When is the birational Torelli problem for CY threefolds true? I am aware from Borisov, Căldăraru, Perry and Ottem, Rennemo that what is known as the birational Torelli problem is false in general for Calabi-Yau threefolds, but I would like to know if there are further conditions one can impose for the statement to hold true for a large class of CY 3-folds.
I have tried to understand the counterexample in the first cited paper to see what goes wrong but I havent been able to pin down what exactly it is that doesn't go as expected. I initially thought that BCP's counterexample is so because for the varieties they consider, birational means isomorphism, but I'm not sure about this intuition.
The version of the problem that I'm interested on asks for derived equivalence instead of polarized Hodge isomorphic structures, the counterexample above fall under this version too. I write down the statement for convenience:

If $M_1$ and $M_{2}$ are smooth deformation equivalent complex Calabi-Yau threefolds such that $D^{b}(M_{1})\cong D^{b}(M_{2})$ then are $M_1$ and $M_2$ birational?

So, when does the above hold true?
I apologize if this is well documented in the literature, I have not been able to find a survey on this.
 A: I think the main idea in both papers you mention is that two birational Calabi-Yau threefolds with Picard number 1 are necessarily isomorphic. The use of this argument in such a situation is quite old and can at least be traced back to The Pfaffian-Grassmannian equivalence by Borisov and Caldararu. In the Ottem-Rennemo paper this classical argument takes the following form : if $X_g$ and $Y_g$ are birational, they must be isomorphic outside a subscheme of codimension at least 2 by the Minimal Model Program in dimension 3. However we have:
$$\mathrm{Pic}(X_g) \simeq \mathrm{Pic}(Y_g) \simeq \mathbb{Z}.H,$$
where $H$ is very ample. This implies that the birational morphism between $X_g$ and $Y_g$ sends $\mathcal{O}_{X_g}(H)$ to $\mathcal{O}_{Y_g}(H)$.
Now, computations of sections for line bundles on $X_g$ and $Y_g$ do not depend on a codimension $2$ subset (because they are both smooth, hence normal). As a consequence, we have a ring isomorphism:
$$ \bigoplus_{k \geq 0} H^0(X_g, \mathcal{O}_{X_g}(kH)) \simeq \bigoplus_{k \geq 0} H^0(Y_g, \mathcal{O}_{Y_g}(kH)).$$
Since $H$ is very ample on both sides, we can deduce that $Y_g$ and $X_g$ are isomorphic. Finally Ottem and Rennemo prove that for generic choice of $g \in G$, the varieties $X_g$ and $Y_g$ are not isomorphic.
This example suggests that in the Picard number 1 situation, you can never expect derived equivalent CY-3 to be birational, unless they are already isomorphic.
On the other hand, if $X$ and $Y$ are derived equivalent smooth projective varieties, a Theorem of Orlov warrants the existence of an object $\mathcal{E} \in \mathcal{D}^b(X \times Y)$ such that the equivalence is given by:
$$ \mathrm{R}p_* \left(\mathrm{L}q^*( \cdot) \otimes^{\mathrm{L}} \mathcal{E} \right) : \mathcal{D}^b(X) \longrightarrow \mathcal{D}^b(Y),$$
where $p : X \times Y \longrightarrow Y$ and $q : X \times Y \longrightarrow X$ are the natural projections. The cohomological morphism:
$$ p_{!} \left( q^{*}(\cdot) \cap \mathrm{ch}(\mathcal{E}) \cap \sqrt{\mathrm{Td}(X)} \right) : H^{\bullet}(X, \mathbb{Q}) \longrightarrow H^{\bullet}(Y, \mathbb{Q}) $$
behaves very nicely, and is perhaps even better, in some sense, than a birational correspondance.
