Adams operations on the virtual K-theory of $\mathbb{P}(1,n)$

We begin by quickly recalling a few facts about the virtual K-theory ring and virtual Adams (or power) operations of a complex orbifold. We will focus upon the case of primary interest to us, $\mathbb{P}(1,n)$, for simplicity of exposition, keeping in mind that these notions can be defined quite generally, e.g. for toric orbifolds. We refer the interested reader to [5, 6] for the general case.

Let $\mathbb{P}(1,n)$ be the weighted projective line regarded as the toric orbifold $[X/G]$ where the smooth manifold $X=\mathbb{C}^{2}-\{(0,0)\}$ has the action of $G$, the complex torus $\mathbb{C}^{*}$, given by the map $G\times X\to X$ taking $(\alpha,(z_{1},z_{2}))\mapsto(\alpha z_{1},\alpha^{n}z_{2})$. The associated inertia orbifold $I\mathbb{P}(1,n)$ is the quotient orbifold $[(I_{G}X)/G]$ where the inertia manifold $I_{G}X\subset G\times X$, consisting of all points $(m,x)$ such that $mx=x$, is a $G$-manifold with the induced action $G\times I_{G}X\to I_{G}X$ given by $(h,(g,x))\mapsto(hgh^{-1},hx)$. Notice that $I\mathbb{P}(1,n)$ contains $\mathbb{P}(1,n)$ via the the $G$-equivariant embedding $X\to I_{G}X$ taking $x\mapsto(1,x)$.

Let $K(I\mathbb{P}(1,n))$ be the Grothendieck group of complex vector bundles on the orbifold $I\mathbb{P}(1,n)$ or, equivalently, $K(I\mathbb{P}(1,n))$ is $K_{G}(I_{G}X)$, the Grothendieck group of $G$-equivariant vector bundles on $I_{G}X$. There are many so-called inertial products on $K(I\mathbb{P}(1,n))$ in the sense of [5] which make $K(I\mathbb{P}(1,n))$ into a commutative, unital ring containing the ordinary K-theory ring $K(\mathbb{P}(1,n))$, called the untwisted sector, as a subring. Recall that the motivating example of an inertial product is the orbifold product which, in K-theory, is due to Adem-Ruan-Zhang [1] and Jarvis-Kaufmann-Kimura [9] (and, in terms of a Lie group presentation, Edidin-Jarvis-Kimura [4]). This orbifold product is a K-theoretic version of an orbifold product originally introduced in cohomology theory by Chen-Ruan [2] as the genus $0$, $3$-pointed, degree $0$ stable maps contribution to (orbifold) Gromov-Witten theory.

The virtual (orbifold) product is an inertial product introduced by Gonzalez-Lupercio-Segovia-Uribe-Xicotencatl [8]. Furthermore, in [6], it was shown that the virtual K-theory ring of any toric orbifold admits virtual Adams (or power) operations, a collection of ring homomorphisms $\widetilde{\psi}^{k}:K(I\mathbb{P}(1,n))\to K(I\mathbb{P}(1,n))$ which satisfy $\psi^{1}=id$, and the conditions $\widetilde{\psi}^{k}\circ\widetilde{\psi}^{\ell}=\widetilde{\psi}^{k\ell}$ for all $k,\ell\geq 1$, yielding the structure of a so-called $\psi$-ring. In fact, the $\psi$-ring structure on $K(I\mathbb{P}(1,n))$ induces a $\lambda$-ring structure on $K(I\mathbb{P}(1,n))_{\mathbb{Q}}$. In addition, on the untwisted sector $K(\mathbb{P}(1,n))$, these virtual Adams operations reduce to the ordinary Adams operations which satisfy

for all $k\geq 1$ when ${\mathscr{L}}$ is the class of any line bundle.

The previous equation motivates the following definition: An element ${\mathscr{L}}$ in the virtual $\psi$-ring $K(I\mathbb{P}(1,n))_{\mathbb{C}}$ is said to be a virtual line element if it is invertible and satisfies Equation (1) for all $k\geq 1$. The multiplicative group of virtual line elements $\mathcal{P}\subset K(I\mathbb{P}(1,n))_{\mathbb{C}}$ contains, in particular, classes of ordinary line bundles on $\mathbb{P}(1,n)$. Virtual line elements share many of the same properties as ordinary line bundles in ordinary equivariant K-theory. For example, they possess a virtual version of an Euler class constructed from the virtual $\lambda$-ring structure and a virtual version of dualization (see [6] for details).

In this paper, we apply non-Abelian localization [3] to $K(I\mathbb{P}(1,n))_{\mathbb{C}}$ and express the virtual product and virtual Adams operations in terms of natural generators on the localization. We use these generators to calculate the group of virtual line elements $\mathcal{P}$, show that $\mathcal{P}$ spans the vector space $K(I\mathbb{P}(1,n))_{\mathbb{C}}$, and then give a simple presentation of the virtual K-theory ring $K(I\mathbb{P}(1,n))_{\mathbb{C}}$ in terms of these virtual line elements. Finally, we show that for a particular crepant resolution, $Z_{n}$, of the total space of the cotangent bundle of $\mathbb{P}(1,n)$, there is a surjective homomorphism from the virtual K-theory ring $K(I\mathbb{P}(1,n))_{\mathbb{C}}$ to ordinary K-theory ring $K(Z_{n})_{\mathbb{C}}$ which respects the $\psi$-ring structures in the spirit of the hyper-Kähler resolution conjecture. Indeed, there is a summand of the virtual K-theory ring $K(I\mathbb{P}(1,n))_{\mathbb{C}}$, isomorphic to the completion with respect to the augmentation ideal, which is isomorphic to $K(Z_{n})_{\mathbb{C}}$ as $\psi$-rings. These results generalize those of [6] where a different presentation of the virtual K-theory ring and the same results for the crepant resolution for $n=2,3$ were obtained without using localization.

Finally, it is worth pointing out that these techniques should apply to general toric orbifolds. It would be very interesting to study these questions in greater generality.

This section provides a brief introduction to theory of inertial pairs on smooth Deligne-Mumford stacks developed in [5, 6]. An inertial pair defines a multiplication on the K-theory and Chow ring of the inertia stack. These rings have compatible power operations for a special case of inertial pairs, which will be of interest here. Let $X$ be a scheme or algebraic space (or smooth manifold) and $G$ a linear algebraic group acting properly on $X$. Then the quotient $[X/G]$ is a smooth Deligne-Mumford stack (or complex orbifold).

An inertial product will be defined on $K_{G}(I_{G}X)$ and $A^{*}_{G}(I_{G}X)$ associated to an inertial pair $(\mathscr{R},\mathscr{S})$ [5], where $\mathscr{R}$ is a $G$-equivariant vector bundle on $\mathbb{I}^{2}_{G}X$ and $\mathscr{S}$ is a non-negative class in $K_{G}(I_{G}X)_{\mathbb{Q}}$, as we will briefly review below. In a special case of inertial pairs associated to vector bundles, we will obtain the virtual orbifold product, which will be the focus of the remaining sections of this paper.

The compatibility condition on $\mathscr{S}$ giving an inertial pair is the identity

where $T_{\mu}$ is the relative tangent bundle. We refer the reader to [5] §3 for additional details concerning the definition of an inertial pair.

The same result holds for Chow and cohomology. Furthermore, there is an inertial Chern character map that gives a homomorphism between the inertial K-theory and inertial Chow rings.

The class $\mathscr{S}$ is also used to define a new grading on the inertial Chow and K-theory.

Proposition 3.11 in [5] states that the inertial Chern character homomorphism preserves the $\mathscr{S}$-degree modulo $\mathbb{Z}$.

With respect to the usual product on K-theory, have the following definition.

The augmented inertial ring above has compatible Adams operations, as shown in [6].

We will also need to consider $\psi$-rings with an augmentation.

Recall that ordinary equivariant K-theory $K_{G}(X)$ is a $\psi$-ring whose Adams operations $\psi^{k}:K_{G}(X)\to K_{G}(X)$ are defined though the $\lambda$-ring structure of $K_{G}(X)$. That is, for all $i\geq 0$, define maps $\lambda^{i}:K_{G}(X)\to K_{G}(X)$ by demanding that for any $G$-equivariant vector bundle $V$, $\lambda^{i}([V]):=[\Lambda^{i}V]$, the class of the $i$-th exterior power of $V$, and also by demanding that the series $\lambda_{t}:=\sum_{i\geq 0}t^{i}\lambda^{i}:K_{G}(X)\to K_{G}(X)[[t]]$ satisfy the multiplicativity relation $\lambda_{t}({\mathscr{F}}_{1}+{\mathscr{F}}_{2})=\lambda_{t}({\mathscr{F}}_{1})\lambda_{t}({\mathscr{F}}_{2})$ for all ${\mathscr{F}}_{1},{\mathscr{F}}_{2}$ in $K_{G}(X)$. These $\lambda$-operations make $K_{G}(X)$ into a so-called $\lambda$-ring. The Adams operations are defined through the equality of power series

In the special case where $\mathscr{L}$ is the class of a line bundle, it follows that

If an element ${\mathscr{F}}$ in $K_{G}(X)$ is the class of a rank $n$ vector bundle then $\lambda_{t}({\mathscr{F}})$ is a degree $n$ polynomial in $t$ and $\lambda^{n}({\mathscr{F}})$ is an invertible element of $K_{G}(X)$. An element ${\mathscr{F}}$ in $K_{G}(X)$ which satisfies these properties is said to be a $\lambda$-positive element of rank $n$. Although a rank $n$ $\lambda$-positive element ${\mathscr{F}}$ need not be the class of a rank $n$ vector bundle, in general, they do share many of the properties of vector bundles, e.g. they possess an Euler class in K-theory, Chow and cohomology.

We will now define the inertial Adams operations. We begin by introducing Bott classes.

We are motivated by equation (5) to introduce the following analog of classes of line bundles in ordinary equivariant K-theory.

Notice that after tensoring with $\mathbb{Q}$, an inertial line element in $K_{G}(I_{G}X)_{\mathbb{Q}}$ is nothing more than a $\lambda$-positive element of rank $1$ with respect to the inertial $\lambda$-ring structure $\widetilde{\lambda}_{t}:K_{G}(I_{G}X)_{\mathbb{Q}}\to K_{G}(I_{G}X)_{\mathbb{Q}}$, where $\widetilde{\lambda}_{t}:=\sum_{i\geq 0}t^{i}\widetilde{\lambda}^{i}$ is defined from the inertial Adams operations $\widetilde{\psi}^{k}$ through Equation (4) but where $\psi^{n}$ is replaced by $\widetilde{\psi}^{n}$, $\lambda_{t}$ is replaced by $\widetilde{\lambda}_{t}$, and all products are understood to be the inertial product.

The virtual orbifold product was defined in [8], and in [5] was shown to arise from a particular inertial pair.

In [6], a variant of the hyper-Kähler resolution conjecture was introduced for inertial K-theory. Let $\mathscr{X}=[X/G]$ be an orbifold where $G$ is diagonalizable, and let $Z$ be a hyper-Kähler resolution of $\mathbb{T}^{*}\mathscr{X}$. There is an isomorphism between $\widehat{K}(I\mathscr{X})_{\mathbb{C}}$ with the virtual orbifold product and $K(Z)_{\mathbb{C}}$ with the usual product.

The statements and results in this section are a synopsis of the results from §7 of [6]. Let $X=\mathbb{C}^{2}-\{(0,0)\}$ have an action of the group $\mathbb{C}^{*}$ given by $\lambda\cdot(z_{1},z_{2})=(\lambda z_{1},\lambda^{n}z_{2})$, for a positive integer $n\geq 2$. Then the quotient space $[X/\mathbb{C}^{*}]$ is the global quotient orbifold $\mathbb{P}(1,n)$. As a vector space, the virtual orbifold K-theory is given by the equivariant K-theory of the inertia manifold $I_{G}X=\coprod_{m=0}^{n-1}X^{m}$, the disjoint union of the fixed point set of $X$ under the action of an element of the group $\mathbb{Z}/n\mathbb{Z}=\{0,\ldots,n-1\}$ using additive notation. We view $\mathbb{Z}/n\mathbb{Z}$ as a subgroup of $\mathbb{C}^{*}$ under the identification $m\mapsto\zeta^{m}$, for $\zeta=exp(2\pi i/n)$. With respect to the usual product, when $m=1$, this gives the ring

and when $m\in\{1,\ldots,n-1\}$,

Denote the identities in each ring above by $1_{m}$ where $m\in\{0,\ldots,n-1\}$. Equation (8) gives the formula for $\mathscr{R}$ and $\mathscr{S}_{m}:=\mathscr{S}|_{X^{m}}=x_{m}$, so the virtual multiplication in this case can be written as

where the Euler class is

The virtual augmentation is given by $\widetilde{\epsilon}(x_{0}^{a})=1_{0}$ and $\widetilde{\epsilon}(x_{m}^{a})=0$ for $m\in\{1,\ldots,n-1\}$. For the definition of the virtual Adams operations, see equation (7). Here, the Bott classes are given by

In Proposition 7.22 of [6], Edidin, Jarvis, and Kimura prove that a subring of the inertial K-theory of $\mathbb{P}(1,2)$ and $\mathbb{P}(1,3)$ is isomorphic to the ordinary K-theory of a toric crepant resolution of the cotangent bundle of $\mathbb{P}(1,2)$ and $\mathbb{P}(1,3)$, respectively. The cotangent bundle to $\mathbb{P}(1,n)$ is the quotient stack $[(X\times\mathbb{A}^{1})/\mathbb{C}^{*}]$ where the action of $\mathbb{C}^{*}$ has weights $(1,n,-(n+1))$. There is a simplicial fan associated to this quotient, and a subdivision of this fan determines the toric crepant resolution of the cotangent bundle. See [6] §7.3 for more details. This resolution has ordinary K-theory

Henceforth, we will assume complex coefficients unless otherwise stated. Edidin and Graham [3] proved the following proposition.

In the case of $\mathbb{P}(1,n)$, we take $Y=I_{G}X$, and the maximal ideals $\mathfrak{m}_{i}\subseteq R(G)\cong\mathbb{C}[x^{\pm 1}]$, for $i\in\mathbb{Z}/n\mathbb{Z}$, are the maximal ideals of characters vanishing on $\zeta^{i}$. Thus, $\mathfrak{m}_{0}=\langle x-1\rangle$ and $\mathfrak{m}_{i}=\langle x-\zeta^{i}\rangle$ where $\zeta=exp(2\pi i/n)$. We define

where the summands $\mathcal{K}_{l}$ are the localizations above

Define generators of the direct sum decomposition of localizations by

The result is a decomposition as a vector space of the K-theory of $\mathbb{P}(1,n)$, as in the diagram below.

where $I_{ml}=\langle x_{ml}-\zeta^{l}1_{ml}\rangle$ for all $m$ and $l$, except for $I_{00}=\langle(x_{00}-1_{00})^{2}\rangle$.