There is a phenomenon which has appeared often over the past fifteen years wherein spaces of holomorphic objects are shown to have the same global structure as their topological counterparts. This is seen in the work of Graeme Segal [S] on holomorphic maps from ${\bf P}^1$ to ${\bf P}^1$ and many generalizations. It appeared also in the recently established conjecture of Atiyah and Jones [AJ] on self-dual gauge fields over the 4-sphere. There is a general theory of such phenomena due to Grauert and Gromov [G]. However, many truely interesting cases lie beyond the domain of these "$h$-principles". In fact, in these cases where results are not universal, a precise measure of the extent to which holomorphic approximation succeeds or fails can yield useful invariants.
One area where all this has taken place is that of algebraic cycles and certain spaces of morphisms on projective varieties. For "good" varieties, such as complex homogeneous spaces, there is global accord between the spaces of algebraic and topological objects (e.g., the groups of algebraic cycles are homotopy equivalent to the corresponding groups of all integral cycles). This yields explicit algebraic models for certain classifying spaces which have led to new results in topology. For general algebraic varieties the failure of holomorphic approximation is explicitly measured by bigraded homology (and cohomology) theories which have been extensively studied in recent years. This talk will survey many of these results due to Boyer, Friedlander, Gabber, Lam, Lima-Filho, Mann, Mazur, Michelsohn, the author, and others. A synopsis follows.
Let $X$ be a polyhedron and consider the topological abelian group ${\scr Z}_0(X)$ generated by the points of $X$. A classical result of Dold and Thom [DT] states that $\pi_*({\scr Z}_0(X))\cong H_*(X;{\bf Z})$.
When $X$ is an algebraic variety, one can speak of "$p$-dimensional points" of $X$, namely, the irreducible $p$-dimensional subvarieties. Over ${\bf C}$ the group ${\scr Z}_p(X)$ freely generated by these points has a natural topology. For spaces with algebraic cell decompositions (e.g., projective spaces, flag manifolds, etc.) it has been proved that $$\pi_*{\scr Z}_p(X)\cong H_{2p+*}(X;{\bf Z}).\tag1$$ By results of F. Almgren [A1], (1) is equivalent to the assertion that the inclusion $${\scr Z}_p(X)\subset3_{2p}(X)$$ into the group $3_{2p}(X)$ of all integral $2p$-cycles is a homotopy equivalent. This shows in particular that there is a homotopy equivalence $${\scr Z}_{n-s}({\bf P}^n)\cong K({\bf Z},0)\times K({\bf Z},2)\times\cdots\times K({\bf Z},2s)\tag2$$ for $0\leq s\leq n$. Let ${\scr G}_{n-s}({\bf P}^)$ be the Grassmannian of linear subspaces of codimension-$s$ in ${\bf P}^n$. Then the natural inclusion $${\scr G_{n-s}({\bf P}^n)\subset {\scr Z}_{n-s}({\bf P}^n)$$ can be interpreted via (2) to be the total Chern class of the tautological $s$-plane bundle over ${\scr G}_{n-s}({\bf P}^n)$. The direct sum $\oplus$ on linear subspaces extends to a complex join operation on general cycles. This structure has helped to settle an old question of Graeme Segal and has led to the construction of new equivariant cohomology theories with Chern classes.
For general varieties one defines $L_pH_k(X)=\pi_{k-2p}({\scr Z}_{n-s}(X))$, for $0\leq 2p\leq k$. This is a covariant functor on the category of varieties with a natural transformation $L_pH_k(\bullet)\to H_k(\bullet; {\bf Z})$. It induces filtrations on $H_k(X;{\bf Z})$ proved to be subordinate to Grothendieck's geometric filtration (the dual to his arithmetic filtration [Gr]). In fact it agrees with an analogously defined correspondence filtration introduced by Friedlander and Mazur [FM]. It is a general fact that $L_0H_k(X)\cong H_k(X;{\bf Z})$ and that $L_pH_{2p}(X)$ is the group of algebraic $p$-cycles on $X$ modulso algebraic equivalence. These groups $L_*H_*(X)$ carry mixed Hodge structures and can be related to Bloch's higher Chow groups. When $X$ is smooth there is an intersection product giving $L_*H_*(X)$ a ring structure.
A result of great usefulness in the theory is the Algebraic Suspension Theorem which establishes a homotopy equivalent ${\scr Z}_p(X)\cong {\scr Z}_{p+1}(\not\Sigma X)$ where $\not\Sigma X$ is the projective cone on $X$.
Constructions in topology and algebraic geometry motivate the consideration of the space ${\rm Mor}(X,{\scrc C}^s({\bf P}^n))$, where ${\scr C}^s({\bf P}^n)$ denotes the Chow variety of positive $(n-2)$-cycles on ${\bf P}^n$. This leads to a group ${\scr Z}^s(X)$ of algebraic $s$-cocyles on $X$ consisting of ${\scr z}^*(X)$ yields a contravariant ring-valued functor $L^*H^*(X)$ with a natural transformation to $H^*(X;{\bf Z})$ and many interesting properties.
For smooth $n$-dimensional varieties $X$ a "Poincaré Duality" theorem has been established. It is proved that there is an isomorphism $$L^sH^k(X){\overset\cong\to\to}L_{n-s}H_{2n-k}(X)$$ compatible under natural transformation with the standard Poincaré Duality map. This result has many applications to the topological structure of spaces of algebraic mappings. Its proof involves establishing a Chow Moving Lemma for families of cycles, which is of some independent interest.
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