Hirzebruch surfaces
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Contents |
1 Introduction
Hirzebruch surfaces were introduced by Hirzebruch (without that name) in his thesis [Hirzebruch1951]. They are algebraic surfaces over the complex numbers. Here we consider them as smooth manifolds. The interest in them comes from Hirzebruch's result that as complex manifolds they are pairwise distinct whereas there are only two diffeomorphism types.
2 Construction and examples
Let for some
be the tautological line bundle bundle over
. Recall that if
is a vector bundle over
(real or complex) taking the fibrewise projective space yields the associated projective bundle
, where
if and only if there is a
in the ground field such that
. The fibres of
are projective spaces
. If
is a smooth vector bundle over a smooth manifold, then P(E) is a smooth manifold, or similarly if
and
are holomorphic, the total space is a complex manifold.
For define
as
, where
is the tensor product of
copies of
with itself and
denotes the trivial complex line bundle. For
the bundle
is by definition
. For
we define
as the corresponding construction with
, the complex conjugated bundle, instead of
. These are the
. They come with a complex structure but we consider them as smooth manifolds.
3 Invariants
As smooth manifolds the Hirzebruch surfaces are -bundles over
. Hence they are closed and, by the orientation coming from the complex structure, oriented 4-dimensional manifolds. Let
denote a fibre
-sphere and let
denote the section of points at infinity.
-
and
.
-
, in particular
.
-
has basis with
and
.
- With respect to the above basis the intersection form on
is given by the following matrix:
- The Euler characteristic is given by
.
- The signature vanishes:
.
- The first Pontrjagin class of
is zero:
.
- The second Stiefel-Whitney class
is given by
and
mod
.
-
is a spinable if and only if
is even.
- For the complex manifold
, the first Chern class
, is given by
and
.
Explanation
- The computation of the homotopy groups of
follows from the homotopy sequence of a fibration and the existence of the section
.
- The homology groups of
can be computed by decomposing
where
is the
disc bundle associated to
and using the Mayer-Vietoris sequence.
- The computation of the intersection form follows by inspecting the embedded
-spheres which represent
and their normal bundles: in particular we apply the fact that the self intersection number of a homology class represented by an embedded oriented submanifold is the evaluation of the Euler class of the normal bundle on the fundamental class the submanifold.
- The signature of
is zero since the Hirzebruch surfaces are the boundary of the associated
-bundle. One can also see this directly from the intersection form.
- The first Pontrjagin class vanishes as its evaluation on the fundamenatal class of
is an oriented bordism invariant.
4 Classification
The intersection form implies that if is diffeomorphic to
, then
mod
. On the other hand, considered as smooth manifolds, Hirzebruch surfaces are the total spaces of the 2-sphere bundle of a 3-dimensional vector bundle over
. these bundles are classified by
(note that
is diffeomorphic to
). Thus there are at most two diffeomorphism types of Hirzebruch surfaces and so we conclude:
![\displaystyle H_n \cong H_m \Longleftrightarrow n = m ~ \mathrm{mod} ~ 2,](/images/math/0/4/9/0499be5a432d38070e2acb8d61aca69b.png)
where means diffeomorphic. By construction
and by an easy consideration
#
, where # is the connected sum and
is
with the opposite orientation.
For more information on Hirzebruch surfaces, in particular why they are pairwise distinct as complex manifolds, see [Hirzebruch1951].
5 Further remarks
The Hirzebruch surfaces are the second stage of the so called Bott towers, which are inductively constructed starting from a point as the total space of a projective bundle associated to , where
is a line bundle over a lower Bott tower (for more details see [Choi&Masuda&Suh2008]). The classification of the Bott towers up to homeomorphism or diffeomorphism is an interesting open problem. In particular one can ask wether the integral cohomology ring determines the homeomorphism or diffeomorphism type.
6 References
- [Choi&Masuda&Suh2008] Template:Choi&Masuda&Suh2008
- [Hirzebruch1951] F. Hirzebruch, Über eine Klasse von einfachzusammenhängenden komplexen Mannigfaltigkeiten, Math. Ann. 124 (1951), 77–86. MR0045384 (13,574e) Zbl 0043.30302