Specializations of symplectic and van Geemen--Sarti involutions on K3 surfaces

1. Problem Statement

The paper investigates the relationship between a K3 surface $X$ and the desingularization $Y$ of its quotient by a symplectic automorphism $\sigma$ of finite order. Specifically, it focuses on:

• Symplectic Involutions (Order 2): The van Geemen–Sarti involution, induced by translation by a 2-torsion section on an elliptic fibration.
• Symplectic Automorphisms (Order 3): Induced by translation by a 3-torsion section.

The central question is: Under what conditions is the Néron–Severi group of the quotient surface isometric to that of the original surface ($NS(X) \simeq NS(Y)$)?

• It is known that for a generic K3 surface with a van Geemen–Sarti involution, $NS(X) \simeq NS(Y)$.
• However, for most specializations (subfamilies) of these surfaces, this isometry fails.
• The paper seeks to identify infinitely many codimension 2 subfamilies where this isometry is preserved, and to characterize the specific geometric and lattice-theoretic conditions under which $X \simeq Y$ (isomorphism) or $NS(X) \simeq NS(Y)$.

2. Methodology

The author employs a combination of lattice theory and algebraic geometry, specifically focusing on elliptic fibrations.

• Lattice Theoretic Framework:

  • Utilizes the standard K3 lattice $\Lambda_{K3} \simeq U^{\oplus 3} \oplus E_8^{\oplus 2}$.
  • Analyzes the action of the involution $\sigma^*$ on $H^2(X, \mathbb{Z})$. For symplectic involutions, this action swaps two copies of $E_8$ and acts as the identity on the orthogonal complement.
  • Uses the Nikulin lattice ($N$) generated by the exceptional curves of the resolution of $X/\sigma$.
  • Constructs specific embeddings of Néron–Severi groups ($NS$) and transcendental lattices ($T_X$) into $\Lambda_{K3}$.
  • Defines a rational map $\gamma$ (induced by the quotient map $\pi: X \to Y$) acting on the transcendental lattice $(T_X)_\mathbb{Q}$. The condition $NS(X) \simeq NS(Y)$ is linked to the invariance of the lattice under $\gamma$.

• Geometric Specializations:

  • Studies families of K3 surfaces with elliptic fibrations.
  • Analyzes specializations where singular fibers are "glued" together (e.g., two $I_2$ fibers merging into an $I_4$ fiber, or $I_1$ into $I_2$).
  • Tracks how these geometric changes affect the Mordell–Weil group, the configuration of singular fibers, and the Picard number ($\rho$).
  • Constructs a sequence of specializations from a generic surface ($\rho=10$) up to a rigid surface with maximal Picard number ($\rho=20$).

3. Key Contributions and Results

A. General Symplectic Involutions (Non-van Geemen–Sarti)

• Theorem 1.1: The author proves the existence of infinitely many 12-dimensional families of projective K3 surfaces admitting a symplectic involution (not necessarily of van Geemen–Sarti type) such that for the generic member, $NS(X) \simeq NS(Y)$.
• This demonstrates that the property $NS(X) \simeq NS(Y)$ is not exclusive to the generic van Geemen–Sarti family but occurs in a countable infinity of other families.

B. The van Geemen–Sarti Family (Order 2)

• Codimension 1 Specializations ($\rho=11$):

  • The paper classifies all codimension 1 subfamilies (Picard number 11).
  • Theorem 1.2: It is proven that for any K3 surface $X_{11}$ in these subfamilies, $NS(X_{11}) \not\simeq NS(Y_{11})$. The isometry is broken at codimension 1.
  • The paper explicitly describes the change in singular fibers (e.g., $9I_2 + 6I_1$ vs. $I_4 + 6I_2 + 8I_1$) and the Mordell–Weil lattice.

• Codimension 2 Specializations ($\rho=12$):

  • Theorem 1.3: There exist infinitely many codimension 2 subfamilies where the generic member $X_{12}$ satisfies $NS(X_{12}) \simeq NS(Y_{12})$.
  • These families are constructed by adding specific classes to the Néron–Severi group that preserve the lattice structure under the quotient map.

• Isomorphism and Complex Multiplication ($X \simeq Y$):

  • The paper identifies a 5-dimensional family (a subfamily of the $\rho=10$ family) where $X \simeq Y$.
  • These surfaces admit Complex Multiplication (CM) by $\mathbb{Q}(\sqrt{-2})$.
  • Theorem 1.4: A lattice-theoretic criterion is established: $NS(Z) \simeq NS(W)$ if the transcendental lattice $T_Z$ is invariant under the map $\gamma$ (the rational quotient map acting on the transcendental lattice).

• Explicit Construction of Specializations (Proposition 6.3):

  • The author constructs a sequence of specializations $X_{12}, X_{14}, X_{16}, X_{18}, X_{20}$ by iteratively gluing fibers.
  • At each step, the map $\gamma$ on the transcendental lattice is explicitly described as a block matrix (e.g., $\begin{pmatrix} 0 & 1 \\ 2 & 0 \end{pmatrix}$).
  • The final specialization $X_{20}$ has $\rho=20$, singular fibers $I_8 + 3I_4$, and $X_{20} \simeq Y_{20}$.

C. Order 3 Symplectic Automorphisms

• The paper extends the analysis to K3 surfaces with a 3-torsion section.
• Theorems 7.4, 7.5, 7.7: Analogous results are proven for order 3 automorphisms.
  • Codimension 1 subfamilies generally break the isometry $NS(X) \simeq NS(Y)$.
  • Codimension 2 subfamilies exist where the isometry is preserved.
  • A 5-dimensional family (analogous to the order 2 case) exists where $X \simeq Y$, involving surfaces with specific singular fiber configurations ($I_{12} + 2I_3 + I_4 + 2I_1$).
  • The map $\gamma$ for order 3 acts as $\begin{pmatrix} 0 & 1 \\ 3 & 0 \end{pmatrix}$ on specific lattice blocks.

4. Significance

• Classification of Isometries: The paper provides a complete classification of when the isometry $NS(X) \simeq NS(Y)$ holds for symplectic involutions, moving beyond the generic case to specific, infinite families of specializations.
• Lattice-Theoretic Criterion: It establishes a powerful, purely lattice-theoretic criterion (invariance under $\gamma$) to determine if a specialization preserves the isometry of Néron–Severi groups, bypassing the need for explicit geometric construction in every case.
• Connection to Complex Multiplication: It links the geometric property $X \simeq Y$ to the arithmetic property of Complex Multiplication, identifying specific CM fields ($\mathbb{Q}(\sqrt{-2})$ for order 2, and analogous structures for order 3).
• Geometric Insight: By explicitly tracking the "gluing" of singular fibers (e.g., $I_2 \to I_4$, $I_4 \to I_8$), the paper offers a concrete geometric understanding of how the quotient map transforms the elliptic fibration structure while preserving the lattice isometry.
• Generalization: The methodology successfully generalizes from order 2 to order 3 automorphisms, suggesting a broader framework for studying symplectic automorphisms of higher orders on K3 surfaces.

In summary, Garbagnati's work bridges the gap between the generic behavior of K3 surfaces and their highly specialized, rigid counterparts, providing a rigorous framework to predict when the quotient operation preserves the algebraic structure of the surface.