Let $X$ be a smooth complex variety of dimension $3$ and $Y$ a (perhaps singular) normal complex variety also of dimension $3$ which is smooth outside a point $y \in Y$. If needed one may assume that both $X$ and $Y$ are projective.
Let $f:X \to Y$ be a proper holomorphic map satisfying the following properties:
- the map $f$ is a biholomorphism between $X-f^{-1}(y)$ and $Y-\{y\}$.
- the pre-image $f^{-1}(y)$ is (set theoretically) a simple normal crossing divisor.
Question 1. Are the conormal bundles of the irreducible components of $f^{-1}(y)$ pseudo-effective ?
Remark. This question is related to Contracting divisors to a point. Motivated by one of the answers there I posted, a few days ago, a previous version of this question asking if the conormal of the irreducible components are ample. Already in the case where $f^{-1}(y)$ is irreducible this does not seem to be true, contrary to what is claimed in one of the answers to the question linked above. The blow-up of contractible rational curves (existence due to Laufer) with conormal bundle $\mathcal O \oplus \mathcal O(2)$ provide counter-examples. An easier counter-example for my previous question (ampleness of conormal bundles of irreducible components of the exceptional divisor) can be obtained as the blow-up of a smooth point on $\mathbb C^3$ and then the blow-up of a line on the exceptional divisor. The last exceptional divisor, say $E$, contains a $(-1)$-curve $C$ such that $N^*_E \cdot C = -1$.
Still it seems reasonable to ask if the conormal bundle carries some form of positivity.
Question 2. Assuming that the answer to Question 1 is positive, can we say something about the nef part of the Zariski decomposition of $N^*_E$ for an irreducible component $E$ of $f^{-1}(y)$? Can we at least guarantee that it is non-trivial ?
Any references dealing with the subject are mostly welcome.
Added later. The answer to Question 1 seems to be positive. Replace $Y$ by a a sufficiently small neighborhood of $y$. Fix an irreducible component $E$ of the exceptional divisor and let $h$ be a non-constant holomorphic function on $Y$ vanishing at $y$. The pull-back of $h$ under $f$ is a holomorphic function vanishing at $E$, i.e. $f^*h \in H^0(X,\mathcal I)$ where $\mathcal I$ is the ideal sheaf defining $E$. If $k$ is the biggest integer such that $f^*h \in H^0(X,\mathcal I^k)$ then $f^*h$ induces a non-zero section of $\mathcal I^k/\mathcal I^{k+1}$ and therefore of ${N^*_E}^{\otimes k}$. It follows that $N^*_E$ is pseudo-effective.
Added even later. Aparently the conormal bundle of $E$ is a big line-bundle. This should be a consequence of Theorem 3.7 of Faisceaux ample sur les espaces analytiques by Vincenzo Ancona which says that there exists an (non necessarily reduced) ideal sheaf $\mathcal I\subset \mathcal O_X$ defining an analytic subspace $Z$ (analogue of scheme) with $Z_{red} = Y$ and with ample conormal sheaf $N^*_Z = \mathcal I/\mathcal I^2$. Notice that the conormal sheaf is not necessarily locally free and ampleness here means that for any coherent sheaf $\mathcal F$ on $Z$ we have surjectivity of the morphism $$ H^0(Z, \mathcal F \otimes Sym^n N^*_Z ) \otimes \mathcal O_Z \to \mathcal F \otimes Sym^n N^*_Z $$ when $n\gg 0$.
Along an open subset $U$ of $E$, $Z$ coincides with a thickening of $U$ (i.e. $Z_{|U} = kE_{|U}$ and the ampleness of $N^*_Z$ allows us to produce global sections of $Sym^n N^*_Z$ ($n\gg0$) which will separate points in $U$ and will induce sections of powers of ${N^*_E}^{\otimes nk}$ with the same property. This is enough to prove that $N^*_E$ is big.
I am still interested in references on the subject and alternative arguments to deal with questions 1 and 2.
