Complete intersections

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[edit] 1 Construction and examples

Let ${{Stub}} == Construction and examples == <wikitex>; Let $f_1, \dots, f_k$ be complex, homogeneous polynomials of degree $d_1, \dots, d_k$ in $n+k+1$ complex variables $z = (z_1, \dots, z_{n+k+1})$. Define $$ X(f_1, \dots, f_k) \coloneq \{ [z] \in \CP^{n+k} | f_i(z) = 0 \text{~for~} i = 1, \dots, k\}.$$ The set $X(f_1, \dots f_k)$ is an [[Wikipedia:Algebraic_variety|algebraic variety]]. It is non-singular if for all $j$ with <script type="math/tex">f_1, \dots, f_k$ be complex, homogeneous polynomials of degree $d_1, \dots, d_k$ in $n+k+1$ complex variables $z = (z_1, \dots, z_{n+k+1})$ . Define

$\displaystyle X(f_1, \dots, f_k) \coloneq \{ [z] \in \CP^{n+k} \,|\, f_i(z) = 0 ~\text{for}~ i = 1, \dots, k\}.$ $\displaystyle X(f_1, \dots, f_k) \coloneq \{ [z] \in \CP^{n+k} \,|\, f_i(z) = 0 ~\text{for}~ i = 1, \dots, k\}.$

The set $X(f_1, \dots f_k)$ is an algebraic variety. It is non-singular if for all $j$ with $0 \leq j \leq n+k$ the affine function

$\displaystyle \Cc^{n+k} \to \Cc^{k}, ~~~ (z_1, \dots, z_{n+k}) \longmapsto (f_1^j(z_1, \dots, z_{n+k}), \dots, f_k^j(z_1, \dots, z_{n+k}))$ $\displaystyle \Cc^{n+k} \to \Cc^{k}, ~~~ (z_1, \dots, z_{n+k}) \longmapsto (f_1^j(z_1, \dots, z_{n+k}), \dots, f_k^j(z_1, \dots, z_{n+k}))$

where $f_i^j = f_1(z_1, \dots, z_{j-1}, 1, z_{j+1}, \dots, z_{n+k})$ , has $(0, \dots , 0)$ as a regular value. In this case $X(f_1, \dots f_k)$ is a smooth, complex variety and in particular defines a closed, smooth, oriented dimensional manifold of real dimension $2n$ . This is called a complete intersection.

By a theorem of Thom [???] the diffeomorphism type of $X(f_1, \dots, f_k)$ depends only upon the multi-degree, $\underline{d} = (d_1, \dots, d_k)$ , and we write $X_n(\underline{d})$ for $X(f_1, \dots, f_k)$ .

For example:

$X_n(1, \dots, 1) = \CP^{n}$ ,
$X_1(d_1, \dots, d_k) = F_{g}$ the oriented surface of genus $g = 1-(d/2)(k+2 - \sum_{j=1}^k d_j)$ ,
$X_2(4)$ is a complex K3 surface,
$X_3(5)$ is a Calabi-Yau 3-fold.

[edit] 2 Invariants

By the Lefschetz hyperplane theorem the inclusion $i: X_n(\underline{d}) \to \mathbb{C}P^{n+k}$ is an n-connected map. Hence:

if n > 1 then $\pi_1(X_n(\underline{d})) = \{ e \}$ ,
$H^{2i+1}(X_n({\underline{d}})) = 0$ if $2i+1 \neq n$ ,
$H^{2i}(X_n({\underline{d}})) \cong \Zz$ for all $i \leq n$ unless $2i=n$ .
$H^{n}(X_n({\underline{d}})) \cong \Zz^{\chi(\underline{d})-n}$ where $\chi(\underline{d})$ is the Euler characteristic of $X_n({\underline{d}})$ which we discuss further below.
When
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is even $x^{n/2}$ generates a summand of $H^n(X_n(\underline{d})$ .

Note that here and throughout integer coefficients are use for (co)homology.

[edit] 2.1 Cohomology ring

Let $L$ be the canonical line bundle over $\mathbb{C}P^{n+k}$ and let $x \in H^2(X_n({\underline{d}}))$ be defined by $x = i^*(c_1(L))$ where $c_1(L)$ is the first Chern class of $L$ . Let $[X_n({\underline{d}})] \in H_{2n}(X_n({\underline{d}}))$ denote the fundamental class of $X_n({\underline{d}})$ and let $d = d_1 d_2 \dots d_k$ be the product of all degrees, called the total degree. We have the following useful identity [???]

$\displaystyle \langle x^n, [X_n({\underline{d}})] \rangle = d.$ $\displaystyle \langle x^n, [X_n({\underline{d}})] \rangle = d.$

Let $n = 2m$ or $2m+1$ and consider the graded ring

$\displaystyle H^*(n, d) :=\Zz[x, y]/\{x^{m+1} = dy, y^2 = 0 \}$ $\displaystyle H^*(n, d) :=\Zz[x, y]/\{x^{m+1} = dy, y^2 = 0 \}$

where the dimensions of $x$ $x$ and $y$ $y$ are $2$ $2$ and $2m+2$ $2m+2$ respectively. Let $H^n = H^n(X_n(\underline{d})$ $H^n = H^n(X_n(\underline{d})$ , considered as a graded ring in dimension

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$n$

If
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is odd:
- the ring $H^*(X_n(\underline{d}))$ is determined by
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  , $d$ and $e(\underline{d})$ ,
- there is a short exact sequence $0 \to H^*(n, d) \to H^*(X_n(\underline{d})) \to H^n \to 0,$
- the intersection form $\lambda : H^n\times H^n \to \Zz$ is of course skew hyperbolic.
If
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is even:
- the ring $H^*(H_n(\underline{d}))$ is determined by
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  , $d$ and the pair $((H^n, \lambda), x^{n/2})$ :
- there is a short exact sequence $0 \to H^*(n, d) \to H^*(X_n(\underline{d})) \to H^n/(x^{n/2}) \to 0$ .
- Some properties of $((H^n, \lambda), x^{n/2})$ are described below.

If $n \geq 3$ is even, then $\lambda : H^n \times H^n \to \Zz$ is indefinite unless $X_n({\underline{d}}) = X_n(1), X_n(2)$ or $X_n(2,2)$ .

[edit] 2.2 Characteristic classes

The stable tangent bundle of $\CP^n$ is isomorphic to $(n+1)L$ , [Milnor&Stasheff1974], and the normal bundle of the inclusion $i : X_n({\underline{d}}) \to \CP^{n+k}$ is given by the identity [???]

$\displaystyle \nu(i) \cong i^*(L^{d_1} \oplus \dots \oplus L^{d_k})$ $\displaystyle \nu(i) \cong i^*(L^{d_1} \oplus \dots \oplus L^{d_k})$

where $L^{d_j} = L \otimes \dots \otimes L$ denotes the $d_j$ -fold tensor product of $L$ with itself. From this one deduces that the stable tangent bundle of $X_n({\underline{d}})$ , $\tau(X_n({\underline{d}}))$ , satisfies the equation

$\displaystyle \tau(X_n({\underline{d}})) \oplus i^*(L^{d_1} \oplus \dots \oplus L^{d_k}) \cong i^*((n+k+1)L).$ $\displaystyle \tau(X_n({\underline{d}})) \oplus i^*(L^{d_1} \oplus \dots \oplus L^{d_k}) \cong i^*((n+k+1)L).$

It follows immediately that the total Chern class and the total Pontrjagin class of $X_n({\underline{d}})$ are given by

$\displaystyle c(X_n({\underline{d}})) = (1+x)^{n+k+1}\prod_{j=1}^k(1+d_jx)^{-1} \in H^{2*}(X_n(\underline{d});\Zz),$ $\displaystyle c(X_n({\underline{d}})) = (1+x)^{n+k+1}\prod_{j=1}^k(1+d_jx)^{-1} \in H^{2*}(X_n(\underline{d});\Zz),$

$\displaystyle p(X_n({\underline{d}})) = (1-x^2)^{n+k+1}\prod_{j=1}^k(1-d_j^2x^2)^{-1} \in H^{4*}(X_n(\underline{d});\Zz).$ $\displaystyle p(X_n({\underline{d}})) = (1-x^2)^{n+k+1}\prod_{j=1}^k(1-d_j^2x^2)^{-1} \in H^{4*}(X_n(\underline{d});\Zz).$

Moreover, the Euler class and Euler characteristic of $X_n(\underline{d})$ is given by

$\displaystyle e(X_n({\underline{d}})) = c_n(X_n({\underline{d}})), ~~~ \chi(\underline{d}) = \langle e(X_n(\underline{d}), [X_n(\underline{d})]\rangle.$ $\displaystyle e(X_n({\underline{d}})) = c_n(X_n({\underline{d}})), ~~~ \chi(\underline{d}) = \langle e(X_n(\underline{d}), [X_n(\underline{d})]\rangle.$

[edit] 3 Classification

The smooth classification of complete intersections for $n \neq 2$ is organised by the following conjecture, often called the Sullivan Conjecture after Dennis Sullivan.

For $n \neq 2$ , complete intersections $X_n({{\underline{d}}_1})$ and $X_n({{\underline{d}}_2})$ are diffeomorphic if and only if all of the following conditions hold

$d_1 = d_2$ ,
$P_i(X_n({{\underline{d}}_1})) = P_i(X_n({{\underline{d}}_2})) \in \Zz$ ,
$e(X_n({{\underline{d}}_1})) = e(X_n({{\underline{d}}_2}))$ .

Note that we regard $p_i(X_n(\underline{d}_\epsilon)), \epsilon = 1, 2$ as multiplies of $x^{2i}$ in order to view $p_i(X_n({\underline{d}}_\epsilon))$ as an element of $\Zz$ .

[edit] 3.1 Classification in low dimensions

For $n=1$ the Sullivan Conjecture holds by the classification of surfaces.
For $n=2$ the topological Sullivan Conjecture holds for $n=2$ by applying [Freedman1982]. The Sullivan Conjecture fails smoothly by [Ebeling1990] and [Libgober&Wood1990].
For $n=3$ the Sullivan Conjecture holds by [Wall1966] and [Jupp1973]. See the page 6-manifolds: 1-connected.
For $n = 4$ the topological Sullivan Conjecture is true [Fang&Klaus1996]. Indeed, Fang and Klaus prove that the smooth conjecture, which is still open in general, holds up to connected sum with a homotopy 8-sphere.
For $n = 5, 6$ and $7$ the topological Sullivan Conjecture is true [Fang&Wang2009].

[edit] 3.2 Further classification theorems

We now discuss further classification results for complete intersections. Let $d = \prod_{p}p^{\nu_p(d)}$ where $p$ is prime.

If $n \geq 3$ and $\nu_p(d) \geq ((2n+1)/2(p-1) + 1)$ for all $p$ such that $p(p-1) \leq n+1$ then the Sullivan Conjecture holds.

Traving's proof uses modified surgery: see [Kreck1999, Section 8] for a summary.

If $n \geq 3$ and $\nu_p(d) = 0$ for all $p$ such that $p \leq (n+3)/2$ then the topological Sullivan Conjecture holds.

Fang's proof proceeds by extending results of [Libgober&Wood1982] on the homotopy classification of complete intersections where

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$n$ is even to the case of

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$n$ 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

Let $X_n(\underline{d}_1)$ and $X_n(\underline{d}_2)$ be homotopy equivalent complete intersections. If $d$ is odd and $n \neq 2^i - 2$ for all $i \in \Zz$ then $X_n(\underline{d}_1)$ and $X_n(\underline{d}_2)$ are homeomorphic to each other if and only if their Pontrjagin classes agree.

[edit] 3.3 Classification up to homotopy

Recall that the signature of, $\sigma(X_n(\underline{d}))$ $\sigma(X_n(\underline{d}))$ , of $X_n(\underline{d})$ $X_n(\underline{d})$ is the signature of its intersection form. If

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$n$ is odd $\sigma(X_n(\underline{d}))=0$ $\sigma(X_n(\underline{d}))=0$ and if

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$n$ is even $\sigma(X_n(\underline{d}))$ $\sigma(X_n(\underline{d}))$ can be computed from the $p_j(X_n(\underline{d}))$ $p_j(X_n(\underline{d}))$ via Hirzebruch's signature theorem.

Let $X_n(\underline{d}_1)$ and $X_n(\underline{d}_2)$ be complete intersections with the same total degree $d$ . Suppose that $\nu_p(d) = 0$ if $p \leq (n+3)/2$ . If $n \neq 2$ then $X_n(\underline{d}_1)$ and $X_n(\underline{d}_2)$ 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 $\mathcal{L}_p(\underline{d})$ to be the unordered sequence

$\displaystyle (\nu_p(d_1), \dots, \nu_p(d_k))$ $\displaystyle (\nu_p(d_1), \dots, \nu_p(d_k))$

obtained by removing entries where $\nu_p(d_j) = 0$ and write $\underline{d}_1 \sim \underline{d}_2$ if $\mathcal{L}_p(\underline{d}_1) = \mathcal{L}_p(\underline{d}_2)$ for all primes $p$ .

If $\underline{d}_1 \sim \underline{d}_2$ and $n > 2$ then $X_n(\underline{d}_1)$ is homotopy equivalent to $X_n(\underline{d}_2)$ 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

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$n$ is small relative to the primes dividing $d$ $d$ , the above theorem leads to a homotopy classification which holds when

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$n$ is large relative to $d$ $d$ . If $\nu_p(d) < 2$ $\nu_p(d) < 2$ for all p then define $N_d : = 3$ $N_d : = 3$ otherwise let

$\displaystyle N_d := \mathrm{Max} \{ 2(p_i-1)p_i^{[(\nu_{p}(d)-2)/2]} | \nu_{p}(d) \geq 2 \}.$ $\displaystyle N_d := \mathrm{Max} \{ 2(p_i-1)p_i^{[(\nu_{p}(d)-2)/2]} | \nu_{p}(d) \geq 2 \}.$

If $X_n(\underline{d}_1)$ and $X_n(\underline{d}_2)$ have the same multidegree $d$ and $n \geq N_d$ then $X_n(\underline{d}_1)$ and $X_n(\underline{d}_2)$ are homotopy equivalent if and only if they have the same signature and Euler characteritic.

[edit] 4 Further discussion

Every smooth algebraic variety of dimension n which is embedded in $\CP^{r}$ is isomorphic to a complete intersection, if $3n>2r$ .

[edit] 4.1 Splitting theorems

[edit] 5 References

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[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 $4$ -dimensional complete intersections, Manuscripta Math. 90 (1996), no.2, 139–147. MR1391206 (97g:57046) Zbl 0866.57015
[Fang&Wang2009] F. Fang and J. Wang, Homeomorphism classification of complex projective complete intersections of dimensions 5, 6 and 7, to appear in Math. Zeit. (2009).

[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 $6$ -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 $6$ -manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601

[edit] 6 External links

The Wikipedia page about complete intersections

\leq j \leq n+k$ the affine function $$ \Cc^{n+k} \to \Cc^{k}, ~~~ (z_1, \dots, z_{n+k}) \longmapsto (f_1^j(z_1, \dots, z_{n+k}), \dots, f_k^j(z_1, \dots, z_{n+k}))$$ where $f_i^j = f_1(z_1, \dots, z_{j-1}, 1, z_{j+1}, \dots, z_{n+k})$, has $(0, \dots , 0)$ as a regular value. In this case $X(f_1, \dots f_k)$ is a smooth, complex variety and in particular defines a closed, smooth, oriented dimensional manifold of real dimension n$. This is called a complete intersection. By a theorem of Thom {{cite|???}} the diffeomorphism type of $X(f_1, \dots, f_k)$ depends only upon the multi-degree, $\underline{d} = (d_1, \dots, d_k)$, and we write $X_n(\underline{d})$ for $X(f_1, \dots, f_k)$. For example: * $X_n(1, \dots, 1) = \CP^{n}$, * $X_1(d_1, \dots, d_k) = F_{g}$ the oriented [[Surface|surface]] of [[Wikipedia:Genus_(mathematics)#Orientable_surface|genus]] $g = 1-(d/2)(k+2 - \sum_{j=1}^k d_j)$, * $X_2(4)$ is a complex [[Wikipedia:K3_suface|K3]] surface, * $X_3(5)$ is a [[Wikipedia:Calabi-Yau|Calabi-Yau]] 3-fold. == Invariants == ; By the [[Wikipedia:Lefschetz_hyperplane_theorem|Lefschetz hyperplane theorem]] the inclusion $i: X_n(\underline{d}) \to \mathbb{C}P^{n+k}$ is an [[Wikipedia:N-connected#n-connected_map|n-connected]] map. Hence: * if n > 1 then $\pi_1(X_n(\underline{d})) = \{ e \}$, * $H^{2i+1}(X_n({\underline{d}})) = 0$ if i+1 \neq n$, * $H^{2i}(X_n({\underline{d}})) \cong \Zz$ for all $i \leq n$ unless i=n$. * $H^{n}(X_n({\underline{d}})) \cong \Zz^{\chi(\underline{d})-n}$ where $\chi(\underline{d})$ is the [[Wikipedia:Euler_characteristic|Euler characteristic]] of $X_n({\underline{d}})$ which we discuss further below. * When $n$ is even $x^{n/2}$ generates a summand of $H^n(X_n(\underline{d})$. Note that here and throughout integer coefficients are use for (co)homology. === Cohomology ring === ; Let $L$ be the [[Wikipedia:Canonical_line_bundle|canonical line bundle]] over $\mathbb{C}P^{n+k}$ and let $x \in H^2(X_n({\underline{d}}))$ be defined by $x = i^*(c_1(L))$ where $c_1(L)$ is the [[Wikipedia:Chern_class|first Chern class]] of $L$. Let $[X_n({\underline{d}})] \in H_{2n}(X_n({\underline{d}}))$ denote the fundamental class of $X_n({\underline{d}})$ and let $d = d_1 d_2 \dots d_k$ be the product of all degrees, called the total degree. We have the following useful identity {{cite|???}} $$\langle x^n, [X_n({\underline{d}})] \rangle = d.$$ Let $n = 2m$ or m+1$ and consider the graded ring $$H^*(n, d) :=\Zz[x, y]/\{x^{m+1} = dy, y^2 = 0 \}$$ where the dimensions of $x$ and $y$ are $ and m+2$ respectively. Let $H^n = H^n(X_n(\underline{d})$, considered as a graded ring in dimension $n$ * If $n$ is odd: ** the ring $H^*(X_n(\underline{d}))$ is determined by $n$, $d$ and $e(\underline{d})$, ** there is a short exact sequence $f_1, \dots, f_k$ be complex, homogeneous polynomials of degree $d_1, \dots, d_k$ $d_1, \dots, d_k$ in $n+k+1$ $n+k+1$ complex variables $z = (z_1, \dots, z_{n+k+1})$ $z = (z_1, \dots, z_{n+k+1})$ . Define