Algebraic Cycles of a Fixed Degree

In this paper, the homotopy groups of Chow variety $C_{p,d}(P^n)$ of effective $p$-cycles of degree $d$ is proved to be stable in the sense that $p$ or $n$ increases. We also obtain a negative answer to a question by Lawson and Michelsohn on homotopy groups for the space of degree two cycles.

The proof of this theorem is based on Theorem 1.3 below and direct calculations under the assumption of a positive answer to Question 1.1.
The proof of Theorem 1.3 is given in section 3.
Acknowledgements: I would like to gratitude Eric Friedlander and Blaine Lawson for their interest and helpful advice on the organization of the paper.
2 Homology groups of the space of algebraic cycles with degree two The p-cycles of degree 1 is linear, so C p,1 (P n ) = G(p + 1, n + 1), where G(p + 1, n + 1) denotes the Grassmannian of (p + 1)-plane in C n+1 . Note that C p,2 (P n ) = SP 2 (G(p + 1, n + 1)) ∪ Q p,n where SP i (X) denotes the i-th symmetric product X and Q p,n consists of effective irreducible p-cycles of degree 2 in P n . An irreducible p-cycle of degree 2 always degenerates if n ≥ p + 2 (cf. [GH]). Hence Q p,n = (P p+1 , c) ∈ G(p + 2, n + 1) × C p,2 (P n )|c is irreducible and c ∈ C p,2 (P p+1 ) and it is a fiber bundle over G(p + 2, n + 1) whose fiber is the space of all irreducible quadric hypersurfaces in P p+1 . Note that all irreducible quadric hypersurfaces in P p+1 is isomorphic to P ( p+3 2 )−1 − SP 2 (P p+1 ), i.e., the complement of non-irreducible quadrics (which is a pair of p-planes) in the space of all quadric hypersurfaces in P p+1 .
In the following computation, we take p = 4, d = 2 as our example.  (F, Q), that is, the Leray spectral sequence degenerates at E 2 .
Proof. By the Leray's Theorem for singular homology, we get the E 2 term since B is simply connected. From the assumption, all odd dimensional homology groups of B and F vanish, so at least one of E 2 p,q and E 2 p−3,q+2 vanishes. This implies that d 2 is a zero map . Hence we get E 3 p,q = E 2 p,q and d 3 : E 2 p,q → E 2 p−3,q+2 . By the same reason, d 3 = d 4 = · · · = 0. Therefore, the Leray spectral sequence degenerates at E 2 , i.e., Proposition 2.2 Let X be a connected CW complex such that π k (X) = Z, 0 < k ≤ 9 and even; 0, k ≤ 9 and odd.
From the proof of Proposition 2.2, we actually prove the following result: Remark 2.4 Let M be a connected CW complex such that π k (X) = 0 for k odd and π k (M) ∼ = Z for k positive even integers. Then ) (with the weak topology) of Eilenberg-MacLane spaces and BU = lim q→∞ BU q . Although the homotopy type of these topological spaces are different, their corresponding Betti numbers coincide. Now we will compute Betti numbers of C 4,2 (P n ) (n ≥ 9) in a different way. Since C p,2 (P n ) − SP 2 (G(p + 1, n + 1)) = Q p,n , we have H i (C p,2 (P n ), SP 2 (G(p + 1, n + 1)) ∼ = H BM i (Q p,n ) for all i, where H BM i denotes the Borel-Moore homology. Let A p,n be the fiber bundle over G(p + 2, n + 1) whose fiber is the space of all quadric hypersurfaces in P p+1 and let B p,n be the fiber bundle over G(p + 2, n + 1) whose fiber is the space of pairs of hyperplanes in P p+1 . From the definition of Q p,n , we have for i ≥ 0 and n ≥ 9.
Lemma 2.5 Let A 4,n , B 4,n be defined as above.
To show the third formula, we note that B 4,n is a fiber bundle over G(6, n + 1) with fibers the space of pairs of hyperplanes in P p+1 , i.e., fibers are isomorphic to SP 2 (P 5 ). By Lemma 2.1, all the odd Betti numbers of B 4,n vanish and the even Betti numbers of B 4,n are given by the formula: The first five Betti numbers of SP 2 (P 5 ) are given as follows (cf. [M]): β i (SP 2 (P 5 )) = 1, 1, 2, 2, 3 for i = 0, 2, 4, 6, 8.

Proof of Theorem 1.3
In this section we will prove Theorem 1.3. The method comes from Lawson's proof to the Complex Suspension Theorem [L1], i.e., the complex suspension to the space of p-cycles yields a homotopy equivalence to the space of (p + 1)-cycles. Here we briefly review the general construction of such a homotopy equivalence. For details, the reader is referred to [L1], [L2] and [F].
Fix a hyperplane P n ⊂ P n+1 and a point P 0 ∈ P n+1 − P n . For any non-negative integer p and d, set (when d = 0, C p,0 (P n ) is defined to be the empty cycle.) In particular, their corresponding homotopy groups are isomorphic, i.e., Fix linear embedding P n+1 ⊂ P n+2 and two points x 0 , x 1 ∈ P n+2 − P n+1 . Each projection p i : P n+2 − {x 0 } → P n+1 (i = 0, 1) gives us a holomorphic line bundle over P n+1 .
Let D ∈ C n+1,e (P n+2 ) be effective divisor of degree e in P n+2 such that x 0 , x 1 are not in D. Any effective cycle c ∈ C p+1,d (P n+1 ) can be lifted to a cycle with support in D, defined as follows: The map Ψ(c, D) := Ψ D is a continuous map with variables c and D. Hence we have a continuous map Φ D : C p+1,d (P n+1 ) → C p+1,de (P n+2 − {x 0 , x 1 }). The composition of Φ D with the projection (p 0 ) * is (p 0 ) * • Φ D = e (multiplication by the integer e in the monoid, e · c = c + · · · + c for e times). The composition of Φ D with the projection (p 1 ) * gives us a transformation of cycles in P n+1 which makes most of them intersecting properly to P n . To see this, we consider the family of divisors tD, 0 ≤ t ≤ 1, given by scalar multiplication by t in the line bundle p 0 : Assume x 1 is not in tD for all t. Then the above construction gives us a family transformation The question is that for a fixed c, which divisors D ∈ C n+1,e (P n+2 ) (x 0 is not in D and x 1 is not in 0≤t≤1 tD) have the property that F tD (c) ∈ T p+1,de (P n+1 ) for all 0 < t ≤ 1.
As a corollary, we get the simply connectedness of C p,d (P n ), which has been obtained using general position arguments by Lawson ([L1], the proof to Lemma 2.6.): Corollary 3.4 ( [L1]) The Chow variety C p,d (P n ) is simply connected for integers p, d, n ≥ 0.
2 Now we study the connectedness of maps induced by the inclusion i : P n ֒→ P n+1 .
Remark 3.6 By using Proposition 3.5, we give another possibly more elementary proof Corollary 3.4. If n = p, then C p,d (P n ) is a point and so it is simply connected. If n = p + 1, then C p,d (P n ) ∼ = P ( n+d d )−1 so it is simply connected. If n − p ≥ 2, then π k (C p,d (P n )) ∼ =π k (C p,d (P n−1 )) ∼ = · · · ∼ = π k (C p,d (P p+1 )) = 0 for k ≤ 1 by using Proposition 3.5 and so C p,d (P n ) is simply connected.
Proposition 3.5 can be used to compute the second homotopy group of Chow varieties.
Proof. Replacing π k by π 2 in Remark 3.6 yields the proof of the first statement. The second statement is a result of the first statement, Corollary 3.4 and the Hurewicz isomorphism theorem.
2 Lawson's idea in the proof of the Complex Suspension Theorem in [L1] can be used to prove Proposition 3.5.
For any non-negative integer p and d, set Proposition 3.5 follows directly from the combination of Lemma 3.8 and 3.9 below: Lemma 3.8 U p,d (P n+1 ) is homotopy equivalent to C p,d (P n ). In particular, their corresponding homotopy groups are isomorphic, i.e., π * (U p,d (P n+1 )) ∼ = π * (C p,d (P n )).
Proof. Let p 0 : P n+1 −P 0 → P n be the canonical projection away from P 0 ∈ P n+1 −P n . Then p 0 induces a deformation retract from U p,d (P n+1 ) to C p,d (P n ).
To see this, note that p 0 is a holomorphic line bundle and let F t : (P n+1 − P 0 ) × C → P n+1 − P 0 denote the scalar multiplication by t ∈ C in this bundle. This map F t is holomorphic (in fact, algebraic) and satisfies F 1 = id P n+1 −P 0 and F 0 = p 0 . Hence F t induces a family of continuous maps (F t ) * : U p,d (P n+1 ) → C p,d (P n ). Therefore, (p 0 ) * is a deformation retraction. 2 Lemma 3.9 The inclusion i : U p,d (P n+1 ) ֒→ C p,d (P n+1 ) is 2(n − p)-connected.
Proof. By definition, it is enough to show that the induced maps on homotopy groups i * : π k (U p,d (P n+1 )) → π k (C p,d (P n+1 )) are isomorphisms for k ≤ 2(n − p). Let f : S k → C p,d (P n+1 ) be a continuous map for k ≤ 2(n − p). We may assume f to be piecewise linear up to homotopy. Then f is homotopy to a map S k → U p,d (P n+1 ). To see this, we note firstly that the union is a set of real codimension ≥ 2(n+1)−2p−k ≥ 2 > 0. So we can find a point Q ∈ P n+1 −P n such that Q is not in x∈S k f (x). Let G t be a family of automorphism of P n+1 mapping P 0 to Q but preserving P n . Composing with the automorphism G t , we obtain the family G t • f : S k → C p,d (P n+1 ) such that G 0 • f = f and G 1 • f : S k → U p,d (P n+1 ). Hence i * is surjective for k ≤ 2(n − p). Similarly, suppose g is a map of pairs g : (D k+1 , S k ) → (C p,d (P n+1 ), U p,d (P n+1 )). Then the map can be deformed through a map of pairs to one with image in U p,d (P n+1 ) if k ≤ 2(n − p). Therefore, i * is injective for k ≤ 2(n − p).