Complete intersections

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[edit] 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.

[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 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.

[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.

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 indefinate 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})

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.

[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.

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.

[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]. Hence by smoothing theory the smooth conjecture, which is still open in general, holds up to connected sum with the exotic 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.

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.

[edit] 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.

[edit] 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.

[edit] 4.1 Splitting theorems

[edit] 5 References

[edit] 6 External links

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