[Paper Review] Chern classes and Gromov--Witten theory of projective bundles

Honglu Fan|arXiv (Cornell University)|May 21, 2017

Homotopy and Cohomology in Algebraic Topology3 citations

TL;DR

This paper establishes that the Gromov–Witten theory of a projective bundle $\mathbb{P}(V)$ over a smooth projective variety $S$ is completely determined by the Gromov–Witten invariants of the base $S$ and the total Chern class $c(V)$ of the vector bundle $V$. Using a master space construction via compactification of $V$ to $\mathbb{P}(V\oplus\mathcal{O})$ and virtual localization, the authors derive a universal formula that computes all invariants of $\mathbb{P}(V)$ from those of $S$ and $c(V)$, resolving a long-standing question in Gromov–Witten theory.

ABSTRACT

We prove that the Gromov--Witten theory (GWT) of a projective bundle can be determined by the Chern classes and the GWT of the base. It completely answers a question raised in a previous paper (arXiv:1607.00740). Its consequences include that the GWT of the blow-up of X at a smooth subvariety Z is uniquely determined by GWT of X, Z plus some topological data.

Motivation & Objective

• To resolve a question posed in prior work about whether the Gromov–Witten invariants of a projective bundle $\mathbb{P}(V)$ are uniquely determined by the invariants of the base $S$ and the total Chern class $c(V)$.
• To develop a general method that computes Gromov–Witten invariants of $\mathbb{P}(V)$ without requiring $V$ to be split or equipped with a torus action.
• To apply the result to refine the blow-up formula in Gromov–Witten theory by eliminating redundant data requirements.
• To establish a universal principle for the functoriality of Gromov–Witten theory under projective bundle constructions.

Proposed method

• Construct a master space $\overline{\mathcal{M}}_{g,n}(\mathbb{P}(V\oplus\mathcal{O}), \beta)$ by compactifying the vector bundle $V$ into $\mathbb{P}(V\oplus\mathcal{O})$ to enable a $\mathbb{C}^*$-action.
• Apply virtual localization to the moduli space of stable maps to $\mathbb{P}(V\oplus\mathcal{O})$, integrating cohomology classes of degree less than the virtual dimension to obtain relations between invariants of $S$ and $\mathbb{P}(V)$.
• Use the resulting relations to recursively determine all Gromov–Witten invariants of $\mathbb{P}(V)$ from those of $S$ and the Chern classes of $V$.
• Define natural isomorphisms $\mathfrak{F}$ and $\Psi$ that identify cohomology and curve class groups of $\mathbb{P}(V_1)$ and $\mathbb{P}(V_2)$ when $c(V_1) = c(V_2)$, ensuring invariance under Chern class equivalence.
• Leverage the degeneration formula and deformation to the normal cone to extend the result to blow-ups, showing that invariants of $\tilde{Y}$ are determined by those of $Y$, $Z$, and $c(N_{Z/Y})$.

Experimental results

Research questions

• RQ1 Can the Gromov–Witten theory of a projective bundle $\mathbb{P}(V)$ be uniquely determined by the Gromov–Witten invariants of the base $S$ and the total Chern class $c(V)$?
• RQ2 Does the invariance of Gromov–Witten invariants under Chern class equivalence hold in all genera, even when the vector bundle is not split?
• RQ3 Can the blow-up formula in Gromov–Witten theory be refined to eliminate the need for input data on the exceptional divisor’s invariants?
• RQ4 Is there a universal principle for the functoriality of Gromov–Witten theory under projective bundle constructions?

Key findings

• The Gromov–Witten invariants of $\mathbb{P}(V)$ are completely determined by the Gromov–Witten invariants of the base $S$ and the total Chern class $c(V)$, regardless of the splitting type of $V$.
• When $c(V_1) = c(V_2)$, the Gromov–Witten invariants of $\mathbb{P}(V_1)$ and $\mathbb{P}(V_2)$ are isomorphic via natural isomorphisms $\mathfrak{F}$ and $\Psi$ on cohomology and curve classes.
• The result implies that the Gromov–Witten theory of the blow-up $\tilde{Y}$ of $Y$ at a smooth subvariety $Z$ is determined solely by the Gromov–Witten invariants of $Y$ and $Z$, along with the Chern classes $c_i(N_{Z/Y})$ and their cohomological pull-back structures.
• The requirement for input invariants of $\mathbb{P}(N_{Z/Y} \oplus \mathcal{O})$ in prior blow-up formulas is redundant, as these are now determined by $c(N_{Z/Y})$ and the invariants of $Z$.
• The method via the master space $\mathbb{P}(V\oplus\mathcal{O})$ and virtual localization enables a uniform computation of invariants without relying on equivariant localization or splitting assumptions.