Are there topological versions of the idea of divisor?

Question

I am trying to extract a particular, more lightweight and more focussed at the same time, case of my recent question Which of the physics dualities are closest in essence to the Spanier-Whitehead duality (with a subquestion)?

From time to time I keep becoming fascinated anew by one of the deepest notions in algebraic geometry - divisors.

Specifically, I would like to understand better the interplay of analysis and elementary arithmetic of integers that occurs there.

At the first sight, forming the group of divisors is a purely "topological" move: you take formal linear combinations of codimension one subvarieties with integer coefficients. Topologists do such things all the time, and with coefficients in arbitrary abelian groups.

However as soon as you add the notion of principal divisor and linear equivalence, two things happen: first, integers become absolutely distinguished among the possible coefficient systems, since the coefficient starts to mean something very analytic and non-topological - order of vanishing/infinity of a function along the subvariety. Second, again from the purely topological point of view, some kind of duality becomes apparent, since involvement of functions on the variety suggests something cohomological, as opposed to homology presumed by considering subvarieties as cycles.

If a topologist is given an $n$-dimensional manifold, then something similar can be done relating $(n-1)$-cycles and $1$-cocycles. But the divisor class group is even more interestingly behaved in the nonsmooth case, and here I am not sure what a topologist would relate this to.

My question then is whether there exists a purely topological version of this subtle mixture of homology and cohomology. Maybe some version of Spanier-Whitehead-like duality or something like that, but the main point is that there must be a single group incorporating homology and cohomology simultaneously for non-manifolds, looking as basic and fundamental as the divisor class group, and the coefficients (in the generalized version, the spectrum) chosen being distinguished among all other possible choices of coefficients.

Turning the same question backwards - is there analog, in algebraic geometry, of taking coefficients other than $\mathbb Z$ for divisors? I am aware of the notion of $\mathbb Q$-divisor but more generally I mean, say, divisors with coefficients in the field of coefficients for varieties defined over that field, or, say, something like $\ell$-adic coefficients, or something similar. Does this have sense in the motivic context, for example? Also, from topological viewpoint, there must be higher versions of that, relating codimension $k$ cycles with zeros/poles of $(k-1)$-forms or some related gadgets (norm residue symbols? higher dimensional local fields? or what?)

Answer 1

Disclaimer. I am no expert at all in algebraic geometry. Therefore much of the following will be oversimplified or maybe even simply wrong. You are still invited to improve it.

EDIT. There is a paper by Totaro where he shows that the cycle class map from the Chow group to singular cohomology factors as $$CH^*(X)\longrightarrow MU^*(X)\otimes_{MU^*}\mathbb Z\longrightarrow H^*(X,\mathbb Z)\;.$$ Here $MU^*$ denotes complex cobordism, as explained below. The fact that Totaro does not construct a map through $MU^*(X)$ seems to indicate that in general, equivalence of cycles can be coarser than complex cobordism. I have not checked the implications for divisors, that is, for $CH^1(X)$.

Let $L\to V$ be a line bundle over a complex variety, let $s$ be a meromorphic section of $L$, and let $D$ be the associated divisor. For simplicity, we assume that $s$ is an algebraic section and that $s$ meets $0$ transversely, so all zeros have multiplicity one. Then $D$ `is' the subvariety $s^{-1}(0)$. If $V$ was smooth, then $D$ is a subvariety of complex codimension $1$, which we regard as a cobordism $2$-cocycle. Its normal bundle is a complex line bundle.

A linear equivalence between two divisors $D_0$, $D_1$ is given by a meromorphic function $f$ such that $D_a=f^{-1}(a)$ for $a=0$, $1$. Consider the graph $\Gamma$ of $f$ in $V\times\mathbb P^1$. If we are still over $\mathbb C$ and everything is sufficiently regular, for a generic real smooth curve $c\colon[0,1]\to\mathbb P^1$ from $0$ to $1$ in $\mathbb P^1$ (now with analytic topology), the (transverse) intersection $W$ of $\Gamma$ and $V\times\operatorname{im}(c)\cong V\times[0,1]$ is then a cobordism between $D_0\times\{0\}$ and $D_1\times\{1\}$. The normal bundle of $W$ in $V\times[0,1]$ naturally carries the structure of a topological complex line bundle, which restricts to the normal bundles of $D_0$, $D_1$ in $V$.

To see that we get a map from the divisor class group to complex cobordism, we recall cocycles and relations. A $k$-cocycle in a manifold $M$ is a submanifold $D\subset M\times\mathbb R^\ell$ of real codimension $(k+\ell)$ together with a complex structure on its normal bundle. If one projects down to $M$, the image may become singular. One can stabilise in $\ell$ by taking $D\times\{c\}\subset M\times\mathbb R^{\ell+m}$.

A cobordism between two $K$-cocycles, represented by submanifolds $D_0$, $D_1\subset M\times\mathbb R^\ell$ is a submanifold $W\subset M\times\mathbb R^\ell\times[0,1]$ such that $\partial W=D_0\times\{0\}\sqcup D_1\times\{1\}$, together with a complex structure on the normal bundle that restricts to the given complex structures of the normal bundles of $D_0$, $D_1\subset M\times\mathbb R^\ell$.

This shows that sufficiently smooth divisors give rise to cocycles, and linear equivalence implies cobordism. It seems that this construction extends to Chow groups, and I am sure it is described somewhere with greater care. It is not entirely clear to me though how to deal with singular subvarieties.

To push the analogy further, to each algebraic line bundle one associated a divisor class $[D]$. This can be regarded as a universal first Chern class, with values in the Chow ring. On the topological side, there is a first Chern class with values in the complex cobordism ring, and it corresponds to $[D]$ under the map above. By Quillen, this class is universal for complex oriented multiplicative cohomology theories.

Conversely, in algebraic geometry, a divisor defines a line bundle. If we can choose $\ell=0$ above, then the Pontryagin-Thom construction identifies a complex $2$-cocycle in $M$ with a homotopy class of maps $M\to\mathbb P^\infty$, and $\mathbb P^\infty$ is also the classifying space for (topological) complex line bundles. There is a direct construction: the normal bundle of $D$ comes with a complex structure. Pull it back to a tubular neighbourhood $\pi\colon U\to D$ of $D$, then $\pi^*\nu\to U$ has a tautological section, which can be used to glue it to trivial bundle over $M\setminus D$, giving a complex line bundle $L\to M$. This construction is unstable however, that is, it does not work for higher values of $\ell$.