq\widetilde{\omega}+i\partial\overline{\partial}(\chi\phi)+tb_{0}$
(84)

is a Kähler metric on $X$ i.e. it is positive definite every-where.

$\textrm{Domain of}\
\phi\supset\overline{Supp\varphi}\supseteq\\{\varphi\equiv
1\\}\supset\overline{Supp\chi}$. See Figure 5.

$R-3$$R-2$$R-1$$R$$R+1$$\chi$$\varphi$$Dom(\phi)$ Figure 5: Supports and
domain in gluing lemma for ansatz

###### Proof.

It is routine by positivity of $\widetilde{\omega}$ in the fiber direction and
positivity of the bump form $tb_{0}$ in the “horizontal” direction. The idea
is that in $Supp\chi$/the gluing region, since $\chi$ is pulled back from
$\Delta^{\star}$, the $(1,1)-$form
$\widetilde{\omega}+i\partial\overline{\partial}(\chi\phi)$ is positive on the
fiber $Y-$direction. The mixed forms with both base and fiber directions can
be bounded from below by Cauchy-Schwarz inequality. Therefore adding the
(horizontal) bump form $tb_{0}$ makes it positive. Here the $t$ is required to
be large enough with respect to the smooth $\phi$ which is only required to
exist locally. It has bounded $C^{k}-$norms on the gluing region (which has
compact closure), for any $k$.

We calculate

$\widetilde{\omega}+i\partial\overline{\partial}(\chi\phi)=\widetilde{\omega}+\chi
i\partial\overline{\partial}\phi+\phi
i\partial\overline{\partial}\chi+2Re(i\partial\chi\wedge\overline{\partial}\phi).$
(85)

The sum of the first two term on the right side above is positive:

$\widetilde{\omega}+\chi
i\partial\overline{\partial}\phi=(1-\chi)\widetilde{\omega}+\chi(\widetilde{\omega}+i\partial\overline{\partial}\phi)>0\
\textrm{is positive definite Hermitian}.$ (86)

We therefore use the norm $|\cdot|$ of the Hermitian metric $H_{\chi}$
(associated to) $\widetilde{\omega}+\chi i\partial\overline{\partial}\phi$.
Let $v$ and $h$ be a vertical (tangent to fiber) and horizontal $(1,0)-$vector
(proportional to $\frac{\partial}{\partial z}$), we compute

$\displaystyle[\widetilde{\omega}+i\partial\overline{\partial}(\chi\phi)][v+h,\overline{v}+\overline{h}]$
$\displaystyle=$ $\displaystyle[\widetilde{\omega}+\chi
i\partial\overline{\partial}\phi][v+h,\overline{v}+\overline{h}]+(\phi
i\partial\overline{\partial}\chi)[h,\overline{h}]+2Re(i\partial\chi\wedge\overline{\partial}\phi)[h,\overline{h}]$
$\displaystyle+2Re(i\partial\chi\wedge\overline{\partial}\phi)[h,\overline{v}]+2Re(i\partial\chi\wedge\overline{\partial}\phi)[v,\overline{h}]$
$\displaystyle\geq$ $\displaystyle|v+h|^{2}-c_{1}|v||h|-c_{2}|h|^{2}.$

In the following bound, the reason for constants $c_{1}$ and $c_{2}$

$\displaystyle(\phi
i\partial\overline{\partial}\chi)[h,\overline{h}]+2Re(i\partial\chi\wedge\overline{\partial}\phi)[h,\overline{h}]$
$\displaystyle+2Re(i\partial\chi\wedge\overline{\partial}\phi)[h,\overline{v}]+2Re(i\partial\chi\wedge\overline{\partial}\phi)[v,\overline{h}]$
$\displaystyle\geq$ $\displaystyle-c_{1}|v||h|-c_{2}|h|^{2}.$

is compactness of $Supp\nabla\chi\cup Supp\partial\chi\cup
Supp\overline{\partial}\chi$. The point is the vanishing of the terms
$(\partial\chi\wedge\overline{\partial}\phi)[v,\overline{v}]$, $(\phi
i\partial\overline{\partial}\chi)[v,\cdot]$ and the conjugates i.e.
$\partial\chi$ and $\overline{\partial}\chi$ vanishes on vectors tangent to
fiber. Cauchy-Schwarz inequality implies

$-c_{1}|v||h|\geq-\frac{1}{2}|v|^{2}-c_{3}|h|^{2}.$

Then when $t>1000(c_{2}+c_{3}+1)$, the bump form $tb_{0}$ makes

$\underline{\omega}\geq\frac{1}{3}H_{\chi}.$

∎

###  8.5 “Un-conditional” solvability of
Laplace/$i\partial\overline{\partial}$ equation in dimension $1$.

As a pre-requisite of $i\partial\overline{\partial}$ Lemma (4.1) and radial
Kähler potential for radial bump $(1,1)$ form on $\mathbb{C}$ (required in
proof of Theorem 1.2), we need the following.

######  Claim 8.3.

For any smooth function $f$ on $\Delta^{\star}$, there is a smooth solution
$h$ to

$\frac{\partial^{2}h}{\partial u\partial\overline{u}}=f,$ (87)

which is equivalent to $\Delta_{\mathbb{C}}h=4f$ because
$\Delta_{\mathbb{C}}=4\frac{\partial^{2}}{\partial u\partial\overline{u}}$.

Suppose $f$ is compactly supported in $\\{\rho<\rho_{1}\\}$ and radial i.e.
$f=f(\rho)$ is a function of $\rho$ only. Then there is a radial solution
$h(\rho)$ such that $h(\rho)=0$ when $\rho>\rho_{1}$.

The first part (no symmetry) is a direct consequence of [54, Theorem 1.4.4]
i.e. the $1-$dimensional $\overline{\partial}-$solvability that implies for
any $f\in C^{\infty}(\Delta^{\star})$,

$\frac{\partial\varsigma}{\partial\overline{u}}=f\ \textrm{admits a solution}\
\varsigma\in C^{\infty}(\Delta^{\star}).$

The complex conjugate of the same solvability yields a smooth function $h$
that solves

$\frac{\partial h}{\partial u}=\varsigma.$

We do not need any ($L^{2}$) integrabilty or asymptotes of $f$ near $0$.

When $f$ is radial, we simply solve the ODE

$\frac{\partial^{2}h}{\partial
u\partial\overline{u}}(\rho)=\frac{1}{4}\Delta_{\mathbb{C}}h(\rho)\ \ i.e.\ \
h^{\prime\prime}+\frac{h^{\prime}}{\rho}=4f$

by

$h=4\log\rho\int^{\rho_{1}}_{\rho}xf(x)dx-4\int^{\rho_{1}}_{\rho}(x\log
x)f(x)dx.$

Consequently, $h=0$ when $\rho>\rho_{1}$ because the integrals are outside the
support of $f$.

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