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Noncompact localization with a proper fixed locus

Statement

Let $T = (C^{\times})^{r}$ , $r \geq 0$ , act algebraically on a smooth separated finite-type complex variety $X$ of pure complex dimension $d$ . Assume $X$ has a finite $T$ -invariant affine open cover and that its fixed-point subscheme $F = X^{T}$ is proper. Use Borel equivariant cohomology $H_{T}^{q} (X; Q) = H^{q} (E_{K} \times_{K} X; Q)$ and the normalized Borel–Moore groups whose smooth $d$ -dimensional fundamental class has degree $2 d$ . Let $χ_{i}$ be the coordinate characters, $u_{i} = c_{1} (L_{χ_{i} ∣_{K}})$ , and $u_{χ} = \sum_{i} n_{i} u_{i}$ for $χ (t_{1}, \dots, t_{r}) = \prod_{i} t_{i}^{n_{i}}$ . The multiplicative set $S_{T}$ is generated by $u_{χ}$ for all nontrivial characters. Throughout, $S_{T}^{- 1} H_{j}^{B M, T} (Y; Q)$ denotes $H_{j}^{T} (Y)$ , the degree- $j$ piece of the localization of the total graded module, and localized cap products act on that total module. Write $F = ∐_{a} F_{a}$ for its finitely many connected components, $i_{a} : F_{a} ↪ X$ for their inclusions, $d_{a} = \dim_{C} F_{a}$ , $c_{a} = d - d_{a}$ , and $ν_{a} = T X ∣_{F_{a}} / T F_{a}$ . Each $p_{a} : F_{a} \to p t$ is proper. The localized pushforward is $p_{*}^{l o c} = p_{F *} \circ (i_{*})^{- 1}$ , where $i_{*} = \sum_{a} i_{a *}$ is the localized fixed-inclusion isomorphism and $p_{F} : F \to p t$ is proper. For every homogeneous $α \in H_{T}^{q} (X; Q)$ , it satisfies $p_{*}^{l o c} (α \cap [X]_{T}) = \sum_{a} p_{a *} (\frac{i_{a}^{*} α}{e_{T} (ν_{a})} \cap [F_{a}]_{T}) in S_{T}^{- 1} H_{2 d - q}^{B M, T} (p t; Q),$ where $p_{a} : F_{a} \to p t$ is proper, $e_{T} (ν_{a})$ is the zero-section Euler class with complex orientation, $[X]_{T}$ and $[F_{a}]_{T}$ are the complex-oriented fundamental classes, and division means multiplication by the localized inverse of $e_{T} (ν_{a})$ . If $X \to p t$ is proper, $p_{*}^{l o c}$ equals the genuine proper pushforward. The construction is independent of the chosen finite invariant affine cover and is compatible with proper equivariant maps between such smooth varieties with proper fixed loci. For finite $F$ , the formula is the finite reciprocal-Euler sum; if $F = \emptyset$ , both sides are zero. A component with $c_{a} = 0$ has denominator $1$ . Each term has degree $2 d - q$ , since $e_{T} (ν_{a})$ has degree $2 c_{a}$ and $[F_{a}]_{T}$ has degree $2 d_{a}$ .

Statement references

[Smooth torus localization and the Euler denominator] Edidin–Graham, Section 4, Theorem 2

[Borel–Moore homology and equivariant cohomology conventions] Ginzburg, Sections 1 and 12

Facts & Assumptions

Given: The torus action, smooth separated variety $X$ , its finite invariant affine cover, proper fixed locus $F$ , normalized Borel–Moore groups, and character-localized coefficient ring.

[A1]

AC is inherited through the equivariant Borel–Moore and Thom suppliers. Intrinsic localized pushforward for a proper fixed locus Fixed-embedding BM self-intersection and Euler units

[F1]

The proper fixed-locus construction is $p_{F *} \circ (i_{*})^{- 1}$ , with $i_{*}$ the character-localized Borel–Moore concentration isomorphism. Intrinsic localized pushforward for a proper fixed locus

[F2]

The finite invariant affine-cover hypothesis makes $i_{*} : S_{T}^{- 1} H_{*}^{B M, T} (F) \to S_{T}^{- 1} H_{*}^{B M, T} (X)$ an isomorphism. BM concentration from a finite invariant affine atlas

[F3]

The fixed locus is smooth, its normal weights are nontrivial, the complex-oriented Gysin map satisfies $i_{a}^{!} i_{a *} β = e_{T} (ν_{a}) \cap β$ , and $e_{T} (ν_{a})$ is a unit after localization when $F_{a}$ is proper. Fixed-embedding BM self-intersection and Euler units

[F4]

Proper pushforward is functorial, closed inclusions are proper, and cap product with an equivariant cohomology class sends $[X]_{T}$ to Borel–Moore degree $2 d - q$ . Localized torus-equivariant Borel–Moore classes

Proof

Given: The data in the Statement.

1.1A1F1F2given

The map $i : F ↪ X$ , its components $i_{a}$ , and the localized groups are intrinsic; the finite affine cover is only a hypothesis used to prove [F2]. Since $i_{*}$ is the same canonical proper pushforward for every such cover, its inverse is unique. Therefore the definition $p_{*}^{l o c} = p_{F *} (i_{*})^{- 1}$ is independent of the cover.

2.1F1F2F4step 1.1

Let $f : X \to Y$ be a proper $T$ -equivariant map between smooth separated finite-type varieties with finite invariant affine covers and proper fixed loci $F_{X}, F_{Y}$ . Its restriction $f^{T} : F_{X} \to F_{Y}$ is proper, and the square formed by $i_{X}, i_{Y}, f^{T}, f$ commutes because $f$ sends fixed points to fixed points. Functoriality of proper Borel–Moore pushforward gives $f_{*} i_{X *} = i_{Y *} f_{*}^{T}$ . After localization both $i_{X *}$ and $i_{Y *}$ are isomorphisms by [F2], so applying their inverses and then the proper traces proves $p_{Y *}^{l o c} f_{*} = p_{X *}^{l o c}$ . If $X \to p t$ itself is proper, then $p_{F *} = p_{X *} i_{*}$ ; composing with $i_{*}^{- 1}$ proves $p_{*}^{l o c} = p_{X *}$ .

2.2F1F3F4step 1.1

Let $β = (i_{*})^{- 1} (α \cap [X]_{T})$ . In a tubular neighborhood, the refined Gysin map is cap with the supported complex Thom class followed by restriction to $F_{a}$ ; associativity and naturality of cap therefore give $i_{a}^{!} (α \cap [X]_{T}) = i_{a}^{*} α \cap i_{a}^{!} [X]_{T} = i_{a}^{*} α \cap [F_{a}]_{T}$ , using the normalization in [F3]. Since $α \cap [X]_{T} = i_{*} β$ , the self-intersection formula [F3] gives $e_{T} (ν_{a}) \cap β_{a} = i_{a}^{*} α \cap [F_{a}]_{T}$ . Properness of $F_{a}$ makes $e_{T} (ν_{a})$ invertible after localization by [F3], hence $β_{a} = (i_{a}^{*} α / e_{T} (ν_{a})) \cap [F_{a}]_{T}$ . Applying $p_{a *}$ and summing the finitely many components yields the displayed formula. The degree is $2 d - q$ : division by the degree- $2 c_{a}$ Euler class lowers the localized cohomological degree of the insertion from $q$ to $q - 2 c_{a}$ , and capping with the degree- $2 d_{a}$ fundamental class gives $2 d_{a} - (q - 2 c_{a}) = 2 d - q$ .

3.1F2F3step 2.2∎

If $F = \emptyset$ , [F2] makes the localized Borel–Moore groups of $X$ zero and the component sum is empty. If $F$ is finite, every $F_{a}$ is a point and $e_{T} (ν_{a})$ is the product of its nonzero tangent-character classes, so each summand is the stated reciprocal-Euler term. If $c_{a} = 0$ , the normal bundle is rank zero and its complex-oriented Euler class is $1$ . These cases agree with the formula and the degree computation above.

Remarks

For a proper equivariant map $f : X \to Y$ as in step 2.1, $f^{T} : F_{X} \to F_{Y}$ is its restriction to the fixed loci, and the fixed-locus square is

The horizontal maps are closed inclusions and the vertical maps are the restrictions of $f$ ; step 2.1 proves the induced pushforward identity independently of the diagram.

The theorem applies to a stable ADHM quotient only after its separate mathematical construction proves smoothness, separatedness, a finite invariant word-slice affine cover, and properness of the partition fixed locus. Those ADHM-specific hypotheses are not assumed or proved here; the present result supplies the localization calculation once they are established.

Noncompact localization with a proper fixed locus

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources