Upper bounds for logarithmic Gromov-Witten invariants of projective space

1. Problem Statement

The paper addresses the problem of estimating genus zero logarithmic Gromov–Witten (LGW) invariants for projective space $\mathbb{P}^k$ relative to its toric boundary $\partial \mathbb{P}^k$.

• Context: LGW theory enumerates stable maps from a curve $C$ to a variety $X$ with prescribed tangency orders $\alpha_{ij}$ to the components of a simple normal crossing divisor $D$.
• Specific Challenge: While these invariants can be computed recursively or via tropical geometry for specific cases, obtaining general upper bounds that are polynomial in the contact orders (tangency degrees) is difficult. Standard compactifications (like the moduli space of logarithmic stable maps) involve complex virtual fundamental classes and boundary contributions that are hard to control globally.
• Goal: To derive an explicit, polynomial upper bound for the primary invariants $\langle \gamma_1, \dots, \gamma_n \rangle_{\mathbb{P}^k|\partial \mathbb{P}^k}^{0, \alpha}$, where the degree of the polynomial depends on the number of marked points $n$ and the contact orders.

2. Methodology

The author employs a geometric strategy that avoids the complexities of virtual cycles by utilizing an alternative compactification and the positivity of intersections.

A. Alternative Compactification via Isomorphism

Instead of working directly with the standard moduli space of logarithmic stable maps, the paper exploits a specific isomorphism available in genus zero when at least one marked point maps to the dense torus $T \subset \mathbb{P}^k$:
$$\mathcal{M}_{0, \alpha}(\mathbb{P}^k, \partial \mathbb{P}^k) \cong \mathcal{M}_{0, n} \times T$$
Since $\mathcal{M}_{0, n}$ is birational to $\mathbb{P}^{n-3}$ and $T$ is an open subset of $\mathbb{P}^k$, the moduli space is embedded as an open subset of the product of projective spaces:
$$\mathcal{M}_{0, \alpha}(\mathbb{P}^k, \partial \mathbb{P}^k) \hookrightarrow \mathbb{P}^{n-3} \times \mathbb{P}^k$$

B. Transversality and Boundary Contributions

The invariants are defined as the degree of the intersection of evaluation cycles.

1. Transverse Intersection: By Kleiman's transversality theorem, for generic insertions, the intersection of evaluation cycles in the interior moduli space consists of reduced points, the count of which is the invariant.
2. Closure in Product Space: The author considers the closures of these cycles in the ambient product space $\mathbb{P}^{n-3} \times \mathbb{P}^k$.
3. Positivity Argument: The intersection in the product space decomposes into:
   • The interior intersection (the actual invariant).
   • Extra components lying in the boundary of the compactification.
   • Key Theorem: Since $\mathbb{P}^{n-3} \times \mathbb{P}^k$ has a globally generated tangent bundle, Theorem 2.6 (based on Fulton-MacPherson intersection theory) guarantees that all intersection multiplicities in the product space are non-negative.
   • Conclusion: The degree of the intersection in the product space serves as an upper bound for the invariant, as the boundary contributions are non-negative and thus can only increase the total degree.

C. Explicit Calculation

The problem reduces to calculating the degree of a product of divisors in $\mathbb{P}^{n-3} \times \mathbb{P}^k$.

• The evaluation maps are analyzed explicitly using homogeneous coordinates.
• The pullback of a hyperplane class under the evaluation map is shown to be a divisor of a specific bidegree.
• The author derives a bound for the bidegree of the pullback divisor $[ev_i^{-1}(H)]$ in terms of the maximum contact orders $\alpha_i^{\max} = \max_j \alpha_{ij}$.

3. Key Contributions

A. The Main Theorem (Theorem 1.1)

The paper establishes the following upper bound for the invariant with insertions $d_i H^{\nu_i}$:
$$\langle d_1 H^{\nu_1}, \dots, d_n H^{\nu_n} \rangle_{\mathbb{P}^k|\partial \mathbb{P}^k}^{0, \alpha} \leq d_1 \cdots d_n \binom{n - 3 + k - \nu}{n - 3} \left( \sum_{i=1}^n \alpha_i^{\max} \right)^{n-3}$$
Where:

• $\alpha_i^{\max}$ is the maximum tangency order for the $i$-th marked point.
• $\nu$ is the maximal codimension of an insertion at a point with no tangency.
• The bound is polynomial in the contact orders with degree $n-3$.

B. Simplification of Moduli Space Geometry

The paper demonstrates that for $\mathbb{P}^k$, the complex logarithmic moduli space can be effectively replaced by a product of projective spaces for the purpose of bounding. This bypasses the need for virtual fundamental classes in the genus zero case, as the space is smooth and the obstruction theory is trivial.

C. Explicit Bidegree Estimates

The author provides a rigorous derivation of the bidegree of the evaluation pullbacks (Proposition 3.6), showing that the degree in the $\mathbb{P}^{n-3}$ factor is bounded by the sum of the maximum contact orders. This is a non-trivial algebraic step involving the balancing condition of the fan and the structure of the rational map defining the evaluation.

4. Results and Examples

The paper validates the bound against known results and provides concrete examples:

• Maximal Contact: For curves maximally tangent to the boundary of $\mathbb{P}^k$ passing through 2 points, the bound is $((k+1)d)^k$.
• Characteristic Numbers ($N_d$): For rational plane curves of degree $d$ through $3d-1$ points, the bound yields $N_d \leq \frac{(3d)^{6d-4}}{(d!)^3}$.
• Tropical Comparisons:
  • 3 Tangency Points: The bound is off only by a constant factor (e.g., for $d^2$ exact vs $9d^2$ bound).
  • 4 Tangency Points: The bound becomes looser as the number of marked points increases (e.g., exact $2d^2(d-1)$ vs bound $81d^4$).
  • The paper notes that the bound is sharpest when the number of marked points is small and degrades as $n$ increases, which is expected as the dimension of the moduli space grows.

5. Significance

• Generalizability: The method is not limited to projective space. The author argues it applies to any smooth toric variety and, more broadly, to moduli spaces birational to varieties with globally generated tangent bundles.
• Computational Utility: It provides a tool for estimating the growth rate of Gromov–Witten invariants without performing full recursive calculations or tropical enumeration, which becomes intractable for high degrees or many marked points.
• Conceptual Insight: The work highlights the power of positivity in algebraic geometry. By embedding the problem into a space where all intersections are positive, the author converts a difficult enumerative problem into a manageable intersection theory calculation on a product of projective spaces.
• Future Directions: The paper suggests this approach could yield explicit bounds for all smooth toric varieties and potentially products of projective spaces, offering a new pathway for bounding invariants in logarithmic geometry.

In summary, Dan Simms provides a novel, geometrically intuitive method to bound logarithmic Gromov–Witten invariants by leveraging the positivity of projective space intersections, resulting in a clean polynomial bound that depends on the contact orders and the number of marked points.