[Paper Review] Examples of topological spaces with arbitrary cohomology jump loci

Botong Wang|arXiv (Cornell University)|Mar 31, 2013

Homotopy and Cohomology in Algebraic Topology3 references3 citations

TL;DR

This paper constructs finite CW complexes with arbitrarily prescribed cohomology jump loci in any given degree $k$, using a geometric construction that attaches cells to a real torus based on Laurent polynomials. The key result shows that for any subvariety $Z \subset (\mathbb{C}^*)^n$ defined over $\mathbb{Z}$ and any $k$, there exists a finite CW complex $M$ such that $\Sigma^k(M) = Z \cup \{\mathds{1}\}$ and all lower jump loci are trivial, demonstrating that quasi-projective varieties face genuine higher homotopy obstructions.

ABSTRACT

Given any subvariety of a complex torus defined over $\mathbb{Z}$ and any positive integer $k$, we construct a finite CW complex $X$ such that the $k$-th cohomology jump locus of $X$ is equal to the chosen subvariety, and the $i$-th cohomology jump loci of $X$ are trivial for $i

Motivation & Objective

• To realize any given subvariety of a complex torus as the $k$-th cohomology jump locus of a finite CW complex.
• To demonstrate that the cohomology jump loci of smooth complex quasi-projective varieties are constrained by higher homotopy type, not just topology.
• To provide a self-contained construction of spaces with prescribed characteristic varieties, extending known results in the literature.
• To show that such examples are not homotopy $k$-equivalent to any quasi-projective variety when $Z$ is not a union of torsion translates of subtori.

Proposed method

• Start with a real torus $M_0 = (S^1)^n$ and attach a $k$-sphere at a basepoint to form $M_1$, inducing a universal cover $N_1$ with $\mathbb{Z}^n$-action.
• For each Laurent polynomial $f_i$ defining the target subvariety $Z$, attach $(k+1)$-cells to $N_1$ such that their boundaries represent cycles in $H_k(N_1, \mathbb{Z})$ determined by the coefficients of $f_i$.
• Construct the total cover $N_2$ by attaching these cells compatibly across all $\mathbb{Z}^n$-translates, preserving the Galois action.
• Form the quotient $M$ by the $\mathbb{Z}^n$-action to obtain a finite CW complex with fundamental group $\pi_1(M) \cong \mathbb{Z}^n$, so $L(M) \cong (\mathbb{C}^*)^n$.
• Use Alexander modules over the group ring $\mathbb{Z}[x_1, x_1^{-1}, \dots, x_n, x_n^{-1}]$ to compute cohomology jump loci via the support of the module $\mathrm{Alex}(G_3) \cong \mathrm{Alex}(G_2) \otimes_R R/(f_1, \dots, f_r)$.
• Show that the support of this module is exactly $Z$, so $\Sigma^k(M) = Z \cup \{\mathds{1}\}$, and lower loci are trivial due to the $(k-1)$-connectivity of $N_1$.

Experimental results

Research questions

• RQ1 Can any subvariety $Z \subset (\mathbb{C}^*)^n$ defined over $\mathbb{Z}$ be realized as the $k$-th cohomology jump locus of a finite CW complex for any $k \geq 1$?
• RQ2 What are the homotopy-theoretic obstructions that prevent a space with prescribed cohomology jump loci from being homotopy equivalent to a quasi-projective variety?
• RQ3 How does the Alexander module structure over the Laurent polynomial ring encode the cohomology jump loci of the constructed space?
• RQ4 To what extent do the cohomology jump loci depend on the homotopy type of the space, particularly in degrees $k \geq 2$?
• RQ5 Can the construction be generalized to include finite abelian groups in the character group $L(M)$, beyond the complex torus case?

Key findings

• For any subvariety $Z \subset (\mathbb{C}^*)^n$ defined over $\mathbb{Z}$ and any $k \geq 1$, there exists a finite CW complex $M$ such that $\Sigma^k(M) = Z \cup \{\mathds{1}\}$ and $\Sigma^i(M)$ is trivial for $i < k$.
• The constructed space $M$ is homotopy $(k-1)$-equivalent to the real torus $(S^1)^n$, but not homotopy $k$-equivalent to any quasi-projective variety when $Z$ is not a union of torsion translates of subtori.
• The cohomology jump locus $\Sigma^k(M)$ is precisely the zero locus of the Laurent polynomials $f_1, \dots, f_r$ defining $Z$, as shown via the support of the Alexander module $\mathrm{Alex}(G_3)$.
• The Alexander module $\mathrm{Alex}(G_3)$ is isomorphic to $\mathrm{Alex}(G_2) \otimes_R R/(f_1, \dots, f_r)$, where $R = \mathbb{Z}[x_1, x_1^{-1}, \dots, x_n, x_n^{-1}]$, and its support is exactly $Z$.
• The construction generalizes to include finite abelian groups in $L(M)$ by replacing $M_0$ with $M_0 \times Y$, where $Y$ is a $K(A,1)$-space for a finite abelian group $A$, and adjusting the group ring accordingly.
• The result confirms that the cohomology jump loci of quasi-projective varieties are constrained by higher homotopy types, as the constructed examples satisfy the locus condition but fail to be $k$-equivalent to any quasi-projective variety.