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Chern classes and compatible power operations in inertial \(K\)-theory. (English) Zbl 1357.14068

The paper is a follow-up to the authors’ previous paper [Ann. K-Theory 1, No. 1, 85–108 (2016; Zbl 1390.55007)] establishing the formalism of inertial products on the Grothendieck group \(K(I\!\mathscr{X})\) of vector bundles on the inertia stack of a smooth Deligne-Mumford quotient stack \(\mathscr{X}=[X/G]\). The notion is motivated by mirror symmetry, and the paper develops analogs of Chern classes and compatible Adams power operations and \(\lambda\) operations of the ordinary \(K\)-theory. When \(G\) is diagonalizable and \(\mathscr{X}\) is strongly Gorenstein these operations induce a rationally augmented \(\lambda\)-ring structure on the inertial \(K\)-theory. Its \(\lambda\)-positive elements share many properties with classes of vector bundles in the ordinary \(K\)-theory, opening the door to orbifold Euler classes, Chow theory, and cohomology. Previously defined orbifold products and virtual orbifold products are particular cases of the inertial products.
The main application is to a \(K\)-theory variant of Ruan’s hyper-Kähler resolution conjecture (HKRC) for orbifolds \(X/G\) with diagonalizable \(G\). Remarkably, there is a summand \(\widehat{K}(I\!\mathscr{X})_{\mathbb{Q}}\), isomorphic as an Abelian group to the Chow group \(A^*(I\!\mathscr{X})_{\mathbb{Q}}\), which inherits any inertial \(\lambda\)-ring structure from \(K(I\!\mathscr{X})_{\mathbb{Q}}\), where the subscripts indicate tensoring with \(\mathbb{Q}\). The modified conjecture is that \(K(I\!\mathscr{X})_{\mathbb{C}}\), with its virtual orbifold product, is \(\lambda\)-ring isomorphic to \(K(Z)_{\mathbb{C}}\), where \(Z\) is a hyper-Kähler resolution of the cotangent bundle \(\mathbb{T}^*\!\mathscr{X}\). The conjecture is proved for the weighted projective lines \(\mathbb{P}(1,2)\) and \(\mathbb{P}(1,3)\) (it is reported that T. Kimura and R. Sweet, “Adams operations on the virtual \(K\)-theory of \(P(1, n)\)”, preprint, to appear in J. Algebra Appl., arXiv:1302.3524] proved it for all \(\mathbb{P}(1,n)\) after the paper was accepted). Moreover, it is shown that there is an isomorphism of Chow rings commuting with the corresponding Chern characters in these cases, and that the semigroup of \(\lambda\)-positive elements induces an exotic integral lattice on \(K(I\!\mathscr{X})_{\mathbb{C}}\) corresponding to the ordinary lattice on \(K(Z)_{\mathbb{C}}\).

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
19L10 Riemann-Roch theorems, Chern characters
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
55N15 Topological \(K\)-theory
14H10 Families, moduli of curves (algebraic)

Citations:

Zbl 1390.55007
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