Valuatively independent bases for the Fermat family of cubic curves

1. Problem Statement and Context

The Core Problem:
The paper addresses a central problem in complex geometry and mirror symmetry: understanding the limiting behavior of Ricci-flat Calabi-Yau metrics on degenerating families of manifolds as the parameter $t \to 0$. Specifically, it focuses on the Metric SYZ Conjecture, which posits that near a large complex structure limit, the manifold admits a fibration by special Lagrangian tori over a base $B$.

The Obstacle:
Proving this conjecture generally requires solving real Monge-Ampère equations. A breakthrough program by Yang Li suggests that if one can solve a corresponding non-Archimedean Monge-Ampère equation, it implies the solution to the real equation. This connection relies on an optimal transport problem formulated on the product of the essential skeleton $S_k(X, \pi)$ and its mirror dual.

The Specific Challenge:
For this optimal transport framework to work, one needs the existence of valuatively independent bases for the spaces of global sections $H^0(X, L^k)$.

• Valuative Independence: A basis $\{\theta_i\}$ is valuatively independent if, for any point $n$ in the essential skeleton and any linear combination, the valuation of the sum is exactly the minimum of the valuations of the individual terms (no cancellation of leading terms occurs).
• The Gap: While locally valuatively independent bases are easy to construct, finding a basis that satisfies this property globally on the entire essential skeleton $S_k(X, \pi)$ is a difficult, previously unsolved problem for general families.

The Specific Case:
The authors tackle this problem explicitly for the Fermat family of cubic curves in $\mathbb{P}^2$, defined by the equation:
$$xyz - t(x^3 + y^3 + z^3) = 0$$
polarized by the restriction of $-K_{\mathbb{P}^2}$.

2. Methodology

The authors employ a combination of non-Archimedean geometry, tropical geometry, and explicit algebraic construction.

A. Geometric Setup

• Essential Skeleton ($S_k$): Identified with the boundary of the dual moment polytope $\partial \Delta^\vee$ of $\mathbb{P}^2$. This is an affine manifold (a circle) with a specific Hessian structure induced by the toric data of $\mathbb{P}^2$.
• Cost Function ($c$): They utilize a canonical cost function $c: S_k \times B \to \mathbb{R}$ derived from the Hessian structure and monodromy data. This function dictates the required vanishing orders of sections along the strata of the central fiber.
• Target Condition: The goal is to construct a basis $\{s_m\}$ indexed by lattice points $m$ such that:
  $$\ell^{-1} \text{val}_n(s_m) = c(m, n)$$
  where $\text{val}_n$ is the valuation at point $n$ on the skeleton.

B. Inductive Construction Strategy

The core of the paper is an explicit, inductive construction of the basis sections $s_m$ (denoted as $S(a,b)$ in the text) as polynomials in the coordinates $x_0, x_1, x_2$ and the parameter $t$.

1. Base Cases: The construction starts with simple monomials and low-degree combinations (e.g., $x_1, x_2, x_1x_2 - tx_0^2$).
2. Recursive Step: For higher degrees, the authors define a "raw" section $R_0(a,b)$ using a recurrence relation involving lower-degree sections.
   • Example: $R_0(a,b) = x_1 S(a-1, b) - t^2 x_0 S(a-2, b+1)$.
3. Correction Mechanism (The "Bad Terms"):
   • The raw section $R_0(a,b)$ generally contains "mixed terms" (monomials) that dominate the desired leading terms for certain regions of the skeleton, violating the valuative independence condition.
   • The authors identify a finite set of "bad" monomial exponents $\Lambda_{a,b}$ that cause these violations.
   • They perform a finite sequence of corrections. For each bad term, they subtract a carefully constructed multiple of a lower-degree section (scaled by powers of $t$) to cancel the unwanted leading term without introducing new dominant terms.
4. Convergence: The process terminates after a finite number of steps (determined by the cardinality of $\Lambda_{a,b}$), yielding the final section $S(a,b)$.

3. Key Contributions

1. Explicit Construction: The paper provides the first explicit construction of a globally valuatively independent basis for a non-trivial degenerating family (the Fermat cubic).
2. Algorithmic Proof: It establishes a rigorous algorithm to correct monomial sections to satisfy global valuation constraints, proving that such corrections are always possible and finite.
3. Verification of Cost Function Hypothesis: The constructed basis satisfies the condition $\ell^{-1} \text{val}_n(s_m) = c(m, n)$, confirming the conjectural picture proposed by Yang Li for this specific family.
4. Uniqueness of Valuations: While the basis elements themselves are not unique (they can be modified by lower-order terms), the paper proves that the valuation functions $\text{val}_n(s_m)$ are uniquely determined by the valuative independence property.

4. Main Results

Theorem 1 & Theorem 3:
For the Fermat family of cubic curves and any integer $k \ge 1$, the space of sections $H^0(X, L^k)$ admits a valuatively independent basis.

Specifically, for any lattice point $m$ on the boundary of the scaled moment polytope, there exists a section $s_m$ such that:

1. $s_m = \sigma_m + t s'_m$ (where $\sigma_m$ is the monomial section).
2. The set $\{s_m\}$ forms a basis for $H^0(X, L^k)$.
3. The valuation of any linear combination is the minimum of the valuations of the terms (no cancellation).
4. The valuation profile matches the canonical cost function: $\text{val}_n(s_m) = k \cdot c(m, n)$.

Proposition 2:
The paper proves that if a basis satisfies the cost function condition (matching the valuations to $c(m,n)$), it is automatically valuatively independent. This reduces the problem of proving independence to the problem of constructing sections with specific valuation profiles.

5. Significance

• Advancement of the SYZ Program: By solving the existence of valuatively independent bases for the Fermat family, the authors remove a major obstruction to applying the non-Archimedean Monge-Ampère approach to the Metric SYZ conjecture in this context.
• Bridge between Geometry and Optimal Transport: The work concretely realizes the link between the geometry of Calabi-Yau degenerations and optimal transport theory, showing how the "cost function" arising from the Hessian structure dictates the algebraic structure of the sections.
• Methodological Blueprint: The inductive correction method (identifying bad terms and subtracting lower-order corrections) provides a potential template for tackling similar problems in higher-dimensional Calabi-Yau varieties or other degenerating families, although the combinatorial complexity would increase significantly.
• Explicit Computability: Unlike many existence proofs in this field, this paper offers a concrete, computable procedure, making the theoretical framework accessible for further numerical or symbolic verification.

In summary, Hultgren and Khalid successfully bridge the gap between abstract non-Archimedean geometry and explicit algebraic construction, proving that the "canonical" bases predicted by mirror symmetry theory indeed exist for the Fermat cubic curves.