[Paper Review] Corrigendum: The symplectic sum formula for Gromov-Witten invariants

Eleny-Nicoleta Ionel, Thomas H. Parker|arXiv (Cornell University)|Oct 14, 2015

Finite Group Theory Research1 references3 citations

TL;DR

This paper corrects a sign error in the curvature term of the symplectic sum formula for Gromov-Witten invariants and revises the Sobolev norm to handle maps with general intersection multiplicity vectors $s = (s_1, \dots, s_\ell)$, where $s_k \geq 2$. The key contribution is a modified norm with a $\rho^{1-s_k}$ weighting near nodes, which restores the validity of key estimates and ensures the invertibility of the linearized operator $D_F$, enabling a consistent proof of the symplectic sum formula for all multiplicity types.

ABSTRACT

We correct an error and an oversight in [IP]. The sign of the curvature in (8.7) is wrong, requiring a new proof of Proposition 8.1. Also, several lemmas addressed only the basic case of maps with intersection multiplicity s=1; the general case follows by applying the pointwise estimates in [IP] with a modified Sobolev norm. These corrections do not affect the results of the paper. We thank M. Tehrani and A. Zinger for pointing out these issues.

Motivation & Objective

• To correct a sign error in the curvature term (8.7) of the symplectic sum formula for Gromov-Witten invariants as presented in [IP].
• To address an oversight in [IP] that only treated maps with intersection multiplicity $s = \mathbf{1}$, extending the analysis to general multiplicity vectors $s = (s_1, \dots, s_\ell)$ with $s_k \geq 2$.
• To redefine the Sobolev norms in [IP, (6.9)] with a modified weighting factor $\rho^{1-s_k}$ on normal components near nodes of multiplicity $s_k \geq 2$, ensuring uniform equivalence and validity of estimates.
• To verify that Lemmas 6.9, 7.1, Proposition 7.3, and Lemma 9.2 remain valid under the new norm using pointwise estimates from [IP].
• To establish the existence of a uniformly bounded approximate inverse for the linearized operator $D_F$ under the revised norm, ensuring the symplectic sum formula holds for all multiplicity types.

Proposed method

• Introduce a revised Sobolev norm $|||\zeta|||_{m,p,s}$ with a $\rho^{1-s_k}$ factor on normal components near nodes with $s_k \geq 2$, defined via weighted $L^p$-integrals over nodal regions.
• Modify the definitions of $|||(.\xi,h)|||_m$ and $|||\eta|||_m$ in [IP, (6.10)–(6.11)] using the new norm to ensure consistency in operator estimates.
• Use pointwise estimates from [IP] to bound differences in $\overline{\partial}$-operators and their derivatives, particularly $|\Phi_F - \Phi_{\tilde{f}}|$ and $|\Phi_{\tilde{f}} - \Phi_f|$, under the new norm.
• Establish uniform bounds on $|||D_F \Gamma_F - \gamma_F D_f|||_0$ using the revised norm, showing decay as $|\lambda|^{1/(5|s|)}$ and $|\log M|^{-1}$, which enables the construction of an approximate inverse.
• Construct the operator $A_F = \Gamma_{F,M} \circ P_f \circ \pi_F$ as a spliced approximate right inverse of $D_F$, proving $|||D_F A_F \eta - \eta|||_0 \leq \frac{1}{2}|||\eta|||_0$ and $|||A_F \eta|||_1 \leq C|||\eta|||_0$.
• Correct minor typographical errors in [IP] that do not affect the mathematical results, including sign errors and incorrect indices in equations.

Experimental results

Research questions

• RQ1 How does a sign error in the curvature term (8.7) of the symplectic sum formula affect the validity of the original proof in [IP]?
• RQ2 What modifications to the Sobolev norm are required to extend the analysis from the basic case $s = \mathbf{1}$ to general multiplicity vectors $s = (s_1, \dots, s_\ell)$ with $s_k \geq 2$?
• RQ3 Can the key estimates in Lemmas 6.9, 7.1, and Proposition 7.3 be preserved under the new norm, and if so, under what conditions?
• RQ4 Does the existence of a uniformly bounded approximate inverse for $D_F$ still hold under the revised norm, and how does this affect the symplectic sum formula?
• RQ5 What is the impact of the revised norm on the operator norm of $D_F \Gamma_F - \gamma_F D_f$, and how does it influence the construction of $A_F$?

Key findings

• The sign error in curvature term (8.7) of [IP] is corrected, requiring a new proof of Proposition 8.1, which is now valid under the revised norm framework.
• The modified Sobolev norm $|||\zeta|||_{m,p,s}$ with $\rho^{1-s_k}$ weighting near nodes of multiplicity $s_k \geq 2$ ensures uniform equivalence to the original norm when $s = \mathbf{1}$, while strengthening estimates in the general case.
• The estimate $|||\Phi_F - \Phi_{\tilde{f}}|||_{0,A_+} \leq c|\lambda|^{1/(3|s|)}$ holds on $A_+$, with the exponent adjusted to $1/(3|s|)$ due to the revised norm and $|\lambda| \sim |\mu|^{s_k}$.
• The operator norm of $D_F \Gamma_F - \gamma_F D_f$ is bounded by $c_M |\lambda|^{1/(5|s|)} + c |\log M|^{-1}$, allowing uniform control via choice of $M$ and $\lambda_0$.
• An approximate inverse $A_F = \Gamma_{F,M} \circ P_f \circ \pi_F$ exists such that $|||D_F A_F \eta - \eta|||_0 \leq \frac{1}{2}|||\eta|||_0$ and $|||A_F \eta|||_1 \leq C|||\eta|||_0$, proving the linearized operator is uniformly invertible under the new norm.
• Minor typographical errors in [IP] are corrected, including sign errors and incorrect indices, without affecting the mathematical conclusions.