##
**Connected components of the moduli of elliptic \(K3\) surfaces.**
*(English)*
Zbl 1420.14087

The paper under review forms a continuation of the author’s analysis of the combinatorial types of complex elliptic \(K3\) surfaces (understood with section). This was started in [I. Shimada, Mich. Math. J. 47, No. 3, 423–446 (2000; Zbl 1085.14509)] where the author developed a classification of all possible combinatorial types. i.e. configuration of reducible fibers (in terms of ADE-types) together with the torsion of the Mordell-Weil group. Presently the moduli of these elliptic \(K3\) surfaces are considered, and the main result computes the number of connected components of each moduli space.

The overall approach consists of the moduli theory of lattice polarized \(K3\) surfaces. Here the ADE-types of the reducible fibres give a natural sublattice \[ L = U\oplus R \] of the Néron-Severi lattice of the \(K3\) surface where the hyperbolic plane \(U\) is generated by the zero section and a fibre and the orthogonal summand \(R\) corresponds to the ADE-types (following Kodaira). Note that \(L\) need not embed primitively into the Néron-Severi lattice; in fact, the non-primitivity is exactly accounted for by the torsion sections. Thus the primitive closure \(L'\) is the generic Néron-Severi lattice on each moduli component (of dimension \(20-\operatorname{rank}(L)\)).

The author’s involved calculations are based on lattice theoretic methods pioneered by Miranda and Morrison in a mostly unpublished manuscript (available electronically). Using these techniques, the author manages to detect precisely which components can be algebraically distinguished, i.e. where the given torsion sections can be configured in different ways to meet the reducible fibres. The classification result of the paper goes as follows:

There are exactly 196 combinatorial types of complex elliptic \(K3\) surfaces with more than one connected moduli component.

Interesting enough, almost half of them (namely 89) comprise extremal elliptic \(K3\) surfaces (i.e. those with maximal Picard number \(\rho=20\), yet finite Mordell-Weil group). Here the results can often be explained through the transcendental lattice (as computed in [I. Shimada and D.-Q. Zhang, Nagoya Math. J. 161, 23–54 (2001; Zbl 1064.14503)]), or through the fields of definition. Among the 107 non-extremal types, 95 have algebraic distinguishable moduli components; for the remaining 12 types, the computations reveal that there are two complex conjugate moduli components.

The overall approach consists of the moduli theory of lattice polarized \(K3\) surfaces. Here the ADE-types of the reducible fibres give a natural sublattice \[ L = U\oplus R \] of the Néron-Severi lattice of the \(K3\) surface where the hyperbolic plane \(U\) is generated by the zero section and a fibre and the orthogonal summand \(R\) corresponds to the ADE-types (following Kodaira). Note that \(L\) need not embed primitively into the Néron-Severi lattice; in fact, the non-primitivity is exactly accounted for by the torsion sections. Thus the primitive closure \(L'\) is the generic Néron-Severi lattice on each moduli component (of dimension \(20-\operatorname{rank}(L)\)).

The author’s involved calculations are based on lattice theoretic methods pioneered by Miranda and Morrison in a mostly unpublished manuscript (available electronically). Using these techniques, the author manages to detect precisely which components can be algebraically distinguished, i.e. where the given torsion sections can be configured in different ways to meet the reducible fibres. The classification result of the paper goes as follows:

There are exactly 196 combinatorial types of complex elliptic \(K3\) surfaces with more than one connected moduli component.

Interesting enough, almost half of them (namely 89) comprise extremal elliptic \(K3\) surfaces (i.e. those with maximal Picard number \(\rho=20\), yet finite Mordell-Weil group). Here the results can often be explained through the transcendental lattice (as computed in [I. Shimada and D.-Q. Zhang, Nagoya Math. J. 161, 23–54 (2001; Zbl 1064.14503)]), or through the fields of definition. Among the 107 non-extremal types, 95 have algebraic distinguishable moduli components; for the remaining 12 types, the computations reveal that there are two complex conjugate moduli components.

Reviewer: Matthias Schütt (Hannover)

### MSC:

14J28 | \(K3\) surfaces and Enriques surfaces |

11E81 | Algebraic theory of quadratic forms; Witt groups and rings |

### Keywords:

elliptic \(K3\) surface; Néron-Severi lattice; lattice polarization; quadratic form; genus theory; combinatorial type### Software:

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