Automorphism Groups of K3 surfaces

A complex K3 surface is a compact complex surface $X$ with vanishing irregularity $h^1(X, \mathcal{O}_X)=0$ and trivial canonical bundle $\mathcal{O}_X\cong \omega_X$.

Much of the theory of (complex) K3 surfaces is governed by its Hodge structure and the $\mathbb{Z}$-lattices $NS(X) \subseteq H^2(X, \mathbb{Z})$.

See [Huy16] for the theory of K3 surfaces.

Automorphisms

K3_surface_automorphism_groupMethod
K3_surface_automorphism_group(S::ZZLat [, ample_class]) -> generators, rational curves, chambers

Compute the automorphism group of a very-general $S$-polarized K3 surface.

Further return representatives of the $\mathrm{Aut}(X)$-orbits of (-2)-curves on X and a fundamental domain for the action of $\mathrm{Aut}(X)$ on the set of nef L|S chambers. This is almost a fundamental domain for $\mathrm{Aut}(X)$ on the nef cone.

Here very general means that $Num(X)$ is isomorphic to S and the image of $\mathrm{Aut}(X) \to H^0(X,\Omega^2_X)$ is $ \pm 1$.

The function returns generators for the image of

\[f\colon \mathrm{Aut}(X) \to O(Num(X))\]

The output is represented with respect to the basis of S.

Note that under our genericity assumptions the kernel of $f$ is of order at most $2$ and it is equal to $2$ if and only if $S$ is $2$-elementary. If an ample class is given, then the generators returned preserve it.

This kind of computation can be very expensive. To print progress information use set_verbose_level(:K3Auto, 2) or higher.

Input

  • S: a hyperbolic lattice
  • ample: a row matrix or a vector given with respect to the ambient space of S.
borcherds_methodFunction
borcherds_method(S::ZZLat, n::Integer; compute_OR=true, entropy_abort=false, max_nchambers=-1)
borcherds_method(L::ZZLat, S::ZZLat, w::QQMatrix; compute_OR=true, entropy_abort=false, max_nchambers=-1)

Compute the symmetry group of a Weyl chamber up to finite index.

Arguments

  • w: initial Weyl row vector represented with respect to the basis of L;
  • L: even, unimodular, hyperbolic lattice of rank n=10,18 or 26;
  • S: a primitive sublattice of L;
  • compute_OR=true: if false take as G all isometries of S extending to L;
  • max_nchambers: break the computation after max_nchambers are found;
  • entropy_abort abort if an automorphism of positive entropy is found.
K3ChamberType
K3Chamber

The $L|S$ chamber induced from a Weyl vector in L.

Let $L$ be an even, unimodular and hyperbolic lattice of rank $10$, $18$ or $26$ and $S$ be a primitive sublattice. Any Weyl vector $w$ of $L$ defines a Weyl chamber $C(w)$ in the positive cone of $L$. The Weyl chamber is a rational locally polyhedral cone with infinitely many facets, i.e. walls. It is the intersection of the positive half-spaces defined by $\Delta_L(w) = \{r \in L | r^2=-2, r.w = 1\}$. We have

\[C(w)=\{x \in \mathcal{P}_L | \forall r \in \Delta_L(w): x.r \geq 0\}\]

The Weyl chamber is a fundamental domain for the action of the Weyl group on the positive cone. We say that $S \otimes \mathbb{R} \cap C(w)$ is the $L|S$-chamber induced by $w$.

Note that two Weyl vectors induce the same chamber if and only if their orthogonal projections to $S$ coincide.

chamberFunction
chamber(data::BorcherdsCtx, weyl_vector::ZZMatrix, [parent_wall::ZZMatrix, walls::Vector{ZZMatrix}])

Return the $L|S$-chamber with the given Weyl vector.

The lattices $L$ and $S$ are stored in data. Via the parent walls we can obtain a spanning tree of the chamber graph.

weyl_vectorMethod
weyl_vector(D::K3Chamber) -> ZZMatrix

Return the Weyl vector defining this chamber.

wallsMethod
walls(D::K3Chamber) -> Vector{ZZMatrix}

Return the walls of the chamber D, i.e. its facets.

The corresponding half space of the wall defined by v in walls(D) is

\[\{x \in S \otimes \mathbb{R} | \langle x,v \rangle \geq 0\}.\]

v is given with respect to the basis of S and is primitive in S.

Note that [Shi15] follows a different convention and takes v primitive in S^\vee.

inner_pointMethod
inner_point(L::ZZLat, S::ZZLat, w::QQMatrix)
inner_point(C::K3Chamber)

Return a reasonably small integer inner point of the given L|S chamber.

raysMethod
rays(D::K3Chamber)

Return the rays of the chamber D.

They are represented as primitive row vectors with respect to the basis of S.

autMethod
aut(E::K3Chamber) -> Vector{ZZMatrix}

Return the stabilizer $\mathrm{Aut}_G(E)$ of $E$ in $G$.

The elements are represented with respect to the basis of $S$.

homMethod
hom(D::K3Chamber, E::K3Chamber) -> Vector{ZZMatrix}

Return the set $\mathrm{Hom}_G(D, E)$ of elements of $G$ mapping D to E.

The elements are represented with respect to the basis of $S$.

adjacent_chamberMethod
adjacent_chamber(D::K3Chamber, v::ZZMatrix) -> K3Chamber

Return return the $L|S$ chamber adjacent to D via the wall defined by v.

separating_hyperplanesMethod
separating_hyperplanes(S::ZZLat, v::QQMatrix, h::QQMatrix, d)

Return $\{x \in S | x^2=d, x.v>0, x.h<0\}$.

Arguments

  • S: a hyperbolic lattice
  • d: a negative integer
  • v,h: vectors of positive square
has_zero_entropyFunction
has_zero_entropy(S::ZZLat; rank_unimod=26) ->

Compute if the symmetry group of a Weyl chamber is elliptic, parabolic or hyperbolic.

Output

  • 1 - elliptic – the symmetry group is finite
  • 0 - parabolic – there is a unique cusp with infinite stabilizer
  • -1 - hyperbolic – positive entropy

This calls borcherds_method and breaks the computation as soon as a symmetry of a Weyl chamber with positive entrop is found.

Contact

Please direct questions about this part of OSCAR to the following people:

You can ask questions in the OSCAR Slack.

Alternatively, you can raise an issue on github.