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I am looking for a compact Calabi-Yau 3-fold which is fibered by both K3 surface and abelian surface (with possibly singular fibers). Are there any examples?

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You can find such an example (which gives you much more indeed) in the following paper:

Oguiso, Keiji. On algebraic fiber space structures on a Calabi-Yau 3-fold. With an appendix by Noboru Nakayama. Internat. J. Math. 4 (1993), no. 3, 439-465.

It is the content of Theorem 4.9 on page 452. It says that there exists a Calabi-Yau $3$-fold which admits a Type $II_0$ fibration and infinitely many different fibrations of each of the three types $I_+$, $I_0$ and $II_+$.

In his notation, roughly speaking, these fibration types are respectively:

  • $I_0$, fibration on $\mathbb P^1$, general fiber an abelian surface.
  • $I_+$, fibration on $\mathbb P^1$, general fiber a $K3$ surface.
  • $II_0$ and $II_+$, fibrations on a rational surface, general fiber an elliptic curve (they are distinguished by the intersection of the second Chern class of $X$ with the class of the divisor defining the fibration).

This $X$ is constructed as a certain blow-up of the quotient $\overline X$ of $Y=S\times G$ by a canonically defined involution, where $G$ is an elliptic curve and $S$ a $K3$ surface with an involution whose fixed points locus is a union of rational curves. I refer to the original paper for more details!

Finally, notice that for Oguiso $X$ is a Calabi-Yau $3$-fold in the most restrictive way, that is $K_X$ is trivial and $X$ is simply connected.

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I believe the following works (although it needs to be double-checked). For each of $i=1,2$, let $\nu_i:S_i \to P_i$ be the blowing up of a projective plane $P_i$, isomorphic to $\mathbb{P}^2$, at the 9 base points of a general pencil of plane cubics. Let $B_i$, isomorphic to $\mathbb{P}^1$, be the parameter space for this pencil of plane cubics, and let $\pi_i:S_i \to B_i$ be the corresponding elliptic fibration. Thus there is a projection $$\pi_1\times \pi_2: S_1 \times S_2 \to B_1 \times B_1,$$ whose geometric generic fiber is isomorphic to an Abelian surface (in fact $\pi_i$ has a section, so also $\pi_1\times \pi_2$ has a section). By my computation, the dualizing sheaf of $S_1\times S_2$ is the pullback $(\pi_1\times \pi_2)^*[\mathcal{O}_{B_1}(-1)\otimes \mathcal{O}_{B_2}(-1)]$, where $\mathcal{O}_{B_i}(+1)$ is the unique ample generator of the Picard group of $B_i$ (usual Serre twisting sheaf).

Now let $f:B_2\to B_1$ be a general choice of isomorphism. The graph is $g:B_2\to B_1\times B_2$. The normal bundle of the divisor $g(B_2)$ is $[\mathcal{O}_{B_1}(-1)\otimes \mathcal{O}_{B_2}(-1)]$. Thus, by adjunction, the inverse image $X$ of this divisor under $\pi_1\times \pi_2$ is a divisor in $S_1\times S_2$ that has trivial dualizing sheaf. By Bertini's theorem, for general choice of $f$, $X$ is smooth. Thus $X$ is a smooth, projective $3$fold with trivial dualizing sheaf. Also the fibration $\pi:X\to g(B_2)$ is an Abelian fibration over $B_2$ (I will identify $g(B_2)$ with $B_2$).

What about the K3 fibration? Let $x$ be any of the $9$ base points of the pencil of plane cubics on $S_1$. Let $\widetilde{P}_1 \to P_1$ be the blowing up along $x$. Linear projection away from $x$ defines a morphism $\rho':\widetilde{P}^1 \to \Lambda$, where $\Lambda$ is isomorphic to $\mathbb{P}^1$ and $\widetilde{P}^1$ is a $\mathbb{P}^1$-bundle over $\Lambda$. Since $S_1$ is the blowing up of $P_1$ along $9$ points that includes $x$, also $S_1$ is a blowing up of $\widetilde{P}_1$, $\mu:S_1\to \widetilde{P}_1$, at the transforms of the remaining $8$ points. Denote by $\rho:S_1 \to \Lambda$ the composition of $\mu$ and $\rho'$. The general fiber of $\rho$ is isomorphic to $\mathbb{P}^1$, but there are $8$ reducible fibers (isomorphic to a union of two copies of $\mathbb{P}^1$ intersecting at a single ordinary double point).

Consider the projection $\rho_1:S_1\to S_2 \to \Lambda$ that is the composition of the projection $\text{pr}_1:S_1\times S_2 \to S_1$ with $\rho$. Denote by $\rho_X:X\to \Lambda$ the restriction of $\rho_1$ to $X$. The claim is that a general fiber of $\rho_X$ is an elliptically fibered K3 surface.

The fiber $F$ of $\rho$ over a general point $t$ of $\Lambda$ is isomorphic to $\mathbb{P}^1$. Moreover, the projection $\pi_1|_F:F\to B_1$ is a degree $2$ cover of $B_1$ branched over $2$ general points. Thus the fiber $X_t$ of $\rho_X$ over $t$ is the fiber product of $\pi_1|_F:F\to B_1$ and $f\circ \pi_2:S_2\to B_1$. For general choice of $t$, $X_t$ is a smooth surface. Of course $S_2\to B_1$ has relative canonical bundle isomorphic to the pullback of $\mathcal{O}_{B_1}(+1)$. Thus $X_t \to F$ has relative canonical isomorphic to the pullback of $\mathcal{O}_{B_1}(+1)$, which is the same as the pullback of $\mathcal{O}_F(+2)$. Since $\omega_F$ is isomorphic to $\mathcal{O}_F(-2)$, this means that $X_t$ has trivial dualizing sheaf. So $X_t\to F$ is an elliptic fibration over $\mathbb{P}^1$ with trivial dualizing sheaf. Thus $X_t$ is an elliptic K3 surface. Thus $X\to \Lambda$ is a fibration over $\mathbb{P}^1$ by K3 surfaces.

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Thank you for the detailed answer, Jason. The construction is quite involved and I have to your post carefully. – BlakeA Aug 7 '13 at 19:51

Let me give a construction which is a bit simpler than the construction suggested by Jason.

Let $S$ be an elliptically fibered K3 with a section. Let $C$ be an elliptic curve. Then take just $X = S \times C$. Then the fibers of $X \to S \to P^1$ are products of two elliptic curves, while the fibers of $X \to C$ are K3 surfaces.

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Sasha, perhaps BlakeA wanted a "strict" Calabi-Yau, while yours is not: it is not simply connected! – diverietti Aug 11 '13 at 14:29
@diverietti: This is, of course, possible. – Sasha Aug 11 '13 at 14:55
:) indeed! I thus proposed a simply connected example! – diverietti Aug 11 '13 at 15:06

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