[Paper Review] Virtual pullbacks in $K$-theory

F. Qu|arXiv (Cornell University)|Aug 8, 2016

Algebraic Geometry and Number Theory18 references3 citations

TL;DR

This paper establishes virtual pullbacks in K-theory for algebraic stacks, proving they yield bivariant classes and satisfy functoriality. It applies these tools to prove a K-theoretic virtual localization formula for schemes and a degeneration formula in Donaldson-Thomas theory, extending classical results from Chow groups to K-theory using deformation spaces and perfect obstruction theories.

ABSTRACT

We consider virtual pullbacks in $K$-theory, and show that they are bivariant classes and satisfy certain functoriality. As applications to $K$-theoretic counting invariants, we include proofs of a virtual localization formula for schemes and a degeneration formula in Donaldson-Thomas theory.

Motivation & Objective

• To develop a theory of virtual pullbacks in K-theory analogous to existing results in Chow groups.
• To prove that K-theoretic virtual pullbacks yield bivariant classes and satisfy functoriality.
• To establish a K-theoretic virtual localization formula for schemes using deformation space techniques.
• To derive a degeneration formula in Donaldson-Thomas theory via virtual pullbacks in K-theory.
• To extend cycle-level formulas to K-theoretic invariants using bivariant machinery and perfect obstruction theories.

Proposed method

• Uses deformation spaces $M^{ullet}_f$ over $Π^1$ to define virtual pullbacks via pullback and pushforward along the normal cone $C_f$.
• Applies the deformation to the normal cone map $\sigma_f: A(Y) \to A(C_f)$ and lifts it to $K_0$ via coherent sheaves.
• Constructs virtual pullbacks $f^!$ as a composition $A(Y) \xrightarrow{\sigma_f} A(C_f) \xrightarrow{\iota_*} A(\mathfrak{E}_f) \to A(X')$, where $\mathfrak{E}_f$ is a vector stack.
• Relies on Kresch's extension of Chow groups to Artin stacks and the theory of perfect obstruction theories to generalize to algebraic stacks.
• Uses bivariant class formalism from [3] and adapts functoriality from [27] to K-theory via [18, Proposition 1].
• Applies the method of [5] to prove the virtual localization formula and adapts arguments from [24, 26] to derive the degeneration formula in DT theory.

Experimental results

Research questions

• RQ1 How can virtual pullbacks in K-theory be defined and shown to satisfy bivariance and functoriality?
• RQ2 Can the virtual localization formula for schemes be extended from Chow groups to K-theory using deformation spaces?
• RQ3 Does the degeneration formula in Donaldson-Thomas theory hold in K-theory, and how can it be derived from virtual pullbacks?
• RQ4 What role do perfect obstruction theories play in constructing virtual pullbacks in K-theory for algebraic stacks?
• RQ5 How do virtual pullbacks interact with proper pushforwards and flat pullbacks in the K-theoretic setting?

Key findings

• Virtual pullbacks in $K_0$ are shown to be bivariant classes, generalizing the bivariant formalism to K-theory.
• The functoriality of virtual pullbacks is established via compatibility with perfect obstruction theories and deformation space constructions.
• A $K$-theoretic virtual localization formula for schemes is proven using the method of [5], extending the cycle-level version to $K_0$.
• A degeneration formula in Donaldson-Thomas theory is derived in $K_0$, stating that $\sum_{k=0}^\infty \sum_{\delta: P(\delta)=P, k(\delta)=k} (-1)^k (\iota_\delta)_* \mathcal{O}_{\mathcal{M}_\delta}^{\mathrm{vir}} = \mathcal{O}_{\mathcal{M}_0^P}^{\mathrm{vir}}$ in $K_0(\mathcal{M}_0^P)$.
• Deformation invariance holds: $i_c^! \mathcal{O}_{\mathcal{M}^P}^{\mathrm{vir}} = \mathcal{O}_{\mathcal{M}_c^P}^{\mathrm{vir}}$ for $c \neq 0$, confirming consistency with smooth fibers.
• The construction is valid even when $\mathfrak{C}_0$ is not quasi-compact, relying on boundedness of $\mathcal{M}_0^P$ and virtual pullbacks in subsubsection 2.6.1.