Effective homology and periods of complex projective hypersurfaces

Technical Summary: Effective Homology and Periods of Complex Projective Hypersurfaces

Problem Statement
The paper addresses the computational challenge of determining the period matrices of smooth complex projective hypersurfaces $X \subset \mathbb{P}^n(\mathbb{C})$. A period matrix represents the pairing between singular homology $H_k(X)$ and algebraic De Rham cohomology $H^k_{DR}(X)$ via integration. While numerical computation of periods is a powerful tool for recovering algebraic invariants (such as the Néron–Severi group, Picard rank, and endomorphism rings) through integer relation detection, existing algorithms face significant hurdles. Specifically, there is a lack of efficient methods to compute an explicit basis for the singular homology of higher-dimensional varieties. Previous approaches, such as those by Sertöz (2019) for smooth hypersurfaces, rely on global deformations to Fermat hypersurfaces, resulting in Picard–Fuchs differential equations that are often computationally prohibitive in size and complexity. Furthermore, direct numerical integration over high-dimensional cycles suffers from exponential complexity with respect to precision.

Methodology
The authors propose a new algorithm that intertwines the computation of an explicit homology basis with the numerical evaluation of periods. The method is grounded in Picard–Lefschetz theory and proceeds by induction on the dimension of the hypersurface.

1. Lefschetz Fibration Setup: The algorithm considers a generic pencil of hyperplanes $\{H_t\}_{t \in \mathbb{P}^1}$ intersecting the hypersurface $X$. This induces a Lefschetz fibration $f: Y \to \mathbb{P}^1$, where $Y$ is the blowup of $X$ along the base locus. The fibers $X_t = X \cap H_t$ are hypersurfaces of dimension $n-1$.
2. Monodromy Action: The core of the homology reconstruction relies on computing the monodromy action of the fundamental group $\pi_1(\mathbb{P}^1 \setminus \Sigma)$ (where $\Sigma$ is the set of critical values) on the homology of a generic fiber $H_{n-1}(X_b)$. This action is computed using the Picard–Lefschetz formula, which relates the monodromy to vanishing cycles.
3. Duality and Differential Systems: Instead of computing monodromy directly on homology, the algorithm computes it via duality with De Rham cohomology. It utilizes the Gauss–Manin connection to derive a system of linear differential equations (Picard–Fuchs equations) satisfied by the periods of the fiber $X_b$.
4. Numerical Analytic Continuation: The algorithm employs numerical analytic continuation of differentially finite functions (using the ore_algebra package in SageMath) to compute the monodromy matrices with rigorous error bounds. This step allows for the reconstruction of the homology basis of $X$ from the thimbles (Lefschetz thimbles) associated with the critical points of the fibration.
5. Period Integration: To compute the periods of $X$, the algorithm integrates the varying periods of the fibers $X_t$ along paths in the base $\mathbb{P}^1$. By expressing the integrand as a rational multiple of solutions to the Gauss–Manin connection, the integration is performed using algorithms for differentially finite functions. This approach achieves quasilinear complexity with respect to the number of digits of precision, a significant improvement over the polynomial complexity of standard quadrature methods.
6. Lattice Reconstruction: For surfaces ($n=2$), the method extends to compute the intersection product on $H_2(X)$ and identify exceptional divisors arising from the blowup, allowing for the rigorous computation of the Néron–Severi lattice.

Key Contributions

• Explicit Homology Basis: The paper introduces the first algorithm capable of computing an explicit basis for the singular homology of smooth projective hypersurfaces in arbitrary dimensions, a prerequisite for period computation that was previously a major obstacle.
• Efficient Algorithm: The proposed method is more efficient than the global deformation approach (Sertöz, 2019) because the Picard–Fuchs equations associated with a pencil of hyperplane sections are generally much smaller than those associated with a global deformation to a Fermat hypersurface.
• Rigorous Error Bounds: The use of numerical analytic continuation provides rigorous error bounds for the computed periods, ensuring that the precision is sufficient to recover exact integer relations.
• SageMath Implementation: The authors provide a complete implementation in SageMath (package lefschetz-family), making these computations accessible.

Results and Experiments
The authors validate their algorithm through extensive experiments:

• Performance: On a standard laptop, the algorithm computes the periods of a smooth quartic surface (a K3 surface) with 300 decimal digits of precision in approximately one hour. This contrasts sharply with previous methods, which often failed to complete within reasonable timeframes or memory limits (e.g., exceeding 500 GB RAM) for similar tasks.
• Precision Scaling: The complexity is quasilinear with respect to precision; doubling the number of digits roughly doubles the computation time.
• Applications to Invariants:
  • Picard Rank: The algorithm successfully computed the generic Picard ranks of various families of quartic surfaces, verifying theoretical results by Bouyer (2018) and conjecturing new generic ranks for symmetric polynomial families.
  • Isomorphism Verification: The method was used to numerically verify that two specific smooth quartic surfaces, known to be isomorphic as abstract varieties but not Cremona equivalent, possess a Hodge isometry between their homology lattices.
  • New Examples: The authors provided explicit defining equations for smooth quartic K3 surfaces with Picard ranks 2, 3, and 5, filling gaps in existing literature.
  • Physics Application: The methodology was adapted to compute periods for a family of singular quartic surfaces arising from the "Tardigrade" Feynman graph, demonstrating the potential to handle non-smooth varieties and elliptic fibrations.

Significance and Claims
The paper claims that its primary contribution is the effective computation of periods for smooth projective hypersurfaces by solving the "explicit homology" problem. The authors emphasize that their method is intrinsic, relying on the geometry of the hypersurface itself rather than a specific deformation to a known case. This intrinsic nature suggests the methodology has the potential to be extended beyond smooth projective hypersurfaces to other varieties, such as the singular K3 surface example provided.

The authors maintain a modest tone regarding the scope, noting that while the current implementation targets smooth hypersurfaces, the underlying principles (Picard–Lefschetz theory and effective homology reconstruction) are robust enough to handle more complex cases, such as nodal surfaces, with ad hoc adjustments. The work bridges the gap between theoretical algebraic geometry and experimental mathematics, enabling the high-precision numerical investigation of algebraic invariants that were previously inaccessible.