[Paper Review] Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties

Samuel Boissière, Marc Nieper-Wißkirchen+1 more|arXiv (Cornell University)|Jan 26, 2010

Nonlinear Waves and Solitons7 references4 citations

TL;DR

This paper introduces higher-dimensional Enriques varieties as quotients of irreducible holomorphic symplectic manifolds—specifically generalized Kummer varieties—by fixed point free automorphisms of finite order. It constructs explicit examples of Enriques varieties in dimensions 4 and 6 using automorphisms of order 3 and 4 on generalized Kummer varieties, and classifies natural automorphisms of these varieties induced by automorphisms of the underlying abelian surface.

ABSTRACT

We define Enriques varieties as a higher dimensional generalization of Enriques surfaces and construct examples by using fixed point free automorphisms on generalized Kummer varieties. We also classify all automorphisms of generalized Kummer varieties that come from an automorphism of the underlying abelian surface.

Motivation & Objective

• To generalize Enriques surfaces to higher dimensions by defining higher-dimensional Enriques varieties as quotients of irreducible holomorphic symplectic manifolds by fixed point free automorphisms.
• To classify automorphisms of generalized Kummer varieties that arise from automorphisms of the underlying abelian surface.
• To provide explicit constructions of Enriques varieties in dimensions 4 and 6 using automorphisms of order 3 and 4 on generalized Kummer varieties.
• To resolve a question posed by Arnaud Beauville regarding the existence of such varieties in higher dimensions.

Proposed method

• Define Enriques varieties as quotients of irreducible holomorphic symplectic manifolds by finite, fixed point free group actions that act purely non-symplectically on the symplectic form.
• Use generalized Kummer varieties $K_n(A)$ associated to an abelian surface $A = E \times E$, where $E$ is an elliptic curve with complex multiplication.
• Construct automorphisms $\psi = t_a \circ h$ on $A$, where $h$ is a diagonal matrix with roots of unity and $t_a$ is translation by a torsion point of order $n$, to define automorphisms on $K_n(A)$.
• Verify the absence of fixed points on $K_n(A)$ by analyzing the sum of orbit points and solving conditions on the first coordinate involving $a_1$ and roots of unity.
• Prove that the group $\langle \psi^{[n]} \rangle$ acts freely on $K_n(A)$ for $n=3,4$ by showing that the required linear conditions on $a_1$ have non-trivial solutions.
• Construct a counterexample in dimension 10 by taking a product $W \times V$ with $W = K_3(A)$ and a Calabi–Yau threefold $V$, using a free involution on $V$ to form a non-split Enriques variety.

Experimental results

Research questions

• RQ1 Can Enriques surfaces be generalized to higher dimensions via quotients of irreducible holomorphic symplectic manifolds by fixed point free automorphisms?
• RQ2 Which automorphisms of generalized Kummer varieties arise from automorphisms of the underlying abelian surface?
• RQ3 Do there exist Enriques varieties of dimension 4 and 6 constructed as quotients of generalized Kummer varieties by fixed point free automorphisms of order 3 and 4?
• RQ4 Can an Enriques variety be constructed that is not a quotient of an irreducible holomorphic symplectic manifold by a free group action?
• RQ5 What are the conditions under which an automorphism of a generalized Kummer variety acts freely on the Hilbert scheme of points?

Key findings

• The paper constructs explicit examples of Enriques varieties of dimension 4 and 6 as quotients of generalized Kummer varieties $K_3(A)$ and $K_4(A)$ by fixed point free automorphisms of order 3 and 4, respectively.
• For $n=3$, choosing $a_1 = 1/3$ ensures that the automorphism $\psi^{[3]}$ acts freely on $K_3(A)$, as the condition $(2 + \zeta_3)a_1 = 0$ is not satisfied in $E$.
• For $n=4$, choosing $a_1 = 1/4$ ensures that $\psi^{[4]}$ acts freely on $K_4(A)$, as the condition $2(1+i)a_1 = 0$ is not satisfied.
• For $n=6$, the automorphism $f^{[6]}$ constructed from a degree-3 automorphism on $K_6(A)$ has no fixed points if $a_1 = 1/3$, yielding a weak Enriques variety.
• A 10-dimensional Enriques variety is constructed as a product $X = W \times V$ with $W = K_3(A)$ and $V$ a Calabi–Yau threefold, where the free action of $f^{[3]} \times \iota$ prevents $Y$ from being a quotient of a single irreducible holomorphic symplectic manifold.
• The constructed Enriques variety in dimension 10 is reducible, isomorphic to $(W / \langle f^{[3]} \rangle) \times (V / \langle \iota \rangle)$, showing that not all Enriques varieties arise as quotients of irreducible holomorphic symplectic manifolds by free group actions.