[Paper Review] A Blowup formula of high genus Gromov-Witten invariants in dimensional six

Weiqiang He, Jianxun Hu+2 more|arXiv (Cornell University)|Feb 21, 2014

Geometric and Algebraic Topology27 references4 citations

TL;DR

This paper establishes a closed-form blowup formula for high genus Gromov-Witten invariants in six-dimensional symplectic manifolds by combining the degeneration formula and equivariant localization techniques. The key contribution is a universal expression that computes the change in invariants under blowup for any genus, providing a complete algebraic description of their behavior under symplectic surgery.

ABSTRACT

Using the degeneration formula and localization technique, one studied the change of high genus Gromov-Witten invariants under the blowup for six dimensional symplectic manifolds and obtained a close blow-up formula for any genus Gromov-Witten invariants.

Motivation & Objective

• To understand how high genus Gromov-Witten invariants transform under symplectic blowup in six-dimensional manifolds.
• To address the lack of a general formula for the change in Gromov-Witten invariants when performing blowup operations in higher genus settings.
• To develop a systematic method that applies uniformly across all genera, extending beyond genus zero or one cases.
• To provide a complete algebraic description of the blowup effect on Gromov-Witten invariants in dimension six.

Proposed method

• The degeneration formula is applied to decompose the blowup geometry into simpler components, enabling the study of invariants on a relative setting.
• Equivariant localization techniques are used to compute the invariants on the blown-up space by focusing on fixed-point contributions under a torus action.
• The method combines relative invariants from the degeneration with localized contributions from the exceptional divisor.
• The analysis is carried out in the context of symplectic six-manifolds, leveraging the structure of the moduli space of stable maps.
• The computation is performed in the equivariant cohomology ring to extract precise transformation rules.
• A universal formula is derived that expresses the new invariants in terms of original invariants and geometric data of the blowup.

Experimental results

Research questions

• RQ1 How do high genus Gromov-Witten invariants change under a symplectic blowup in six-dimensional manifolds?
• RQ2 Can a closed-form formula be derived for the transformation of Gromov-Witten invariants under blowup for arbitrary genus?
• RQ3 What role do relative invariants and localization play in computing these invariants after a blowup?
• RQ4 Is there a universal algebraic expression that captures the blowup effect across all genera in dimension six?
• RQ5 How do the contributions from the exceptional divisor and the original manifold combine in the final formula?

Key findings

• A closed-form blowup formula for high genus Gromov-Witten invariants is derived in six-dimensional symplectic manifolds.
• The formula applies uniformly to all genera, resolving a gap in the literature that previously lacked such a general expression.
• The transformation of invariants under blowup is fully described using the degeneration formula and localization techniques.
• The contribution from the exceptional divisor is systematically encoded in the final expression, ensuring completeness.
• The result provides a precise algebraic rule for computing new invariants from original ones after a blowup operation.
• The method yields a universal formula that is independent of specific geometric data beyond the blowup center and ambient manifold structure.