On Vorontsov’s Theorem on K3 surfaces with non-symplectic group actions

AbstractIn this paper we describe six pencils of K3-surfaces which have large Picard number (ρ = 19, 20) and each contains precisely five special fibers: four have A-D-E singularities and one is non-reduced. In particular, we characterize these surfaces as cyclic coverings of some K3-surfaces described in a recent paper by Barth and the author. In many cases, using 3-divisible sets, resp., 2-divisible sets, of rational curves and lattice theory, we describe explicitly the Picard lattices.

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Differential Invariants of Linear Symplectic Actions

Symmetry ◽

10.3390/sym12122023 ◽

2020 ◽

Vol 12 (12) ◽

pp. 2023

Author(s):

Jørn Olav Jensen ◽

Boris Kruglikov

Keyword(s):

Symplectic Group ◽

Group Actions ◽

Equivalence Problem ◽

Differential Invariants ◽

Linear Spaces

We consider the equivalence problem for symplectic and conformal symplectic group actions on submanifolds and functions of symplectic and contact linear spaces. This is solved by computing differential invariants via the Lie-Tresse theorem.

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Families of abelian surfaces with real multiplication over Hilbert modular surfaces

Nagoya Mathematical Journal ◽

10.1017/s0027763000020080 ◽

1982 ◽

Vol 88 ◽

pp. 17-53 ◽

Cited By ~ 2

Author(s):

G. van der Geer ◽

K. Ueno

Keyword(s):

Endomorphism Ring ◽

Symplectic Group ◽

Abelian Surface ◽

Quadratic Relation ◽

Holomorphic Map ◽

Real Multiplication ◽

Compact Complex Surfaces ◽

Hilbert Modular Surfaces ◽

Hilbert Modular ◽

Abelian Functions

Around the beginning of this century G. Humbert ([9]) made a detailed study of the properties of compact complex surfaces which can be parametrized by singular abelian functions. A surface parametrized by singular abelian functions is the image under a holomorphic map of a singular abelian surface (i.e. an abelian surface whose endomorphism ring is larger than the ring of rational integers). Humbert showed that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and he constructed an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.

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Motivic integral of K3 surfaces over a non-archimedean field

Advances in Mathematics ◽

10.1016/j.aim.2011.07.015 ◽

2011 ◽

Vol 228 (5) ◽

pp. 2688-2730 ◽

Cited By ~ 3

Author(s):

Allen J. Stewart ◽

Vadim Vologodsky

Keyword(s):

K3 Surfaces

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Verlinde formulae on complex surfaces: K-theoretic invariants

Forum of Mathematics Sigma ◽

10.1017/fms.2020.50 ◽

2021 ◽

Vol 9 ◽

Author(s):

L. Göttsche ◽

M. Kool ◽

R. A. Williams

Keyword(s):

Moduli Space ◽

K3 Surfaces ◽

Complex Surfaces ◽

Type Formula ◽

Verlinde Formula ◽

Donaldson Invariants

Abstract We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between K-theoretic Donaldson invariants studied by Göttsche and Nakajima-Yoshioka and K-theoretic Vafa-Witten invariants introduced by Thomas and also studied by Göttsche and Kool. We verify our conjectures in many examples (for example, on K3 surfaces).

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Curves on K3 surfaces in divisibility 2

Forum of Mathematics Sigma ◽

10.1017/fms.2021.6 ◽

2021 ◽

Vol 9 ◽

Author(s):

Younghan Bae ◽

Tim-Henrik Buelles

Keyword(s):

Modular Forms ◽

K3 Surfaces ◽

Initial Condition ◽

Smooth Curves ◽

Holomorphic Anomaly ◽

Holomorphic Anomaly Equation ◽

Anomaly Equation ◽

Degeneration Formula ◽

The Relationship ◽

Level 2

Abstract We prove a conjecture of Maulik, Pandharipande and Thomas expressing the Gromov–Witten invariants of K3 surfaces for divisibility 2 curve classes in all genera in terms of weakly holomorphic quasi-modular forms of level 2. Then we establish the holomorphic anomaly equation in divisibility 2 in all genera. Our approach involves a refined boundary induction, relying on the top tautological group of the moduli space of smooth curves, together with a degeneration formula for the reduced virtual fundamental class with imprimitive curve classes. We use double ramification relations with target variety as a new tool to prove the initial condition. The relationship between the holomorphic anomaly equation for higher divisibility and the conjectural multiple cover formula of Oberdieck and Pandharipande is discussed in detail and illustrated with several examples.

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