[Paper Review] Torus localization and wall crossing for cosection localized virtual cycles

Huai-Liang Chang, Young‐Hoon Kiem+1 more|arXiv (Cornell University)|Jan 31, 2015

Algebraic Geometry and Number Theory32 references14 citations

TL;DR

This paper establishes virtual pullback, torus localization, and wall crossing formulas for cosection-localized virtual cycles in algebraic geometry, generalizing classical results to cases with cosections. It removes technical assumptions from prior work, proving that the localized virtual fundamental class behaves well under equivariant actions and wall crossings, with applications to Gromov-Witten and Fan-Jarvis-Ruan-Witten invariants of Calabi-Yau threefolds.

ABSTRACT

Since its introduction in 1995 by Li-Tian and Behrend-Fantechi, the theory of virtual fundamental class has played a key role in algebraic geometry, defining important invariants such as the Gromov-Witten invariant and the Donaldson-Thomas invariant. Quite a few methods for handling the virtual fundamental classes were discovered such as torus localization, degeneration, virtual pullback and cosection localization. Often combining these methods turns out to be quite effective. In this paper, we prove virtual pullback, torus localization and wall crossing formulas for cosection localized virtual cycles.

Motivation & Objective

• To extend virtual pullback, torus localization, and wall crossing formulas to the setting of cosection-localized virtual cycles.
• To remove technical assumptions—specifically, the need for an equivariant global embedding into a smooth Deligne-Mumford stack—previously required in torus localization.
• To generalize classical formulas for ordinary virtual cycles to the cosection-localized case, where the virtual cycle is supported on the complement of the cosection's vanishing locus.
• To provide tools for computing invariants in Landau-Ginzburg theory and Calabi-Yau threefold invariants via cosection localization.
• To establish a wall crossing formula for cosection-localized virtual cycles in $$\mathbb{C}^*\u0024$-wall crossings, linking $M_+$ and $M_-$ via the fixed locus $F$.

Proposed method

• Proves a cosection-localized virtual pullback formula by adapting Manolache’s method, ensuring rational equivalences lie in substacks compatible with localized Gysin maps.
• Develops a new torus localization formula that avoids requiring a global equivariant embedding or resolution of the perfect obstruction theory, instead relying only on a resolution of the virtual normal bundle on the fixed locus.
• Constructs a master space $\mathfrak{M} = [X \times \mathbb{P}^1 - \Sigma_- \times \{0\} - \Sigma_+ \times \{\infty\}]/\mathbb{C}^*$ to analyze wall crossings.
• Uses the induced $T$-equivariant perfect obstruction theory and cosection on $\mathfrak{M}$ to define localized virtual cycles on $M_+$, $M_-$, and $F$.
• Applies the localized Gysin map to the intrinsic normal cone to define $[X]^{\mathrm{vir}}_{\mathrm{loc}}$, ensuring compatibility with deformation invariance.
• Takes the residue at $t=0$ of the localized virtual cycle on the master space to derive the wall crossing formula.

Experimental results

Research questions

• RQ1 Can the virtual pullback formula be extended to the case of cosection-localized virtual cycles?
• RQ2 Does the torus localization formula hold without the assumption of an equivariant global embedding into a smooth Deligne-Mumford stack?
• RQ3 How does the wall crossing formula behave for cosection-localized virtual cycles in simple $\mathbb{C}^*$-wall crossings?
• RQ4 What is the relationship between the localized virtual cycles on $M_+$ and $M_-$ and the fixed locus $F$?
• RQ5 Can the localized virtual cycle on the master space $\mathfrak{M}$ be used to derive a wall crossing formula for the localized virtual fundamental class?

Key findings

• The virtual pullback formula for cosection-localized virtual cycles holds, generalizing Manolache’s result to the localized setting.
• The torus localization formula is proven without requiring an equivariant global embedding or a global resolution of the perfect obstruction theory, only a resolution of the virtual normal bundle on the fixed locus.
• The wall crossing formula for cosection-localized virtual cycles is established as $[M_+]^{\mathrm{vir}}_{\mathrm{loc}} - [M_-]^{\mathrm{vir}}_{\mathrm{loc}} = \mathrm{res}_{t=0}\frac{[F]^{\mathrm{vir}}_{\mathrm{loc}}}{e(N^{\mathrm{vir}})}$ in equivariant Chow theory.
• In the example of $V = \mathbb{C}^d$ with trivial action and tautological cosection, $[V]^{\mathrm{vir}}_{\mathrm{loc}} = (-1)^d [O]$, consistent with known results.
• When the cosection is surjective, $[V]^{\mathrm{vir}}_{\mathrm{loc}} = 0$, but the localized torus localization formula still gives $(-1)^d [O]$, showing the formula does not apply in such cases.
• The master space construction allows the wall crossing formula to be derived via residue extraction from the localized virtual cycle on $\mathfrak{M}$.