[Paper Review] Stable pairs with descendents on local surfaces I: the vertical component

Martijn Kool, Richard Thomas|arXiv (Cornell University)|May 9, 2016

Algebraic Geometry and Number Theory20 references3 citations

TL;DR

This paper computes the full stable pair theory with descendents on the local Calabi-Yau 3-fold $X = K_S$, where $S$ is a surface with a smooth canonical divisor $C$. Using $\mathbb{C}^*$-localization and cosection localization, it reduces the computation to vertical thickenings of $C$ indexed by strict partitions, and derives a closed product formula for the generating series of invariants in curve classes $[C]$ and $2[C]$, verifying a conjecture of Maulik-Pandharipande via the descendent-MNOP correspondence.

ABSTRACT

We study the full stable pair theory --- with descendents --- of the Calabi-Yau 3-fold $X=K_S$, where $S$ is a surface with a smooth canonical divisor $C$. By both $\mathbb C^*$-localisation and cosection localisation we reduce to stable pairs supported on thickenings of $C$ indexed by partitions. We show that only strict partitions contribute, and give a complete calculation for length-1 partitions. The result is a surprisingly simple closed product formula for these "vertical" thickenings. This gives all contributions for the curve classes $[C]$ and $2[C]$ (and those which are not an integer multiple of the canonical class). Here the result verifies, via the descendent-MNOP correspondence, a conjecture of Maulik-Pandharipande, as well as various results about the Gromov-Witten theory of $S$ and spin Hurwitz numbers.

Motivation & Objective

• To compute the full stable pair theory with descendents on the Calabi-Yau 3-fold $X = \mathrm{Tot}(K_S)$ for surfaces $S$ with a smooth canonical divisor $C$.
• To reduce the computation of stable pair invariants to the fixed locus of the $\mathbb{C}^*$-action, focusing on components supported on thickenings of $C$.
• To show that only strict partitions contribute to the invariants in curve classes $d\cdot [\mathsf{k}]$, where $\mathsf{k}$ is the canonical class.
• To derive a closed-form product formula for the generating series of vertical contributions in curve classes $[C]$ and $2[C]$, verifying a conjecture of Maulik-Pandharipande.

Proposed method

• Apply $\mathbb{C}^*$-equivariant localization to the moduli space of stable pairs on $X = K_S$, reducing the computation to the fixed locus under the $\mathbb{C}^*$-action on the fibers of $K_S$.
• Use cosection localization to further restrict the computation to stable pairs supported on thickenings of the canonical divisor $C$, indexed by partitions $\boldsymbol{\lambda} \vdash d$.
• Characterize the fixed components as those with support defined by ideals of the form $I_C(-\lambda_1 S) + \cdots + I_C^l$, where $I_C$ is the ideal sheaf of $\pi^*C$.
• Show that only strict partitions (with $\lambda_0 > \lambda_1 > \cdots > \lambda_{l-1} > 0$) contribute to the invariants.
• Express the virtual normal bundle and obstruction theory in terms of tautological classes on the nested Hilbert scheme of points on $C$, enabling explicit computation.
• Use a generating function identity from Pixton and Zagier to evaluate the integrals and derive a closed product formula for the vertical contributions.

Experimental results

Research questions

• RQ1 Which curve classes on $X = K_S$ support non-vanishing stable pair invariants with descendents, and under what conditions do they vanish?
• RQ2 How can the stable pair invariants on $X = K_S$ be localized to components supported on thickenings of the canonical divisor $C$?
• RQ3 What is the precise contribution of the 'vertical' components (supported on $\pi^{-1}(C)$) to the full stable pair invariants in curve classes $[C]$ and $2[C]$?
• RQ4 Can the descendent-MNOP correspondence be used to verify the Maulik-Pandharipande conjecture on Gromov-Witten invariants of $S$ via stable pair computations on $X = K_S$?
• RQ5 What is the explicit closed-form formula for the generating series of vertical stable pair invariants with descendents?

Key findings

• Only strict partitions $\boldsymbol{\lambda} = (\lambda_0 > \cdots > \lambda_{l-1} > 0)$ with $|\boldsymbol{\lambda}| = d$ contribute to the stable pair invariants in curve class $d\cdot [\mathsf{k}]$, while non-strict partitions vanish.
• For length-1 partitions $\boldsymbol{\lambda} = (d)$, the vertical contribution to the generating series is given by a closed product formula involving trigonometric and rational functions of the equivariant parameter $t$ and the curve class $d\cdot [\mathsf{k}]$.
• The generating series for vertical invariants in curve class $d\cdot [\mathsf{k}]$ is expressed as a product over formal variables $v_j$, involving $\sin(n\,du/2)$ and Pochhammer-like denominators.
• The leading term of the generating series matches the Gromov-Witten invariants of $S$ in curve class $d\cdot [\mathsf{k}]$, confirming the descendent-MNOP correspondence for these cases.
• The result verifies the Maulik-Pandharipande conjecture on the Gromov-Witten theory of $S$ via the vertical stable pair invariants on $X = K_S$, particularly for $d=1$ and $d=2$.
• A key identity from Pixton and Zagier is used to evaluate the integrals, with the leading coefficient $A_\alpha = \frac{1}{(2\alpha-1)!!}$ appearing in the expansion of a generating function involving $\sin(nx)/\sin^n x$.