[Paper Review] Non-homeomorphic conjugate complex varieties

Ichiro Shimada|ArXiv.org|Jan 4, 2007

Algebraic Geometry and Number Theory11 references14 citations

TL;DR

This paper presents a novel method to construct non-homeomorphic conjugate complex varieties using genus theory of lattices, particularly focusing on singular K3 surfaces and their transcendental lattices. The key contribution is the explicit construction of arithmetic Zariski pairs of maximizing sextic curves with simple singularities, where conjugate curves are not homeomorphic despite having isomorphic topological invariants under Galois conjugation.

ABSTRACT

We present a method to produce examples of non-homeomorphic conjugate complex varieties based on the genus theory of lattices. As an application, we give examples of arithmetic Zariski pairs.

Motivation & Objective

• To develop a systematic method for constructing non-homeomorphic conjugate complex varieties.
• To address the scarcity of explicit examples of conjugate varieties that are not homeomorphic.
• To apply arithmetic geometry of K3 surfaces and transcendental lattices to detect topological non-equivalence under Galois conjugation.
• To establish the existence of arithmetic Zariski pairs among maximizing sextic curves with simple singularities.

Proposed method

• Define the topological invariant $(B_U, eta_U)$ for the complement $U = X \setminus Y$ of a smooth projective variety $X$ and a union of codimension-$n$ subvarieties $Y$.
• Show that $(B_U, eta_U)$ is isomorphic to the torsion-free part of the orthogonal complement of the homology classes of $Y_i$ in $H_{2n}(X)$, denoted $\Lambda_{(X,Y)}$.
• Use the transcendental lattice $T[C]$ of a singular $K3$ surface $Y_C$ associated to a double cover branched over a sextic curve $C$.
• Relate the topological invariant of the complement $U_C = Y_C \setminus D_C$ to $T[C]$, so that non-isomorphic $T[C]$ implies non-homeomorphic $U_C$ and $U_{C^\sigma}$.
• Apply genus theory of binary quadratic forms to detect non-isomorphic lattices $T[C]$ and $T[C^\sigma]$ under Galois conjugation.
• Use computer-aided computation via Yang’s algorithm to verify that the genus of $T[C]$ contains more than one class for specific configurations, yielding arithmetic Zariski pairs.

Experimental results

Research questions

• RQ1 Can we construct explicit examples of conjugate complex varieties that are not homeomorphic using arithmetic lattice theory?
• RQ2 Under what conditions do Galois conjugates of a complex variety fail to be homeomorphic?
• RQ3 How can the topological invariant $B_U$ of the complement of a divisor in a $K3$ surface detect non-homeomorphism under conjugation?
• RQ4 Do arithmetic Zariski pairs exist among maximizing sextic curves with simple singularities?
• RQ5 Can genus theory of lattices be used to systematically generate such non-homeomorphic conjugate pairs?

Key findings

• The paper constructs explicit examples of non-homeomorphic conjugate complex varieties using genus theory of lattices.
• For 34 configurations of Dynkin types $R$, the paper exhibits arithmetic Zariski pairs $(C, C^\sigma)$ of maximizing sextics with simple singularities.
• The transcendental lattices $T[C]$ and $T[C^\sigma]$ are non-isomorphic for these pairs, as verified by genus theory and computer-aided computation.
• The lattices $T[C]$ and $T[C^\sigma]$ are presented as $L[2a,b,2c]$, with distinct invariants such as $L[6,2,8]$ vs. $L[2,0,22]$, confirming non-isomorphism.
• The existence of such pairs is guaranteed when the genus of $T[C]$ contains more than one isomorphism class, which holds for all 34 cases listed in Table 5.1.
• The method confirms that non-homeomorphic conjugate varieties arise precisely when the genus of the transcendental lattice is non-trivial, providing a classification mechanism.