Let $X^n$ be a compact Kähler manifold with $K\_X$ semi-ample, i.e., a sufficiently high power of $K\_X$ is basepoint free. The associated pluricanonical system $| K\_X^{\ell} |$ furnishes a birational map $$f : X \dashrightarrow \mathbb{P}^{\dim H^0(X, K\_X^{\ell})-1}$$ onto some normal projective variety $Y \subset \mathbb{P}^{\dim H^0(X, K\_X^{\ell})-1}$ of dimension $\kappa(X)$. Here, $\kappa(X)$ denotes the *Kodaira dimension of $X$.* The semi-ampleness of $K\_X$ further implies that $K\_X^{\ell} \simeq f^{\ast} \mathcal{O}(1)$. In particular, for every $y \in Y$ that is not contained in the discriminant locus of $f$, $K\_X^{\ell} \vert\_{f^{-1}(y)} \simeq \mathcal{O}\_{X\_y}$. Since $f$ is a submersion near $f^{-1}(y)$, we have the adjunction-type relation $K\_{f^{-1}(y)} \simeq K\_X \vert\_{f^{-1}(y)}$ and therefore the fibres of $f$ are Calabi--Yau manifolds of dimension $n- \kappa(X)$.


In the Kähler geometry literature, it is common to refer to this map $f$ as a Calabi--Yau fibration. My question may be extremely obvious, but nevertheless:


**Question:** Is this a fibration in the sense of homotopy theory, i.e., does this map satisfy the homotopy lifting property?



Fibrations in complex geometry
I would say that the answer is in general *no*.


Think of an elliptic surface $X$ with Kodaira dimension $1$ and whose elliptic fibration contains a cuspidal curve. Then the general fibre is not homotopically equivalent to the special one (the former is homeomorphic to $S^1 \times S^1$, the latter to $S^1$, in particular their fundamental groups are different), whereas all the fibres of a Hurewicz fibration have the same homotopy type.


**Edit.** Actually, any elliptic surface $X$ with Kodaira dimension $1$ and whose elliptic fibration contains a nodal curve also provides a counterexample. In fact, a nodal cubic is homeomorphic to a torus "with one cycle shrunk away”; in particular, it has the homotopy type of $S^1 \vee S^2$ and the previous argument applies.


