Explicit equations of a fake projective plane (1802.06333v3)

Abstract: Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to those of the usual projective plane. They come in complex conjugate pairs and have been classified as quotients of the two-dimensional ball by explicitly written arithmetic subgroups. In this paper we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order seven automorphism.

• The paper presents a detailed derivation of 84 explicit cubic equations that define a fake projective plane with a non-abelian G21 automorphism group.
• It employs advanced computational algebraic geometry methods and software tools like Magma and Mathematica to rigorously validate the theoretical framework.
• The study enhances understanding of the algebraic and topological properties of complex surfaces, paving the way for further exploration in the field.

Explicit Equations of a Fake Projective Plane

This paper, authored by Lev A. Borisov and JongHae Keum, provides a detailed investigation into the explicit description of fake projective planes through the derivation of their equations. Fake projective planes are complex algebraic surfaces that, while exhibiting the same Betti numbers as the complex projective plane ($\mathbb{CP}^2$), are not isomorphic to it. This peculiar property makes them of significant interest in the paper of algebraic geometry and complex surfaces.

Summary

The authors focus on the complex surfaces, known as fake projective planes, which are characterized by distinct topological and geometric properties. These surfaces are distinguished by having Chern numbers $c_1^2 = 3c_2 = 9$, self-intersection of the canonical class $K^2 = 9$, a geometric genus $p_g = 0$, and odd behavior under the symmetry transformation due to their complex conjugate pairs. A specific subset, those that have a non-commutative automorphism group of order 21, is examined intensively.

Throughout the paper, the authors present a methodical approach to compute explicit polynomial equations that define a particular conjugate pair of fake projective planes equipped with an automorphism group isomorphic to the non-abelian group $G_{21}$. Using complex geometry and computational algebra techniques, they explore the implicit framework within which these planes reside as quotients of the two-dimensional complex ball by certain arithmetic subgroups.

Key Developments

1. Theoretical Framework: The authors build upon prior classification work that identifies fake projective planes as complex ball quotients. They carefully examine the configuration of these surfaces in terms of their algebraic and geometric properties.
2. Automorphism and Symmetry: The paper identifies the specific structure of the automorphism group, detailing its properties as being among $\{1\}$, $\mathbb{Z}_3$, $\mathbb{Z}_3^2$, and $G_{21}$, where $G_{21}$ is noted for having higher geometric symmetries.
3. Explicit Equations: The main achievement of this research is the derivation of 84 explicit cubic equations that define the discussed fake projective plane embedded in $\mathbb{P}^9$, the 9-dimensional complex projective space. This explicit construction is both an intricate computational task and a notable advancement in understanding these complex algebraic structures.
4. Mathematical and Computational Techniques: To reach their results, the authors employed sophisticated techniques in computational algebraic geometry, including the use of software packages like Magma and Mathematica, to verify their theoretical conclusions numerically.

Implications and Future Work

The implications of this work are manifold. The explicit nature of the derived equations allows for further exploration into the geometry of fake projective planes. Additionally, understanding the explicit mappings and symmetries of these surfaces could have ramifications for related fields such as topology and mathematical physics, where similar constructions play a role in string theory and other advanced frameworks.

The authors’ approach sets a precedent for future research that could explore other classes of algebraic surfaces or further explore the arithmetic and geometric properties underpinning fake projective planes. Future studies might focus on automorphism groups of even higher order or explore the application of these techniques to unresolved problems in algebraic surface classification.

In summary, this paper provides a definitive mathematical exploration into a class of algebraic surfaces, expanding the toolkit available to researchers engaged in the paper of algebraic geometry and complex surfaces. The explicit equations, alongside the analytical approach, contribute significantly to the field's understanding of complex algebraic structures with exotic properties.