[Paper Review] Localizing virtual structure sheaves by cosections

Young‐Hoon Kiem, Jun Li|arXiv (Cornell University)|May 26, 2017

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper introduces a K-theoretic cosection localization for virtual structure sheaves on Deligne-Mumford stacks with perfect obstruction theories and cosections of the obstruction sheaf. By defining a localized Gysin map on the support of the cosection, it constructs a virtual structure sheaf in $K_0(X(σ))$ that lifts the virtual cycle to algebraic K-theory, enabling deformation-invariant K-theoretic invariants and extending results to Gromov-Witten and Fan-Jarvis-Ruan-Witten invariants.

ABSTRACT

We construct a cosection localized virtual structure sheaf when a Deligne-Mumford stack is equipped with a perfect obstruction theory and a cosection of the obstruction sheaf.

Motivation & Objective

• To extend cosection localization from Chow groups to algebraic K-theory by constructing a K-theoretic virtual structure sheaf.
• To define a localized virtual Euler characteristic in $K_0$-theory that remains well-defined even when the moduli space is not proper.
• To lift results from [6,7] on cosection-localized virtual cycles to the K-theoretic setting, particularly for FJRW invariants.
• To establish a K-theoretic version of the Landau-Ginzburg/Calabi-Yau correspondence via the virtual Euler characteristic.
• To prove independence of the construction from the choice of global resolution and invariance under deformation.

Proposed method

• Define the cosection localized Gysin map $0^{!}_{E_1,\sigma}: K_0(E_1(\sigma)) \to K_0(X(\sigma))$ using the support of the cone $C_1$ under the cosection.
• Construct the localized virtual structure sheaf as $[\mathscr{O}^{\mathrm{vir}}_{X,\mathrm{loc}}] = 0^{!}_{E_1,\sigma}[\mathscr{O}_{C_1}] \in K_0(X(\sigma))$.
• Use the commutativity of pushforward and localized Gysin map to reduce the problem to blowups along $X(\sigma)$, where exact sequences are deformed to split cases.
• Leverage the fact that $E_1(\sigma)$ is the union of $E_1|_{X(\sigma)}$ and $\ker(\sigma: E_1|_U \to \mathscr{O}_U)$ for $U = X \setminus X(\sigma)$.
• Apply deformation invariance and compatibility with the virtual fundamental class to ensure independence of the resolution choice.
• Establish comparison with Chiodo's K-theory class $Ke(E^\vee, \tau^\vee)$ via reduction to split exact sequences on blowups.

Experimental results

Research questions

• RQ1 Can the cosection localization of virtual cycles be lifted from Chow groups to algebraic K-theory?
• RQ2 Is the resulting K-theoretic virtual structure sheaf independent of the choice of global resolution of the perfect obstruction theory?
• RQ3 Can the localized virtual Euler characteristic be defined and computed in $K_0$-theory even when the moduli space is not proper?
• RQ4 Does the K-theoretic version of the Landau-Ginzburg/Calabi-Yau correspondence hold via the localized virtual Euler characteristic?
• RQ5 How does the localized virtual structure sheaf relate to existing K-theoretic invariants such as Chiodo's class?

Key findings

• The cosection localized virtual structure sheaf $[\mathscr{O}^{\mathrm{vir}}_{X,\mathrm{loc}}] \in K_0(X(\sigma))$ is well-defined and independent of the choice of global resolution $[E^{-1} \to E^0]$.
• The construction satisfies $\imath_*[\mathscr{O}^{\mathrm{vir}}_{X,\mathrm{loc}}] = [\mathscr{O}^{\mathrm{vir}}_X] \in K_0(X)$, ensuring compatibility with the global virtual structure sheaf.
• The localized virtual structure sheaf is deformation invariant, preserving its class under deformations of the moduli space.
• The cosection localized virtual Euler characteristic $\chi^{\mathrm{vir}}_{\mathrm{loc}}(X,\beta)$ is defined as $\sum_i (-1)^i \dim H^i(X(\sigma), \beta \cdot \mathscr{O}^{\mathrm{vir}}_{X,\mathrm{loc}})$, even when $X$ is not proper.
• Chiodo's K-theory class $Ke(E^\vee, \tau^\vee)$ coincides with $[\mathscr{O}^{\mathrm{vir}}_{X,\mathrm{loc}}]$ in $K_0(S)$, establishing a direct link to existing K-theoretic constructions.
• For the Gromov-Witten model on the Fermat quintic, $\chi^{\mathrm{vir}}_{\mathrm{loc}}(\overline{M}_g(\mathbb{P}^4,d)^p) = (-1)^{5d - g + 1} \chi^{\mathrm{vir}}(N)$, proving a K-theoretic version of the LG/CY correspondence.