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Supported Chow convolution and its action for the A1 family

Statement

For $0 \leq u, v \leq w$ use the smooth $M_{v}$ and projective maps $μ_{v}$ of Arbitrary-framing A1 varieties and their conormal Hecke correspondence, and put $R_{u v} = M_{u} \times_{End (W)} M_{v}$ . A cycle on $R_{u v}$ is regarded as a cycle with this support in the smooth product. Define $D ⋆ C = (p_{13})_{*} Δ_{M_{v}}^{!} (C \times D), C \in A_{c} (R_{u v})_{Q}, D \in A_{d} (R_{v t})_{Q},$ where the diagonal identifies the two middle factors. Its degree is $c + d - 2 a_{v}$ and its support is in $R_{u t}$ . For $α \in A_{k} (L_{u})_{Q}$ define $C ⋆ α = (p_{v})_{*} Δ_{M_{u}}^{!} (α \times C) \in A_{k + c - 2 a_{u}} (L_{v})_{Q} .$ All pushforwards here are proper on the indicated supports. Convolution is associative, the diagonal fundamental classes are units, and $D ⋆ (C ⋆ α) = (D ⋆ C) ⋆ α$ . A cycle of degree $a_{u} + a_{v}$ therefore acts between the top zero-fiber Chow groups. For $α = [L_{u}]$ , its action is equivalently the refined intersection of $C$ with $L_{u} \times M_{v} ↪ M_{u} \times M_{v}$ , followed by pushforward with support to $L_{v}$ . These operations commute with the $GL (W)$ -actions on ordinary Chow groups.

Facts & Assumptions

Given: The common-endomorphism fiber products and Chow operations in the statement; every diagonal below is taken in its smooth ambient variety, not in a possibly singular $R_{u v}$ .

[F1]

Every $μ_{v}$ is projective, $\dim M_{v} = 2 a_{v}$ , and $L_{v} = μ_{v}^{- 1} (0)$ (Arbitrary-framing A1 varieties and their conormal Hecke correspondence).

[F2]

Refined Gysin maps for regular embeddings exist after every base change, commute with proper pushforward, and commute and compose with one another and smooth pullback (Refined Gysin pullback for regular embeddings, Refined Gysin operations commute and compose).

[F3]

Proper pushforward is functorial on Chow groups (Proper pushforward of cycles and the norm formula).

Proof

1.1F1F2F3algebra

The projections of $R_{u v}$ to either factor are projective by base change from $μ_{v}$ or $μ_{u}$ . The intersection support defining $D ⋆ C$ lies in $M_{u} \times_{End (W)} M_{v} \times_{End (W)} M_{t}$ ; forgetting the middle factor is projective by base change from $μ_{v}$ , so $p_{13}$ is proper there. The support for $C ⋆ α$ lies in $L_{u} \times L_{v}$ and projects properly to $L_{v}$ because $L_{u}$ is projective. The diagonals of the smooth $M_{v}$ and $M_{u}$ are regular of codimensions $2 a_{v}$ and $2 a_{u}$ . Refined Gysin and pushforward therefore define the stated homomorphisms with exactly the displayed degree shifts.

2.1F2F3step 1.1algebra

For three kernels on consecutive pairs, place their external product on the six-factor product, and impose the two identifications of the repeated middle varieties by refined diagonal Gysins. The two regular embeddings use independent factors; their refined operations commute and their composite is the common double-diagonal operation by [F2]. Either parenthesization of convolution is this operation followed by forgetting the two middle factors. To move each intermediate pushforward past the remaining Gysin, use the proper base-change compatibility in [F2] on the common-endomorphism support; properness was checked in step 1.1. Functoriality [F3] then identifies the two expressions, proving associativity without a transversality assertion for the kernels.

3.1F2F3step 2.1algebra

Place $α$ , $C$ and $D$ on the corresponding five-factor product. The same argument imposes first the source and then the middle diagonal, or in the opposite order, with the same final projection to $L_{t}$ . This proves the action identity. For a diagonal kernel the imposed middle equality is a regular section of a smooth projection; its refined Gysin cancels that projection's flat pullback by [F2], so the resulting operation is the identity. This proves the unit assertions as well.

4.1F1F2F3step 3.1algebra∎

Since $L_{u}$ and $M_{u}$ are smooth and $L_{u} ↪ M_{u}$ is a zero section, the diagonal intersection with $[L_{u}] \times C$ equals the base-changed zero-section Gysin on $C$ . In local smooth coordinates this is the graph equality followed by the equations for that zero section; Gysin composition [F2] identifies it with the single regular embedding $L_{u} \times M_{v} ↪ M_{u} \times M_{v}$ . The resulting classes remain supported on $L_{v}$ by step 1.1. Finally transport by $h \in GL (W)$ preserves the common-endomorphism supports, all diagonals and all projection maps by [F1]. Proper and Gysin naturality [F2]–[F3] therefore intertwine the induced ordinary Chow actions. This proves the final assertions.

Depends on

Used by

Arbitrary-framing A1 Hecke cycles realize the highest-weight sl2 module Theorem

Dependency tree · two levels

39 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hiraku Nakajima, Introduction to quiver varieties—for ring and representation theorists (standard reference, not scraped)
The Stacks Project, Chow Homology and Chern Classes (standard reference, not scraped)