1304.1820v1 - arXiv:1304.1820v1[math.DG 5 Apr 2013...

arXiv:1304.1820v1
[math.DG]
5 Apr 2013
GROMOV-HAUSDORFF COLLAPSING OF CALABI-YAU
MANIFOLDS
MARK GROSS
∗
, VALENTINO TOSATTI
†
, AND YUGUANG ZHANG
‡
Abstract. This paper is a sequel to [12]. We further study Gromov-
Hausdorff collapsing limits of Ricci-flat K¨ ahler metrics on abelian fibered
Calabi-Yau manifolds. Firstly, we show that in the same setup as [12],
if the dimension of the base manifold is one, the limit metric space is
homeomorphic to the base manifold. Secondly, if the fibered Calabi-Yau
manifolds are Lagrangian fibrations of holomorphic symplectic mani-
folds, the metrics on the regular parts of the limits are special K¨ ahler
metrics. By combining these two results, we extend [13] to any fibered
projective K3 surface without any assumption on the type of singular
fibers.
1. Introduction
In this paper we continue our study in [12] of the structure of collapsed
Gromov-Hausdorff limits of Ricci-flat K¨ ahler metrics on compact Calabi-
Yau manifolds. Let M be a projective Calabi-Yau manifold of complex
dimension m , with Ω a nowhere vanishing holomorphic m -form on M . Let
N be a projective manifold of dimension 0 <n<m , and f : M → N be
a holomorphic fibration (i.e., a surjective holomorphic map with connected
fibers) whose general fibre is an abelian variety. Let α be an ample class on
M , and let N
0
⊂ N be the Zariski open subset such that, for any y ∈ N
0
,
M
y
= f
− 1
( y ) is smooth (and therefore Calabi-Yau). Let D = N \ N
0
be
the discriminant locus of the map f . Let α
0
be an ample class on N , and
˜ ω
t
∈ f
∗
α
0
+ tα be the Ricci-flat K¨ ahler metric given by Yau’s Theorem [40]
for t ∈ (0 , 1], which satisfies the complex Monge-Amp` ere equation
(1.1) ˜ ω
m
t
= c
t
t
m − n
( − 1)
m
2
2
Ω ∧ Ω .
By [35] and [12], ˜ ω
t
converges smoothly to f
∗
ω on f
− 1
( K ) for any compact
K ⊂ N
0
as t → 0, and on N
0
, where ω is the K¨ ahler metric on N
0
with
Ric( ω ) = ω
W P
obtained in [35] and [29] (see also [28]), and ω
W P
is a Weil-Petersson semi-
positive form on N
0
coming from the variation of the complex structures of
the fibers M
y
. Furthermore, the Ricci-flat metrics ˜ ω
t
have locally uniformly
∗
Supported in part by NSF grant DMS-1105871.
†
Supported in part by a Sloan Research Fellowship and NSF grant DMS-1236969.
‡
Supported in part by NSFC-11271015.
1

2 M. GROSS, V. TOSATTI, AND Y. ZHANG
bounded curvature on f
− 1
( N
0
). Thanks to [34] and [41], the diameter of
the metrics ˜ ω
t
satisfies
diam
˜ ω
t
( M ) lessorequalslant C,
for a constant C> 0 independent of t . Gromov’s pre-compactness theorem
(cf. [10]) then implies that for any sequence t
k
→ 0, a subsequence of ( M, ˜ ω
t
k
)
converges to a compact metric space ( X,d
X
) in the Gromov-Hausdorff sense
(a priori different subsequences could result in non-isometric limits). In our
earlier work [12] we proved that ( N
0
,ω ) can be locally isometrically embed-
ded into ( X,d
X
), with open dense image X
0
⊂ X , via a homeomorphism
φ : N
0
→ X
0
. The following questions remain open (see [36, 37]).