[Paper Review] The Evaluation Space of Logarithmic Stable Maps

Dan Abramovich, Qile Chen+2 more|arXiv (Cornell University)|Dec 24, 2010

Algebraic Geometry and Number Theory18 references21 citations

TL;DR

This paper constructs the evaluation space $Ω X$ for logarithmic stable maps, enabling the definition of logarithmic evaluation maps and virtual fundamental classes on moduli stacks of minimal log stable maps. The key contribution is the establishment of logarithmic Gromov-Witten invariants via these tools, particularly in the context of log-smooth Deligne-Faltings log schemes with simple normal crossing divisors.

ABSTRACT

The evaluation stack for minimal logarithmic stable maps is constructed, parameterizing families of standard log points in the target log scheme. This construction provides the ingredients necessary to define appropriate evaluation maps for minimal log stable maps and establish the logarithmic Gromov-Witten theory of a log-smooth Deligne- Faltings log scheme.

Motivation & Objective

• To define a universal parameter space for families of standard log points in a fine saturated log scheme, enabling evaluation maps in logarithmic Gromov-Witten theory.
• To construct the logarithmic algebraic stack $(\wedge X, \mathcal{M}_{\wedge X})$ representing the category of standard log points in a log scheme $(X, \mathcal{M}_X)$.
• To define logarithmic evaluation maps $\text{ev}_i: \mathcal{K}_\Gamma(X, \mathcal{M}_X) \to \wedge X$ for minimal log stable maps.
• To establish a virtual fundamental class on the moduli stack $\mathcal{K}_\Gamma(X, \mathcal{M}_X)$ under log smoothness assumptions.
• To define logarithmic Gromov-Witten invariants via pullbacks of cohomology classes along evaluation maps and intersection with the virtual fundamental class.

Proposed method

• Introduces the category $\wedge'X$ of families of standard log points in a fine saturated log scheme $(X, \mathcal{M}_X)$, fibered over $\mathfrak{LogSch}^{\text{fs}}$.
• Proves that $\wedge'X$ is representable by a logarithmic algebraic stack $(\wedge X, \mathcal{M}_{\wedge X})$, establishing the evaluation space.
• Constructs natural morphisms $\text{ev}_i: \mathcal{K}_\Gamma(X, \mathcal{M}_X) \to \wedge X$ by restricting log stable maps to marked points, where the log structure becomes standard.
• Uses a perfect obstruction theory $E. \to \mathbb{L}_{\mathcal{K}_\Gamma(X, \mathcal{M}_X)/\mathcal{T}or_{\mathbb{C}}}$ to define the virtual fundamental class $[\mathcal{K}_\Gamma(X, \mathcal{M}_X)]^{\text{vir}}$ via Manolache's refined pullback.
• Applies the refined pullback $\alpha^{!}_{\mathbb{E}}$ to the fundamental class $[\mathcal{T}or_{\mathbb{C}}]$, yielding the virtual class in the log-smooth case.
• Defines logarithmic Gromov-Witten invariants as $\left<\gamma_1,\ldots,\gamma_n\right>^{(X,\mathcal{M}_X)}_\Gamma := \left(\prod_{i=1}^n \text{ev}_i^*\gamma_i\right) \cap [\mathcal{K}_\Gamma(X, \mathcal{M}_X)]^{\text{vir}}$.

Experimental results

Research questions

• RQ1 How can one construct a universal parameter space for standard log points in a log scheme to support evaluation maps in logarithmic Gromov-Witten theory?
• RQ2 Under what conditions does the category of families of standard log points in a fine saturated log scheme admit a representing logarithmic algebraic stack?
• RQ3 How can evaluation maps be defined for minimal log stable maps into a log-smooth Deligne-Faltings log scheme?
• RQ4 What conditions ensure the existence of a virtual fundamental class on the moduli stack of minimal log stable maps?
• RQ5 How can logarithmic Gromov-Witten invariants be defined using evaluation maps and virtual fundamental classes?

Key findings

• The category $\wedge'X$ of families of standard log points in a fine saturated log scheme $(X, \mathcal{M}_X)$ is representable by a logarithmic algebraic stack $(\wedge X, \mathcal{M}_{\wedge X})$.
• Logarithmic evaluation maps $\text{ev}_i: \mathcal{K}_\Gamma(X, \mathcal{M}_X) \to \wedge X$ are constructed by restricting log stable maps to marked points, where the log structure becomes standard.
• In the log-smooth case, the moduli stack $\mathcal{K}_\Gamma(X, \mathcal{M}_X)$ admits a virtual fundamental class $[\mathcal{K}_\Gamma(X, \mathcal{M}_X)]^{\text{vir}}$ via a perfect obstruction theory and Manolache's refined pullback.
• The virtual fundamental class is obtained as $[\mathcal{K}_\Gamma(X, \mathcal{M}_X)]^{\text{vir}} = \alpha^{!}_{\mathbb{E}}[\mathcal{T}or_{\mathbb{C}}]$, where $\alpha$ is the structure map to the stack of fs log structures over $\mathbb{C}$.
• Logarithmic Gromov-Witten invariants are defined via the formula $\left<\gamma_1,\ldots,\gamma_n\right>^{(X,\mathcal{M}_X)}_\Gamma := \left(\prod_{i=1}^n \text{ev}_i^*\gamma_i\right) \cap [\mathcal{K}_\Gamma(X, \mathcal{M}_X)]^{\text{vir}}$.
• The construction provides a foundational framework for logarithmic Gromov-Witten theory in the setting of log-smooth Deligne-Faltings log schemes with simple normal crossing divisors.