Strong Tube Theorem

Let $X$ be a smooth hyperplane section of a $2n$-dimensional complex projective manifold, and let $X_{t_0}$ be the smooth hyperplane section over $t_0\in \mathbb{O}_X^{\operatorname{sm}}$. Let $\alpha\in H_{2n-2}^{\operatorname{van}}(X_{t_0},\mathbb{Z})$ be an elementary vanishing cycle, and let $G_\alpha$ be its stabilizer in $\pi_1(\mathbb{O}_X^{\operatorname{sm}},t_0)$.

Strong Tube Theorem. The image of the topological Abel--Jacobi homomorphism

$$\operatorname{a}_{X*}^{\operatorname{top}}:G_\alpha\to H_{2n-1}^{\operatorname{van}}(X,\mathbb{Z})$$

is $mH_{2n-1}^{\operatorname{van}}(X,\mathbb{Z})$ for some positive integer $m$.

This strengthens Schnell's tube theorem by requiring a single elementary vanishing cycle to generate the vanishing homology up to finite index. The paper presents it as an open conjecture.