# Fibrations in complex geometry

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?

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.
• It is rather elementary. The cuspidal cubic has affine equation $y^2=x^3$, and the projection $(x, \, y) \mapsto y$ induces a homeomorphism onto $\mathbb{P}^1$, which is topologically $S^2$. Regarding the nodal cubic, in can be seen as $\mathbb{P}^1$ with two points identified. Hence, topologically, we are identifying two points on $S^2$; this is clearly the same thing as shrinking a cycle on a torus, and the corresponding quotient space is homotopically equivalent to a sphere with a circle attached at a point. Aug 14, 2020 at 21:24