[Paper Review] A virtual fundamental class construction for the moduli space of torus equivariant morphisms

Andrei Mustaţǎ|arXiv (Cornell University)|Sep 22, 2014

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper constructs a virtual fundamental class for the moduli space of $({b C}^*)^n$-equivariant morphisms from stable toric varieties to a smooth projective variety $X$, using the inverse limit of Deligne-Mumford stacks associated to GIT quotients and their flips. The construction yields a dimension-$\dim X - n$ class in the Chow group that is invariant under equivariant deformations of $X$, and agrees with existing definitions in the $n=1$ case via comparison to stable map moduli spaces.

ABSTRACT

Let X be a smooth projective variety with the action of the n dimensional torus. The article describes the moduli space of torus equivariant morphisms from stable toric varieties into X as the inverse limit of the GIT quotients of X and their flips when these spaces are enhanced with a naturally associated Deligne-Mumford stack structure. This description is used for constructing a class in the Chow group of the moduli space of dimension dim(X)-n which is invariant to equivariant deformations of X.

Motivation & Objective

• To define a virtual fundamental class for the moduli space of $({\mathbb{C}}^*)^n$-equivariant morphisms from stable toric varieties to a smooth projective variety $X$, despite the absence of a perfect obstruction theory.
• To establish a geometric realization of this moduli space as the inverse limit of Deligne-Mumford stacks associated to GIT quotients and their flips under the torus action.
• To construct a Chow class of dimension $\dim X - n$ that is invariant under equivariant deformations of $X$, ensuring functoriality and compatibility with known cases.
• To provide a framework that generalizes to reductive group actions, with potential applications to Gromov-Witten invariants and stable map moduli spaces.

Proposed method

• The moduli space is realized as the inverse limit of Deligne-Mumford stacks associated to GIT quotients of $X$ and their flips under the $({\mathbb{C}}^*)^n$-action.
• The universal family over the inverse limit is constructed via the moduli space of genus-zero stable maps to $X$, linking equivariant morphisms to stable map theory.
• A relative intrinsic normal cone $\mathcal{C}_Z$ is embedded into the cohomology sheaf $h^1/h^0(R^f\pi_* e^*T_Z)$, enabling the virtual class construction.
• The virtual class is pulled back from the inverse limit of $[BT]$-stacks via a morphism involving the fixed part of the direct image of the tangent bundle.
• Tautological rings are defined for subcategories of GIT quotients and their flips, with pull-back and push-forward maps established via Proposition 5.2.
• The construction is verified to coincide with the standard virtual fundamental class in the $n=1$ case, as shown in Theorem 6.3.

Experimental results

Research questions

• RQ1 Can a virtual fundamental class be constructed for the moduli space of $({\mathbb{C}}^*)^n$-equivariant morphisms into a smooth projective variety $X$, even when a perfect obstruction theory does not exist?
• RQ2 How does the inverse limit of Deligne-Mumford stacks associated to GIT quotients and their flips relate to the moduli space of equivariant morphisms?
• RQ3 Does the constructed virtual class transform correctly under equivariant deformations of $X$, and is it compatible with the virtual class in the stable map moduli space when $n=1$?
• RQ4 Can the framework be extended to actions of reductive groups beyond tori, such as $SL_{k+1}$, and what would be the implications for stable map moduli spaces?
• RQ5 What is the structure of tautological rings in the inverse limit of GIT quotients, and how do they behave under pull-back and push-forward?

Key findings

• The moduli space of $({\mathbb{C}}^*)^n$-equivariant morphisms is isomorphic to the inverse limit of Deligne-Mumford stacks associated to GIT quotients and their flips, under a natural stack structure.
• A virtual fundamental class of dimension $\dim X - n$ is constructed in the Chow group, which is invariant under equivariant deformations of $X$.
• In the case $n=1$, the virtual class coincides with the standard virtual fundamental class from the moduli space of stable maps, as verified in Theorem 6.3.
• The relative intrinsic normal cone $\mathcal{C}_Z$ embeds into $h^1/h^0(R^f\pi_* e^*T_Z)$, enabling the virtual class to be pulled back from the base stack $\underleftarrow{\lim} [BT]$.
• Tautological rings on the inverse limit admit pull-back and push-forward maps, as established in Proposition 5.2, with an algorithm for computation provided in Remark 5.5.
• The construction suggests a pathway to defining moduli spaces of stable maps for reductive group actions, such as $SL_{k+1}$ on spaces of homogeneous polynomials, via similar inverse limit techniques.