Complete intersections

1 Construction and examples


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

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

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

For example:

• $X_n(1, \dots, 1) = \CP^{n}$$X_n(1, \dots, 1) = \CP^{n}$,
• $X_1(d_1, \dots, d_k) = F_{g}$$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)$$g = 1-(d/2)(k+2 - \sum_{j=1}^k d_j)$,
• $X_2(4)$$X_2(4)$ is a complex K3 surface,
• $X_3(5)$$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}$$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 \}$$\pi_1(X_n(\underline{d})) = \{ e \}$,
• $H^{2i+1}(X_n({\underline{d}})) = 0$$H^{2i+1}(X_n({\underline{d}})) = 0$ if $2i+1 \neq n$$2i+1 \neq n$,
• $H^{2i}(X_n({\underline{d}})) \cong \Zz$$H^{2i}(X_n({\underline{d}})) \cong \Zz$ for all $i \leq n$$i \leq n$ unless $2i=n$$2i=n$.
• $H^{n}(X_n({\underline{d}})) \cong \Zz^{\chi(\underline{d})-n}$$H^{n}(X_n({\underline{d}})) \cong \Zz^{\chi(\underline{d})-n}$ where $\chi(\underline{d})$$\chi(\underline{d})$ is the Euler characteristic of $X_n({\underline{d}})$$X_n({\underline{d}})$ which we discuss further below.
• When $n$$n$ is even $x^{n/2}$$x^{n/2}$ generates a summand of $H^n(X_n(\underline{d})$$H^n(X_n(\underline{d})$.

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

2.1 Cohomology ring

Let $L$$L$ be the canonical line bundle over $\mathbb{C}P^{n+k}$$\mathbb{C}P^{n+k}$ and let $x \in H^2(X_n({\underline{d}}))$$x \in H^2(X_n({\underline{d}}))$ be defined by $x = i^*(c_1(L))$$x = i^*(c_1(L))$ where $c_1(L)$$c_1(L)$ is the first Chern class of $L$$L$. Let $[X_n({\underline{d}})] \in H_{2n}(X_n({\underline{d}}))$$[X_n({\underline{d}})] \in H_{2n}(X_n({\underline{d}}))$ denote the fundamental class of $X_n({\underline{d}})$$X_n({\underline{d}})$ and let $d = d_1 d_2 \dots d_k$$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$$n = 2m$ or $2m+1$$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$$x$ and $y$$y$ are $2$$2$ and $2m+2$$2m+2$ respectively. Let $H^n = H^n(X_n(\underline{d})$$H^n = H^n(X_n(\underline{d})$, considered as a graded ring in dimension $n$$n$

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

Proposition 2.1 [Libgober&Wood1981]. If $n \geq 3$$n \geq 3$ is even, then $\lambda : H^n \times H^n \to \Zz$$\lambda : H^n \times H^n \to \Zz$ is indefinate unless $X_n({\underline{d}}) = X_n(1), X_n(2)$$X_n({\underline{d}}) = X_n(1), X_n(2)$ or $X_n(2,2)$$X_n(2,2)$.

2.2 Characteristic classes

The stable tangent bundle of $\CP^n$$\CP^n$ is isomorphic to $(n+1)L$$(n+1)L$, [Milnor&Stasheff1974], and the normal bundle of the inclusion $i : X_n({\underline{d}}) \to \CP^{n+k}$$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$$L^{d_j} = L \otimes \dots \otimes L$ denotes the $d_j$$d_j$-fold tensor product of $L$$L$ with itself. From this one deduces that the stable tangent bundle of $X_n({\underline{d}})$$X_n({\underline{d}})$, $\tau(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}})$$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})$$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$$n \neq 2$ is organised by the following conjecture, often called the Sullivan Conjecture after Dennis Sullivan.

Conjecture 3.1. For $n \neq 2$$n \neq 2$, complete intersections $X_n({{\underline{d}}_1})$$X_n({{\underline{d}}_1})$ and $X_n({{\underline{d}}_2})$$X_n({{\underline{d}}_2})$ are diffeomorphic if and only if all of the following conditions hold

• $d_1 = d_2$$d_1 = d_2$,
• $P_i(X_n({{\underline{d}}_1})) = P_i(X_n({{\underline{d}}_2})) \in \Zz$$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}))$$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$$p_i(X_n(\underline{d}_\epsilon)), \epsilon = 1, 2$ as multiplies of $x^{2i}$$x^{2i}$ in order to view $p_i(X_n({\underline{d}}_\epsilon))$$p_i(X_n({\underline{d}}_\epsilon))$ as an element of $\Zz$$\Zz$.

3.1 Classification in low dimensions

• For $n=1$$n=1$ the Sullivan Conjecture holds by the classification of surfaces.
• For $n=2$$n=2$ the topological Sullivan Conjecture holds for $n=2$$n=2$ by applying [Freedman1982]. The Sullivan Conjecture fails smoothly by [Ebeling1990] and [Libgober&Wood1990].
• For $n=3$$n=3$ the Sullivan Conjecture holds by [Wall1966] and [Jupp1973]. See the page 6-manifolds: 1-connected.
• For $n = 4$$n = 4$ the topological Sullivan Conjecture is true [Fang&Klaus1996]. Hence by smoothing theory the smooth conjecture, which is still open in general, holds up to connected sum with the exotic 8-sphere.
• For $n = 5, 6$$n = 5, 6$ and $7$$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)}$$d = \prod_{p}p^{\nu_p(d)}$ where $p$$p$ is prime.

Theorem 3.2 [Traving1985]. If $n \geq 3$$n \geq 3$ and $\nu_p(d) \geq ((2n+1)/2(p-1) + 1)$$\nu_p(d) \geq ((2n+1)/2(p-1) + 1)$ for all $p$$p$ such that $p(p-1) \leq n+1$$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$$n \geq 3$ and $\nu_p(d) = 0$$\nu_p(d) = 0$ for all $p$$p$ such that $p \leq (n+3)/2$$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$$n$ is even to the case of $n$$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)$$X_n(\underline{d}_1)$ and $X_n(\underline{d}_2)$$X_n(\underline{d}_2)$ be homotopy equivalent complete intersections. If $d$$d$ is odd and $n \neq 2^i - 2$$n \neq 2^i - 2$ for all $i \in \Zz$$i \in \Zz$ then $X_n(\underline{d}_1)$$X_n(\underline{d}_1)$ and $X_n(\underline{d}_2)$$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}))$$\sigma(X_n(\underline{d}))$, of $X_n(\underline{d})$$X_n(\underline{d})$ is the signature of its intersection form. If $n$$n$ is odd $\sigma(X_n(\underline{d}))=0$$\sigma(X_n(\underline{d}))=0$ and if $n$$n$ is even $\sigma(X_n(\underline{d}))$$\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.

Theorem 3.5 [Libgober&Wood1982] and [Fang1997]. Let $X_n(\underline{d}_1)$$X_n(\underline{d}_1)$ and $X_n(\underline{d}_2)$$X_n(\underline{d}_2)$ be complete intersections with the same total degree $d$$d$. Suppose that $\nu_p(d) = 0$$\nu_p(d) = 0$ if $p \leq (n+3)/2$$p \leq (n+3)/2$. If $n \neq 2$$n \neq 2$ then $X_n(\underline{d}_1)$$X_n(\underline{d}_1)$ and $X_n(\underline{d}_2)$$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})$$\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$$\nu_p(d_j) = 0$ and write $\underline{d}_1 \sim \underline{d}_2$$\underline{d}_1 \sim \underline{d}_2$ if $\mathcal{L}_p(\underline{d}_1) = \mathcal{L}_p(\underline{d}_2)$$\mathcal{L}_p(\underline{d}_1) = \mathcal{L}_p(\underline{d}_2)$ for all primes $p$$p$.

Theorem 3.6 [Astey&Gitler&Micha&Pastor2003]. If $\underline{d}_1 \sim \underline{d}_2$$\underline{d}_1 \sim \underline{d}_2$ and $n > 2$$n > 2$ then $X_n(\underline{d}_1)$$X_n(\underline{d}_1)$ is homotopy equivalent to $X_n(\underline{d}_2)$$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$$n$ is small relative to the primes dividing $d$$d$, the above theorem leads to a homotopy classification which holds when $n$$n$ is large relative to $d$$d$. If $\nu_p(d) < 2$$\nu_p(d) < 2$ for all p then define $N_d : = 3$$N_d : = 3$ otherwise let

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

Theorem 3.7 [Astey&Gitler&Micha&Pastor2003]. If $X_n(\underline{d}_1)$$X_n(\underline{d}_1)$ and $X_n(\underline{d}_2)$$X_n(\underline{d}_2)$ have the same multidegree $d$$d$ and $n \geq N_d$$n \geq N_d$ then $X_n(\underline{d}_1)$$X_n(\underline{d}_1)$ and $X_n(\underline{d}_2)$$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}$$\CP^{r}$ is isomorphic to a complete intersection, if $3n>2r$$3n>2r$.