Complete intersections

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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
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
$$ X(f_1, \dots, f_k) \coloneq \{ [z] \in \CP^{n+k} | f_i(z) = 0 ~\text{for}~ i = 1, \dots, k\}.$$
+
$$ X(f_1, \dots, f_k) \coloneq \{ [z] \in \CP^{n+k} \,|\, f_i(z) = 0 ~\text{for}~ i = 1, \dots, k\}.$$
The set $X(f_1, \dots f_k)$ is an [[Wikipedia:Algebraic_variety|algebraic variety]]. It is non-singular if for all $j$ with $0 \leq j \leq n+k$ the affine function
The set $X(f_1, \dots f_k)$ is an [[Wikipedia:Algebraic_variety|algebraic variety]]. It is non-singular if for all $j$ with $0 \leq j \leq n+k$ the affine function
$$ \Cc^{n+k} \to \Cc^{k}, ~~~ (z_1, \dots, z_{n+k}) \longmapsto (f_1^j(z_1, \dots, z_{n+k}), \dots, f_k^j(z_1, \dots, z_{n+k}))$$
$$ \Cc^{n+k} \to \Cc^{k}, ~~~ (z_1, \dots, z_{n+k}) \longmapsto (f_1^j(z_1, \dots, z_{n+k}), \dots, f_k^j(z_1, \dots, z_{n+k}))$$

Latest revision as of 05:28, 7 January 2020

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Contents

1 Construction and examples

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

\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}))

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.

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 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.

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.

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 \}

where the dimensions of x and y are 2 and 2m+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 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 n is even:
    • the ring H^*(H_n(\underline{d})) is determined by n, 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.

Proposition 2.1 [Libgober&Wood1981]. 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).

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})

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).

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  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.


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.

Conjecture 3.1. 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.

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].

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.

Theorem 3.2 [Traving1985]. 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.

Theorem 3.3 [Fang1997]. 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 n is even to the case of 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

Theorem 3.4 [Fang1997]. 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.

3.3 Classification up to homotopy

Recall that the signature of, \sigma(X_n(\underline{d})), of X_n(\underline{d}) is the signature of its intersection form. If n is odd \sigma(X_n(\underline{d}))=0 and if n is even \sigma(X_n(\underline{d})) can be computed from the p_j(X_n(\underline{d})) via Hirzebruch's signature theorem.

Theorem 3.5 [Libgober&Wood1982] and [Fang1997]. 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))

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.

Theorem 3.6 [Astey&Gitler&Micha&Pastor2003]. 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 n is small relative to the primes dividing d, the above theorem leads to a homotopy classification which holds when n is large relative to d. If \nu_p(d) < 2 for all p then define 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 \}.

Theorem 3.7 [Astey&Gitler&Micha&Pastor2003]. 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.

4 Further discussion

Conjecture 4.1 Hartshorne. Every smooth algebraic variety of dimension n which is embedded in \CP^{r} is isomorphic to a complete intersection, if 3n>2r.

4.1 Splitting theorems


5 References

6 External links

\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 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\}.

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}))

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.

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 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.

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.

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 \}

where the dimensions of x and y are 2 and 2m+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 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 n is even:
    • the ring H^*(H_n(\underline{d})) is determined by n, 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.

Proposition 2.1 [Libgober&Wood1981]. 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).

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})

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).

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  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.


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.

Conjecture 3.1. 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.

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].

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.

Theorem 3.2 [Traving1985]. 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.

Theorem 3.3 [Fang1997]. 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 n is even to the case of 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

Theorem 3.4 [Fang1997]. 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.

3.3 Classification up to homotopy

Recall that the signature of, \sigma(X_n(\underline{d})), of X_n(\underline{d}) is the signature of its intersection form. If n is odd \sigma(X_n(\underline{d}))=0 and if n is even \sigma(X_n(\underline{d})) can be computed from the p_j(X_n(\underline{d})) via Hirzebruch's signature theorem.

Theorem 3.5 [Libgober&Wood1982] and [Fang1997]. 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))

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.

Theorem 3.6 [Astey&Gitler&Micha&Pastor2003]. 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 n is small relative to the primes dividing d, the above theorem leads to a homotopy classification which holds when n is large relative to d. If \nu_p(d) < 2 for all p then define 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 \}.

Theorem 3.7 [Astey&Gitler&Micha&Pastor2003]. 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.

4 Further discussion

Conjecture 4.1 Hartshorne. Every smooth algebraic variety of dimension n which is embedded in \CP^{r} is isomorphic to a complete intersection, if 3n>2r.

4.1 Splitting theorems


5 References

6 External links

\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 $n$ is even: ** the ring $H^*(H_n(\underline{d}))$ is determined by $n$, $d$ and the pair $((H^n, \lambda), x^{n/2})$: ** there is a short exact sequence 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\}.

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}))

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.

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 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.

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.

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 \}

where the dimensions of x and y are 2 and 2m+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 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 n is even:
    • the ring H^*(H_n(\underline{d})) is determined by n, 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.

Proposition 2.1 [Libgober&Wood1981]. 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).

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})

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).

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  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.


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.

Conjecture 3.1. 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.

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].

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.

Theorem 3.2 [Traving1985]. 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.

Theorem 3.3 [Fang1997]. 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 n is even to the case of 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

Theorem 3.4 [Fang1997]. 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.

3.3 Classification up to homotopy

Recall that the signature of, \sigma(X_n(\underline{d})), of X_n(\underline{d}) is the signature of its intersection form. If n is odd \sigma(X_n(\underline{d}))=0 and if n is even \sigma(X_n(\underline{d})) can be computed from the p_j(X_n(\underline{d})) via Hirzebruch's signature theorem.

Theorem 3.5 [Libgober&Wood1982] and [Fang1997]. 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))

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.

Theorem 3.6 [Astey&Gitler&Micha&Pastor2003]. 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 n is small relative to the primes dividing d, the above theorem leads to a homotopy classification which holds when n is large relative to d. If \nu_p(d) < 2 for all p then define 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 \}.

Theorem 3.7 [Astey&Gitler&Micha&Pastor2003]. 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.

4 Further discussion

Conjecture 4.1 Hartshorne. Every smooth algebraic variety of dimension n which is embedded in \CP^{r} is isomorphic to a complete intersection, if 3n>2r.

4.1 Splitting theorems


5 References

6 External links

\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. {{beginthm|Proposition|{{cite|Libgober&Wood1981}}}} 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 $X_n(2,2)$. {{endthm}}
=== Characteristic classes === ; The stable tangent bundle of $\CP^n$ is isomorphic to $(n+1)L$, {{cite|Milnor&Stasheff1974}}, and the [[Wikipedia:Normal_bundle#General_definition|normal bundle]] of the inclusion $i : X_n({\underline{d}}) \to \CP^{n+k}$ is given by the identity {{cite|???}} $$\nu(i) \cong i^*(L^{d_1} \oplus \dots \oplus L^{d_k})$$ where $L^{d_j} = L \otimes \dots \otimes L$ denotes the $d_j$-fold [[Wikipedia:Vector_bundle#Operations_on_vector_bundles|tensor product]] of $L$ with itself. From this one deduces that the stable [[Wikipedia:Tangent_bundle|tangent bundle]] of $X_n({\underline{d}})$, $\tau(X_n({\underline{d}}))$, satisfies the equation $$ \tau(X_n({\underline{d}})) \oplus i^*(L^{d_1} \oplus \dots \oplus L^{d_k}) \cong i^*((n+k+1)L).$$ It follows immediately that the total [[Wikipedia:Chern_class|Chern class]] and the total [[Wikipedia:Pontrjagin_class|Pontrjagin class]] of $X_n({\underline{d}})$ are given by $$ c(X_n({\underline{d}})) = (1+x)^{n+k+1}\prod_{j=1}^k(1+d_jx)^{-1} \in H^{2*}(X_n(\underline{d});\Zz),$$ $$ p(X_n({\underline{d}})) = (1-x^2)^{n+k+1}\prod_{j=1}^k(1-d_j^2x^2)^{-1} \in H^{4*}(X_n(\underline{d});\Zz).$$ Moreover, the [[Wikipedia:Euler_class|Euler class]] and Euler characteristic of $X_n(\underline{d})$ is given by $$ e(X_n({\underline{d}})) = c_n(X_n({\underline{d}})), ~~~ \chi(\underline{d}) = \langle e(X_n(\underline{d}), [X_n(\underline{d})]\rangle.$$ == 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. {{beginthm|Conjecture}} 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 * $d_1 = d_2$, * $P_i(X_n({{\underline{d}}_1})) = P_i(X_n({{\underline{d}}_2})) \in \Zz$, * $e(X_n({{\underline{d}}_1})) = e(X_n({{\underline{d}}_2}))$. {{endthm}} 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 $\Zz$. === Classification in low dimensions === ; * For $n=1$ the Sullivan Conjecture holds by the classification of [[Surface|surfaces]]. * For $n=2$ the topological Sullivan Conjecture holds for $n=2$ by applying {{cite|Freedman1982}}. The Sullivan Conjecture fails smoothly by {{cite|Ebeling1990}} and {{cite|Libgober&Wood1990}}. * For $n=3$ the Sullivan Conjecture holds by {{cite|Wall1966}} and {{cite|Jupp1973}}. See the page [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. * For $n = 4$ the topological Sullivan Conjecture is true {{cite|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 $ the topological Sullivan Conjecture is true {{cite|Fang&Wang2009}}. === Further classification theorems === ; We now discuss further classification results for complete intersections. Let $d = \prod_{p}p^{\nu_p(d)}$ where $p$ is prime. {{beginthm|Theorem|{{cite|Traving1985}}}} 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. {{endthm}} Traving's proof uses modified surgery: see {{cite|Kreck1999|Section 8}} for a summary. {{beginthm|Theorem|{{cite|Fang1997}}}} 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. {{endthm}} Fang's proof proceeds by extending results of {{cite|Libgober&Wood1982}} on the homotopy classification of complete intersections where $n$ is even to the case of $n$ 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 {{beginthm|Theorem|{{cite|Fang1997}}}} 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. {{endthm}} === Classification up to homotopy === ; Recall that the [[Wikipedia:Signature_(topology)|signature]] of, $\sigma(X_n(\underline{d}))$, of $X_n(\underline{d})$ is the signature of its intersection form. If $n$ is odd $\sigma(X_n(\underline{d}))=0$ and if $n$ is even $\sigma(X_n(\underline{d}))$ can be computed from the $p_j(X_n(\underline{d}))$ via Hirzebruch's [[Wikipedia:Hirzebruch_signature_theorem#L_genus_and_the_Hirzebruch_signature_theorem|signature theorem]]. {{beginthm|Theorem|{{cite|Libgober&Wood1982}} and {{cite|Fang1997}}}} 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. {{endthm}} Finally we summarise theorems of {{cite|Astey&Gitler&Micha&Pastor2003}}. Define $\mathcal{L}_p(\underline{d})$ to be the unordered sequence $$ (\nu_p(d_1), \dots, \nu_p(d_k))$$ 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 $p$. {{beginthm|Theorem|{{cite|Astey&Gitler&Micha&Pastor2003}}}} 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. {{endthm}} In contrast to the results of Fang and Ligober and Wood which hold when $n$ is small relative to the primes dividing $d$, the above theorem leads to a homotopy classification which holds when $n$ is large relative to $d$. If $\nu_p(d) < 2$ for all p then define $N_d : = 3$ otherwise let $$ N_d := \mathrm{Max} \{ 2(p_i-1)p_i^{[(\nu_{p}(d)-2)/2]} | \nu_{p}(d) \geq 2 \}.$$ {{beginthm|Theorem|{{cite|Astey&Gitler&Micha&Pastor2003}}}} 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. {{endthm}} == Further discussion == ; {{beginthm|Conjecture|Hartshorne}} Every smooth algebraic variety of dimension n which is embedded in $\CP^{r}$ is isomorphic to a complete intersection, if n>2r$. {{endthm}} === Splitting theorems === ; == References == {{#RefList:}} == External links == * The Wikipedia page about [[Wikipedia:Complete intersection|complete intersections]] [[Category:Manifolds]] 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\}.

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}))

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.

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 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.

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.

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 \}

where the dimensions of x and y are 2 and 2m+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 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 n is even:
    • the ring H^*(H_n(\underline{d})) is determined by n, 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.

Proposition 2.1 [Libgober&Wood1981]. 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).

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})

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).

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  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.


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.

Conjecture 3.1. 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.

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].

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.

Theorem 3.2 [Traving1985]. 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.

Theorem 3.3 [Fang1997]. 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 n is even to the case of 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

Theorem 3.4 [Fang1997]. 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.

3.3 Classification up to homotopy

Recall that the signature of, \sigma(X_n(\underline{d})), of X_n(\underline{d}) is the signature of its intersection form. If n is odd \sigma(X_n(\underline{d}))=0 and if n is even \sigma(X_n(\underline{d})) can be computed from the p_j(X_n(\underline{d})) via Hirzebruch's signature theorem.

Theorem 3.5 [Libgober&Wood1982] and [Fang1997]. 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))

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.

Theorem 3.6 [Astey&Gitler&Micha&Pastor2003]. 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 n is small relative to the primes dividing d, the above theorem leads to a homotopy classification which holds when n is large relative to d. If \nu_p(d) < 2 for all p then define 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 \}.

Theorem 3.7 [Astey&Gitler&Micha&Pastor2003]. 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.

4 Further discussion

Conjecture 4.1 Hartshorne. Every smooth algebraic variety of dimension n which is embedded in \CP^{r} is isomorphic to a complete intersection, if 3n>2r.

4.1 Splitting theorems


5 References

6 External links

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