[Paper Review] An algebraic proof of the hyperplane property of the genus one GW-invariants of quintics

Huai-Liang Chang, Jun Li|arXiv (Cornell University)|Jun 23, 2012

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper provides an algebro-geometric proof of the hyperplane property for genus one Gromov-Witten invariants of the quintic threefold, using Guffin-Sharpe-Witten theory with p-fields to algebraically separate the primary and ghost contributions in the virtual cycle. It confirms that the ordinary genus one invariants equal the reduced invariants plus 1/12 of the genus zero invariants, resolving a key result originally proven via analytic methods.

ABSTRACT

Li-Zinger's hyperplane theorem states that the genus one GW-invariants of the quintic threefold is the sum of its reduced genus one GW-invariants and 1/12 multiplies of its genus zero GW-invariants. We apply the Guffin-Sharpe-Witten's theory (GSW theory) to give an algebro-geometric proof of the hyperplane theorem, including separation of contributions and computation of 1/12.

Motivation & Objective

• To provide a purely algebraic proof of the hyperplane property for genus one Gromov-Witten invariants of the quintic threefold, which was previously established using analytic methods.
• To algebraically separate the virtual cycle of genus one stable maps to the quintic into primary and ghost components via the Guffin-Sharpe-Witten (GSW) theory with p-fields.
• To compute the contribution of the ghost component explicitly, showing it equals 1/12 of the genus zero invariants.
• To establish a framework using p-fields and cosection localization that can be extended to higher genus invariants of quintics and other complete intersections.

Proposed method

• Utilizes the Guffin-Sharpe-Witten (GSW) theory, realizing Gromov-Witten invariants as those of stable maps to $\mathbb{P}^4$ with p-fields, which simplifies the moduli space structure.
• Constructs the moduli space $\overline{M}_1(\mathbb{P}^4,d)^p$ of genus one stable maps with p-fields, equipped with a perfect obstruction theory.
• Introduces a cosection $\sigma$ of the obstruction sheaf using the quintic polynomial $x_1^5 + \cdots + x_5^5$, whose degeneracy locus is $\overline{M}_1(Q,d)$.
• Applies Kiem-Li's cosection localized virtual cycle construction to define $[\overline{M}_1(\mathbb{P}^4,d)^p]^{\mathrm{vir}}_\sigma \in A_0(\overline{M}_1(Q,d))$.
• Relates the localized invariant $N_1(d)^p_{\mathbb{P}^4}$ to the ordinary genus one invariant $N_1(d)_Q$ via $N_1(d)^p_{\mathbb{P}^4} = (-1)^{5d} \cdot N_1(d)_Q$.
• Computes the ghost contribution via localization on a family of rational curves in $\mathbb{P}^4$, using Chern class calculations on $\overline{W}_h = \mathbb{P}(L_B \oplus \mathcal{O}_B)$ and evaluating the Euler class of a vector bundle quotient.

Experimental results

Research questions

• RQ1 How can the hyperplane property for genus one Gromov-Witten invariants of the quintic threefold be proven using algebraic geometry rather than analytic methods?
• RQ2 What is the algebraic origin of the 1/12 correction term in the relation between ordinary and reduced genus one invariants?
• RQ3 Can the Guffin-Sharpe-Witten theory with p-fields be used to decompose the virtual cycle into primary and ghost components in a purely algebro-geometric way?
• RQ4 How does the localized virtual cycle construction via cosection theory capture the contribution of the ghost component?
• RQ5 Can this method be generalized to higher genus Gromov-Witten invariants of quintics and other complete intersections?

Key findings

• The ordinary genus one Gromov-Witten invariants of the quintic threefold satisfy $N_1(d)_Q = N_1(d)^{\mathrm{red}}_Q + \frac{1}{12}N_0(d)_Q$, confirming the Li-Zinger hyperplane theorem algebraically.
• The ghost component of the virtual cycle contributes exactly $\frac{1}{12}$ of the genus zero invariants, as computed via localization on a rational curve family in $\mathbb{P}^4$.
• The contribution of the ghost component is calculated to be $-\frac{1}{12}$ in the localized Euler class computation, which matches the expected correction term.
• The reduced genus one invariants are defined via a proper birational modification $\tilde{\mathcal{X}}_{\mathrm{pri}}$ of the primary component, ensuring the direct image sheaf is locally free.
• The method successfully separates the primary and ghost parts of the virtual cycle using p-fields and cosection localization, enabling an algebro-geometric proof of the hyperplane property.
• The result establishes a foundation for extending this approach to higher genus invariants of quintics and other complete intersections in products of projective spaces.