Chow ring of weighted projective space

Question

Is the structure of the Chow ring of the weighted projective space $\mathbb{P}(a_0, \ldots a_n)$ known?

The integral cohomology ring of $\mathbb{P}(a_0, \ldots a_n)$ is apparently described in the introduction of The equivariant cohomology ring of weighted projective spaces, eqn. (1.4). It's temping to hope that the Chow ring isomorphic to this ring, as is the case in standard projective spaces (and Grassmannians, etc.).

Thanks!

Edit: While I'm prepared to dive into the geometry of toric varieties, it would be nice to have this specific case answered without recourse to the general theory.

Answer 1

Your question in your answer about the meaning of Chow cohomology classes does not have an easy or clear answer. Most algebraic geometers agree that the intersection product on Chow homology gives the correct and natural ring to study for intersecting cycles on a smooth variety. For singular varieties, it is less clear what should be the natural object to consider. At the very least, it should be a ring that acts on Chow homology, is contravariantly functorial for arbitrary morphisms in a way that is compatible with the projection formula, and contains the Chern class of vector bundles. Chow cohomology is defined abstractly as the universal such object (i.e. any functor from varieties to rings that satisfies these properties admits a unique natural transformation to Chow cohomology). Another way of stating the universal property of Chow cohomology, assuming resolution of singularities, is that it is the Kan extension of the Chow ring from smooth varieties to all varieties. A good place to learn the basics of this approach is Chapter 17 in the book "Intersection Theory" by Fulton here.

Fulton, William. Intersection theory. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 2. Springer-Verlag, Berlin, 1998. xiv+470 pp.

One might hope that someday there will be a description of a Chow cohomology theory in which the elements are something more geometric, such as cycles transverse to singularities modulo rational equivalences transverse to singularities. On Whitney stratified topological spaces (which are a good analogue of singular algebraic varieties), there is such a description of singular cohomology in terms of cycles transverse to singularities modulo boundaries transverse to singularities, in the paper of Goresky here.

Goresky, R. Mark. Whitney stratified chains and cochains. Trans. Amer. Math. Soc. 267 (1981), no. 1, 175–196.

People have tried to imitate his construction in the algebraic setting, to give a more geometric Chow cohomology theory, but without success.

Also, your guess about the graded group structure on the Chow ring of weighted projective space is correct; it is one copy of the integers in each degree up to the dimension.

Answer 2

My question was a bit misguided, so I will try to clarify things as well as give some answers. A primary reference is Fulton, Intersection Theory on Toric Varieties.

Because any weighted projective space $P = \mathbb{P}(a_0, \ldots, a_n)$ which is non-trivial (i.e., one of the $a_i$ is greater than $1$) is singular, here we must first distinguish between the Chow homology groups $A_k(P)$, whose members are rational equivalence classes of $k$-dimensional cycles in $P$, and the Chow cohomology groups $A^k(P)$, whose members are homomorphisms acting on Chow homology (and decreasing degree by $k$). The Chow cohomology groups $A^k(P)$ assemble into a graded ring $A^*(P)$, whose product is given by composition; this ring acts on the module $A_*(P)$.

The Fulton paper describes both $A_k(P)$ and $A^k(P)$ in the case that $P$ is a general toric variety. Specializing to the case of the weighted projective space, a number of simplifications occur, though these do not come easily; I will try to present the structure of these groups in a way that depends as little as possible on the theory of toric varieties. We assume that $a_0 = 1$.

For notation, let $N$ denote the set $\left\{ 0, 1, \ldots, n \right\}$, and denote by $\Delta^{(k)}$ the set of subsets of $N$ of cardinality $n - k$.

Then we have that $A_k(P)$ is generated as a $\mathbb{Z}$-module by the set

\begin{equation}\left\{ [V(\sigma)] \mid \sigma \in \Delta^{(k)} \right\} \end{equation}

i.e., by the rational equivalence classes of certain subvarieties $V(\sigma)$ of $P$, as $\sigma$ ranges through the $n-k$-element subsets in $N$ (to explain what the $V(\sigma)$ are requires the theory of toric varieties; if one only wants to know the structure of this group, the identities of the $V(\sigma)$ can be suppressed).

Meanwhile, the submodule of relations is generated (as a subgroup of the free group generated over the above set) by the elements:

\begin{equation}\left\{ a_p [V(\tau \cup q)] - a_q [V(\tau \cup p)] \mid \tau \in \Delta^{(k+1)}, p, q \in N \backslash \tau \right\} \end{equation}

i.e., by the combinations described, as $\tau$ ranges through all $n-k-1$-element subsets of $N$ and $p$ and $q$ range, for each such $\tau$, through the elements of the set $N \backslash \tau$. It's difficult to know whether, in general, a more simplified presentation exists, though in the trivial case $a_0 = \cdots = a_n = 1$ we recover the fact $A_k(\mathbb{P}^n) = \mathbb{Z}$.

Finally, the Chow cohomology group $A^k(P)$ is torsion free, generated freely as a $\mathbb{Z}$-module by the set:

\begin{equation}\left\{ c \colon \Delta^{(k)} \rightarrow \mathbb{Z} \mid a_p \cdot c(\tau \cup q) = a_q \cdot c(\tau \cup p), \forall \tau \in \Delta^{(k+1)}, p, q \in N \backslash \tau \right\} \end{equation}

of integer-valued functions $c$ on $\Delta^{(k)}$ satisfying the "linearity" property described.

I believe each group $A^k(P)$ above is actually isomorphic to $\mathbb{Z}$, generated in each case by the function $c$ which assigns to each subset $\sigma = \left\{i_1, \ldots, i_{n-k} \right\} \subset \left\{0, \ldots, n\right\}$ the integer $c(\sigma) = a_{i_1} \cdot \ldots \cdot a_{i_{n-k}}$. Thus the situation in ordinary projective space does generalize, in some sense, to the weighted case.

The ring structure of $A^*(P)$ and its action on $A_*(P)$ are complicated; they are described in the paper.

On the other hand, I'm not sure how to make sense of "intersection theory" when it's no longer cycles that are doing the intersecting. What are the meanings of the elements of $A^*(P)$? I would appreciate insight on this. Thanks to all the commenters on the original post.