Suppose that $X$ is an algebraic variety with divisor $D$ such that the logarithmic tangent bundle $\mathcal{T}_X(-\log D)$ is locally free. Suppose moreover that $\iota\colon Y\to X$ is a regular embedding of varieties with $Y$ smooth so that the normal bundle $\mathcal{N}_{Y/X}$ is also locally free and the natural map $\iota^\ast\mathcal{T}_X\to \mathcal{N}_{Y/X}$ is surjective.

Question: Are there natural geometric conditions on $D$ and $Y$ that guarantee that the restriction of the map $\iota^\ast \mathcal{T}_X\to \mathcal{N}_{Y/X}$ to a map $\iota^\ast\mathcal{T}_X(-\log D)\to \mathcal{N}_{Y/X}$ remains surjective? Moreover if the answer is yes, does it appear in the literature?

I suspect that the answer is along the lines that the map is surjective when $D$ and $Y$ intersect transversely for some suitable notion of transverse intersection in this context but I'd be grateful for something precise.

My motivation comes from the analogous question where 'smooth algebraic variety' is replaced in my first sentence by 'smooth rigid analytic variety' but I'm guessing I'm more likely to get an answer as stated and that it will translate easily. I don't mind assuming that $X$ is smooth as well as $Y$.

Suppose that $X$ is an algebraic variety with divisor $D$ such that the logarithmic tangent bundle $\mathcal{T}_X(-\log D)$ is locally free. Suppose moreover that $\iota\colon Y\to X$ is a regular embedding of varieties with $Y$ smooth so that the normal bundle $\mathcal{N}_{Y/X}$ is also locally free and the natural map $\iota^\ast\mathcal{T}_X\to \mathcal{N}_{Y/X}$ is surjective.

Question: Are there natural geometric conditions on $D$ and $Y$ that guarantee that the restriction of the map $\iota^\ast \mathcal{T}_X\to \mathcal{N}_{Y/X}$ to a map $\iota^\ast\mathcal{T}_X(-\log D)\to \mathcal{N}_{Y/X}$ remains surjective? Moreover if the answer is yes, does it appear in the literature?

I suspect that the answer is along the lines that the map is surjective when $D$ and $Y$ intersect transversely for some suitable notion of transverse intersection in this context but I'd be grateful for something precise.

My motivation comes from the analogous question where 'algebraic variety' is replaced in my first sentence by 'rigid analytic variety' but I'm guessing I'm more likely to get an answer as stated and that it will translate easily. I don't mind assuming that $X$ is smooth as well as $Y$.

Suppose that $X$ is a smooth algebraic variety with free divisor $D$ so that the logarithmic tangent bundle $\mathcal{T}_X(-\log D)$ is locally free. Suppose moreover that $\iota\colon Y\to X$ is a regular embedding of varieties so that the normal bundle $\mathcal{N}_{Y/X}$ is also locally free and the natural map $\mathcal{T}_X\to \mathcal{N}_{Y/X}$ is surjective.

Question: Are there natural geometric conditions on $D$ and $Y$ that guarantee that the restriction of the map $\mathcal{T}_X\to \mathcal{N}_{Y/X}$ to a map $\mathcal{T}_X(-\log D)\to \mathcal{N}_{Y/X}$ remains surjective? Moreover if the answer is yes, does it appear in the literature?

I suspect that the answer is along the lines that the map is surjective when $D$ and $Y$ intersect transversely for some suitable notion of transverse intersection in this context but I'd be grateful for something precise.

My motivation comes from the analogous question where 'smooth algebraic variety' is replaced in my first sentence by 'smooth rigid analytic variety' but I'm guessing I'm more likely to get an answer as stated and that it will translate easily.

Suppose that $X$ is an algebraic variety with divisor $D$ such that the logarithmic tangent bundle $\mathcal{T}_X(-\log D)$ is locally free. Suppose moreover that $\iota\colon Y\to X$ is a regular embedding of varieties with $Y$ smooth so that the normal bundle $\mathcal{N}_{Y/X}$ is also locally free and the natural map $\iota^\ast\mathcal{T}_X\to \mathcal{N}_{Y/X}$ is surjective.

Question: Are there natural geometric conditions on $D$ and $Y$ that guarantee that the restriction of the map $\iota^\ast \mathcal{T}_X\to \mathcal{N}_{Y/X}$ to a map $\iota^\ast\mathcal{T}_X(-\log D)\to \mathcal{N}_{Y/X}$ remains surjective? Moreover if the answer is yes, does it appear in the literature?

I suspect that the answer is along the lines that the map is surjective when $D$ and $Y$ intersect transversely for some suitable notion of transverse intersection in this context but I'd be grateful for something precise.

My motivation comes from the analogous question where 'smooth algebraic variety' is replaced in my first sentence by 'smooth rigid analytic variety' but I'm guessing I'm more likely to get an answer as stated and that it will translate easily. I don't mind assuming that $X$ is smooth as well as $Y$.