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 $Y$.



When is a map from a logarithmic tangent bundle to a normal bundle surjective?
I guess my answer will be in the category of complex manifolds, therefore we definitely need to assume that $X$ is smooth. I am not that familiar with this language, so I apologize for any stupid mistakes coming from this. 


Assume that $D$ is simple normal crossing and that $Y$ is smooth. Denote by $X\_0 = X \setminus D$, $X\_1 = D \setminus \text{Sing}(D)$, and inductively $X\_{k+1}$ is the non-singular part of $\text{Sing}(\overline{X\_k})$. Then the sections of $\mathcal{T}\_X(- \log D)$ are those holomorphic vector fields on $X$, which are tangent to all strata $X\_k$. I think a sufficient criterion for surjectivity is:


**Y intersects all the strata $X\_k$ transversally (in the sense $T\_p Y + T\_p X\_k = T\_p X$ for all $p \in Y \cap X\_k$)**


Indeed, to check whether the morphism $i^\* \mathcal{T}\_X(- \log D) \to \mathcal{N}\_{Y/X}$ of locally free sheaves on $Y$ is surjective, it suffices to check in the fibres. At a point $p \in Y \cap X\_k$ the divisor $D$ has equation $z\_1 z\_2 \ldots z\_k =0$ in local coordinates and a local basis of $\mathcal{T}\_X(- \log D)$ is given by 
$$z\_1 \frac{\partial}{\partial z\_1}, \ldots, z\_k \frac{\partial}{\partial z\_k}, \frac{\partial}{\partial z\_{k+1}} \ldots, \frac{\partial}{\partial z\_{\dim X}}.$$ 
The assumption, that $Y$ intersects $X\_k$ transversally means that $T\_p Y$ together with $\frac{\partial}{\partial z\_{k+1}}|\_p \ldots, \frac{\partial}{\partial z\_{\dim X}}|\_p$ span the vector space $T\_p X$, thus the sections $\frac{\partial}{\partial z\_{k+1}} \ldots, \frac{\partial}{\partial z\_{\dim X}}$ restrict to generators of the fibre of $\mathcal{N}\_{Y/X}$ at $p$. 


If we drop the assumption that $Y$ is smooth, we might be able to phrase the transversality condition in terms of the matrix of derivatives of a regular sequence $x\_1, \ldots, x\_l$ defining $\iota$ along tangent directions to $X\_k$ having rank $l$. [For this see also the discussion in the comments.]


