Non-algebraic flops

Question

I am looking at Exercise 84 on Kollár's Exercises in the birational geometry of algebraic varieties about non-algebraic flops:

Let $X \subset \mathbb{P}^4$ be a general smooth quintic hypersurface. It is know that for every $d \geq 1$, $X$ contains a smooth rational curve $\mathbb{P}^1 \cong C_{d} \subset X$ of degree $d$ with normal bundle $\mathcal{O}(-1)^{\oplus 2}$. Prove that the flop of $C_{d}$ exists if we work with compact complex manifolds. Denote the flop by $\phi_{d}: X \dashrightarrow X_{d}$ and let $H_d$ be the image of the hyperplane class. Compute the self intersection $(H_{d})^{3}$ and conclude that the $X_{d}$ are not homeomorphic to each other and not projective.

If I'm correct the intersection number $(H_{d})^{3} = 5 - 5d^3$ and the picard number $\rho(X_{d}) = 1$ so $X_{d}$ is not projective. However, I think the variety $X_{d}$ is constructed by first blowing up the curve $C_d$ with exceptional divisor $E \cong \mathbb{P}^1 \times \mathbb{P}^1$, and then contract the other negative extremal ray by the minimal model program. Doesn't this construction always give projective varieties? I feel like I messed up with something in this contradiction.

Any help is appreciated!

Answer 1

After reading some analytic theory on contractions, it turns out the point here is that the other fibre of the exceptional divisor $E$ is not extremal, so one cannot use the Minimal Model Program here. Instead one should use the following analytic contraction theorem of Nakano (Main Theorem, On the Inverse of Monoidal Transformation, Shigeo Nakano):

Let $\tilde{X}$ be a complex analytic manifold of complex dimension $n \geq 3$ and $S$ an analytic submanifold of $\tilde{X}$ of codimension 1. Suppose that $S$ has a structure of an analytic fibre bundle over an analytic manifold $M$ with a projective $(r-1)$-space as the standard fibre and that $r > 1$. Denote $L_a$ the fibre over $a \in M$ in the bundle $S \rightarrow M$. Then, in order that there exists an $n$-dimensional analytic manifold $X$ containing $M$ and a holomorphic map $\pi: \tilde{X} \rightarrow X$ in such a way that $(\tilde{X},\pi)$ is the monoidal transformation of $X$ with center $M$ and $S = \pi^{-1}(M)$, it is necessary and sufficient that the following conditions are satisfied:(1) for any $a \in M$, $\mathcal{O}(S)|_{L_a} \cong \mathcal{O}_{L_a}(-1)$; (2) each $L_a$ has a neighbourhood $V$ in $\tilde{X}$ such that $\mathcal{O}(K_V) = \mathcal{O}(S)^{k}$, where $k$ is a non-negative integer.

One can verify condition (1) by computing the normal bundle of $S$, and condition (2) from the fact that $X$ is smooth Calabi-Yau variety.

Answer 2

This is a very interesting question. I was also puzzled by this for a while. Here is a proof of why the fiber $F_2$ of the other contraction is not an extremal curve.

Let $\tilde{X}$ denote the blowup of $X$ at one rational curve $C$ of degree $d$. The exceptional divisor $E\cong \mathbb P^1\times \mathbb P^1$ has two rulings $F_1$ and $F_2$. $F_1$ is a fiber of $\tilde{X}\to X$ and $F_2$ is the fiber of the other contraction.

Since $X$ contains 2875 lines, one can take one of the lines $L$ disjoint from $C$. Now we have three curve classes $[L], [F_1]$ and $[F_2]$ for $\tilde{X}$. Since $H^4(X,\mathbb Z)$ has rank 1 (generated by $[L]$), $H^4(\tilde{X},\mathbb Z)$ has rank 2 according to the blowup formula for cohomology, so there has to be a relation between three classes.

We claim that $$[F_2]=d[L]+[F_1]. \label{1}\tag{1}$$ This shows $[F_2]$ is indeed not extremal in the cone of curves.

It suffices to show the intersection number with all the surface classes on the both sides of equation coincide. The Picard group $\text{Pic}(\tilde{X})$ is generated by $E$ and the strict transform $\tilde{H}$ of a general hyperplane section $H$. A direct computation shows:

$$\tilde{H}\cdot L= 1, ~E\cdot L= 0,$$ $$\tilde{H}\cdot F_1= 0, ~E\cdot F_1= -1,$$ $$\tilde{H}\cdot F_2= d, ~E\cdot F_2= -1.$$

Therefore the equation $\eqref{1}$ is established.

(p.s., I still can't see homologically why the relation $\eqref{1}$ holds, i.e., how to find a 3-chain bounding the three curves.)