Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds
File(s)
Accepted version
Author(s)
Corti, A
Haskins, M
Nordström, J
Pacini, T
Type
Journal Article
Abstract
We prove the existence of asymptotically cylindrical (ACyl) Calabi–Yau 3–folds
starting with (almost) any deformation family of smooth weak Fano 3–folds. This
allow us to exhibit hundreds of thousands of new ACyl Calabi–Yau 3–folds; previously
only a few hundred ACyl Calabi–Yau 3–folds were known. We pay particular
attention to a subclass of weak Fano 3–folds that we call semi-Fano 3–folds. SemiFano
3–folds satisfy stronger cohomology vanishing theorems and enjoy certain
topological properties not satisfied by general weak Fano 3–folds, but are far more
numerous than genuine Fano 3–folds. Also, unlike Fanos they often contain P
1
s with
normal bundle O.1/
˚ O.1/,
giving rise to compact rigid holomorphic curves in
the associated ACyl Calabi–Yau 3–folds.
We introduce some general methods to compute the basic topological invariants of
ACyl Calabi–Yau 3–folds constructed from semi-Fano 3–folds, and study a small
number of representative examples in detail. Similar methods allow the computation
of the topology in many other examples.
All the features of the ACyl Calabi–Yau 3–folds studied here find application in [17]
where we construct many new compact G2 –manifolds using Kovalev’s twisted
connected sum construction. ACyl Calabi–Yau 3–folds constructed from semi-Fano
3–folds are particularly well-adapted for this purpose.
starting with (almost) any deformation family of smooth weak Fano 3–folds. This
allow us to exhibit hundreds of thousands of new ACyl Calabi–Yau 3–folds; previously
only a few hundred ACyl Calabi–Yau 3–folds were known. We pay particular
attention to a subclass of weak Fano 3–folds that we call semi-Fano 3–folds. SemiFano
3–folds satisfy stronger cohomology vanishing theorems and enjoy certain
topological properties not satisfied by general weak Fano 3–folds, but are far more
numerous than genuine Fano 3–folds. Also, unlike Fanos they often contain P
1
s with
normal bundle O.1/
˚ O.1/,
giving rise to compact rigid holomorphic curves in
the associated ACyl Calabi–Yau 3–folds.
We introduce some general methods to compute the basic topological invariants of
ACyl Calabi–Yau 3–folds constructed from semi-Fano 3–folds, and study a small
number of representative examples in detail. Similar methods allow the computation
of the topology in many other examples.
All the features of the ACyl Calabi–Yau 3–folds studied here find application in [17]
where we construct many new compact G2 –manifolds using Kovalev’s twisted
connected sum construction. ACyl Calabi–Yau 3–folds constructed from semi-Fano
3–folds are particularly well-adapted for this purpose.
Date Issued
2013-07-15
Date Acceptance
2013-03-04
Citation
Geometry & Topology, 2013, 17 (4), pp.1955-2059
ISSN
1465-3060
Start Page
1955
End Page
2059
Journal / Book Title
Geometry & Topology
Volume
17
Issue
4
Copyright Statement
© 2013 Mathematical Sciences Publishers.
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Grant Number
EP/G007241/1
EP/G06170X/1
EP/L001527/1
Subjects
Science & Technology
Physical Sciences
Mathematics
MATHEMATICS
CANONICAL SINGULARITIES
BIRATIONAL MAPS
RICCI CURVATURE
MINIMAL MODELS
GENERAL TYPE
BLOWING-UP
THREEFOLDS
POINTS
NUMBER
VARIETIES
math.AG
math.DG
14J30, 53C29 (Primary) 14E15, 14J28, 14J32, 14J45, 53C25 (Secondary)
Geological & Geomatics Engineering
0101 Pure Mathematics
Publication Status
Published