Harnack-Thom Theorem for higher cycle groups and Picard varieties

We generalize the Harnack-Thom theorem to relate the ranks of the Lawson homology groups with $\Z_2$-coefficients of a real quasiprojective variety with the ranks of its reduced real Lawson homology groups. In the case of zero-cycle group, we recover the classical Harnack-Thom theorem and generalize the classical version to include real quasiprojective varieties. We use Weil's construction of Picard varieties to construct reduced real Picard groups, and Milnor's construction of universal bundles to construct some weak models of classifying spaces of some cycle groups. These weak models are used to produce long exact sequences of homotopy groups which are the main tool in computing the homotopy groups of some cycle groups of divisors. We obtain some congruences involving the Picard number of a nonsingular real projective variety and the rank of its reduced real Lawson homology groups of divisors.


Introduction
In [3,6,10], Friedlander and Lawson constructed Lawson homology and morphic cohomology, which serve as an enrichment of singular homology and singular cohomology, respectively, for complex projective varieties. In [18], the author constructed parallel theories for real projective varieties which are called reduced real Lawson homology and reduced real morphic cohomology. They enjoy many nice properties such as the Lawson suspension property, the homotopy invariance property, the bundle projection property, the splitting properties and for each theory there exists a localization long exact sequence. By using the Friedlander-Lawson moving lemma (see [7]), it is shown that there is a duality theorem between Lawson homology and morphic cohomology (see [8]) and a duality theorem between reduced real Lawson homology and reduced real morphic cohomology (see [18]). Furthermore, this duality is compatible with Poincaré duality.
The Harnack theorem says that a nonsingular totally real curve of degree d in RP 2 has at most g(d) + 1 connected components where g(d) = (d−1)(d−2) 2 . Later on Thom generalized Harnack's result to a statement which says that for a real projective variety X, the total Betti number B(X), B(ReX) and the Euler characteristic χ(X), χ(ReX) in Z 2 -coefficients of X and the real points ReX of X respectively satisfy the following relations(see [2,9,19]): B(ReX) ≤ B(X), B(ReX) ≡ B(X)mod 2, χ(ReX) ≡ χ(X)mod 2 In section 2 we give an overview of Lawson homology and reduced real Lawson homology. In section 3 we prove a splitting theorem which is the core of the proof of our main theorem. In section 4 we extend the classical Harnack-Thom theorem to a statement involving the ranks of Lawson homology groups with Z 2 -coefficients and the ranks of reduced real Lawson homology groups. For 0-cycle groups, we recover the Harnack-Thom theorem and generalize it to real quasiprojective varieties, in which case we need to use Borel-Moore homology instead of singular homology. To construct some nontrivial examples, we apply Weil's construction of Picard varieties to construct reduced real Picard groups in section 5. In section 6 we prove a vanishing theorem for the reduced real Lawson homology groups of divisors and under some mild conditions, we get the following result by applying our main theorem from section 4: where ρ(X) is the Picard number of X.
The results of this paper suggest that Lawson homology and reduced real Lawson homology are useful enrichments of singular homology.

Review of Lawson homology and reduced real Lawson homology
Let us recall some basic properties of Lawson homology and reduced real Lawson homology (see [3,10,18]). For a projective variety X, denote the set of effective p-cycles of degree d by C p,d (X). By the Chow Theorem (see [16]), C p,d (X) can be realized as a complex projective variety. With the analytic topology on C p,d (X), we get a compact topological space K p,d (X) = d1+d2≤d C p,d1 (X)×C p,d2 (X)/ ∼ where ∼ is the equivalence relation defined by (a, b) ∼ (c, d) if and only if a + d = b + c. These spaces form a filtration: K p,0 (X) ⊂ K p,1 (X) ⊂ K p,2 (X) ⊂ · · · = Z p (X) where Z p (X) is the naive group completion of the monoid C p (X) = d≥0 C p,d (X). We give Z p (X) the weak topology defined by this filtration, i.e., U ⊂ Z p (X) is open if and only if U ∩ K p,d (X) is open for all d. We define the n-th Lawson homology group of p-cycles to be L p H n (X) = π n−2p Z p (X) the (n − 2p)-th homotopy group of Z p (X). We define the n-th Lawson homology group with Z 2 -coefficients of p-cycles to be L p H n (X; Z 2 ) = π n−2p Z p (X) 2Z p (X) .
For the zero-cycle group, by the Dold-Thom Theorem, we have an isomorphism L 0 H n (X; Z 2 ) = H n (X; Z 2 ) between Lawson homology and singular homology. For a quasiprojective variety U , there exist projective varieties X and Y where Y ⊂ X such that U = X − Y . The Lawson homology group of U is defined to be L p H n (U ) = π n−2p Z p (X) Z p (Y ) .
It is proved in [12] that this definition is independent of the choice of X and Y . We define Lawson homology with Z 2 -coefficients of U to be For the zero-cycle group, by the Dold-Thom Theorem (see [4] Proposition 1.6), we have an isomorphism where H BM * denotes Borel-Moore homology. A real projective variety X ⊂ P n is a complex projective variety which is invariant under conjugation. Equivalently, it is a complex projective variety defined by some real polynomials. Conjugation induces a Z 2 -action on Z p (X). Let Z p (X) R be the subgroup of p-cycles on X which are invariant under this action and let Z p (X) av be the subgroup consisting of cycles of the form c + c where c ∈ Z p (X) and c is the conjugate cycle of c. These two subgroups are endowed with the subspace topology. Define the reduced real p-cycle group to be It is shown in [18] Proposition 2.3 that Z p (X) av is a closed subgroup of Z p (X) R and in the Appendix of [18] that all these cycle groups Z p (X), Z p (X) R , Z p (X) av , R p (X) are CW-complexes. We define the n-th reduced real Lawson homology group of p-cycles to be RL p H n (X) = π n−p R p (X).
For zero cycles, it is shown in [18] Proposition 2.7 that RL 0 H n (X) = H n (ReX; Z 2 ), the singular homology group of the real points of X. We define the reduced real Lawson homology group of a real quasiprojective variety U = X − Y to be where X, Y are real projective varieties and Y ⊂ X. It is shown in [18] that this definition is independent of the choice of X, Y . For the group of zero cycles, we

The Splitting Theorem
Let us recall that the real part RP (C) of a cycle C, roughly speaking, is the part consisting of irreducible real subvarieties and the averaged part AP (C) of a cycle C is the part consisting of conjugate pairs of complex cycles. The imaginary part is the part left after canceling out the real and averaged parts. We give the precise definition in the following: Definition 3.1. For any f ∈ Z p (X), let f = i∈I n i V i be in the reduced form, i.e., each V i is an irreducible subvariety of X and V i = V j if and only if i = j. Let which is called the real part of f . Let J = {i ∈ I|V i is not real and V i is also a component of f } and for i ∈ J, let m i be the maximum value of the coefficients of V i and V i . Define the averaged part to be and the imaginary part to be It is easy to see that f is a real cycle if and only if IP (f ) = 0 and a real cycle g is an averaged cycle if and only if RP (g) is divisible by 2.
In the following, we will assume that X is a real projective variety.
Proposition 3.2. The following sequence is exact: Proof. It is easy to check that i(f + 2Z p (X) R ) = f + 2Z p (X) is well defined and injective and (1 + c * )(f + 2Z p (X)) = f +f + 2Z p (X) R is well defined and surjective. The map 1 + c * sends the image of i to 0, thus the only thing we need to prove is for Definition 3.3. Let Q p (X) be the collection of all averaged cycles c such that there exists a sequence {v i } ⊂ Z p (X) R where v i = RP (v i ) for all i and v i converges to c. It is not difficult to see that Q p (X) is a topological subgroup of Z p (X) av . Let ZQ p (X) R = 2Z p (X) R + Q p (X) denote the internal sum of 2Z p (X) R and Q p (X). Then ZQ p (X) R is again a topological subgroup of Z p (X) av . The group Q p (X) is the intersection of the closure of the group formed by irreducible real p-subvarieties with the averaged p-cycle group. Thus ZQ p (X) R is a closed subgroup.
Proof. The free abelian group Z 0 (ReX) generated by real points of X is closed in Z 0 (X), so if c ∈ Q 0 (X), then c ∈ 2Z 0 (X) R (see Proposition 2.7 in [18]).
The following example was given by Lawson to show that the set of 1-cycles formed by irreducible real subvarieties may not be closed which contrasts to the case of 0-cycles, i.e., ZQ p (X) R may not equal to 2Z p (X) R if p > 0.
Example 3.5. In P 2 , consider the sequence of irreducible real subvarieties V ǫ = zero locus of X 2 + Y 2 − ǫZ 2 . As ǫ converges to 0, V ǫ converges to the cycle formed by two lines X = iY and X = −iY which is an averaged cycle but not in 2Z p (X) R .
Lemma 3.7. Define AP : Then AP is continuous.
Proof. There is a filtration called the canonical real filtration where each K i is compact and the topology of Z p (X) R is given by the weak topology induced from this filtration. Thus the filtration Hence for x ∈ A, there is y ∈ π −1 (A) such that π(y) = x. Since Z p (X) R is a CW-complex (actually it is also a metric space), there is y n ∈ π −1 (A) such that y n → y. If AP maps convergent sequences to convergent sequences, we see that AP (π(y n )) converges to AP (x). Since π(y n ) ∈ A and π(y) = x, we have AP (x) ∈ AP (A) which implies that AP is continuous.
Suppose that f i +ZQ p (X) R converges to ZQ p (X) R . Since A = {f i +ZQ p (X) R }∪ {ZQ p (X) R } is compact and Zp(X) R ZQp(X) R is Hausdorff, by Lemma 2.2 in [18], A ⊂ K n + ZQ p (X) R for some n. Thus there exists g i ∈ K n such that under the quotient map q, q(g i ) = f i + ZQ p (X) R for all i. The set K n is compact, thus {g i } has a convergent subsequence.
Let {g ij } be a subsequence of {g i } which converges to g. Since g ij + ZQ p (X) R converges to ZQ p (X) R , we have g ∈ ZQ p (X) R . The set {g ij } ⊂ K n and each g ij is a real cycle which implies that {AP (g ij )} ⊂ K n and hence {AP (g ij )} has a convergent subsequence. Let {AP (g ijk )} be a subsequence of {AP (g ij )} which converges to a real cycle h. Since {g ijk } is a subsequence of {g ij }, it converges to g, hence The cycle g is in ZQ p (X) R thus h ∈ ZQ p (X) R . Passing to the quotient, we see that For any convergent subsequence {g ij } of {g i }, AP (g ij + ZQ p (X) R ) = AP (g ij ) + ZQ p (X) R converges to the point ZQ p (X) R . Consequently, this implies that AP (g i + ZQ p (X) R ) = AP (f i + ZQ p (X) R ) converges to ZQ p (X) R . So AP is continuous.
Then RP is continuous.
Proof. We proceed as in the proof above. The canonical real filtration which defines the topology of Zp(X) av and the filtration By an argument similar to that of the previous proof, it suffices to prove that RP maps convergent sequences to convergent sequences. Suppose that The cycle h is a real cycle and thus h ∈ Z p (X) av . Furthermore, {RP (g ijk )} −→ h, so by definition, h ∈ Q p (X). Passing to the quotient, we see that By the two Lemmas above, ψ and φ are continuous and it is easy to check they are inverse to each other.

The Generalized Harnack-Thom Theorem
While it is easy to produce an exact sequence H ֒→ G −→ G/H of topological groups, it is cumbersome to verify that it is a locally trivial principal H-bundle, and worse, it may not be in general. But the long exact homotopy sequence induced by a fibration is extremely useful in homotopy group calculation. We use Milnor's construction of universal bundles to construct some weak models of the classifying spaces of some cycle groups. They are used to produce long exact sequences of homotopy groups. To make everything work out, we need to work in the category of compactly generated topological spaces CG (see [17]).
We recall that a space X is compactly generated if and only if X is Hausdorff and each subset A of X with the property that A ∩ C is closed for every compact subset C of X is itself closed. Since the topology of our cycle groups is defined by a filtration of compact Hausdorff spaces, all groups we are dealing with are in CG. To make sure the quotient G/H is in this category, we need H to be a normal closed subgroup of G.
Let us recall Milnor's construction of universal G-bundles. We adopt the notation from page 36 of [5]. For a topological group G, let C G = (G × I)/(G × {0}) be the cone on G, and the n-th join, G * n , is the subspace of C G × · · · × C G of points ((g 0 , t 0 ), ..., (g n , t n )) such that . For a topological group G ∈ CG, as in [5], we give E(G) = ∪ n G * n the weak topology determined by G * n instead of Milnor's strong topology for arbitrary topological groups. Then we have a continuous action of G in E(G) given by ((g 0 , t 0 ), ..., (g n , t n )) · g = ((g 0 g, t 0 ), ..., (g n g, t n )).
Set B(G) = E(G)/G and let p G : E(G) → B(G) be the quotient map. Then The space B(G) is called the classifying space of G and we have π k+1 (B(G)) ∼ = π k (G). We say that a space T is a weak model of BG if T is weak homotopy equivalent to BG, i.e, they have the same homotopy groups.
The following result is the main tool that we use to produce long exact sequences of homotopy groups. A similar argument for topological groups which are CWcomplexes can be found in Theorem 2.4.12 of [1].
Let H, G ∈ CG be two topological abelian groups and H be a closed subgroup of G. Then we have a long exact sequence of homotopy groups: From this result, when we have a short exact sequence 0 → H → G → K → 0 of topological abelian groups such that K is isomorphic to G/H, it induces a long exact sequence of homotopy groups. By abuse of terminology, we will call this long exact sequence the homotopy sequence induced by the short exact sequence.

Proof. From the principal H-bundle
given by the first projection which induces a long exact sequence of homotopy groups Note that since H, G may not be CW-complexes, the homomorphisms in the long exact sequence may not be induced by maps between H and G.
Since π 1 , π 2 are open maps, φ is also an open map. The proof of (4) and (5) are similar to the proof of (3).
In the following Proposition, we use the notation T n to denote the n-th homotopy group of T where T is any of the groups A, ..., G. We note that all these groups are Z 2 -spaces so their homotopy groups are vector spaces over Z 2 . Proposition 4.3. We have the following short exact sequences: ( They induce long exact sequences: Proof. (1) By Proposition 3.2, A is isomorphic to ker(1 + c * ). The map 1 + c * is surjective and closed by Proposition 4.2, thus C is isomorphic as a topological group to B/ker(1 + c * ). Hence we have the first exact sequence and by Proposition 4.1, we have the first homotopy sequence. A similar argument works for (2), (3) and (4).
Since every topological abelian group is a product of Eilenberg-Mac lane spaces, we are able to compute the homotopy types of topological abelian groups from knowledge of their homotopy groups alone.
Example 4.4. The homotopy types of the seven groups mentioned above for 1cycles on Definition 4.5. Suppose that X is a real quasiprojective variety. We define the L ptotal Betti number of X with Z 2 -coefficients to be We define the real L p -total Betti number to be β(p)( Theorem 4.6. Suppose that X is a real projective variety. If B(p)(X) and B(p)(X) R are finite, then If in addition G is weakly contractible, then Proof. To simplify the notation, we use the same notation as in Proposition 4.3 but with different meaning. We use M n to denote the rank of the n-th homotopy group of M , Kerg n and Img n the rank of the kernel and the rank of the image over Z 2 of a homomorphism g n respectively.
From the finiteness assumption on B(p)(X) and B(p)(X) R , we know that (2) From the long exact sequence 1, we have A n = Imc n+1 + Ima n = C n+1 − kerc n+1 + B n − Imb n and from the long exact sequence 2, we have A n = Ima ′ n + Imc ′ n = E n − kerc ′ n−1 + c n − kerc ′ n . Simplifying the equation and taking sums, we get (3) If G is weakly contractible, then π k (G) = 0 for all k. From the long exact sequence 3 and 4, we have A n = D n and C n = F n . Since by Theorem 3.9, D n = E n + F n , we have A n = C n + E n for all n. From the long exact sequence 1, we have A n = C n+1 − kerc n+1 + B n − Imb n . Thus C n + E n ≤ C n+1 + B n . Taking the sum over all n, we have E n ≤ B n For zero-cycles, to simplify the notation, we simply write B(X) = B(0)(X), B(ReX) = β(0)(X), χ(X) = χ 0 (X) and χ(ReX) = Rχ 0 (ReX) which are the standard total Betti numbers and Euler characteristic of X and ReX in Z 2 -coefficients.
where X/Z 2 is the orbit space of X under the action of conjugation. Thus B(0)(X) and B(0)(X) R are finite. The result now follows from the Theorem above.
Suppose that Y ⊂ X are real projective varieties and U = X − Y is a real quasiprojective variety. Let A, B, ..., G be the cycle groups of X defined as above and let We can check as in Proposition 2.8 of [18] that T ′ is embedded as a closed subgroup of T thus we identify T ′ with its image in T for any group T above. To simplify the notations, we will use (T ′ ) to mean the image of T ′ in T .
Lemma 4.8. We have two short exact sequences of topological abelian groups: We prove that the first sequence is exact and similar argument works for the second one. Injectivity: let a ∈ Z p (X) R and a+ 2Z p (X)+ ( 2Zp(Y ) ), then a = 2b + c where b ∈ Z p (X), c ∈ Z p (Y ) and we may assume b, c have no common components and the conjugation of each component of c is not a component of b. Since a is real, b, c are real hence a is 0 in A A ′ . It is trivial that the image of i is contained in the kernel of 1 + c * and the map 1 + c * is surjective. Suppose that for a ∈ Z p (X), a + a + 2Z p (X) R + ( By a similar calculation as in Theorem 4.6, we get  ( Proof. By the Dold-Thom theorem of Z 2 -coefficients, we have For zero-cycles, the equation   [18]) and the Duality Theorem between reduced real morphic cohomology and reduced real Lawson homology(see Theorem 5.14 in [18]), we get R 1 (P 1 × P 1 ) = R 0 (P 1 ) × R 1 (P 1 ).
The reduced real Lawson homology groups of a variety naturally depend on its real structure. Two real projective varieties may be isomorphic as complex projective varieties, but they may not be isomorphic as real projective varieties. Thus reduced real Lawson homology groups may be used to distinguish two real projective varieties.
Example 4.13. Let X be the smooth quadric defined by the equation x 2 + y 2 + z 2 = 0 in P 2 . The variety X is complex algebraically isomorphic to P 1 but not real algebraically isomorphic to P 1 since X has no real point. Therefore all the reduced real Lawson homology groups of zero-cycles on X are trivial but B(0)(X) = χ 0 (X) = 2.

A construction of Weil
Throughout this section, X is a nonsingular projective variety of dimension m. Let G be the fundamental group of X and G ′ be the commutator-group of G. Then H = G/G ′ = H 1 (X, Z) is the first homology group of X with integral coefficients.
Definition 5.1. Let p : (X, y) −→ (X, x) be a covering map whereX, X are complex manifolds. The covering is said to be abelian if p * π 1 (X, y) = G ′ .
For an abelian covering, the group of deck transformations is isomorphic to H. Every element σ ∈ H determines an automorphism ofX, transforming each point s ofX into a point σ(ŝ) lying over the same point s in X.
Let T be the torsion subgroup of H. A multiplicator-set which is 1 on T is called special. Let Θ X be the group of all special multiplicator-sets ǫ where |ǫ(σ)| = 1 for all σ ∈ H. A divisor Z on X is defined by a meromorphic multiplicative function φ onX, as explained in Page 873 of [20], by taking the zero locus of φ, and the multiplicator-set of φ is special if and only if Z is algebraically equivalent to zero. It is proved in [20] that Θ X is an abelian variety for a nonsingular projective variety X and the real dimension of Θ X is equal to the rank of H. Definition 5.2. For X a nonsingular projective variety, the abelian variety Θ X is called the Picard variety of X.
Let Z m−1 (X) alg be the group of divisors on X which are algebraically equivalent to zero. Definition 5.3. (Weil's construction) Define a group homomorphism w : Z m−1 (X) alg −→ Θ X by w(Z) = ǫ where ǫ ∈ Θ X is the special multiplicator-set of φ and Z is the divisor defined by φ.
For the reader's convenience, we recall a definition from [20].
Definition 5.4. An analytic family of divisors on a nonsingular projective variety X parametrized by a nonsingular projective variety S is an algebraic cycle V on S × X such that V s := P r * (V • (s × X)) is a divisor on X (where P r : S × X −→ X is the projection and • is the intersection product). A mapping f : Z m−1 (X) alg −→ Θ X is said to be analytic if for any analytic family of divisors algebraically equivalent to zero on X, parametrized by S, the map f • λ : S −→ Θ X is an analytic map where λ : S −→ Z m−1 (X) alg is the parametrization.
Let Z m−1 (X) lin be the group of divisors on X which are linearly equivalent to zero. The following is the "Main Theorem" in Weil's paper [20].
Theorem 5.5. The surjective group homomorphism w : Z m−1 (X) alg −→ Θ X in Weil's construction is analytic and the kernel of w is Z m−1 (X) lin . There is a bijective parametrization Λ : Θ X −→ Zm−1(X) alg Zm−1(X) lin . The main result we need is that the topology on Θ X is actually same as the topology on Zm−1(X) alg Zm−1(X) lin .
Corollary 5.6. The map w : Z m−1 (X) alg −→ Θ X in Weil's construction is continuous and therefore it induces a topological group isomorphism w : Proof. We may form a topology on Z m−1 (X) alg by declaring that a set U ⊂ Z m−1 (X) alg is open if and only for all parametrizations λ : S −→ Z m−1 (X) alg , λ −1 (U ) is open. By Theorem 2.16 in [13], this topology coincides with the Chow topology. Combining this with Theorem 5.5, we have that w is a continuous map. From Weil's construction, w • Λ =the identity map. Since Θ X is compact, Λ is a topological group isomorphism, which implies that w is a topological group isomorphism. All nonsingular projective varieties defined by real polynomials in P n have a natural conjugation which is induced by the standard conjugation of P n .
Definition 5.8. IfX and X are real complex manifolds and the covering map p :X −→ X satisfies p(z) = p(z) for all z ∈X, then the covering is said to be real. Lemma 5.9. Suppose that p : (X, y) −→ (X, x) is a covering map whereX is a complex manifold and X is real complex manifold. Then the conjugation on X induces a conjugation onX such that the covering is real.
Proof. For each point t ∈ X, take a small connected open neighborhood U t and a biholomorphic local trivialization φ t : p −1 (U t ) −→ U t × F where F is the fibre which is discrete. We may take U t small enough and make Ut = U t for all t. We define a conjugation on U t × F by (w, b) = (w, b). Suppose that z is a point in the fibre over . It is then easy to verify that φ −1 t φ t (z) = φ −1 s φ s (z) in the overlap of U t and U s . So z is well defined. Since on U t ×F for any t, the map sending z to z is antiholomorphic, thus the map we just defined is a conjugation onX.
We remark that this conjugation depends on the choice of local trivializations, for instance in the case of trivial covering spaceX = X × F . Definition 5.10. Suppose that p : (X,x) −→ (X, x) is a real abelian covering and σ is a deck transformation induced by a loop [f ] ∈ π 1 (X, x). Let γ be a path from x to x and let g = γ −1 * f * γ be the loop at x, defined by traveling from x to x along γ, going around x along the conjugation of f and then traveling back to x along γ with opposite direction. Let σ be the deck transformation defined by g. If we take another path γ ′ from x to x and let g ′ = γ ′ −1 * f * γ ′ , then it is easy to show that g ′ g −1 is an element in the commutator group, thus g ′ defines a same deck transformation as g does. We can check that for all z ∈X. We say that σ is real if σ = σ.
From the theory of covering spaces, we know that σ is real if and only [f ] = [g] in π 1 (X, x)/p * (π 1 (X,x)).
If X is a nonsingular real projective manifold and φ is a multiplicative function onX, define φ(z) := φ(z) which is also a multiplicative function onX.
Lemma 5.11. Suppose that p : (X,x) −→ (X, x) is a real abelian covering and X is a real projective manifold. Let Z be a divisor of X. If φ is a multiplicative function defining Z, with multiplicator-set ǫ, then φ is a multiplicative function defining Z, with multiplicator-set ǫ.
Proof. φ(σ(y)) = φ(σ(y)) = φ(σ(y)) = φ(y)ǫ(σ) = φ(y)ǫ(σ) It was shown by Weil in [20] that a divisor Z on X is linearly equivalent to 0 if and only if for a multiplicative function φ defining Z, the special multiplicator-set ǫ of φ is 1. Therefore, since Z is defined by φ with multiplicator-set ǫ, this implies that Z is also linearly equivalent to 0. So the conjugation on Z m−1 (X) alg passes to Zm−1(X) alg Zm−1(X) lin . By Lemma 5.11, it is clear that we have the following result.
Definition 5.13. Suppose that p : (X,x) −→ (X, x) is a real abelian covering and X is a real projective manifold. We say that X is real symmetric if all the deck transformations of X are real.
Proposition 5.14. Suppose that p : (X,x) −→ (X, x) is a real abelian covering and X is real symmetric. If a divisor Z is algebraically equivalent to 0, then the averaged divisor Z + Z is linearly equivalent to 0.
Corollary 5.15. A projective curve X is not real symmetric if the genus g of X is greater than 0.
Proof. Let p ∈ X and D = p + p. For a divisor E on X, let L(E) be the dimension of H 0 (X, [E]) where [E] is the line bundle associated to E, and let |E| be the linear system associated to E. If g = 1, by Riemann-Roch theorem, we have L(D) = 2. If g > 1, L(D) ≥ 1 and L(K − D) ≥ 1 where K is a canonical divisor on X, then by Clifford's theorem, L(D) ≤ 2. Assume that for every q ∈ X, q + q is linearly equivalently to D. Then dim|D| = L(D) − 1 = 1. Consider the set C 0,2 (X) = SP 2 (X) of effective divisors of degree 2 where SP 2 (X) is the 2-fold symmetric product of X. We have C 0,2 (X) R = SP 2 (X) R = {q + q|q ∈ X} and by the assumption we have C 0,2 (X) R ⊂ |D| = P 1 . Since the map X −→ SP 2 (X) R defined by a −→ a + a is a homeomorphism, it gives an embedding of X into P 1 which is impossible. Therefore, there exists q ∈ X such that q + q / ∈ |D|. Since p − q is algebraically equivalent to zero but (p − q) + (p − q) = (p + p) − (q + q) is not linearly equivalent to zero, this contradicts to the conclusion of Corollary 5.14. Hence, X is not real symmetric.
Lemma 5.16. If D is a real divisor which is linearly equivalent to 0, then there is a real rational function F such that D = (F ), the divisor defined by F .
Proof. Let D = D 1 − D 2 where D 1 and D 2 are effective real divisors. Since D is linearly equivalent to zero, there exists a rational functions F = f g such that D = (F ). Suppose that (f ) = D 1 +D 3 , (g) = D 2 +D 3 . Since D 3 +D 3 is a real divisor, we can take a real homogeneous polynomial h such that (h) = D 3 + D 3 + D 4 . We show that we can find a real homogeneous polynomial which defines the divisor (f h). Proof. Let (x, y) ∈ C 2 \{0}, D = (F ) where F is a real rational function. Let V x,y be the divisor defined by x + yF . We have V 0,1 = D and V 1,0 = 0. Let γ : [0, 1] −→ C 2 \{0} be the path given by γ(t) = (t, 1 − t). Then each V γ(t) is real and this gives a path in Z m−1 (X) R joining D and 0.
It follows from this result that we do not have to distinguish between real and complex linear equivalence in Z m−1 (X) R . Denote Corollary 5.18. Suppose that X is a real nonsingular projective variety of dimension m. We have the following inclusions: It is easy to check the following result.
Lemma 5.20. Suppose that X is a nonsingular real projective variety of dimension m. Then The inclusion map R m−1 (X) lin ֒→ R m−1 (X) 0 is a closed embedding. We will abusively denote the image of R m−1 (X) lin in R m−1 (X) 0 by R m−1 (X) lin .
Let P ic 0 (X) be the group of holomorphic line bundles on X whose first Chern class are zero. There is an isomorphism where u maps a divisor Z to the line bundle associated to Z. We give a topology on P ic 0 (X) by making u a homeomorphism. For L ∈ P ic 0 (X), L = [c] for some c ∈ Z m−1 (X) alg . We define L = [c]. Then the map u is real. We have the following commutative diagram and each map is a real topological group isomorphism: Θ X u•w r r r r r Definition 5.21. We say that a holomorphic line bundle L on a nonsingular projective variety X is real if L is the line bundle associated to some real divisor, and L is averaged if L is the line bundle associated to some averaged divisor. Denote P ic 0 (X) R to be the 0-component of real line bundles and P ic 0 (X) av to be the 0component of averaged line bundles. We define the reduced real Picard group of X to be RP ic 0 (X) = P ic 0 (X) R P ic 0 (X) av which is a topological abelian group.
The real isomorphism u gives us the following result.
Theorem 6.1. Suppose that X is a nonsingular projective variety of dimension m. Then We make a similar calculation for the real case by the method developed in this paper.
Proposition 6.2. Suppose that X is a nonsingular real projective variety of dimension m. Then where N S(X) R is the real Neron-Severi group which is defined to be π 0 Z m−1 (X) R .
Proof. Suppose that X ⊂ P n . Let be the real coordinate ring of X where I R (X) ⊂ R[z 0 , ..., z n ] is the ideal of real polynomials vanishing over X. Let K[X] R = ⊕ ∞ k=0 I k where I k is the real vector space generated by homogeneous polynomials of degree k of X. Define and by Lemma 5.16, we have a filtration where PI d is the real projectivisation of I d , thus I ∼ = K(Z 2 ⊕ Z 2 , 1), ∆ ∼ = K(Z 2 , 1). For (f 1 , g 1 ) ∈ PI d1 × PI d1 , (f 2 , g 2 ) ∈ PI d2 × PI d2 , we define (f 1 , g 1 ) · (f 2 , g 2 ) := (f 1 f 2 , g 1 g 2 ) which induces a monoid structure on I and ∆.
Let I, ∆ be the naive group completions of I and ∆ respectively. Since all ∆ d , PI d are compact CW-complexes, the monoids ∆ and I are free, strongly properly cgraded (see [14] for the definitions), by Theorem 4.4' of [14], I, ∆ are homotopy equivalent to their homotopy theoretic group completions respectively. Hence π k I = π k I, and π k ∆ = π k ∆ for k > 0 and π 0 I = π 0 ∆ = Z. Since (I, ∆) is a properly c-filtered free pair of monoids, by Theorem 5.2 of [14], we have a fibration which implies that I/ ∆ ∼ = K(Z 2 , 1). There is a surjective monoid homomorphism φ : I −→ Z m−1 (X) lin R defined by φ(f, g) = ( f g ). We extend it to a group homomorphism φ : = P ic 0 (X) R , and the group P ic 0 (X) R is a closed subgroup of P ic 0 (X), thus a real torus, from the homotopy sequence induced by the short exact sequence otherwise. This completes the proof. Proposition 6.3. For a nonsingular real projective variety X, N S(X) R is finitely generated.
Proof. Let m be the dimension of X and let By Proposition 5.17, H is embedded as a closed subgroup of P ic 0 (X) = Zm−1(X) alg
Proof. Suppose that a, b ∈ C m−1 (X) and a + a = b + b. Write a = n i=1 n i V i where each V i is an irreducible subvariety and n i > 0. Since a − b ∈ T , we may assume that a and b have no common irreducible subvariety components. From the relation a + a = b + b, we see that each V i must be a component of b. Thus b = a.
The following observation is the main tool that we are going to use to compute R m−1 (X). Proposition 6.5. We have the following exact sequences of topological groups: where Sa(c) = c−c and Av(c) = c+c, the groups T and Z m−1 (X) av are isomorphic as a topological group to Zm−1(X) Zm−1(X) R and Zm−1(X) T respectively.
Proof. A direct verification shows that the sequences are exact. To show that T is isomorphic as a topological group to Zm−1(X) Zm−1(X) R , it suffices to prove that Sa is a closed map. Let K 1 ⊂ K 2 ⊂ · · · ⊂ Z m−1 (X) be the canonical filtration. The topology of T is the subspace topology of Z m−1 (X). For C ⊂ Z m−1 (X) a closed subset, C ∩ K n is compact and Sa(C ∩ K n ) = Sa(C) ∩ K 2n which is closed for any n, so Sa is a closed map. The map Av is a closed map which is proved in Proposition 4.2. Lemma 6.6. Suppose that X is a nonsingular real projective variety of dimension m.
(2) Let T 0 be the zero-component of T . Since T is a closed subgroup of Z m−1 (X), Zm−1(X) lin ֒→ Zm−1(X) alg Zm−1(X) lin = P ic 0 (X) is a closed embedding and hence T 0 Zm−1(X) lin is a closed Lie subgroup of P ic 0 (X) which implies that π k T Zm−1(X) lin is free for k > 0. By a similar calculation of the homotopy type of Z m−1 (X) lin R in Proposition 6.2, we get Z m−1 (X) lin ∼ = K(Z, 2). From the homotopy sequence induced by the short exact sequence Then π 0 Z m−1 (X) = Zm−1(X) Zm−1(X) lin is free from the hypothesis. From the homotopy sequence induced by the short exact sequence above, we see that π 0 T is also free. Theorem 6.7. For a nonsingular real projective X of dimension m, π k R m−1 (X) = 0 for k > 2.
Proof. By the weak Lefschetz theorem and the exponential sequence on X, we have N S(X) ∼ = Z and H 1 (X, C) = 0, so ρ(X) = 1. The result then follows from Corollary 6.10.