Complete intersections
Contents |
1 Construction and examples
Let be complex, homogeneous polynomials of degree
in
complex variables
. Define
![\displaystyle X(f_1, \dots, f_k) \coloneq \{ [z] \in \CP^{n+k} | f_i(z) = 0 \text{~for~} i = 1, \dots, k\}.](/images/math/4/8/d/48d767881e8626164661eb292ab83ecd.png)
The set is an algebraic variety. It is non-singular if for all
with
the affine function
![\displaystyle \Cc^{n+k} \to \Cc^{k}, ~~~ (z_1, \dots, z_{n+k}) \longmapsto (f_1^j(z_1, \dots, z_{n+k}), \dots, f_k^j(z_1, \dots, z_{n+k}))](/images/math/8/3/2/832c31943fc43d6fd25ed2cb651d38fc.png)
where , has
as a regular value. In this case
is a smooth, complex variety and in particular defines a closed, smooth, oriented dimensional manifold of real dimension
. This is called a complete intersection.
By a theorem of Thom [{???]} the diffeomorphism type of depends only upon the multi-degree,
, and we write
for
.
For example:
-
,
-
the oriented surface of genus
,
-
is a complex K3 surface,
-
is a Calabi-Yau 3-fold.
2 Invariants
By the Lefschetz hyperplane theorem the inclusion is an n-connected map. Hence:
- if n > 1 then
,
-
if
,
-
for all
unless
.
-
where
is the Euler characteristic of
which we discuss further below.
- When
is even
generates a summand of
.
Note that here and throughout integer coefficients are use for (co)homology.
2.1 Cohomology ring
Let be the canonical line bundle over
and let
be defined by
where
is the first Chern class of
. Let
denote the fundamental class of
and let
be the product of all degrees, called the total degree. We have the following useful identity [???]
![\displaystyle \langle x^n, [X_n({\underline{d}})] \rangle = d.](/images/math/c/4/2/c42040a33bb3081f0d64c0ef9cba1e3b.png)
Let or
and consider the graded ring
![\displaystyle H^*(n, d) :=\Zz[x, y]/\{x^{m+1} = dy, y^2 = 0 \}](/images/math/a/2/8/a2875558e0fa4f20ba7f73dbf2170c41.png)
where the dimensions of and
are
and
respectively. Let
, considered as a graded ring in dimension
- If
is odd:
- the ring
is determined by
,
and
,
- there is a short exact sequence
- the intersection form
is of course skew hyperbolic.
- the ring
- If
is even:
- the ring
is determined by
,
and the pair
:
- there is a short exact sequence
.
- Some properties of
are described below.
- the ring
Proposition 2.1 [Libgober&Wood1981].
If is even, then
is indefinate unless
or
.
2.2 Characteristic classes
The stable tangent bundle of is isomorphic to
, [Milnor&Stasheff1974], and the normal bundle of the inclusion
is given by the identity [???]
![\displaystyle \nu(i) \cong i^*(L^{d_1} \oplus \dots \oplus L^{d_k})](/images/math/9/6/d/96d13cacbdcae3dbcb1cfa973b190b59.png)
where denotes the
-fold tensor product of
with itself. From this one deduces that the stable tangent bundle of
,
, satisfies the equation
![\displaystyle \tau(X_n({\underline{d}})) \oplus i^*(L^{d_1} \oplus \dots \oplus L^{d_k}) \cong i^*((n+k+1)L).](/images/math/f/f/b/ffb5f6d4135a353f956aedad9a699dda.png)
It follows immediately that the total Chern class and the total Pontrjagin class of are given by
![\displaystyle c(X_n({\underline{d}})) = (1+x)^{n+k+1}\prod_{j=1}^k(1+d_jx)^{-1} \in H^{2*}(X_n(\underline{d});\Zz),](/images/math/4/2/3/4238ed3fed927ec647060b57003c47da.png)
![\displaystyle p(X_n({\underline{d}})) = (1-x^2)^{n+k+1}\prod_{j=1}^k(1-d_j^2x^2)^{-1} \in H^{4*}(X_n(\underline{d});\Zz).](/images/math/a/f/2/af262d8be56d51bc908dea24b90be462.png)
Moreover, the Euler class and Euler characteristic of is given by
![\displaystyle e(X_n({\underline{d}})) = c_n(X_n({\underline{d}})), ~~~ \chi(\underline{d}) = \langle e(X_n(\underline{d}), [X_n(\underline{d})]\rangle.](/images/math/f/e/0/fe059d8703b858339f44a160448043ea.png)
3 Classification
The smooth classification of complete intersections for is organised by the following conjecture, often called the Sullivan Conjecture after Dennis Sullivan.
Conjecture 3.1. For , complete intersections
and
are diffeomorphic if and only if all of the following conditions hold
-
,
-
,
-
.
Note that we regard as multiplies of
in order to view
as an element of
.
3.1 Classification in low dimensions
- For
the Sullivan Conjecture holds by the classification of surfaces.
- For
the topological Sullivan Conjecture holds for
by applying [Freedman1982]. The Sullivan Conjecture fails smoothly by [Ebeling1990] and [Libgober&Wood1990].
- For
the Sullivan Conjecture holds by [Wall1966] and [Jupp1973]. See the page simply-connected 6-manifolds.
- For
the topological Sullivan Conjecture is true [Fang&Klaus1996]. Hence by smoothing theory the smooth conjecture, which is still open in general, holds up to connected sum with the exotic 8-sphere.
3.2 Further classification theorems
We now discuss further classification results for complete intersections. Let where
is prime.
Theorem 3.2 [Traving1985].
If and
for all
such that
then the Sullivan Conjecture holds.
Traving's proof uses modified surgery: see [Kreck1999, Section 8] for a summary.
Theorem 3.3 [Fang1997].
If and
for all
such that
then the topological Sullivan Conjecture holds.
Fang's proof proceeds by extending results of [Libgober&Wood1982] on the homotopy classification of complete intersections where is even to the case of
odd. He then solves the homeomorphism classification by exhibiting characteristic varieties for complete intersections and using them to calculate classical surgery obstructions. In particular he proves
Theorem 3.4 [Fang1997].
Let and
be homotopy equivalent complete intersections. If
is odd and
for all
then
and
are homeomorphic to each other if and only if their Pontrjagin classes agree.
3.3 Classification up to homotopy
Recall that the signature of, , of
is the signature of its intersection form. If
is odd
and if
is even
can be computed from the
via Hirzebruch's signature theorem.
Theorem 3.5 [Libgober&Wood1982] and [Fang1997]. Let and
be complete intersections with the same total degree
. Suppose that
if
. If
then
and
are homotopy equivalent if and only if they have the same signature and Euler characteristic.
Finally we summarise theorems of [Astey&Gitler&Micha&Pastor2003]. Define to be the unordered sequence
![\displaystyle (\nu_p(d_1), \dots, \nu_p(d_k))](/images/math/4/2/2/4228fb192d2ce47623abd1b05003cd4b.png)
obtained by removing entries where and write
if
for all primes
.
Theorem 3.6 [Astey&Gitler&Micha&Pastor2003].
If and
then
is homotopy equivalent to
if and only if they have the same Euler characteristic and signature.
In contrast to the results of Fang and Ligober and Wood which hold when is small relative to the primes dividing
, the above theorem leads to a homotopy classification which holds when
is large relative to
. If
for all p then define
otherwise let
![\displaystyle N_d := \mathrm{Max} \{ 2(p_i-1)p_i^{[(\nu_{p}(d)-2)/2]} | \nu_{p}(d) \geq 2 \}.](/images/math/6/3/6/636ce5ebc2851f61f05adf3e72d0c8ec.png)
Theorem 3.7 [Astey&Gitler&Micha&Pastor2003].
If and
have the same multidegree
and
then
and
are homotopy equivalent if and only if they have the same signature and Euler characteritic.
4 Further discussion
4.1 Splitting theorems
5 References
- [???] Template:???
- [Astey&Gitler&Micha&Pastor2003] L. Astey, S. Gitler, E. Micha and G. Pastor, On the homotopy type of complete intersections, Topology 44 (2005), no.1, 249–260. MR2104011 (2005i:14058) Zbl 1072.14063
- [Ebeling1990] W. Ebeling, An example of two homeomorphic, nondiffeomorphic complete intersection surfaces, Invent. Math. 99 (1990), no.3, 651–654. MR1032884 (91c:57038) Zbl 0707.14045
- [Fang&Klaus1996] F. Fang and S. Klaus, Topological classification of
-dimensional complete intersections, Manuscripta Math. 90 (1996), no.2, 139–147. MR1391206 (97g:57046) Zbl 0866.57015
- [Fang1997] F. Fang, Topology of complete intersections, Comment. Math. Helv. 72 (1997), no.3, 466–480. MR1476060 (98k:14071) Zbl 0896.14028
- [Freedman1982] M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no.3, 357–453. MR679066 (84b:57006) Zbl 0528.57011
- [Jupp1973] P. E. Jupp, Classification of certain
-manifolds, Proc. Cambridge Philos. Soc. 73 (1973), 293–300. MR0314074 (47 #2626) Zbl 0249.57005
- [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
- [Libgober&Wood1981] A. S. Libgober and J. W. Wood, On the topological structure of even-dimensional complete intersections, Trans. Amer. Math. Soc. 267 (1981), no.2, 637–660. MR626495 (83e:57017) Zbl 0475.57013
- [Libgober&Wood1982] A. S. Libgober and J. W. Wood, Differentiable structures on complete intersections. I, Topology 21 (1982), no.4, 469–482. MR670748 (84a:57037) Zbl 0504.57015
- [Libgober&Wood1990] A. S. Libgober and J. W. Wood, Uniqueness of the complex structure on Kähler manifolds of certain homotopy types, J. Differential Geom. 32 (1990), no.1, 139–154. MR1064869 (91g:32039) Zbl 0711.53052
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Traving1985] C. Traving, Zur Diffeomorphieklassifikation vollständiger Durchschnitte, Diplomarbeit, Mainz (1985).
- [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain
-manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
- [[Template:{???|[{???]]] {{{???}}
This page has not been refereed. The information given here might be incomplete or provisional. |
![d_1, \dots, d_k](/images/math/b/8/0/b80fc20baa8171018aa1a3764483e5ce.png)
![n+k+1](/images/math/4/3/a/43a2a7a3c6f8abbb4276b0e238b69b85.png)
![z = (z_1, \dots, z_{n+k+1})](/images/math/f/8/f/f8fd4df65be7da46a74fe70ba0779447.png)
![\displaystyle X(f_1, \dots, f_k) \coloneq \{ [z] \in \CP^{n+k} | f_i(z) = 0 \text{~for~} i = 1, \dots, k\}.](/images/math/4/8/d/48d767881e8626164661eb292ab83ecd.png)
The set is an algebraic variety. It is non-singular if for all
with
the affine function
![\displaystyle \Cc^{n+k} \to \Cc^{k}, ~~~ (z_1, \dots, z_{n+k}) \longmapsto (f_1^j(z_1, \dots, z_{n+k}), \dots, f_k^j(z_1, \dots, z_{n+k}))](/images/math/8/3/2/832c31943fc43d6fd25ed2cb651d38fc.png)
where , has
as a regular value. In this case
is a smooth, complex variety and in particular defines a closed, smooth, oriented dimensional manifold of real dimension
. This is called a complete intersection.
By a theorem of Thom [{???]} the diffeomorphism type of depends only upon the multi-degree,
, and we write
for
.
For example:
-
,
-
the oriented surface of genus
,
-
is a complex K3 surface,
-
is a Calabi-Yau 3-fold.
2 Invariants
By the Lefschetz hyperplane theorem the inclusion is an n-connected map. Hence:
- if n > 1 then
,
-
if
,
-
for all
unless
.
-
where
is the Euler characteristic of
which we discuss further below.
- When
is even
generates a summand of
.
Note that here and throughout integer coefficients are use for (co)homology.
2.1 Cohomology ring
Let be the canonical line bundle over
and let
be defined by
where
is the first Chern class of
. Let
denote the fundamental class of
and let
be the product of all degrees, called the total degree. We have the following useful identity [???]
![\displaystyle \langle x^n, [X_n({\underline{d}})] \rangle = d.](/images/math/c/4/2/c42040a33bb3081f0d64c0ef9cba1e3b.png)
Let or
and consider the graded ring
![\displaystyle H^*(n, d) :=\Zz[x, y]/\{x^{m+1} = dy, y^2 = 0 \}](/images/math/a/2/8/a2875558e0fa4f20ba7f73dbf2170c41.png)
where the dimensions of and
are
and
respectively. Let
, considered as a graded ring in dimension
- If
is odd:
- the ring
is determined by
,
and
,
- there is a short exact sequence
- the intersection form
is of course skew hyperbolic.
- the ring
- If
is even:
- the ring
is determined by
,
and the pair
:
- there is a short exact sequence
.
- Some properties of
are described below.
- the ring
Proposition 2.1 [Libgober&Wood1981].
If is even, then
is indefinate unless
or
.
2.2 Characteristic classes
The stable tangent bundle of is isomorphic to
, [Milnor&Stasheff1974], and the normal bundle of the inclusion
is given by the identity [???]
![\displaystyle \nu(i) \cong i^*(L^{d_1} \oplus \dots \oplus L^{d_k})](/images/math/9/6/d/96d13cacbdcae3dbcb1cfa973b190b59.png)
where denotes the
-fold tensor product of
with itself. From this one deduces that the stable tangent bundle of
,
, satisfies the equation
![\displaystyle \tau(X_n({\underline{d}})) \oplus i^*(L^{d_1} \oplus \dots \oplus L^{d_k}) \cong i^*((n+k+1)L).](/images/math/f/f/b/ffb5f6d4135a353f956aedad9a699dda.png)
It follows immediately that the total Chern class and the total Pontrjagin class of are given by
![\displaystyle c(X_n({\underline{d}})) = (1+x)^{n+k+1}\prod_{j=1}^k(1+d_jx)^{-1} \in H^{2*}(X_n(\underline{d});\Zz),](/images/math/4/2/3/4238ed3fed927ec647060b57003c47da.png)
![\displaystyle p(X_n({\underline{d}})) = (1-x^2)^{n+k+1}\prod_{j=1}^k(1-d_j^2x^2)^{-1} \in H^{4*}(X_n(\underline{d});\Zz).](/images/math/a/f/2/af262d8be56d51bc908dea24b90be462.png)
Moreover, the Euler class and Euler characteristic of is given by
![\displaystyle e(X_n({\underline{d}})) = c_n(X_n({\underline{d}})), ~~~ \chi(\underline{d}) = \langle e(X_n(\underline{d}), [X_n(\underline{d})]\rangle.](/images/math/f/e/0/fe059d8703b858339f44a160448043ea.png)
3 Classification
The smooth classification of complete intersections for is organised by the following conjecture, often called the Sullivan Conjecture after Dennis Sullivan.
Conjecture 3.1. For , complete intersections
and
are diffeomorphic if and only if all of the following conditions hold
-
,
-
,
-
.
Note that we regard as multiplies of
in order to view
as an element of
.
3.1 Classification in low dimensions
- For
the Sullivan Conjecture holds by the classification of surfaces.
- For
the topological Sullivan Conjecture holds for
by applying [Freedman1982]. The Sullivan Conjecture fails smoothly by [Ebeling1990] and [Libgober&Wood1990].
- For
the Sullivan Conjecture holds by [Wall1966] and [Jupp1973]. See the page simply-connected 6-manifolds.
- For
the topological Sullivan Conjecture is true [Fang&Klaus1996]. Hence by smoothing theory the smooth conjecture, which is still open in general, holds up to connected sum with the exotic 8-sphere.
3.2 Further classification theorems
We now discuss further classification results for complete intersections. Let where
is prime.
Theorem 3.2 [Traving1985].
If and
for all
such that
then the Sullivan Conjecture holds.
Traving's proof uses modified surgery: see [Kreck1999, Section 8] for a summary.
Theorem 3.3 [Fang1997].
If and
for all
such that
then the topological Sullivan Conjecture holds.
Fang's proof proceeds by extending results of [Libgober&Wood1982] on the homotopy classification of complete intersections where is even to the case of
odd. He then solves the homeomorphism classification by exhibiting characteristic varieties for complete intersections and using them to calculate classical surgery obstructions. In particular he proves
Theorem 3.4 [Fang1997].
Let and
be homotopy equivalent complete intersections. If
is odd and
for all
then
and
are homeomorphic to each other if and only if their Pontrjagin classes agree.
3.3 Classification up to homotopy
Recall that the signature of, , of
is the signature of its intersection form. If
is odd
and if
is even
can be computed from the
via Hirzebruch's signature theorem.
Theorem 3.5 [Libgober&Wood1982] and [Fang1997]. Let and
be complete intersections with the same total degree
. Suppose that
if
. If
then
and
are homotopy equivalent if and only if they have the same signature and Euler characteristic.
Finally we summarise theorems of [Astey&Gitler&Micha&Pastor2003]. Define to be the unordered sequence
![\displaystyle (\nu_p(d_1), \dots, \nu_p(d_k))](/images/math/4/2/2/4228fb192d2ce47623abd1b05003cd4b.png)
obtained by removing entries where and write
if
for all primes
.
Theorem 3.6 [Astey&Gitler&Micha&Pastor2003].
If and
then
is homotopy equivalent to
if and only if they have the same Euler characteristic and signature.
In contrast to the results of Fang and Ligober and Wood which hold when is small relative to the primes dividing
, the above theorem leads to a homotopy classification which holds when
is large relative to
. If
for all p then define
otherwise let
![\displaystyle N_d := \mathrm{Max} \{ 2(p_i-1)p_i^{[(\nu_{p}(d)-2)/2]} | \nu_{p}(d) \geq 2 \}.](/images/math/6/3/6/636ce5ebc2851f61f05adf3e72d0c8ec.png)
Theorem 3.7 [Astey&Gitler&Micha&Pastor2003].
If and
have the same multidegree
and
then
and
are homotopy equivalent if and only if they have the same signature and Euler characteritic.
4 Further discussion
4.1 Splitting theorems
5 References
- [???] Template:???
- [Astey&Gitler&Micha&Pastor2003] L. Astey, S. Gitler, E. Micha and G. Pastor, On the homotopy type of complete intersections, Topology 44 (2005), no.1, 249–260. MR2104011 (2005i:14058) Zbl 1072.14063
- [Ebeling1990] W. Ebeling, An example of two homeomorphic, nondiffeomorphic complete intersection surfaces, Invent. Math. 99 (1990), no.3, 651–654. MR1032884 (91c:57038) Zbl 0707.14045
- [Fang&Klaus1996] F. Fang and S. Klaus, Topological classification of
-dimensional complete intersections, Manuscripta Math. 90 (1996), no.2, 139–147. MR1391206 (97g:57046) Zbl 0866.57015
- [Fang1997] F. Fang, Topology of complete intersections, Comment. Math. Helv. 72 (1997), no.3, 466–480. MR1476060 (98k:14071) Zbl 0896.14028
- [Freedman1982] M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no.3, 357–453. MR679066 (84b:57006) Zbl 0528.57011
- [Jupp1973] P. E. Jupp, Classification of certain
-manifolds, Proc. Cambridge Philos. Soc. 73 (1973), 293–300. MR0314074 (47 #2626) Zbl 0249.57005
- [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
- [Libgober&Wood1981] A. S. Libgober and J. W. Wood, On the topological structure of even-dimensional complete intersections, Trans. Amer. Math. Soc. 267 (1981), no.2, 637–660. MR626495 (83e:57017) Zbl 0475.57013
- [Libgober&Wood1982] A. S. Libgober and J. W. Wood, Differentiable structures on complete intersections. I, Topology 21 (1982), no.4, 469–482. MR670748 (84a:57037) Zbl 0504.57015
- [Libgober&Wood1990] A. S. Libgober and J. W. Wood, Uniqueness of the complex structure on Kähler manifolds of certain homotopy types, J. Differential Geom. 32 (1990), no.1, 139–154. MR1064869 (91g:32039) Zbl 0711.53052
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Traving1985] C. Traving, Zur Diffeomorphieklassifikation vollständiger Durchschnitte, Diplomarbeit, Mainz (1985).
- [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain
-manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
- [[Template:{???|[{???]]] {{{???}}
This page has not been refereed. The information given here might be incomplete or provisional. |
![d_1, \dots, d_k](/images/math/b/8/0/b80fc20baa8171018aa1a3764483e5ce.png)
![n+k+1](/images/math/4/3/a/43a2a7a3c6f8abbb4276b0e238b69b85.png)
![z = (z_1, \dots, z_{n+k+1})](/images/math/f/8/f/f8fd4df65be7da46a74fe70ba0779447.png)
![\displaystyle X(f_1, \dots, f_k) \coloneq \{ [z] \in \CP^{n+k} | f_i(z) = 0 \text{~for~} i = 1, \dots, k\}.](/images/math/4/8/d/48d767881e8626164661eb292ab83ecd.png)
The set is an algebraic variety. It is non-singular if for all
with
the affine function
![\displaystyle \Cc^{n+k} \to \Cc^{k}, ~~~ (z_1, \dots, z_{n+k}) \longmapsto (f_1^j(z_1, \dots, z_{n+k}), \dots, f_k^j(z_1, \dots, z_{n+k}))](/images/math/8/3/2/832c31943fc43d6fd25ed2cb651d38fc.png)
where , has
as a regular value. In this case
is a smooth, complex variety and in particular defines a closed, smooth, oriented dimensional manifold of real dimension
. This is called a complete intersection.
By a theorem of Thom [{???]} the diffeomorphism type of depends only upon the multi-degree,
, and we write
for
.
For example:
-
,
-
the oriented surface of genus
,
-
is a complex K3 surface,
-
is a Calabi-Yau 3-fold.
2 Invariants
By the Lefschetz hyperplane theorem the inclusion is an n-connected map. Hence:
- if n > 1 then
,
-
if
,
-
for all
unless
.
-
where
is the Euler characteristic of
which we discuss further below.
- When
is even
generates a summand of
.
Note that here and throughout integer coefficients are use for (co)homology.
2.1 Cohomology ring
Let be the canonical line bundle over
and let
be defined by
where
is the first Chern class of
. Let
denote the fundamental class of
and let
be the product of all degrees, called the total degree. We have the following useful identity [???]
![\displaystyle \langle x^n, [X_n({\underline{d}})] \rangle = d.](/images/math/c/4/2/c42040a33bb3081f0d64c0ef9cba1e3b.png)
Let or
and consider the graded ring
![\displaystyle H^*(n, d) :=\Zz[x, y]/\{x^{m+1} = dy, y^2 = 0 \}](/images/math/a/2/8/a2875558e0fa4f20ba7f73dbf2170c41.png)
where the dimensions of and
are
and
respectively. Let
, considered as a graded ring in dimension
- If
is odd:
- the ring
is determined by
,
and
,
- there is a short exact sequence
- the intersection form
is of course skew hyperbolic.
- the ring
- If
is even:
- the ring
is determined by
,
and the pair
:
- there is a short exact sequence
.
- Some properties of
are described below.
- the ring
Proposition 2.1 [Libgober&Wood1981].
If is even, then
is indefinate unless
or
.
2.2 Characteristic classes
The stable tangent bundle of is isomorphic to
, [Milnor&Stasheff1974], and the normal bundle of the inclusion
is given by the identity [???]
![\displaystyle \nu(i) \cong i^*(L^{d_1} \oplus \dots \oplus L^{d_k})](/images/math/9/6/d/96d13cacbdcae3dbcb1cfa973b190b59.png)
where denotes the
-fold tensor product of
with itself. From this one deduces that the stable tangent bundle of
,
, satisfies the equation
![\displaystyle \tau(X_n({\underline{d}})) \oplus i^*(L^{d_1} \oplus \dots \oplus L^{d_k}) \cong i^*((n+k+1)L).](/images/math/f/f/b/ffb5f6d4135a353f956aedad9a699dda.png)
It follows immediately that the total Chern class and the total Pontrjagin class of are given by
![\displaystyle c(X_n({\underline{d}})) = (1+x)^{n+k+1}\prod_{j=1}^k(1+d_jx)^{-1} \in H^{2*}(X_n(\underline{d});\Zz),](/images/math/4/2/3/4238ed3fed927ec647060b57003c47da.png)
![\displaystyle p(X_n({\underline{d}})) = (1-x^2)^{n+k+1}\prod_{j=1}^k(1-d_j^2x^2)^{-1} \in H^{4*}(X_n(\underline{d});\Zz).](/images/math/a/f/2/af262d8be56d51bc908dea24b90be462.png)
Moreover, the Euler class and Euler characteristic of is given by
![\displaystyle e(X_n({\underline{d}})) = c_n(X_n({\underline{d}})), ~~~ \chi(\underline{d}) = \langle e(X_n(\underline{d}), [X_n(\underline{d})]\rangle.](/images/math/f/e/0/fe059d8703b858339f44a160448043ea.png)
3 Classification
The smooth classification of complete intersections for is organised by the following conjecture, often called the Sullivan Conjecture after Dennis Sullivan.
Conjecture 3.1. For , complete intersections
and
are diffeomorphic if and only if all of the following conditions hold
-
,
-
,
-
.
Note that we regard as multiplies of
in order to view
as an element of
.
3.1 Classification in low dimensions
- For
the Sullivan Conjecture holds by the classification of surfaces.
- For
the topological Sullivan Conjecture holds for
by applying [Freedman1982]. The Sullivan Conjecture fails smoothly by [Ebeling1990] and [Libgober&Wood1990].
- For
the Sullivan Conjecture holds by [Wall1966] and [Jupp1973]. See the page simply-connected 6-manifolds.
- For
the topological Sullivan Conjecture is true [Fang&Klaus1996]. Hence by smoothing theory the smooth conjecture, which is still open in general, holds up to connected sum with the exotic 8-sphere.
3.2 Further classification theorems
We now discuss further classification results for complete intersections. Let where
is prime.
Theorem 3.2 [Traving1985].
If and
for all
such that
then the Sullivan Conjecture holds.
Traving's proof uses modified surgery: see [Kreck1999, Section 8] for a summary.
Theorem 3.3 [Fang1997].
If and
for all
such that
then the topological Sullivan Conjecture holds.
Fang's proof proceeds by extending results of [Libgober&Wood1982] on the homotopy classification of complete intersections where is even to the case of
odd. He then solves the homeomorphism classification by exhibiting characteristic varieties for complete intersections and using them to calculate classical surgery obstructions. In particular he proves
Theorem 3.4 [Fang1997].
Let and
be homotopy equivalent complete intersections. If
is odd and
for all
then
and
are homeomorphic to each other if and only if their Pontrjagin classes agree.
3.3 Classification up to homotopy
Recall that the signature of, , of
is the signature of its intersection form. If
is odd
and if
is even
can be computed from the
via Hirzebruch's signature theorem.
Theorem 3.5 [Libgober&Wood1982] and [Fang1997]. Let and
be complete intersections with the same total degree
. Suppose that
if
. If
then
and
are homotopy equivalent if and only if they have the same signature and Euler characteristic.
Finally we summarise theorems of [Astey&Gitler&Micha&Pastor2003]. Define to be the unordered sequence
![\displaystyle (\nu_p(d_1), \dots, \nu_p(d_k))](/images/math/4/2/2/4228fb192d2ce47623abd1b05003cd4b.png)
obtained by removing entries where and write
if
for all primes
.
Theorem 3.6 [Astey&Gitler&Micha&Pastor2003].
If and
then
is homotopy equivalent to
if and only if they have the same Euler characteristic and signature.
In contrast to the results of Fang and Ligober and Wood which hold when is small relative to the primes dividing
, the above theorem leads to a homotopy classification which holds when
is large relative to
. If
for all p then define
otherwise let
![\displaystyle N_d := \mathrm{Max} \{ 2(p_i-1)p_i^{[(\nu_{p}(d)-2)/2]} | \nu_{p}(d) \geq 2 \}.](/images/math/6/3/6/636ce5ebc2851f61f05adf3e72d0c8ec.png)
Theorem 3.7 [Astey&Gitler&Micha&Pastor2003].
If and
have the same multidegree
and
then
and
are homotopy equivalent if and only if they have the same signature and Euler characteritic.
4 Further discussion
4.1 Splitting theorems
5 References
- [???] Template:???
- [Astey&Gitler&Micha&Pastor2003] L. Astey, S. Gitler, E. Micha and G. Pastor, On the homotopy type of complete intersections, Topology 44 (2005), no.1, 249–260. MR2104011 (2005i:14058) Zbl 1072.14063
- [Ebeling1990] W. Ebeling, An example of two homeomorphic, nondiffeomorphic complete intersection surfaces, Invent. Math. 99 (1990), no.3, 651–654. MR1032884 (91c:57038) Zbl 0707.14045
- [Fang&Klaus1996] F. Fang and S. Klaus, Topological classification of
-dimensional complete intersections, Manuscripta Math. 90 (1996), no.2, 139–147. MR1391206 (97g:57046) Zbl 0866.57015
- [Fang1997] F. Fang, Topology of complete intersections, Comment. Math. Helv. 72 (1997), no.3, 466–480. MR1476060 (98k:14071) Zbl 0896.14028
- [Freedman1982] M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no.3, 357–453. MR679066 (84b:57006) Zbl 0528.57011
- [Jupp1973] P. E. Jupp, Classification of certain
-manifolds, Proc. Cambridge Philos. Soc. 73 (1973), 293–300. MR0314074 (47 #2626) Zbl 0249.57005
- [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
- [Libgober&Wood1981] A. S. Libgober and J. W. Wood, On the topological structure of even-dimensional complete intersections, Trans. Amer. Math. Soc. 267 (1981), no.2, 637–660. MR626495 (83e:57017) Zbl 0475.57013
- [Libgober&Wood1982] A. S. Libgober and J. W. Wood, Differentiable structures on complete intersections. I, Topology 21 (1982), no.4, 469–482. MR670748 (84a:57037) Zbl 0504.57015
- [Libgober&Wood1990] A. S. Libgober and J. W. Wood, Uniqueness of the complex structure on Kähler manifolds of certain homotopy types, J. Differential Geom. 32 (1990), no.1, 139–154. MR1064869 (91g:32039) Zbl 0711.53052
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Traving1985] C. Traving, Zur Diffeomorphieklassifikation vollständiger Durchschnitte, Diplomarbeit, Mainz (1985).
- [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain
-manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
- [[Template:{???|[{???]]] {{{???}}
This page has not been refereed. The information given here might be incomplete or provisional. |
![d_1, \dots, d_k](/images/math/b/8/0/b80fc20baa8171018aa1a3764483e5ce.png)
![n+k+1](/images/math/4/3/a/43a2a7a3c6f8abbb4276b0e238b69b85.png)
![z = (z_1, \dots, z_{n+k+1})](/images/math/f/8/f/f8fd4df65be7da46a74fe70ba0779447.png)
![\displaystyle X(f_1, \dots, f_k) \coloneq \{ [z] \in \CP^{n+k} | f_i(z) = 0 \text{~for~} i = 1, \dots, k\}.](/images/math/4/8/d/48d767881e8626164661eb292ab83ecd.png)
The set is an algebraic variety. It is non-singular if for all
with
the affine function
![\displaystyle \Cc^{n+k} \to \Cc^{k}, ~~~ (z_1, \dots, z_{n+k}) \longmapsto (f_1^j(z_1, \dots, z_{n+k}), \dots, f_k^j(z_1, \dots, z_{n+k}))](/images/math/8/3/2/832c31943fc43d6fd25ed2cb651d38fc.png)
where , has
as a regular value. In this case
is a smooth, complex variety and in particular defines a closed, smooth, oriented dimensional manifold of real dimension
. This is called a complete intersection.
By a theorem of Thom [{???]} the diffeomorphism type of depends only upon the multi-degree,
, and we write
for
.
For example:
-
,
-
the oriented surface of genus
,
-
is a complex K3 surface,
-
is a Calabi-Yau 3-fold.
2 Invariants
By the Lefschetz hyperplane theorem the inclusion is an n-connected map. Hence:
- if n > 1 then
,
-
if
,
-
for all
unless
.
-
where
is the Euler characteristic of
which we discuss further below.
- When
is even
generates a summand of
.
Note that here and throughout integer coefficients are use for (co)homology.
2.1 Cohomology ring
Let be the canonical line bundle over
and let
be defined by
where
is the first Chern class of
. Let
denote the fundamental class of
and let
be the product of all degrees, called the total degree. We have the following useful identity [???]
![\displaystyle \langle x^n, [X_n({\underline{d}})] \rangle = d.](/images/math/c/4/2/c42040a33bb3081f0d64c0ef9cba1e3b.png)
Let or
and consider the graded ring
![\displaystyle H^*(n, d) :=\Zz[x, y]/\{x^{m+1} = dy, y^2 = 0 \}](/images/math/a/2/8/a2875558e0fa4f20ba7f73dbf2170c41.png)
where the dimensions of and
are
and
respectively. Let
, considered as a graded ring in dimension
- If
is odd:
- the ring
is determined by
,
and
,
- there is a short exact sequence
- the intersection form
is of course skew hyperbolic.
- the ring
- If
is even:
- the ring
is determined by
,
and the pair
:
- there is a short exact sequence
.
- Some properties of
are described below.
- the ring
Proposition 2.1 [Libgober&Wood1981].
If is even, then
is indefinate unless
or
.
2.2 Characteristic classes
The stable tangent bundle of is isomorphic to
, [Milnor&Stasheff1974], and the normal bundle of the inclusion
is given by the identity [???]
![\displaystyle \nu(i) \cong i^*(L^{d_1} \oplus \dots \oplus L^{d_k})](/images/math/9/6/d/96d13cacbdcae3dbcb1cfa973b190b59.png)
where denotes the
-fold tensor product of
with itself. From this one deduces that the stable tangent bundle of
,
, satisfies the equation
![\displaystyle \tau(X_n({\underline{d}})) \oplus i^*(L^{d_1} \oplus \dots \oplus L^{d_k}) \cong i^*((n+k+1)L).](/images/math/f/f/b/ffb5f6d4135a353f956aedad9a699dda.png)
It follows immediately that the total Chern class and the total Pontrjagin class of are given by
![\displaystyle c(X_n({\underline{d}})) = (1+x)^{n+k+1}\prod_{j=1}^k(1+d_jx)^{-1} \in H^{2*}(X_n(\underline{d});\Zz),](/images/math/4/2/3/4238ed3fed927ec647060b57003c47da.png)
![\displaystyle p(X_n({\underline{d}})) = (1-x^2)^{n+k+1}\prod_{j=1}^k(1-d_j^2x^2)^{-1} \in H^{4*}(X_n(\underline{d});\Zz).](/images/math/a/f/2/af262d8be56d51bc908dea24b90be462.png)
Moreover, the Euler class and Euler characteristic of is given by
![\displaystyle e(X_n({\underline{d}})) = c_n(X_n({\underline{d}})), ~~~ \chi(\underline{d}) = \langle e(X_n(\underline{d}), [X_n(\underline{d})]\rangle.](/images/math/f/e/0/fe059d8703b858339f44a160448043ea.png)
3 Classification
The smooth classification of complete intersections for is organised by the following conjecture, often called the Sullivan Conjecture after Dennis Sullivan.
Conjecture 3.1. For , complete intersections
and
are diffeomorphic if and only if all of the following conditions hold
-
,
-
,
-
.
Note that we regard as multiplies of
in order to view
as an element of
.
3.1 Classification in low dimensions
- For
the Sullivan Conjecture holds by the classification of surfaces.
- For
the topological Sullivan Conjecture holds for
by applying [Freedman1982]. The Sullivan Conjecture fails smoothly by [Ebeling1990] and [Libgober&Wood1990].
- For
the Sullivan Conjecture holds by [Wall1966] and [Jupp1973]. See the page simply-connected 6-manifolds.
- For
the topological Sullivan Conjecture is true [Fang&Klaus1996]. Hence by smoothing theory the smooth conjecture, which is still open in general, holds up to connected sum with the exotic 8-sphere.
3.2 Further classification theorems
We now discuss further classification results for complete intersections. Let where
is prime.
Theorem 3.2 [Traving1985].
If and
for all
such that
then the Sullivan Conjecture holds.
Traving's proof uses modified surgery: see [Kreck1999, Section 8] for a summary.
Theorem 3.3 [Fang1997].
If and
for all
such that
then the topological Sullivan Conjecture holds.
Fang's proof proceeds by extending results of [Libgober&Wood1982] on the homotopy classification of complete intersections where is even to the case of
odd. He then solves the homeomorphism classification by exhibiting characteristic varieties for complete intersections and using them to calculate classical surgery obstructions. In particular he proves
Theorem 3.4 [Fang1997].
Let and
be homotopy equivalent complete intersections. If
is odd and
for all
then
and
are homeomorphic to each other if and only if their Pontrjagin classes agree.
3.3 Classification up to homotopy
Recall that the signature of, , of
is the signature of its intersection form. If
is odd
and if
is even
can be computed from the
via Hirzebruch's signature theorem.
Theorem 3.5 [Libgober&Wood1982] and [Fang1997]. Let and
be complete intersections with the same total degree
. Suppose that
if
. If
then
and
are homotopy equivalent if and only if they have the same signature and Euler characteristic.
Finally we summarise theorems of [Astey&Gitler&Micha&Pastor2003]. Define to be the unordered sequence
![\displaystyle (\nu_p(d_1), \dots, \nu_p(d_k))](/images/math/4/2/2/4228fb192d2ce47623abd1b05003cd4b.png)
obtained by removing entries where and write
if
for all primes
.
Theorem 3.6 [Astey&Gitler&Micha&Pastor2003].
If and
then
is homotopy equivalent to
if and only if they have the same Euler characteristic and signature.
In contrast to the results of Fang and Ligober and Wood which hold when is small relative to the primes dividing
, the above theorem leads to a homotopy classification which holds when
is large relative to
. If
for all p then define
otherwise let
![\displaystyle N_d := \mathrm{Max} \{ 2(p_i-1)p_i^{[(\nu_{p}(d)-2)/2]} | \nu_{p}(d) \geq 2 \}.](/images/math/6/3/6/636ce5ebc2851f61f05adf3e72d0c8ec.png)
Theorem 3.7 [Astey&Gitler&Micha&Pastor2003].
If and
have the same multidegree
and
then
and
are homotopy equivalent if and only if they have the same signature and Euler characteritic.
4 Further discussion
4.1 Splitting theorems
5 References
- [???] Template:???
- [Astey&Gitler&Micha&Pastor2003] L. Astey, S. Gitler, E. Micha and G. Pastor, On the homotopy type of complete intersections, Topology 44 (2005), no.1, 249–260. MR2104011 (2005i:14058) Zbl 1072.14063
- [Ebeling1990] W. Ebeling, An example of two homeomorphic, nondiffeomorphic complete intersection surfaces, Invent. Math. 99 (1990), no.3, 651–654. MR1032884 (91c:57038) Zbl 0707.14045
- [Fang&Klaus1996] F. Fang and S. Klaus, Topological classification of
-dimensional complete intersections, Manuscripta Math. 90 (1996), no.2, 139–147. MR1391206 (97g:57046) Zbl 0866.57015
- [Fang1997] F. Fang, Topology of complete intersections, Comment. Math. Helv. 72 (1997), no.3, 466–480. MR1476060 (98k:14071) Zbl 0896.14028
- [Freedman1982] M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no.3, 357–453. MR679066 (84b:57006) Zbl 0528.57011
- [Jupp1973] P. E. Jupp, Classification of certain
-manifolds, Proc. Cambridge Philos. Soc. 73 (1973), 293–300. MR0314074 (47 #2626) Zbl 0249.57005
- [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
- [Libgober&Wood1981] A. S. Libgober and J. W. Wood, On the topological structure of even-dimensional complete intersections, Trans. Amer. Math. Soc. 267 (1981), no.2, 637–660. MR626495 (83e:57017) Zbl 0475.57013
- [Libgober&Wood1982] A. S. Libgober and J. W. Wood, Differentiable structures on complete intersections. I, Topology 21 (1982), no.4, 469–482. MR670748 (84a:57037) Zbl 0504.57015
- [Libgober&Wood1990] A. S. Libgober and J. W. Wood, Uniqueness of the complex structure on Kähler manifolds of certain homotopy types, J. Differential Geom. 32 (1990), no.1, 139–154. MR1064869 (91g:32039) Zbl 0711.53052
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Traving1985] C. Traving, Zur Diffeomorphieklassifikation vollständiger Durchschnitte, Diplomarbeit, Mainz (1985).
- [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain
-manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
- [[Template:{???|[{???]]] {{{???}}
This page has not been refereed. The information given here might be incomplete or provisional. |