Doubling construction of Calabi-Yau fourfolds from toric Fano fourfolds
Abstract
We give a differential-geometric construction of Calabi-Yau fourfolds by the `doubling' method, which was introduced in \cite{DY14} to construct Calabi-Yau threefolds. We also give examples of Calabi-Yau fourfolds from toric Fano fourfolds. Ingredients in our construction are \emph{admissible pairs}, which were first dealt with by Kovalev in \cite{K03}. Here in this paper an admissible pair
(
X
¯
¯
¯
¯
,D)
consists of a compact Kähler manifold
X
¯
¯
¯
¯
and a smooth anticanonical divisor
D
on
X
¯
¯
¯
¯
. If two admissible pairs
(
X
¯
¯
¯
¯
1
,
D
1
)
and
(
X
¯
¯
¯
¯
2
,
D
2
)
with
dim
C
X
¯
¯
¯
¯
i
=4
satisfy the \emph{gluing condition}, we can glue
X
¯
¯
¯
¯
1
∖
D
1
and
X
¯
¯
¯
¯
2
∖
D
2
together to obtain a compact Riemannian
8
-manifold
(M,g)
whose holonomy group
Hol(g)
is contained in
Spin(7)
. Furthermore, if the
A
ˆ
-genus of
M
equals
2
, then
M
is a Calabi-Yau fourfold, i.e., a compact Ricci-flat Kähler fourfold with holonomy
SU(4)
. In particular, if
(
X
¯
¯
¯
¯
1
,
D
1
)
and
(
X
¯
¯
¯
¯
2
,
D
2
)
are identical to an admissible pair
(
X
¯
¯
¯
¯
,D)
, then the gluing condition holds automatically, so that we obtain a compact Riemannian
8
-manifold
M
with holonomy contained in
Spin(7)
. Moreover, we show that if the admissible pair is obtained from \emph{any} of the toric Fano fourfolds, then the resulting manifold
M
is a Calabi-Yau fourfold by computing
A
ˆ
(M)=2
.