# Embedding Theorems for Calabi–Yau Conifolds

Yohsuke Imagi

###### Abstract

We prove that compact Calabi–Yau varieties with certain isolated singularities
are projective. In dimension 3 we do this by analysis, supposing given
conifold metrics. In higher dimensions it follows more readily from Ohsawa’s
degenerate spectral sequence.

##  1 Introduction

Compact Calabi–Yau conifolds are the metric completions of non-compact Ricci-
flat Kähler manifolds which satisfy some certain decay estimates at finitely
many ends (corresponding to the singularities); see Definition 3.1 below. The
definition is suitable for gluing constructions in geometric analysis, as
performed by Chan [6, 7]. Examples of compact Calabi–Yau conifolds are given
by Hein and Sun [18]. In the present paper we prove

######  Theorem 1.1.

Let $X$ be a compact strictly-singular Calabi–Yau conifold of $($complex
pure$)$ dimension $3.$ Then $X$ is projective.

######  Remark 1.2.

Compact Calabi–Yau conifolds in our sense are allowed to be non-singular, in
which case they may be the products of complex tori and holomorphic symplectic
manifolds, including those non-projective. These are excluded by the strictly
singular condition in the theorem above.

We prove the theorem by extending the method in the non-singular case. We show
first that the following holds (which holds in every dimension $n\geqslant
3$):

######  Theorem 1.3.

Let $X$ be a compact strictly-singular Calabi–Yau conifold of dimension
$n\geqslant 3.$ Then no non-zero $L^{2}$ harmonic $(n-1,0)$ forms exist on the
regular locus $X^{{\rm reg}}.$

We generalize then the fact that compact Kähler manifolds with no non-zero
holomorphic $(2,0)$ forms are projective. Note that the singular version of
the Kodaira embedding theorem is proved already by Grauert [15, §3, Satz3];
see also Theorem 3.11. Using this we prove

######  Lemma 1.4.

Let $X$ be a compact normal Kähler space whose singularities are isolated and
of depth $\geqslant 3.$ Suppose that every element of $H^{2}(X^{{\rm
reg}},{{\mathbb{C}}})$ may be represented on $X^{{\rm reg}}$ by some closed
$(1,1)$ form. Then $X$ is projective.

Theorem 1.1 then follows readily from Theorem 1.3 with $n=3$ and Lemma 1.4 in
dimension $3.$

In dimension $>3$ we prove a slightly different theorem. We do not specify any
particular Kähler forms on $X$ in what follows.

######  Theorem 1.5.

Let $X$ be a compact normal Kähler space of pure dimension $\geqslant 4$ whose
singularities are isolated, rational and such that no non-zero holomorphic
$(2,0)$ forms exist on $X^{{\rm reg}}.$ Then $X$ is projective.

For the proof we use a result of Ohsawa [27] (which we recall in Theorem 6.1).
In the circumstances above the singularities have co-dimension $\geqslant 4$
so Ohsawa’s result applies to the Hodge to de Rham spectral sequence
$(E^{pq}_{r})_{r=1}^{\infty}$ with $p+q=2;$ that is,
$\bigoplus_{p+q=2}H^{q}(X^{{\rm reg}},\Omega^{p}_{X})\Rightarrow H^{2}(X^{{\rm
reg}},{{\mathbb{C}}})$ degenerates at $E_{1}.$ It is then not hard to deduce
Theorem 1.5 from Lemma 1.4 and the hypothesis that holomorphic $(2,0)$ forms
vanish.

We give a corollary to Theorems 1.1 and 1.5 which has to do with quasi-étale
covers and irreducible Calabi–Yau varieties in the Beauville–Bogomolov
decomposition theorem (proved finally by Bakker, Guenancia and Lehn [2,
Theorem A] in the Kähler case). We explain a little more in §7.

######  Corollary 1.6.

Let $X$ be either a compact irreducible Calabi–Yau conifold of dimension $3$
or a compact irreducible Calabi–Yau variety of dimension $\geqslant 4$ whose
singularities are isolated. Then $X$ and its quasi-étale covers are
projective.

######  Remark 1.7.

It is possible in Theorems 1.1 and 1.5 that some quasi-étale covers of $X$
contain the factors of complex tori and holomorphic symplectic varieties which
may accordingly be non-projective. These are excluded by the irreducible
Calabi–Yau condition in the corollary above.

We begin in §2 with the treatment of Kähler cones, the local model of
conifolds. The key result we prove is the following, which is stronger than
the corresponding result (Corollary 2.6) for Riemannian cones.

######  Theorem 1.8 (Corollary 2.13).

Let $C$ be a Kähler $n$-cone and $p,q\geqslant 0$ integers with
$p+q\leqslant\nobreak n-1.$ Then no non-zero homogeneous harmonic $(p,q)$ form
on $C^{{\rm reg}}$ has order in $(p+q-2n,-p-q).$

In §3 we turn to conifolds. We prove in Corollary 3.10 that every compact
Calabi–Yau conifold $X$ in the sense of Definition 3.1 has the natural
structure of a normal Kähler space. This is not very immediate from the
differential geometry definition of $X.$ We have first to make $X$ into a
normal complex space and then to extend the Kähler form to $X.$

In §4 we prove Theorem 1.3 and a few other vanishing theorems. Note that
harmonic $(n-1,0)$ forms on a Calabi–Yau manifold corresponds to harmonic
$(0,1)$ forms under the bundle isomorphisms given by the Kähler form and the
complex volume form. Theorem 1.3 may then be regarded as a vanishing theorem
for harmonic $1$-forms, which may in turn be compared with

######  Theorem 1.9 (Chan [7, Proposition 4.2]).

Let $X$ be a compact strictly-singular Riemannian conifold of real dimension
$n\geqslant 3.$ Let $X^{{\rm reg}}$ have non-negative Ricci curvature. Then
$H^{1}(X^{{\rm reg}},{\mathbin{\mathbb{R}}})=0.$

######  Remark 1.10.

We give a new proof of this which looks clearer than the original; see
Corollary 4.7.

Theorem 1.9 is in fact more about closed and co-closed $1$-forms than about
harmonic $1$-forms. We prove therefore in Theorem 4.5 a slightly different
result on harmonic $1$-forms. We prove then using Theorem 1.8 a stronger
result for Kähler conifolds; see Corollary 4.8. This will imply Theorem 1.3.

In §5 we prove Lemma 1.4 and Theorem 1.1. In §6 we prove Theorem 1.5 which may
be read independently of the preceding sections except the proof of Lemma 1.4.
In other words, most of the paper is devoted to proving and explaining Theorem
1.1. We end in §7 by proving Corollary 1.6.

Acknowledgements. The author was supported by the grant 21K13788 of the Japan
Society for the Promotion of Science.

##  2 Kähler Cones

We begin by defining Riemannian cones.

######  Definition 2.1.

A Riemannian $n$-cone is the metric completion $C$ of some $n$-manifold
$C^{{\rm reg}}=(0,\infty)\times C^{{\rm lk}}$ equipped with the cone metric
${\rm d}r^{2}+r^{2}g^{{\rm lk}}$ where $(C^{{\rm lk}},g^{{\rm lk}}),$ called
the link, is a compact Riemannian manifold without boundary and $r$ the
projection $(0,\infty)\times C^{{\rm lk}}\to(0,\infty),$ called the radius
function. This implies that $C\setminus C^{{\rm reg}}$ contains exactly one
point, which we call the vertex and denote by vx. We say that $C$ is strictly
singular if this is not isometric to the flat ${\mathbin{\mathbb{R}}}^{n}.$

We define harmonic and homogeneous forms on Riemannian cones.

######  Definition 2.2.

Let $C$ be a Riemannian cone and $\phi$ a $p$-form on $C^{{\rm reg}}.$ We put
$\Delta:={\rm d}{\rm d}^{*}+{\rm d}^{*}{\rm d},$ the Laplacian of the
Riemannian manifold $(C^{{\rm reg}},{\rm d}r^{2}+r^{2}g^{{\rm lk}}).$ We say
that $\phi$ is harmonic if $\Delta\phi=0.$ For $\alpha\in{{\mathbb{C}}}$ we
say that $\phi$ is homogeneous of order $\alpha$ if $\phi:=e^{(\alpha+p)\log
r}(\phi^{\prime}+{\rm d}\log r\wedge\phi^{\prime\prime})$ for some $p$-form
$\phi^{\prime}$ on $C^{{\rm lk}}$ and some $(p-1)$ form $\phi^{\prime\prime}$
on $C^{{\rm lk}}.$

We prove

######  Proposition 2.3.

Let $C$ be a Riemannian cone, $\alpha$ a complex number and $\phi$ an
order-$\alpha$ homogeneous harmonic $p$-form on $C^{{\rm reg}}.$ Then $\phi=0$
or $\alpha\in{\mathbin{\mathbb{R}}}.$

###### Proof.

Put $\beta:=\alpha+p$ and write
$\Delta\phi=r^{\beta-2}[(\Delta\phi)^{\prime}+{\rm d}\log
r\wedge(\Delta\phi)^{\prime\prime}]$ where
$(\Delta\phi)^{\prime},(\Delta\phi)^{\prime\prime}$ are forms on $C^{{\rm
lk}}.$ Put $n:=\mathop{\rm dim}\nolimits C^{{\rm reg}}.$ Then we can show by
computation that

$\begin{split}(\Delta\phi)^{\prime}:&=\Delta\phi^{\prime}-\beta(\beta+n-2-2p)\phi^{\prime}-2{\rm
d}\phi^{\prime\prime},\\\
(\Delta\phi)^{\prime\prime}:&=\Delta\phi^{\prime\prime}-(\beta-2)(\beta+n-2p)\phi^{\prime\prime}-2{\rm
d}^{*}\phi^{\prime}\end{split}$ (2.1)

where $\Delta,{\rm d}$ and ${\rm d}^{*}$ on the right-hand side are computed
on $(C^{{\rm lk}},g^{{\rm lk}}).$ The details of the computation are given by
Chan [7, Proposition 3.3(2)] for $n$ even, which is in fact valid for every
$n.$ Since $\Delta\phi=0$ it follows by (2.1) that

$\displaystyle\Delta\phi^{\prime}$
$\displaystyle=\beta(\beta+n-2-2p)\phi^{\prime}+2{\rm d}\phi^{\prime\prime},$
(2.2) $\displaystyle\Delta\phi^{\prime\prime}$
$\displaystyle=(\beta-2)(\beta+n-2p)\phi^{\prime\prime}+2{\rm
d}^{*}\phi^{\prime}.$ (2.3)

Applying ${\rm d}$ to (2.2) we find that ${\rm d}{\rm d}^{*}{\rm
d}\phi^{\prime}=\beta(\beta+n-2-2p){\rm d}\phi^{\prime}.$ So if ${\rm
d}\phi^{\prime}\neq 0$ then $\beta(\beta+n-2-2p)$ is a non-negative real
number, which implies $\beta\in{\mathbin{\mathbb{R}}}$ and accordingly
$\alpha\in{\mathbin{\mathbb{R}}}$ as we have to prove. Suppose therefore that
${\rm d}\phi^{\prime}=0.$ Put $\psi:={\rm d}\phi^{\prime\prime}$ so that (2.2)
becomes

$\Delta\phi^{\prime}=\beta(\beta+n-2-2p)\phi^{\prime}+2\psi.$ (2.4)

Applying ${\rm d}$ to (2.3) and using ${\rm d}\phi^{\prime\prime}=\psi,$ ${\rm
d}\phi^{\prime}=0$ and (2.4) we find that

$\begin{split}\Delta\psi&=(\beta-2)(\beta+n-2p)\psi+2\Delta\phi^{\prime}\\\
&=(\beta-2)(\beta+n-2p)\psi+2\beta(\beta+n-2-2p)\phi^{\prime}+4\psi\\\
&=2\beta(\beta+n-2-2p)\phi^{\prime}+[(\beta-2)(\beta+n-2p)+4]\psi.\end{split}$
(2.5)

This and (2.4) imply $(\Delta\phi^{\prime},\Delta\psi)=(\phi^{\prime},\psi)M$
where

$M:=\begin{pmatrix}\beta(\beta+n-2-2p)&2\beta(\beta+n-2-2p)\\\
2&(\beta-2)(\beta+n-2p)+4\end{pmatrix}.$ (2.6)

This matrix is diagonalizable; and in fact, $P^{-1}MP=D$ where

$P=\begin{pmatrix}\beta+n-2-2p&\beta\\\ 1&-1\end{pmatrix}\text{ and
}D:=\begin{pmatrix}\beta(\beta+n-2p)&0\\\
0&(\beta-2)(\beta+n-2-2p)\end{pmatrix}.$

So $(\Delta\phi^{\prime},\Delta\psi)P=(\phi^{\prime},\psi)PD$ and looking at
the first component we see that

$\Delta[(\beta+n-2-2p)\phi^{\prime}+\psi]=\beta(\beta+n-2p)[(\beta+n-2-2p)\phi^{\prime}+\psi].$
(2.7)

Thus if $(\beta+n-2-2p)\phi^{\prime}+\psi\neq 0$ then $\beta(\beta+n-2p)$ is a
non-negative real number, which implies $\beta\in{\mathbin{\mathbb{R}}}$ as we
have to prove. Suppose therefore that $(\beta+n-2-2p)\phi^{\prime}+\psi=0.$
Then by (2.2) we have

$\Delta\phi^{\prime}=\beta(\beta+n-2-2p)\phi^{\prime}-2(\beta+n-2-2p)\phi^{\prime}=(\beta-2)(\beta-2+n-2p)\phi^{\prime}.$
(2.8)

So if $\phi^{\prime}\neq 0$ then $(\beta-2)(\beta-2+n-2p)$ is a non-negative
real number, which implies $\beta-2\in{\mathbin{\mathbb{R}}}$ and accordingly
$\alpha\in{\mathbin{\mathbb{R}}}$ as we have to prove. Suppose therefore that
$\phi^{\prime}=0.$ Then by (2.3) we have
$\Delta\phi^{\prime\prime}=(\beta-2)(\beta+n-2p)\phi^{\prime\prime}.$ So if
$\phi^{\prime\prime}\neq 0$ then $(\beta-2)(\beta-2+n-2p)$ is a non-negative
real number, which implies $\beta\in{\mathbin{\mathbb{R}}}$ and accordingly
$\alpha\in{\mathbin{\mathbb{R}}}$ as we have to prove. Otherwise
$\phi^{\prime\prime}=0$ and accordingly $\phi=0,$ which completes the proof. ∎

######  Remark 2.4.

The proof shows also that the set of $\alpha$ for which $C^{{\rm reg}}$ has
some non-zero order-$\alpha$ homogeneous harmonic $p$-forms is a discrete
subset of ${\mathbin{\mathbb{R}}}$ (because the eigenvalues of the Laplacians
are discrete).

From the computation above we get the following three corollaries:

######  Corollary 2.5.

Let $C$ be a Riemannian $n$-cone, $p>\frac{n}{2}$ an integer and
$\alpha\in(-p,p-n)$ a real number. Let $\phi$ be an order-$\alpha$ homogeneous
harmonic $p$-form on $C^{{\rm reg}}$ and write
$\phi=r^{p+\alpha}(\phi^{\prime}+{\rm d}\log r\wedge\phi^{\prime\prime})$
where $\phi^{\prime},\phi^{\prime\prime}$ are forms on $C^{{\rm lk}}.$ Then
${\rm d}\phi^{\prime\prime}=(2+p-n-\alpha)\phi^{\prime}.$

###### Proof.

We put $\beta:=p+\alpha\in(0,2p-n)$ and follow the proof of Proposition 2.3.
Applying again ${\rm d}$ to (2.2) we find ${\rm d}{\rm d}^{*}{\rm
d}\phi^{\prime}=\beta(\beta+n-2-2p){\rm d}\phi^{\prime}.$ But now
$\beta(\beta+n-2-2p)<0$ so ${\rm d}\phi^{\prime}=0.$ Put again $\psi:={\rm
d}\phi^{\prime\prime}.$ Then (2.7) holds; that is,

$\Delta[(\beta+n-2-2p)\phi^{\prime}+\psi]=\beta(\beta+n-2p)[(\beta+n-2-2p)\phi^{\prime}+\psi].$
(2.9)

But now $\beta(\beta+n-2p)<0$ so $(\beta+n-2-2p)\phi^{\prime}+\psi=0$ as we
have to prove. ∎

######  Corollary 2.6.

Let $C$ be a Riemannian $n$-cone, $p>\frac{n}{2}+1$ an integer and
$\alpha\in(2-p,p-n)$ a real number. Then no non-zero order-$\alpha$
homogeneous harmonic $p$-form exists on $C^{{\rm reg}}.$

###### Proof.

As $p,\alpha$ satisfy the hypotheses of Corollary 2.5 we can use its result;
that is, writing again $\phi=r^{\beta}(\phi^{\prime}+{\rm d}\log
r\wedge\phi^{\prime\prime})$ we have ${\rm
d}\phi^{\prime\prime}=(2+p-\alpha-n)\phi^{\prime}=(2+2p-\beta-n)\phi^{\prime}$
with $\beta:=p+\alpha\in(2,2p-n).$ Equation (2.4) holds too with $\psi:={\rm
d}\phi^{\prime\prime}$ and

$\Delta\phi^{\prime}=\beta(\beta+n-2-2p)\phi^{\prime}-2(\beta+n-2-2p)\phi^{\prime}=(\beta-2)(\beta+n-2-2p)\phi^{\prime}.$

But $\beta\in(2,2p-n)$ and $(\beta-2)(\beta+n-2-2p)<0$ so $\phi^{\prime}=0.$
Equation (2.3) implies then
$\Delta\phi^{\prime\prime}=(\beta-2)(\beta+n-2p)\phi^{\prime\prime}.$ But
again $\beta\in(2,2p-n)$ so $\phi^{\prime\prime}=0.$ Thus $\phi=0.$ ∎

######  Corollary 2.7.

Let $C$ be a Riemannian $n$-cone, $p<\frac{n}{2}-1$ an integer and
$\alpha\in(2+p-n,-p)$ a real number. Then no non-zero order-$\alpha$
homogeneous harmonic $p$-form exists on $C^{{\rm reg}}.$

###### Proof.

Put $q:=n-p.$ Then $\alpha\in(2-q,q-n).$ Suppose first that $C^{{\rm reg}}$ is
orientable. Then we can define the Hodge dual $*\phi$ as a homogeneous
harmonic $q$-form of order $\alpha,$ to which we can apply Corollary 2.6. So
$*\phi=0$ and $\phi=0.$ If $C^{{\rm reg}}$ is unorientable then the result we
have just obtained applies to the pull-back of $\phi$ to the double cover of
$C^{{\rm reg}};$ that is, the pull-back vanishes and accordingly so does
$\phi.$ ∎

######  Remark 2.8.

We have proved Corollaries 2.6 and 2.7 for their own sake. These are weaker
than we shall need in the Kähler case.

Lichnerowicz and Obata prove the following: let $(C^{{\rm lk}},g^{{\rm lk}})$
be a compact Riemannian $(n-1)$ manifold with $n\geqslant 3$ and $\mathop{\rm
Ric}\nolimits g^{{\rm lk}}\geqslant(n-2)g^{{\rm lk}};$ then the $0$-form
Laplacian over this has first non-zero eigenvalue $\geqslant n-1$ and the
equality holds if and only if the cone on $C^{{\rm lk}}$ is isometric to
${\mathbin{\mathbb{R}}}^{n}.$ This implies the following. For (ii) we use also
a result of Cheeger–Tian [8, Theorem 7.27].

######  Corollary 2.9.

Let $C$ be a strictly-singular Riemannian $n$-cone with $n\geqslant 3$ and let
$C^{{\rm reg}}$ have non-negative Ricci curvature. Then (i) no non-zero
homogeneous closed and co-closed $1$-form on $C^{{\rm reg}}$ has order in
$[-n,0]$ and (ii) no non-zero homogeneous harmonic $1$-form on $C^{{\rm reg}}$
has order $0.$

###### Proof.

Since $C^{{\rm reg}}$ has non-negative Ricci curvature it follows that
$\mathop{\rm Ric}\nolimits g^{{\rm lk}}\geqslant(n-2)g^{{\rm lk}}$ on $C^{{\rm
lk}}.$ So $b^{1}(C^{{\rm lk}})=0$ and the claim is equivalent to no non-
constant homogeneous harmonic $0$-form on $C^{{\rm reg}}$ has order in
$[1-n,1].$ To prove this let the cone have a non-constant order-$\alpha$
homogeneous harmonic $0$-form on $C^{{\rm reg}}.$ Using (2.2) with
$\phi^{\prime\prime}=0$ we see then that $\alpha(\alpha+n-2)$ is an eigenvalue
of the $0$-form Laplacian over $C^{{\rm lk}}.$ Hence we get, according to
Lichnerowicz and Obata, $\alpha(\alpha+n-2)>n-1;$ the inequality is strict
because $C$ is strictly singular. This implies either $\alpha>1$ or
$\alpha<1-n,$ proving (i).

Suppose now that $\phi$ is a homogeneous harmonic $1$-form on $C^{{\rm reg}}$
of order in $0.$ Cheeger–Tian [8, Theorem 7.27] prove that $\phi={\rm d}\psi$
for some harmonic function $\psi$ of order $1.$ In particular, $\phi$ is
closed and co-closed; and (i) implies therefore $\phi=0,$ proving (ii). ∎

We turn now to the treatment of Kähler cones.

######  Definition 2.10.

Let $C$ be a Riemannian $2n$-cone and $\omega$ an order-$0$ homogeneous
$2$-form on $C^{{\rm reg}}.$ We call $(C,\omega),$ or only $C$ for short, a
Kähler $n$-cone if the Riemannian manifold $(C^{{\rm reg}},{\rm
d}r^{2}+r^{2}g^{{\rm lk}})$ is a Kähler manifold with Kähler form $\omega.$
The link $(C^{{\rm lk}},g^{{\rm lk}})$ is accordingly a Sasakian manifold. The
Kähler $n$-cone $C$ has the structure of a normal complex space [10, Theorem
3.1] which we always give to $C.$ We say that $C$ is strictly singular if this
is not isometric to the flat ${{\mathbb{C}}}^{n}$ with the standard complex
structure.

######  Remark 2.11.

The last condition is equivalent to $C$ strictly singular as a Riemannian
cone, which may be proved as follows. If $C$ is isometric to the flat
${\mathbin{\mathbb{R}}}^{2n}$ then the complex structure on $C^{{\rm reg}}$
will be parallel with respect to the flat metric so we can identify
${\mathbin{\mathbb{R}}}^{2n}$ with ${{\mathbb{C}}}^{n}.$ Hence it follows that
$C$ is isometric to ${\mathbin{\mathbb{R}}}^{n}$ if and only if isometric to
${{\mathbb{C}}}^{n}.$ Accordingly $C$ is strictly singular as a Riemannian
cone if and only if strictly singular as a Kähler cone.

We come to the main results of this section.

######  Proposition 2.12.

Let $C$ be a Kähler $n$-cone and $p,q\geqslant 0$ integers with $p+q>n.$ Then
no non-zero homogeneous harmonic $(p,q)$ form on $C^{{\rm reg}}$ has order in
$(-p-q,p+q-2n).$

###### Proof.

Put $l:=\log r.$ Define a ${{\mathbb{C}}}$-vector sub-bundle $E\subset
T^{*}C^{{\rm reg}}\otimes_{\mathbin{\mathbb{R}}}{{\mathbb{C}}}$ by the
orthogonal decomposition

$T^{*}C^{{\rm
reg}}\otimes_{\mathbin{\mathbb{R}}}{{\mathbb{C}}}={{\mathbb{C}}}\partial
l\oplus{{\mathbb{C}}}{\bar{\partial}}l\oplus E={{\mathbb{C}}}{\rm
d}l\oplus{{\mathbb{C}}}{\rm d}^{c}l\oplus E.$ (2.10)

Note that $J$ acts upon $E$ and denote by $E^{10},E^{01}\subset E$ the sub-
bundles with eigenvalues $i,-i$ respectively. For
$p,q\in{\mathbin{\mathbb{Z}}}$ put
$E^{pq}:={\textstyle{\bigwedge}}^{p}E^{10}\otimes_{{\mathbb{C}}}{\textstyle{\bigwedge}}^{q}E^{01}$
so that

${\textstyle{\bigwedge}}^{pq}C^{{\rm reg}}=E^{pq}\oplus({{\mathbb{C}}}\partial
l\otimes E^{p-1,q})\oplus({{\mathbb{C}}}{\bar{\partial}}l\otimes
E^{p,q-1})\oplus({{\mathbb{C}}}(\partial l\wedge{\bar{\partial}}l)\otimes
E^{p-1,q-1}).$ (2.11)

Suppose now that $\phi=r^{\beta}(\phi^{\prime}+{\rm d}\log
r\wedge\phi^{\prime\prime})$ is a homogeneous harmonic $(p,q)$ form on
$C^{{\rm reg}}$ with $\beta\in(0,2p+2q-2n);$ here
$\phi^{\prime},\phi^{\prime\prime}$ are forms on the link $C^{{\rm lk}}.$ We
prove that $\phi$ vanishes. Write

$r^{-\beta}\phi=\phi^{pq}+2\partial
l\wedge\phi^{p-1,q}+2{\bar{\partial}}l\wedge\phi^{p,q-1}+2i\partial
l\wedge{\bar{\partial}}l\wedge\phi^{p-1,q-1}$ (2.12)

according to (2.11). Since $2\partial l={\rm d}l+i{\rm d}^{\rm c}l,$
$2{\bar{\partial}}l={\rm d}l-i{\rm d}^{\rm c}l$ and $2i\partial
l\wedge{\bar{\partial}}l={\rm d}l\wedge{\rm d}^{\rm c}l$ it follows then that

$\displaystyle\phi^{\prime}:=$ $\displaystyle\phi^{pq}+i{\rm d}^{\rm
c}l\wedge\phi^{p-1,q}-i{\rm d}^{\rm c}l\wedge\phi^{p,q-1},$ (2.13)
$\displaystyle\phi^{\prime\prime}:=$
$\displaystyle\phi^{p-1,q}+\phi^{p,q-1}+{\rm d}^{\rm c}l\wedge\phi^{p-1,q-1}.$
(2.14)

Corollary 2.5 implies then that ${\rm
d}\phi^{\prime\prime}=(2+2p+2q-2n-\beta)\phi^{\prime}.$ Thus

$(2+2p+2q-2n-\beta)\phi^{\prime}={\rm d}\phi^{\prime\prime}={\rm
d}(\phi^{p-1,q}+\phi^{p,q-1})-{\rm d}^{\rm c}l\wedge{\rm d}\phi^{p-1,q-1}+{\rm
d}{\rm d}^{\rm c}l\wedge\phi^{p-1,q-1}.$ (2.15)

On the other hand, we can show by computation that

${\rm d}{\rm d}^{\rm c}l=\frac{2}{r^{2}}\left({\rm d}r\wedge J{\rm
d}r+\frac{1}{4}{\rm d}{\rm d}^{\rm c}r^{2}\right).$ (2.16)

Since $\frac{1}{4}{\rm d}{\rm d}^{\rm c}r^{2}$ is the Kähler form on $C^{{\rm
reg}}$ it follows that $\frac{1}{4}{\rm d}{\rm d}^{\rm
c}r^{2}(\partial_{r},J\partial_{r})=1$ and that ${\rm d}{\rm d}^{\rm
c}l(\partial_{r},J\partial_{r})=\frac{4}{r^{2}}\neq 0.$ But $\phi^{\prime}$
vanishes in the component ${{\mathbb{C}}}{\rm d}l\otimes{{\mathbb{C}}}{\rm
d}^{\rm c}l\otimes E^{p-1,q-1}$ and (2.15) implies then $\phi^{p-1,q-1}=0.$
Returning to (2.14) we see now that
$\phi^{\prime\prime}=\phi^{p-1,q}+\phi^{p,q-1}.$ So ${\rm
d}\phi^{\prime\prime}=(2+2p+2q-2n-\beta)\phi^{\prime}$ vanishes in the
component ${{\mathbb{C}}}{\rm d}^{\rm c}l\otimes(E^{p-1,q}\oplus E^{p,q-1});$
that is, $\phi^{p-1,q}=\phi^{p,q-1}=0$ and summing up these we find
$\phi^{\prime\prime}=0.$ Now $(2+2p+2q-2n-\beta)\phi^{\prime}={\rm
d}\phi^{\prime\prime}=0.$ Since $2+2p+2q-2n-\beta\neq 0$ it follows then that
$\phi^{\prime}=0.$ Thus $\phi=0.$ ∎

######  Corollary 2.13 (Theorem 1.8).

Let $C$ be a Kähler $n$-cone and $p,q\geqslant 0$ integers with
$p+q\leqslant\nobreak n-1.$ Then no non-zero homogeneous harmonic $(p,q)$ form
on $C^{{\rm reg}}$ has order in $(p+q-2n,-p-q).$

###### Proof.

Put $s:=n-p$ and $t:=n-q.$ Let $\phi$ be a homogeneous harmonic $(p,q)$ form
on $C^{{\rm reg}}$ of order $\in(p+q-2n,-p-q)=(-s-t,s+t-2n).$ Proposition 2.12
applies then to the Hodge dual $*\phi,$ which thus vanishes; and accordingly
so does $\phi.$ ∎

The following lemma will be needed only at the end of §4.

######  Lemma 2.14.

Let $(C;g,\omega)$ be a Kähler $n$-cone and $\phi$ a holomorphic $p$-form on
$C$ near the vertex ${\rm vx}.$ Then the norm $|\phi|_{g}$ computed with
respect to $g$ is bounded near ${\rm vx}\in C;$ that is, there exists a
neighbourhood $U\subseteq C$ of ${\rm vx}$ with $\sup_{U\setminus\\{{\rm
vx}\\}}|\phi|^{2}_{g}<\infty.$

###### Proof.

Recall that $C$ has a holomorphic ${{\mathbb{C}}}^{*}$ action which extends
the dilation by $(0,\infty).$ Van Coevering [10, Theorem 3.1] proves that
there exists a ${{\mathbb{C}}}^{*}$ equivariant embedding
$C\to{{\mathbb{C}}}^{k}$ where $t\in{{\mathbb{C}}}^{*}$ acts upon
${{\mathbb{C}}}^{k}$ by
$t\cdot(x_{1},\dots,x_{k})\to(t^{\nu_{1}}x_{1},\dots,t^{\nu_{k}}x_{k})$ for
some integers $\nu_{1},\dots,\nu_{k}\geqslant 1.$ Denote by $h:={\rm
d}x_{1}{\rm d}\bar{x}_{1}+\dots+{\rm d}x_{k}{\rm d}\bar{x}_{k}$ the flat
metric on ${{\mathbb{C}}}^{k}.$ Let $A\subseteq C^{{\rm reg}}$ be a
neighbourhood of $\\{1\\}\times C^{{\rm lk}}\subset C^{{\rm reg}}.$ Since
$\nu_{1},\dots,\nu_{k}\geqslant 1$ it follows then that
$t_{*}(h|_{A})\geqslant t^{-2}h|_{tA}.$ On the other hand, $h|_{A}$ is
equivalent to $g|_{A}$ so that there exists $M>0$ independent of $t$ and such
that $t_{*}(g|_{A})\geqslant Mt^{-2}h|_{tA}.$ Since $t_{*}g=t^{-2}g$ it
follows then that $g|_{tA}\geqslant Mh|_{tA}.$ As $t$ is arbitrary we have
$g\geqslant Mh$ or $g_{ab}\geqslant Mh_{ab}$ in the index notation. So
$g^{ab}\leqslant\nobreak M^{-1}h^{ab}$ and $|\phi|_{g}\leqslant\nobreak
M^{-p}|\phi|_{h}$ as $\phi$ is a $p$-form. Since $\phi$ is defined on the
vertex it is clear that $\sup|\phi|_{h}<\infty$ for some neighbourhood
$U\subseteq C$ of ${\rm vx}.$ Hence it follows that $\sup_{U\setminus\\{{\rm
vx}\\}}|\phi|^{2}_{g}<\infty.$ ∎

We define Calabi–Yau cones now.

######  Definition 2.15.

A Calabi–Yau $n$-cone is a Kähler $n$-cone $(C,\omega)$ such that $C^{{\rm
reg}}$ has an order-$0$ homogeneous holomorphic $(n,0)$ form $\Omega$ such
that
$\frac{\omega^{n}}{n!}=(-1)^{\frac{n(n-1)}{2}}(\frac{i}{2})^{n}\Omega\wedge\overline{\Omega}.$

######  Remark 2.16.

Let $\Omega$ be as above and denote by $\nabla$ the Levi-Civita connection of
the Kähler manifold $(C^{{\rm reg}},\omega).$ Since ${\bar{\partial}}\Omega=0$
it follows then that $\nabla\Omega=\phi\otimes\Omega$ for some $(1,0)$ form on
$C^{{\rm reg}}$ and that
$0=\nabla(\Omega\wedge\overline{\Omega})=(\phi+\overline{\phi})\Omega\wedge\overline{\Omega}.$
Thus $\phi+\overline{\phi}=0$ but $\phi$ is a $(1,0)$ form so $\phi=0;$ that
is, $\nabla\Omega=0.$ This implies that the induced connection on the
canonical bundle of $C^{{\rm reg}}$ is flat and that $(C^{{\rm reg}},\omega)$
is Ricci-flat manifold. The link $C^{{\rm lk}}$ is accordingly a
Sasaki–Einstein manifold of Einstein constant $2n-2,$ which is positive for
$n>1.$ The Myers theorem implies then that $C^{{\rm lk}}$ has finite
fundamental group. The equation $\nabla\Omega=0$ implies also that the
holonomy group of $(C^{{\rm reg}},\omega)$ is contained in
$\operatorname{SU}(n).$ We prove more in the following proposition.

We look into the holonomy group of $C^{{\rm reg}}.$

######  Proposition 2.17.

Let $C$ be a Calabi–Yau $n$-cone. Then one of the following holds: (i) the
universal cover of $C^{{\rm reg}}$ is the flat
${{\mathbb{C}}}^{n}\setminus\\{0\\};$ (ii) $C^{{\rm reg}}$ has either holonomy
$\operatorname{SU}(n);$ or (iii) $C^{{\rm reg}}$ has restricted holonomy
$\operatorname{Sp}(\frac{n}{2})$ with $n\in 2{\mathbin{\mathbb{Z}}}.$

###### Proof.

Denote by $C^{\prime}$ the universal cover of $C^{{\rm reg}}$ and give this
the pull-back of the cone metric ${\rm d}r^{2}+r^{2}g^{{\rm lk}}.$ Gallot [13,
Proposition 3.1] proves that if the holonomy representation of $C^{\prime}$ is
reducible then (i) above holds (which is in fact true for any Riemannian cones
not necessarily of even dimension). Suppose therefore that the holonomy
representation of $C^{\prime}$ is irreducible. De Rham’s theorem implies then
that $C^{\prime}$ is an irreducible Riemannian manifold [22, Theorem 3.2.7].
Note that $C^{\prime}$ is not flat but Ricci-flat. We use now the fact that
such Riemannian manifolds are not homogeneous spaces [1]; see also Besse [4,
Theorem 7.61]. Thus $C^{\prime}$ is non-homogeneous and accordingly non-
symmetric. There is a theorem [22, Theorem 3.4.1] (due to Berger and others)
which classifies the holonomy groups of simply-connected irreducible non-
symmetric Riemannian manifolds. Recall from Remark 2.16 that the holonomy
group $G^{\prime}$ of $C^{\prime}$ is contained in $\operatorname{SU}(n).$ The
classification result implies then that $G^{\prime}$ is either
$\operatorname{SU}(n)$ or $\operatorname{Sp}(\frac{n}{2})$ with $n\in
2{\mathbin{\mathbb{Z}}}.$ The holonomy group $G$ of the original $C^{{\rm
reg}}$ contains this $G^{\prime}.$ Suppose first that
$G^{\prime}=\operatorname{SU}(n).$ Recall again from Remark 2.16 that
$G\leqslant\nobreak\operatorname{SU}(n).$ Thus $G=\operatorname{SU}(n).$
Otherwise $G^{\prime}=\operatorname{Sp}(\frac{n}{2})$ with $n\in
2{\mathbin{\mathbb{Z}}},$ in which case (iii) holds as $G^{\prime}$ is the
restricted holonomy group of $C^{{\rm reg}}$ (which may also be defined as the
identity component of the full holonomy group $G$). ∎

We recall the algebraic geometry facts we shall need. We begin with

######  Definition 2.18.

Let $(X,x)$ be the germ of a complex space and $q$ an integer. We say that
$(X,x)$ has depth $\geqslant q$ if the local cohomology group
$H^{p}_{x}(X,{\mathbin{\cal O}}_{X})$ vanishes for $p\leqslant\nobreak q-1.$

######  Example 2.19.

Let $(X,x)$ be the germ of a complex space of pure dimension $n.$ If this is
normal with $n\geqslant 2$ then it has depth $\geqslant 2.$ It is
Cohen–Macaulay if and only if it has depth $\geqslant n$ [20, Proposition
3.5.13].

We prove then

######  Proposition 2.20.

Let $C$ be a Calabi–Yau $n$-cone. Then (i) $(C,{\rm vx})$ is Cohen–Macaulay
and (ii) if $n\geqslant 3$ then for every Stein neighbourhood $U\subseteq C$
we have $H^{q}(U\setminus\\{{\rm vx}\\},{\mathbin{\cal O}}_{C})=0$ for
$q=1,\cdots,n-2.$

###### Proof.

Since $\Omega\wedge\overline{\Omega}$ has finite integral near the vertex it
follows according to Burns [5, Proposition 3.2] that the vertex is a rational
singularity. It is known that rational singularities are Cohen–Macaulay [20,
Theorem 6.2.14]. For (ii) we use the local cohomology exact sequence
$H^{q}_{\rm vx}(C,{\mathbin{\cal O}}_{C})\to H^{q}(U,{\mathbin{\cal
O}}_{C})\to H^{q}(U\setminus\\{{\rm vx}\\},{\mathbin{\cal O}}_{C})\to
H^{q+1}_{\rm vx}(C,{\mathbin{\cal O}}_{C}).$ Part (i) implies then an
isomorphism $H^{q}(U\setminus\\{{\rm vx}\\},{\mathbin{\cal O}}_{C})\cong
H^{q}(U,{\mathbin{\cal O}}_{C})$ which thus vanishes because $U$ is Stein. ∎

######  Remark 2.21.

Since $C$ is normal it follows also that $\Omega^{n}_{C^{{\rm reg}}}$ extends
to $C$ as an invertible sheaf so that $C$ is 1-Gorenstein [20, Definition
6.2.1]. Rational is therefore equivalent to canonical [20, Corollary 6.2.15].

##  3 Kähler Conifolds

The following definition is equivalent to that of Chan [6, Definition 4.6]
except the difference in presentation. Our definition of Riemannian conifolds
is the same as those of Hein–Sun [18, Definition 2.2] and Joyce [21,
Definition 2.1].

######  Definition 3.1.

If $C$ is a Riemmanian $n$-cone then a punctured neighbourhood of the vertex
${\rm vx}\in C$ is the set $U\setminus\\{\rm vx\\}$ where $U$ is some ordinary
neighbourhood of ${\rm vx}\in C.$ For another topological space $X$ and a
point $x\in X$ we define in the same way the punctured neighbourhoods of $x\in
X.$

Let $(X^{{\rm reg}},g)$ be a Riemannian $n$-manifold and $X$ its metric
completion. Suppose that $X$ is compact and that $X^{\rm sing}:=X\setminus
X^{{\rm reg}}$ is finite. We call $(X,g)$ a Riemannian $n$-conifold if for
every $x\in X^{\rm sing}$ we can choose a Riemannian $n$-cone $(C_{x},g_{x})$
such that there exist a punctured neighbourhood in $C_{x}^{{\rm reg}}$ of the
vertex, a punctured neighbourhood in $X^{{\rm reg}}$ of $x,$ and a
diffeomorphism between these two under which the two Riemannian metrics
$g,g_{x}$ approach each other with order $>0;$ that is, there exists
$\epsilon>0$ such that for $k=0,1,2,\cdots$ we have
$|\nabla^{k}(g-g_{x})|=O(r^{\epsilon})$ where $r$ is the radius function on
$C_{x}^{{\rm reg}}$ and where $\nabla,|\,\,|$ are computed with respect to the
cone metric ${\rm d}r^{2}+r^{2}g^{{\rm lk}}.$ The estimate
$|g-g_{x}|=O(r^{\epsilon})$ implies that the diffeomorphism between the two
punctured neighbourhoods extend uniquely to a homeomorphism between the full
neighbourhoods; and that $(C_{x},g_{x})$ is unique up to isomorphism, which we
call the tangent cone to $X$ at $x.$ We say that a Riemannian conifold $(X,g)$
is strictly singular if this has at least one strictly-singular tangent cone;
otherwise $g$ extends smoothly to $X^{\rm sing}$ (see for instance Bando,
Kasue and Nakajima [3, Theorem 5.1]).

Suppose now that $(X^{{\rm reg}},g)$ is a Kähler manifold of complex dimension
$n$ with Kähler form $\omega$ and that every tangent cone $(C_{x},g_{x})$ a
Kähler $n$-cone with Kähler form $\omega_{x}.$ We call $(X;g,\omega),$ or only
$X$ for short, a Kähler $n$-conifold if there exist a germ biholomorphism
$(C_{x},{\rm vx})\cong(X,x)$ under which $\omega$ approaches $\omega_{x}$ with
order $>0$ at the vertex; that is, there exists $\epsilon>0$ such that for
$k=0,1,2,\cdots$ we have $|\nabla^{k}(\omega-\omega_{x})|=O(r^{\epsilon-k}).$
We say that a Käler conifold $(X,g)$ is strictly singular if this has at least
one strictly-singular tangent cone.

Suppose finally that $X$ has a nowhere-vanishing holomorphic $(n,0)$ form
$\Omega$ such that on $X^{{\rm reg}}$ we have
$\frac{\omega^{n}}{n!}=(-1)^{\frac{n(n-1)}{2}}\frac{i^{n}}{2^{n}}\Omega\wedge\overline{\Omega},$
and that every $(C_{x};g_{x},\omega_{x})$ is a Calabi–Yau cone with $(n,0)$
form $\Omega_{x}.$ We call $(X;\omega,\Omega),$ or only $X$ for short, a
Calabi–Yau $n$-conifold if for every $x\in X^{\rm sing}$ there exist a germ
biholomorphism $(C_{x},{\rm vx})\cong(X,x)$ under which $\Omega$ approaches
$\Omega_{x}$ with order $>0$ at the vertex; that is, there exists $\epsilon>0$
such that for $k=0,1,2,\cdots$ we have
$|\nabla^{k}(\Omega-\Omega_{x})|=O(r^{\epsilon-k}).$

######  Remark 3.2.

Let $(X;\omega,\Omega)$ be a Calabi–Yau conifold with some $x\in X^{\rm sing}$
whose tangent cone $C_{x}$ has holonomy group $\operatorname{SU}(n).$ The
semi-continuity of holonomy groups implies then that points near $x$ have
holonomy group $\operatorname{SU}(n).$ So the connected component containing
them has holonomy $\operatorname{SU}(n).$

We say that a complex $n$-form $\Omega$ on a $2n$-manifold $Y$ is a complex
volume form on $Y$ if there exists a complex structure on $Y$ with respect to
which $\Omega$ is a nowhere-vanishing holomorphic $(n,0)$ form. We prove the
following theorem. The reader in a hurry can proceed safely to Corollary 3.6
because the proof will not be needed for the later treatment.

######  Theorem 3.3.

Let $(C,g)$ be a Riemannian $2n$-cone, $\Omega_{0}$ an order-$0$ homogeneous
complex volume form and $\Omega$ a complex volume form which approaches
$\Omega_{0}$ with order $>0$ at the vertex ${\rm vx}\in C.$ Then there exists
a diffeomorphism $f$ between punctured neighbourhoods in $C^{{\rm reg}}$ of
${\rm vx}$ which differs from the identity only by $O(r^{1+\epsilon})$ and
such that $f^{*}\Omega=\Omega_{0}.$

###### Proof.

We use Moser’s technique for complex volume forms (as Goto [16, Proposition
1.12] does on compact manifolds). We define a family
$(\Omega_{t})_{t\in[0,1]}$ of complex volume forms and a family
$(v_{t})_{t\in[0,1]}$ of vector fields such that $\frac{{\rm
d}\Omega_{t}}{{\rm d}t}+{\rm d}(v_{t}\lrcorner\Omega_{t})=0.$ This is not so
simple as the Moser technique in symplectic geometry because pointwise complex
volume forms need not form an open set in
${\textstyle{\bigwedge}}^{n}_{\mathbin{\mathbb{R}}}{{\mathbb{C}}}^{n}$ whereas
symplectic forms do in
${\textstyle{\bigwedge}}^{2}_{\mathbin{\mathbb{R}}}{{\mathbb{C}}}^{n}.$ We
define $\Omega_{t}$ by re-scaling $\Omega$ in a careful way.

Define a smooth function
$\lambda:{\mathbin{\mathbb{R}}}\to{\mathbin{\mathbb{R}}}$ by
$\lambda(t):=\exp\left(1-\frac{1}{t}\right)$ for $t>0$ and $\lambda(t):=0$ for
$t\leqslant\nobreak 0.$ For ${\tau}>0$ we denote by the same ${\tau}$ the
dilation map $C^{{\rm reg}}\to C^{{\rm reg}}$ defined by
$(r,s)\mapsto({\tau}r,s)$ under the identification $C^{{\rm
reg}}=(0,\infty)\times C^{{\rm lk}}.$ For $t\in(0,1]$ define
$\Omega_{t}:=\lambda(t)^{-m}\lambda(t)^{*}\Omega$ where $\lambda(t)^{*}$ is
the pull-back by the dilation map $\lambda(t):C^{{\rm reg}}\to C^{{\rm reg}}.$
It is clear that $\Omega_{t}$ is smooth for $t\in(0,1]$ and we prove

######  Lemma 3.4.

$\Omega_{t}$ is $C^{1}$ with respect to $t\in[0,1]$ including $t=0.$

###### Proof.

We begin by showing that the first derivative $\frac{{\rm d}\Omega_{t}}{{\rm
d}t}$ exists and vanishes at $t=0.$ By hypothesis that there exists
$\epsilon>0$ such that $|\Omega-\Omega_{0}|=O(r^{\epsilon})$ with respect to
the cone metric on $C^{{\rm reg}}=(0,\infty)\times C^{{\rm lk}}.$ Introduce
now the cylinder metric $({\rm d}\log r)^{2}+g^{{\rm lk}},$ with respect to
which we have then $|\Omega-\Omega_{0}|=O(r^{n+\epsilon}).$ So
$|\lambda(t)^{*}(\Omega-\Omega_{0})|=O(\lambda(t)^{n+\epsilon}r^{n+\epsilon}).$
Since $\Omega_{0}$ is homogeneous of order $0$ it follows that
$\Omega_{t}-\Omega_{0}=\lambda(t)^{-n}\lambda(t)^{*}(\Omega-\Omega_{0})$ and
hence that

$\left|\frac{\Omega_{t}-\Omega_{0}}{t}\right|=\frac{\lambda(t)^{\epsilon}}{t}O(r^{n+\epsilon})$
(3.1)

with respect to the cylinder metric. The definition of $\lambda(t)$ implies
therefore that (3.1) converges to $0$ as $t$ tends to $0.$ Thus $\frac{{\rm
d}\Omega_{t}}{{\rm d}t}$ exists and vanishes at $t=0.$

We show then that $\frac{{\rm d}\Omega_{t}}{{\rm d}t}$ is continuous at $t=0.$
Put $l:=\log r$ and $\Omega-\Omega_{0}=:{\rm d}l\wedge A^{l}+B^{l}$ where
$A^{l},B^{l}$ are smooth families of $(n-1)$ forms and $n$-forms respectively
both defined on $C^{{\rm lk}}.$ Put $\partial_{l}=\frac{\partial}{\partial
l}.$ Since $\Omega-\Omega_{0}$ vanishes at the vertex with order $n+\epsilon$
with respect to the cylinder metric it follows that for $k=0,1,2,\cdots$ we
have $|\partial_{l}^{k}A^{l}|=|\partial_{l}^{k}B^{l}|=O(r^{(n+\epsilon)})$ so
that

$|\partial_{l}^{k}A^{l+1-\frac{1}{t}}|=|\partial_{l}^{k}B^{l+1-\frac{1}{t}}|=O(\lambda(t)^{n+\epsilon}r^{(n+\epsilon)})$
(3.2)

with respect to the cylinder metric. On the other hand,

$\displaystyle\frac{{\rm d}\Omega_{t}}{{\rm d}t}$ $\displaystyle=\frac{{\rm
d}}{{\rm d}t}(\Omega_{t}-\Omega_{0})=\frac{{\rm d}}{{\rm
d}t}\lambda(t)^{-n}\lambda(t)^{*}(\Omega-\Omega_{0})=\frac{{\rm d}}{{\rm
d}t}\lambda(t)^{-n}({\rm d}l\wedge A^{l+1-\frac{1}{t}}+B^{l+1-\frac{1}{t}})$
$\displaystyle=\frac{1}{t^{2}}\lambda(t)^{-n}(n{\rm d}l\wedge
A^{l+1-\frac{1}{t}}+nB^{l+1-\frac{1}{t}}-{\rm
d}l\wedge\partial_{l}A^{l+1-\frac{1}{t}}-\partial_{l}B^{l+1-\frac{1}{t}}).$

Hence using (3.2) with $k=0,1$ we see that with respect to the cylinder metric
we have

$\left|\frac{{\rm d}\Omega_{t}}{{\rm
d}t}\right|=\frac{\lambda(t)^{\epsilon}}{t^{2}}O(r^{n+\epsilon})$ (3.3)

which converges to $0$ as $t$ tends to $0.$ Since $\frac{{\rm
d}\Omega_{t}}{{\rm d}t}$ vanishes at $t=0$ it follows therefore that
$\frac{{\rm d}\Omega_{t}}{{\rm d}t}$ is continuous at $t=0,$ which completes
the proof. ∎

######  Remark 3.5.

Putting $\Omega_{t}^{\prime}:=\frac{{\rm d}\Omega_{t}}{{\rm d}t}$ we have

$\left|\frac{\Omega_{t}^{\prime}-\Omega_{0}^{\prime}}{t}\right|=\left|\frac{\Omega_{t}^{\prime}}{t}\right|=\frac{\lambda(t)^{\epsilon}}{t^{3}}O(r^{n+\epsilon})$
(3.4)

so $\frac{{\rm d}^{2}\Omega_{t}}{{\rm d}t^{2}}$ exists and vanishes at $t=0.$
It is also likely that the higher order estimates hold. But we shall not need
these.

Fix $t\in[0,1]$ and put
$\alpha_{t}:=\frac{\partial}{\partial_{r}}\lrcorner\frac{{\rm
d}\Omega_{t}}{{\rm d}t}$ which is an $(n-1)$ form on a punctured neighbourhood
of ${\rm vx}\in C.$ From (3.3) we see then that $\frac{{\rm d}\Omega_{t}}{{\rm
d}t}$ vanishes ($t$-uniformly) at the vertex with order $>n+\epsilon$ with
respect to the cylinder metric and with order $\epsilon$ with respect to the
cone metric. So $\alpha_{t}$ has order $>\epsilon$ with respect to the cone
metric.

Put $\frac{{\rm d}\Omega_{t}}{{\rm d}t}:={\rm d}r\wedge\alpha_{t}+\beta_{t}$
for some $n$-form $\beta_{t}$ which contains no ${\rm d}r.$ Since $\Omega_{t}$
is closed it follows that so is $\frac{{\rm d}\Omega_{t}}{{\rm d}t}$ and hence
that ${\rm d}^{{\rm lk}}\alpha_{t}=\frac{\partial\beta_{t}}{\partial r}.$ This
implies that

$\frac{{\rm d}\Omega_{t}}{{\rm d}t}={\rm d}\left(\int_{0}^{r}\alpha_{t}{\rm
d}r\right)$ (3.5)

where the integral is well defined because $\alpha_{t}$ has order $>0$ with
respect to the cone metric.

Note that $\Omega_{t}$ is an $(n,0)$ form with respect to the complex
structure $J$ determined by $\Omega_{t}.$ So $\frac{{\rm d}\Omega_{t}}{{\rm
d}t}$ is an $(n,0)$ form too with respect to $J.$ Accordingly $\alpha_{t}$ is
an $(n-1,0)$ form with respect to $J$ and so is $-(\int_{0}^{r}\alpha_{t}{\rm
d}r).$ Hence we get on a punctured neighbourhood of ${\rm vx}\in C^{{\rm
reg}}$ a unique vector field $v_{t}$ such that
$v_{t}\lrcorner\Omega_{t}=-(\int_{0}^{r}\alpha_{t}{\rm d}r).$ Since
$\alpha_{t}$ has order $\epsilon$ with respect to the cone metric it follows
that the integral term and $v_{t}$ have order $1+\epsilon$ with respect to the
cone metric. So the time-one flow of $v_{t},$ which we denote by $f,$ is a
diffeomorphism between punctured neighbourhoods in $C^{{\rm reg}}$ of vx that
differs from the identity only by $O(r^{1+\epsilon})$ with respect to the cone
metric. Recall from (3.5) that $\frac{{\rm d}\Omega_{t}}{{\rm d}t}={\rm
d}(\int_{0}^{r}\alpha_{t}{\rm d}r)=-{\rm d}(v_{t}\lrcorner\Omega_{t}).$ Hence
it follows by Moser’s technique that $f^{*}\Omega=\Omega_{0},$ which completes
the proof. ∎

######  Corollary 3.6.

Every compact Calabi–Yau conifold $(X;\omega,\Omega)$ has the structure of a
normal complex space such that for every tangent cone
$(C_{x};\omega_{x},\Omega_{x})$ at $x\in X^{\rm sing}$ there exists a germ
biholomorphism $(X,x)\cong(C_{x},{\rm vx})$ under which $\Omega,\Omega_{x}$
agree entirely and $\omega,\omega_{x}$ agree up to order $>0.$

######  Remark 3.7.

Chan [6, Theorem 4.9] uses Moser’s technique for symplectic forms and proves
that there exists a diffeomorphism between punctured neighbourhoods of $x\in
X$ and ${\rm vx}\in C_{x}$ under which $\omega,\omega_{x}$ agree entirely and
$\Omega,\Omega_{x}$ agree up to order $>0.$

Given Calabi–Yau conifolds we always make them into normal complex spaces by
using Corollary 3.6. We show that these are Kähler spaces. We begin by
recalling the relevant definitions.

######  Definition 3.8.

Let $X$ be a reduced complex space. We say that $X$ is projective if $X$ may
be embedded into some ${\mathbb{CP}}^{n}.$ By a Kähler form on $X$ we mean a
Kähler form on the regular locus $X^{{\rm reg}}$ such that for every singular
point $x\in X^{\rm sing}$ there exist a neighbourhood $U\subseteq X$ of $x,$
an embedding $f:U\to{{\mathbb{C}}}^{n}$ mapping $x$ to
$0\in{{\mathbb{C}}}^{n},$ and a smooth strictly plurisubharmonic function
$\phi:f(U)\to{\mathbin{\mathbb{R}}}$ such that $f_{*}\omega={\rm d}{\rm
d}^{c}\phi|_{f(U)\setminus 0}.$ Here strictly plurisubharmonic means that
there exist an open set $V\subseteq{{\mathbb{C}}}^{n}$ and a smooth function
$\psi:V\to{\mathbin{\mathbb{R}}}$ such that $\psi|_{f(U)}=\phi$ and ${\rm
d}{\rm d}^{\rm c}\psi$ is positive definite at every point of $V.$ We call $X$
a Kähler space if $X$ has Kähler forms.

Kähler forms extend automatically in the following circumstances:

######  Lemma 3.9 (Fujiki [12, Lemma 1]).

Let $X$ be a normal complex space with a unique singular point $x.$ Let
$\phi:X^{{\rm reg}}\to{\mathbin{\mathbb{R}}}$ be a smooth strictly
plurisubharmonic function. Then for any sufficiently small neighbourhood
$U\subseteq X$ of $x$ there exists on $X$ some Kähler form which agrees with
${\rm d}{\rm d}^{\rm c}\phi$ outside $U.$ ∎

######  Corollary 3.10.

Compact Calabi–Yau conifolds are Kähler complex spaces. ∎

We recall also

######  Theorem 3.11 (Grauert[15, §3, Satz3]).

Let $X$ be a compact normal complex space with a Kähler form $\omega$ such
that the de Rham cohomology class $[\omega|_{X^{{\rm reg}}}]\in H^{2}(X^{{\rm
reg}},{\mathbin{\mathbb{R}}})$ lies in the image of the natural homomorphism
$H^{2}(X^{{\rm reg}},{\mathbin{\mathbb{Q}}})\to H^{2}(X^{{\rm
reg}},{\mathbin{\mathbb{R}}}).$ Then $X$ is projective. ∎

We end with

######  Lemma 3.12.

Let $X$ be a compact normal Kähler space whose singularities are rational.
Then for $p\in{\mathbin{\mathbb{Z}}}$ the ${{\mathbb{C}}}$-vector spaces
$H^{0}(X^{{\rm reg}},\Omega^{p}_{X})$ and $H^{p}(X,{\mathbin{\cal O}}_{X})$
have the same dimension.

###### Proof.

Let $Y\to X$ be a resolution of singularities by successive blow-ups. Since
the singularities of $X$ are rational we get then a ${{\mathbb{C}}}$-vector
space isomorphism $H^{p}(X,{\mathbin{\cal O}}_{X})\cong H^{p}(Y,{\mathbin{\cal
O}}_{Y}).$ On the other hand, as $Y$ is Kähler [11, Lemma 4.4] the
${{\mathbb{C}}}$-vector spaces $H^{p}(Y,{\mathbin{\cal
O}}_{Y}),H^{0}(Y,\Omega^{p}_{Y})$ have the same dimension. Kebekus–Schnell
[23, Corollary 1.7] prove that $H^{0}(Y,\Omega^{p}_{Y})$ is isomorphic to
$H^{0}(X^{{\rm reg}},\Omega^{p}_{X}),$ which completes the proof. ∎

##  4 Proof of the Vanishing Theorems

We begin by defining weighted Sobolev spaces and exceptional values.

######  Definition 4.1.

Let $(X,g)$ be a compact Kähler $n$-conifold. For
$p,q\in{\mathbin{\mathbb{Z}}}$ denote by $\Lambda^{p,q}_{X^{{\rm reg}}}$ the
sheaf over $X^{{\rm reg}}$ of $C^{\infty}$ $(p,q)$ forms. For
$s\in{\mathbin{\mathbb{Z}}}$ put $\Lambda^{s}_{X^{{\rm
reg}}}:=\bigoplus_{p+q=s}\Lambda^{p,q}_{X^{{\rm reg}}}$ Choose a smooth
function $\rho:X^{{\rm reg}}\to(0,\infty)$ which agrees near each $x\in X^{\rm
sing}$ with the radius function on $C_{x}^{{\rm reg}}.$ For $k=0,1,2,\dots,$
$\alpha\in{\mathbin{\mathbb{R}}}$ and $p,q\in{\mathbin{\mathbb{Z}}}$ we denote
by $W^{k2}_{\alpha}(\Lambda^{p,q}_{X^{{\rm reg}}})$ the set of $k$ times
weakly differentiable locally-$L^{2}$ $(p,q)$ forms $\phi$’s on $X^{{\rm
reg}}$ with $(|\phi|_{W^{k2}_{\alpha}})^{2}:=\int_{X^{{\rm
reg}}}\sum_{j=0}^{k}\rho^{-2n}|\rho^{-j-\alpha}\nabla^{j}\phi|^{2}{\rm
d}\mu<\infty$ where $|\,\,|,\nabla$ and ${\rm d}\mu$ are computed with respect
to $g.$ For $k=\infty$ we put $W^{\infty 2}_{\alpha}(\Lambda^{p,q}_{X^{{\rm
reg}}}):=\bigcap_{k=0}^{\infty}W^{k2}_{\alpha}(\Lambda^{p,q}_{X^{{\rm
reg}}}).$ For $k=0,1,2,\dots,$ $\alpha\in{\mathbin{\mathbb{R}}}$ and
$p\in{\mathbin{\mathbb{Z}}}$ we denote by
$W^{k2}_{\alpha}(\Lambda^{p}_{X^{{\rm reg}}})$ the set of $k$ times weakly
differentiable locally-$L^{2}$ $p$-forms with finite $W^{k2}_{\alpha}$ norm
defined by the same formula. This is well defined for $X$ a Riemannian
$n$-conifold which need not be Kähler. For $k=\infty$ we put $W^{\infty
2}_{\alpha}(\Lambda^{p}_{X^{{\rm reg}}})$ in the same way.

Define ${\rm d}^{*}$ over $X^{{\rm reg}}$ by using $g.$ We call
$\alpha\in{\mathbin{\mathbb{R}}}$ an exceptional value of the elliptic
operator ${\rm d}+{\rm d}^{*}$ over $X^{{\rm reg}}$ if there exist $x\in
X^{\rm sing}$ and some non-zero order-$\alpha$ homogeneous ${\rm d}$ closed
and co-closed $p$-form on $C_{x}^{{\rm reg}}$ where $C_{x}$ is the tangent
cone to $X$ at $x.$ In the same way we call $\alpha$ an exceptional value of
the Laplacian $\Delta={\rm d}{\rm d}^{*}+{\rm d}^{*}{\rm d}$ if there exists
$x\in X^{\rm sing}$ such that $C_{x}^{{\rm reg}}$ has a non-zero
order-$\alpha$ homogeneous harmonic $(p,q)$ form on $X^{{\rm reg}}.$

We summarize the results we shall need about the operators $\Delta$ and ${\rm
d}+{\rm d}^{*}.$

######  Proposition 4.2.

Let $(X,g)$ be a compact Kähler $n$-conifold, $p,q,k$ integers with
$k\geqslant 2$ and $\alpha,\beta$ real numbers. Then
(i) there exists $c>0$ such that for every $\phi\in
L^{2}_{\alpha}(\Lambda^{p,q}_{X^{{\rm reg}}})$ with $\Delta\phi\in
W^{k-2\,2}_{\alpha-2}(\Lambda^{p,q}_{X^{{\rm reg}}})$ we have $\phi\in
W^{k2}_{\alpha}(\Lambda^{p,q}_{X^{{\rm reg}}})$ and
$|\phi|_{W^{k2}_{\alpha}}\leqslant\nobreak
c(|\Delta\phi|_{W^{k-2\,2}_{\alpha-2}}+|\phi|_{L^{2}_{\alpha}}).$
(ii) The Laplacian $\Delta:W^{k2}_{\alpha}(\Lambda^{p,q}_{X^{{\rm reg}}})\to
W^{k-2\,2}_{\alpha-2}(\Lambda^{p,q}_{X^{{\rm reg}}})$ is Fredholm if and only
if $\alpha$ is not exceptional; and when this holds, each $\phi\in
W^{k-2\,2}_{\alpha-2}(\Lambda^{p,q}_{X^{{\rm reg}}})$ lies in the image of
$\Delta$ if and only if $\int_{X^{{\rm reg}}}(u,v){\rm d}\mu=0$ for every
$v\in L^{2}_{2-\alpha-2n}(\Lambda^{p,q}_{X^{{\rm reg}}})$ with $\Delta v=0.$
(iii) If the closed interval $[\alpha,\beta]$ contains no exceptional values
of $\Delta$ then $\mathop{\rm ker}\Delta_{\alpha}=\mathop{\rm
ker}\Delta_{\beta}.$
(iv) The operator $({\rm d}+{\rm d}^{*})_{\alpha}={\rm d}+{\rm
d}^{*}:W^{k2}_{\alpha}(\Lambda^{p}_{X^{{\rm
reg}}})\to\bigoplus_{p\in{\mathbin{\mathbb{Z}}}}W^{k-2\,2}_{\alpha-2}(\Lambda^{p+1}_{X^{{\rm
reg}}}\oplus\Lambda^{p-1}_{X^{{\rm reg}}})$ is Fredholm if and only if
$\alpha$ is not exceptional. If the closed interval $[\alpha,\beta]$ contains
no exceptional values of ${\rm d}+{\rm d}^{*}$ then $\mathop{\rm ker}({\rm
d}+{\rm d}^{*})_{\alpha}=\mathop{\rm ker}({\rm d}+{\rm d}^{*})_{\beta}.$

###### Proof.

Part (i) is taken from Lockhart–McOwen [24, (2.4)]. They prove in the same
paper [25, Theorem 6.2] that the former part of (ii) holds with
$\alpha\in{{\mathbb{C}}}$ in place of $\alpha\in{\mathbin{\mathbb{R}}}.$
Proposition 2.3 implies then that this holds with
$\alpha\in{\mathbin{\mathbb{R}}}.$ The latter part of (ii) is proved by Joyce
[21, Theorem 2.14] for $p=q=0$ and the proof extends to every $p,q.$ Part
(iii) follows from Lockhart–McOwen [25, Lemma 7.3 and §8]. They prove in the
same paper [25, Theorem 6.2] that the former part of (iv) holds with
$\alpha\in{{\mathbb{C}}}$ in place of $\alpha\in{\mathbin{\mathbb{R}}}.$ As
closed and co-closed forms are harmonic it follows again from Proposition 2.3
that this holds with $\alpha\in{\mathbin{\mathbb{R}}}.$ The latter part of
(iv) follows from Lockhart–McOwen [25, Lemma 7.3 and §8]. ∎

We recall also

######  Theorem 4.3 (Lockhart [24, (0.16)]).

Let $X$ be a compact Riemannian $n$-conifold and $p<\frac{n}{2}$ an integer.
Then every element of $H^{p}(X^{{\rm reg}},{{\mathbb{C}}})$ may be represented
uniquely by some $\phi\in L^{2}_{-n/2}(\Lambda^{p}_{X^{{\rm reg}}})$ with
${\rm d}\phi={\rm d}^{*}\phi=0.$ ∎

######  Remark 4.4.

The weighted space $L^{2}_{-n/2}$ agrees with the unweighted $L^{2}$ space.

We come now to the proof of vanishing theorems.

######  Theorem 4.5.

Let $(X,g)$ be a compact strictly-singular Riemannian $n$-conifold with
$n\geqslant 3$ and let $X^{{\rm reg}}$ have non-negative Ricci curvature. Then
no non-zero elements of $L^{2}_{1-\frac{n}{2}}(\Lambda^{1}_{X^{{\rm reg}}})$
are harmonic.

###### Proof.

Let $\phi\in L^{2}_{1-\frac{n}{2}}(\Lambda^{1}_{X^{{\rm reg}}})$ be harmonic.
Denote by $\nabla$ the Levi-Civita connection of $(X^{{\rm reg}},g)$ Then
$\nabla\phi$ has order $\geqslant-\frac{n}{2}.$ So $\nabla\phi$ lies in the
unweighted $L^{2}$ space and we can define the $L^{2}$ inner product
$\phi\cdot\nabla^{*}\nabla\phi=|\nabla\phi|_{L^{2}}^{2}.$ Weizenbock formula
implies then that

$0=\phi\cdot\Delta\phi=\phi\cdot(\nabla^{*}\nabla\phi+R_{a}^{b}\phi_{b})=|\nabla\phi|_{L^{2}}^{2}+\int_{X^{{\rm
reg}}}R^{ab}\phi_{a}\phi_{b}{\rm d}\mu\geqslant|\nabla\phi|_{L^{2}}^{2}$ (4.1)

where $R_{a}^{b},R^{ab}$ are the Ricci curvature tensors in the index
notation. The estimate (4.1) implies $\nabla\phi=0.$ In particular, the norm
$|\phi|_{g}$ with respect to $g$ is constant; and accordingly, $\phi\in
L^{2}_{-\epsilon}(\Lambda^{1,0}_{X^{{\rm reg}}})$ for every $\epsilon>0.$
Corollary 2.9(ii) and Proposition 4.2(iii) imply then that $\phi\in
L^{2}_{\epsilon}(\Lambda^{1,0}_{X^{{\rm reg}}})$ for some $\epsilon>0.$ But
$|\phi|_{g}$ is constant, which thus vanishes. So $\phi=0.$ ∎

######  Remark 4.6.

The proof will fail if $X$ is non-singular, in which case the statement is
false too; for instance, if $X$ is a flat torus then $X^{{\rm reg}}$ has non-
zero harmonic $1$-forms and the order condition holds automatically.

######  Corollary 4.7 (Chan [7, Proposition 4.2]).

Let $X$ be a compact strictly-singular Riemannian $n$-conifold with
$n\geqslant 3$ and let $X^{{\rm reg}}$ have non-negative Ricci curvature. Then
$H^{1}(X^{{\rm reg}},{\mathbin{\mathbb{R}}})=0.$

###### Proof.

As $n\geqslant 3$ we can use Theorem 4.3; that is, every element of
$H^{1}(X^{{\rm reg}},{\mathbin{\mathbb{R}}})$ may be represented uniquely by
some ${\rm d}$ closed and co-closed $L^{2}$ $1$-form $\phi$ on $X^{{\rm
reg}}.$ Also since $n\geqslant 3$ it follows by Corollary 2.9(i) and
Proposition 4.2(iv) that $\phi$ has order $>0.$ Theorem 4.5 implies therefore
$\phi=0.$ ∎

We prove a stronger result for Kähler conifolds:

######  Corollary 4.8.

Let $X$ be a compact strictly-singular Calabi–Yau $n$-conifold with
$n\geqslant 3$ $($which is now the complex dimension$).$ Then no non-zero
$L^{2}$ harmonic $(1,0)$ forms exist on $X^{{\rm reg}}.$

###### Proof.

Let $\phi\in L^{2}_{-n}(\Lambda^{1,0}_{X^{{\rm reg}}})$ be harmonic (again
$L^{2}_{-n}$ agrees with the unweighted $L^{2}$). Note that
$[-n,1-n]\subset(1-2n,-1).$ Corollary 2.13 and Proposition 4.2(iii) imply then
that $\phi$ has order $1-n.$ Hence it follows by Theorem 4.5 that $\phi=0.$ ∎

Here is a paraphrase of Corollary 4.8:

######  Corollary 4.9 (Theorem 1.3).

Let $(X;\omega,\Omega)$ be a compact strictly-singular Calabi–Yau $n$-conifold
with $n\geqslant 3.$ Then no non-zero $L^{2}$ harmonic $(n-1,0)$ forms exist
on the regular locus $X^{{\rm reg}}.$

###### Proof.

Let $\phi\in L^{2}_{-n}(\Lambda^{n-1,0}_{X^{{\rm reg}}})$ be harmonic. Then
there are on $X^{{\rm reg}}$ a vector field $v\in L^{2}_{-n}(TX^{{\rm reg}})$
defined by $v\lrcorner\,\Omega=\phi$ and a $(0,1)$ form $\psi\in
L^{2}_{-n}(\Lambda^{0,1}_{X^{{\rm reg}}})$ defined by
$\psi:=v\lrcorner\,\omega.$ Since $X^{{\rm reg}}$ is Ricci-flat it follows by
Weizenbock’s formula that $0=\Delta\phi=\nabla^{*}\nabla\phi.$ As $\Omega$ is
parallel we have then
$0=\nabla^{*}\nabla\phi=\nabla^{*}\nabla(v\lrcorner\,\Omega)=(\nabla^{*}\nabla
v)\lrcorner\,\Omega.$ Thus $\nabla^{*}\nabla v=0.$ As $\omega$ is also
parallel we have $0=(\nabla^{*}\nabla
v)\lrcorner\,\omega=\nabla^{*}\nabla(v\lrcorner\,\omega)=\nabla^{*}\nabla\psi.$
Using again the Weizenbock formula we get $\Delta\psi=0.$ Applying Corollary
4.8 to the complex conjugate $\overline{\psi}$ we find $\overline{\psi}=0$ and
hence $\psi=0.$ As $\psi,v$ are obtained from $\phi$ though pointwise
isomorphisms on $X^{{\rm reg}}$ we have $\phi=0.$ ∎

The following will be used to prove Theorem 1.1.

######  Corollary 4.10.

Let $X$ be a compact strictly-singular Calabi–Yau $3$-conifold. Then every
element of $H^{2}(X^{{\rm reg}},{{\mathbb{C}}})$ may be represented on
$X^{{\rm reg}}$ by some closed $(1,1)$ form.

###### Proof.

By Theorem 4.3 every element of $H^{2}(X^{{\rm reg}},{{\mathbb{C}}})$ may be
represented on $X^{{\rm reg}}$ by some $\phi\in
L^{2}_{-n}(\Lambda^{2}_{X^{{\rm reg}}})$ with ${\rm d}\phi={\rm d}^{*}\phi=0.$
In particular, $\phi$ is harmonic and accordingly so are the $(2,0)$ part and
$(0,2)$ parts of $\phi.$ But these vanish by Corollary 4.9, which completes
the proof. ∎

It seems appropriate to compute now some sheaf cohomology groups. But we shall
not need this directly to prove Theorem 1.1 or 1.5 so the reader in a hurry
can skip it.

######  Proposition 4.11.

Let $X$ be a compact strictly-singular Calabi–Yau $n$-conifold with
$n\geqslant 3.$ Then $H^{0}(X^{{\rm
reg}},\Omega^{1}_{X})=H^{1}(X,{\mathbin{\cal O}}_{X})=H^{1}(X^{{\rm
reg}},{\mathbin{\cal O}}_{X})=0.$ Moreover for $n=3$ we have $H^{0}(X^{{\rm
reg}},\Omega^{2}_{X})=H^{2}(X,{\mathbin{\cal O}}_{X})=0$ and there exists an
exact sequence $0\to H^{2}(X^{{\rm reg}},O_{X})\to\textstyle\bigoplus_{x\in
X^{\rm sing}}{\mathbin{\cal O}}_{X,x}\to{{\mathbb{C}}}\to 0.$

###### Proof.

Lemma 2.14 and Corollary 4.8 imply that no holomorphic $(1,0)$ forms exist on
$X^{{\rm reg}};$ that is, $H^{0}(X^{{\rm reg}},\Omega^{1}_{X})=0.$ Lemma 3.12
implies then that $H^{1}(X,{\mathbin{\cal O}}_{X})=0.$ Since $X^{\rm sing}$ is
Cohen–Macaulay and of depth $\geqslant n\geqslant 3$ it follows by the local
cohomology exact sequence that $H^{1}(X^{{\rm reg}},{\mathbin{\cal
O}}_{X})\cong H^{1}(X^{{\rm reg}},{\mathbin{\cal O}}_{X})=0.$

Suppose now that $n=3.$ Arguing as in the proof of Corollary 4.9 we see then
that no non-zero holomorphic $(2,0)$ forms exist on $X^{{\rm reg}};$ that is,
$H^{0}(X^{{\rm reg}},\Omega^{1}_{X})=0.$ Lemma 3.12 implies then that
$H^{2}(X,{\mathbin{\cal O}}_{X})=0.$ We compute now $H^{2}(X^{{\rm
reg}},{\mathbin{\cal O}}_{X})$ and use therefore the local cohomology exact
sequence

$0=H^{2}(X,{\mathbin{\cal O}}_{X})\to H^{2}(X^{{\rm reg}},O_{X})\to
H^{3}_{X^{\rm sing}}(X,{\mathbin{\cal O}}_{X})\to H^{3}(X,{\mathbin{\cal
O}}_{X})\to H^{3}(X^{{\rm reg}},{\mathbin{\cal O}}_{X}).$

There are also a Matlis duality isomorphism $H^{n}_{X^{\rm
sing}}(X,{\mathbin{\cal O}}_{X})\cong\bigoplus_{x\in X^{\rm
sing}}{\mathbin{\cal O}}_{X,x}$ [19, Theorem 4.4(1)], a Serre duality
isomorphism $H^{3}(X,{\mathbin{\cal O}}_{X})\cong
H^{0}(X,\Omega^{3}_{X})=H^{0}(X,{\mathbin{\cal O}}_{X})\cong{{\mathbb{C}}}$
explained by Hartshorne [17, p249] with references in the analytic case, and
another Serre duality isomorphism $H^{3}(X^{{\rm reg}},{\mathbin{\cal
O}}_{X})\cong H^{0}_{\rm c}(X^{{\rm reg}},\Omega^{3}_{X})=0.$ So the exact
sequence becomes $0\to H^{2}(X^{{\rm reg}},O_{X})\to\textstyle\bigoplus_{x\in
X^{\rm sing}}{\mathbin{\cal O}}_{X,x}\to{{\mathbb{C}}}\to 0.$ ∎

##  5 Proof of Lemma 1.4 and Theorem 1.1

For $Y$ a manifold and for
${\mathbin{\mathbb{K}}}={\mathbin{\mathbb{Q}}},{\mathbin{\mathbb{R}}}$ denote
by ${}_{\rm c}H^{2}(Y,{\mathbin{\mathbb{K}}})$ the image of the
${\mathbin{\mathbb{K}}}$-linear map $H^{2}_{\rm
c}(Y,{\mathbin{\mathbb{K}}})\to H^{2}(Y,{\mathbin{\mathbb{K}}})$ from the
compactly-supported cohomology group to the ordinary cohomology group. We
prove

######  Lemma 5.1.

Let $X$ be a compact complex space whose singularities are isolated and of
depth $\geqslant 3.$ Suppose that every element of $H^{2}(X^{{\rm
reg}},{{\mathbb{C}}})$ may be represented on $X^{{\rm reg}}$ by some closed
$(1,1)$ form. Then there exists an open neighbourhood $U\subseteq X$ of
$X^{\rm sing}$ such that every element of ${}_{\rm c}H^{2}(X^{{\rm
reg}},{\mathbin{\mathbb{R}}})$ may be represented on $X^{{\rm reg}}$ by some
real $(1,1)$ form supported on $X\setminus U.$

###### Proof.

As $X$ is triangularizable [28] and locally contractible there exists an open
neighbourhood $U\subseteq X$ of the finite set $X^{\rm sing}$ such that
$H^{2}_{\rm c}(X^{{\rm reg}},{\mathbin{\mathbb{R}}})\cong
H^{2}(X,U;{\mathbin{\mathbb{R}}})=H^{2}(X^{{\rm reg}},U^{{\rm
reg}};{\mathbin{\mathbb{R}}});$ note that $U^{{\rm reg}}=U\cap X^{{\rm reg}}.$
On the other hand, as Stein neighbourhoods of $X^{\rm sing}$ are also a
fundamental system, making $U$ smaller if we need, we can suppose
$H^{1}(U^{{\rm reg}},{\mathbin{\cal O}}_{X})=0$ (by the same reasoning as in
Proposition 2.20(ii)). Take now any element $\Phi\in{}_{\rm c}H^{2}(X^{{\rm
reg}},{\mathbin{\mathbb{R}}}).$ Using the hypothesis represent $\Phi$ on
$X^{{\rm reg}}$ by some closed $(1,1)$ form $\phi.$ Since $H^{2}_{\rm
c}(X^{{\rm reg}},{\mathbin{\mathbb{R}}})\cong H^{2}(X^{{\rm reg}},U^{{\rm
reg}};{\mathbin{\mathbb{R}}})$ it follows then that $\phi$ is exact on
$U^{{\rm reg}}.$ Since $H^{1}(U^{{\rm reg}},{\mathbin{\cal O}}_{X})=0$ it
follows in turn that $\phi|_{U^{{\rm reg}}}$ has the ${\rm d}{\rm d}^{\rm c}$
lemma property; that is, we can write $\phi={\rm d}{\rm d}^{c}\psi$ where
$\psi$ is a smooth function $U^{{\rm reg}}\to{\mathbin{\mathbb{R}}}.$ Using a
cut-off function and making $U^{{\rm reg}}$ smaller if we need, choose a
smooth function $\chi:X^{{\rm reg}}\to{\mathbin{\mathbb{R}}}$ with $\chi=\psi$
on $U^{{\rm reg}}.$ Define then a $(1,1)$ form $\theta\in\Phi$ by
$\theta:=\phi-{\rm d}{\rm d}^{\rm c}\chi,$ which is supported on $X\setminus
U$ because on this we have $\phi={\rm d}{\rm d}^{\rm c}\psi={\rm d}{\rm
d}^{\rm c}\chi.$ The lemma thus holds. ∎

We re-state and prove Lemma 1.4 now.

######  Corollary 5.2 (Lemma 1.4).

Let $X$ be a compact Kähler space whose singularities are isolated and of
depth $\geqslant 3.$ Suppose that every element of $H^{2}(X^{{\rm
reg}},{{\mathbb{C}}})$ may be represented on $X^{{\rm reg}}$ by some closed
$(1,1)$ form. Then $X$ is projective.

###### Proof.

We show first that there exists a natural ${\mathbin{\mathbb{Q}}}$-linear map
$\gamma:{}_{\rm c}H^{2}(X^{{\rm reg}},{\mathbin{\mathbb{Q}}})\to{}_{\rm
c}H^{2}(X^{{\rm reg}},{\mathbin{\mathbb{R}}})$ with dense image. Choose
$U\subset X$ as in the proof of Lemma 5.1. Then there exists a commutative
diagram

${H^{1}(U^{{\rm reg}},{\mathbin{\mathbb{Q}}})}$${H^{2}_{\rm c}(X^{{\rm
reg}},{\mathbin{\mathbb{Q}}})}$${H^{2}(X^{{\rm
reg}},{\mathbin{\mathbb{Q}}})}$${H^{1}(U^{{\rm
reg}},{\mathbin{\mathbb{R}}})}$${H^{2}_{\rm c}(X^{{\rm
reg}},{\mathbin{\mathbb{R}}})}$${H^{2}(X^{{\rm
reg}},{\mathbin{\mathbb{R}}})}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$
(5.1)

of natural homomorphisms where the rows are exact and the vertical arrows are
induced by the inclusion ${\mathbin{\mathbb{Q}}}\to{\mathbin{\mathbb{R}}}.$
The diagram (5.1) induces then a natural homomorphism $\gamma:{}_{\rm
c}H^{2}(X^{{\rm reg}},{\mathbin{\mathbb{Q}}})\to{}_{\rm c}H^{2}(X^{{\rm
reg}},{\mathbin{\mathbb{R}}})$ which fits into the commutative diagram

${H^{1}(U^{{\rm reg}},{\mathbin{\mathbb{Q}}})}$${H^{2}_{\rm c}(X^{{\rm
reg}},{\mathbin{\mathbb{Q}}})}$${{}_{\rm c}H^{2}(X^{{\rm
reg}},{\mathbin{\mathbb{Q}}})}$${0}$${H^{1}(U^{{\rm
reg}},{\mathbin{\mathbb{R}}})}$${H^{2}_{\rm c}(X^{{\rm
reg}},{\mathbin{\mathbb{R}}})}$${{}_{\rm c}H^{2}(X^{{\rm
reg}},{\mathbin{\mathbb{R}}})}$${0.}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$

Note that if we tensor the top row with ${\mathbin{\mathbb{R}}}$ over
${\mathbin{\mathbb{Q}}}$ then $\alpha,\beta$ become isomorphisms. Hence it
follows by the five lemma that $\gamma$ induces an isomorphism ${}_{\rm
c}H^{2}(X^{{\rm
reg}},{\mathbin{\mathbb{Q}}})\otimes_{\mathbin{\mathbb{Q}}}{\mathbin{\mathbb{R}}}\to{}_{\rm
c}H^{2}(X^{{\rm reg}},{\mathbin{\mathbb{R}}}).$ This implies in turn that
$\gamma:{}_{\rm c}H^{2}(X^{{\rm reg}},{\mathbin{\mathbb{Q}}})\to{}_{\rm
c}H^{2}(X^{{\rm reg}},{\mathbin{\mathbb{R}}})$ has dense image.

Let $\omega$ be a Kähler form on $X.$ Then $\omega$ is exact near $X^{\rm
sing}$ and defines accordingly a de Rham class $[\omega]\in{}_{\rm
c}H^{2}(X^{{\rm reg}},{\mathbin{\mathbb{R}}}).$ Since $\mathop{\rm
im}\gamma\subseteq{}_{\rm c}H^{2}(X^{{\rm reg}},{\mathbin{\mathbb{R}}})$ is
dense we get an arbitrarily small $\Phi\in{}_{\rm c}H^{2}(X^{{\rm
reg}},{\mathbin{\mathbb{R}}})$ with $[\omega]+\Phi\in\mathop{\rm im}[{}_{\rm
c}H^{2}(X^{{\rm reg}},{\mathbin{\mathbb{Q}}})\to{}_{\rm c}H^{2}(X^{{\rm
reg}},{\mathbin{\mathbb{R}}})].$ Using the hypothesis and Lemma 5.1 choose a
finite-dimensional ${\mathbin{\mathbb{R}}}$-vector space $V$ of real $(1,1)$
forms on $X^{{\rm reg}}$ which vanish on $U$ such that the natural projection
$V\to H_{\rm c}^{2}(X^{{\rm reg}},{\mathbin{\mathbb{R}}})$ is an isomorphism.
Then $\Phi=[\phi]\in H^{2}_{\rm c}(X^{{\rm reg}},{\mathbin{\mathbb{R}}})$ for
some $\phi\in V.$ Note that $\phi$ is as small as $\Phi.$ So we can suppose
that $\omega+\phi$ is a positive definite $(1,1)$ form on $X\setminus U.$ On
the other hand, $\omega+\phi=\omega$ on $U,$ which is clearly a Kähler form.
Thus $\omega+\phi$ is a Kähler form on $X.$ Recall now that
$[\omega+\phi]=[\omega]+\Phi\in\mathop{\rm im}\gamma.$ This lies in the image
of the composite map ${}_{\rm c}H^{2}(X^{{\rm reg}},{\mathbin{\mathbb{Q}}})\to
H^{2}(X^{{\rm reg}},{\mathbin{\mathbb{Q}}})\to H^{2}(X^{{\rm
reg}},{\mathbin{\mathbb{R}}}).$ So we can use Theorem 3.11 to show that $X$ is
projective. ∎

Corollaries 4.10 and 5.2 imply Theorem 1.1. ∎

##  6 Proof of Theorem 1.5

We begin by recalling

######  Theorem 6.1 (Ohsawa [27, Theorem 1]; see also Namikawa [26, Lemma
2.6]).

Let $X$ be a compact Kähler space of pure dimension and $Y\subseteq X$ a
complex subspace which contains $X^{\rm sing}.$ Then the Hodge to de Rham
spectral sequence $H^{q}(X\setminus Y,\Omega^{p}_{X})\Rightarrow{\rm
gr}^{p}H^{p+q}(X\setminus Y,{{\mathbb{C}}})$ degenerates for
$p+q\leqslant\nobreak k-2$ where $k$ is the co-dimension of $Y.$

This implies

######  Corollary 6.2.

Let $X$ be a compact normal Kähler space of pure dimension $n\geqslant 4$
whose singularities are isolated, rational and such that no non-zero
holomorphic $(2,0)$ forms exist on $X^{{\rm reg}}.$ Then every element of
$H^{2}(X^{{\rm reg}},{{\mathbb{C}}})$ may be represented on $X^{{\rm reg}}$ by
some closed $(1,1)$ form.

###### Proof.

Theorem 6.1 implies that for $p,q$ with $p+q\leqslant\nobreak 2$ the spectral
sequence $H^{q}(X^{{\rm reg}},\Omega^{p}_{X})\Rightarrow{\rm
gr}^{p}H^{p+q}(X^{{\rm reg}},{{\mathbb{C}}})$ degenerates at $E_{1}^{pq}.$ So
$H^{1}(X^{{\rm reg}},\Omega^{1}_{X})$ consists of ${\rm d}$-closed $(1,1)$
forms on $X^{{\rm reg}}$ and the natural projection $H^{1}(X^{{\rm
reg}},\Omega^{1}_{X})\to{\rm gr}^{1}H^{2}(X^{{\rm reg}},{{\mathbb{C}}})$ is a
${{\mathbb{C}}}$-vector space isomorphism. On the other hand, Lemma 3.12
implies $H^{2}(X^{{\rm reg}},{\mathbin{\cal O}}_{X})\cong
H^{2}(X,{\mathbin{\cal O}}_{X}).$ Since $X^{\rm sing}$ is Cohen–Macaulay and
of depth $\geqslant n\geqslant 4$ it follows that $H^{2}(X,{\mathbin{\cal
O}}_{X})\cong H^{0}(X^{{\rm reg}},\Omega^{2}_{X}),$ which vanishes by
hypothesis. Thus $H^{2}(X^{{\rm reg}},{\mathbin{\cal O}}_{X})=0$ and
accordingly ${\rm gr}^{1}H^{2}(X^{{\rm reg}},{{\mathbb{C}}})=H^{2}(X^{{\rm
reg}},{{\mathbb{C}}}).$ The degenerate spectral sequence induces then an
isomorphism $H^{1}(X^{{\rm reg}},\Omega^{1}_{X})\cong H^{2}(X^{{\rm
reg}},{{\mathbb{C}}}),$ completing the proof. ∎

Corollaries 5.2 and 6.2 imply Theorem 1.5. ∎

##  7 Proof of Corollary 1.6

We begin by recalling

######  Definition 7.1.

Let $X,Y$ be connected normal complex spaces of the same pure dimension, and
$f:Y\to X$ a morphism. We call $f$ a cover if it is finite and surjective. We
say that $f$ is quasi-étale if there exists a closed subspace $Z\subset Y$ of
co-dimension $\geqslant 2$ such that $f|_{Y\setminus Z}$ is étale.

######  Remark 7.2.

The category of quasi-étale covers of $X$ is equivalent to that of étale
covers of $X^{{\rm reg}};$ see for instance Claudon, Graf, Guenancia and
Naumann [9] (after Definition 2.7 in their paper).

We recall also

######  Definition 7.3.

Let $X$ be a compact normal Kähler space of pure dimension $n\geqslant 3,$
whose singularities are rational and such that the canonical sheaf
$\Omega^{n}_{X}$ has a nowhere-vanishing global section. We call $X$ an
irreducible Calabi–Yau variety if for every quasi-étale cover $Y\to X$ we have
$H^{0}(Y^{{\rm reg}},\Omega^{p}_{Y})=0$ for $p=1,\dots,n-1.$

######  Remark 7.4.

Results of Claudon, Graf, Guenancia and Naumann [9, Theorems A and C] (and
Greb, Guenancia and Kebekus [14, Proposition F] in the projective case) imply
that the condition above holds if and only if the singular Ricci-flat Kähler
form in each Kähler class of $X$ has holonomy $\operatorname{SU}(n)$ on
$X^{{\rm reg}}.$

We finally prove

######  Corollary 7.5 (Corollary 1.6).

Let $X$ be either a compact irreducible Calabi–Yau conifold of dimension $3$
or a compact irreducible Calabi–Yau variety of dimension $\geqslant 4$ whose
singularities are isolated. Then $X$ and its quasi-étale covers are
projective.

###### Proof.

Let $X$ be a compact irreducible Calabi–Yau conifold of dimension $3,$ and
$Y\to X$ a quasi-étale cover. The latter defines an étale cover of $X^{{\rm
reg}}$ and the pull-back of the conifold metric on $X^{{\rm reg}}$ defines a
conifold metric on $Y.$ Thus $Y$ is a compact irreducible Calabi–Yau conifold
which is perhaps non-singular. If so the irreducible Calabi–Yau condition
implies that $Y$ is projective. Otherwise (if it is strictly singular) Theorem
1.1 implies that $Y$ is projective. Suppose now that $X$ is a compact
irreducible Calabi–Yau variety of dimension $\geqslant 4$ whose singularities
are isolated, and $Y\to X$ a quasi-étale cover. Then $Y$ satisfies the
hypothesis of Theorem 1.5 and is accordingly projective. ∎

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