Holomorphic homotopy conjecture

Question

Let $X$ be a smooth projective variety over the complex numbers, and let $\text{Coh}(X)$ be the category of coherent sheaves on $X$. Consider the dg-category $\text{Perf}(X)$ of perfect complexes on $X$, which is a full dg-subcategory of the dg-category of complexes of objects in $\text{Coh}(X)$. Let $\text{H}_{\text{qe}}$ be the homotopy category of $\text{Perf}(X)$.

Let $Y$ be another smooth projective variety over the complex numbers, and similarly define $\text{Coh}(Y)$, $\text{Perf}(Y)$, and its homotopy category $\text{H}_{\text{qe}}(Y)$.

Suppose we have a fully faithful dg-functor $F: \text{Perf}(X) \to \text{Perf}(Y)$ that induces an equivalence of triangulated categories $\text{H}_{\text{qe}}(X) \to \text{H}_{\text{qe}}(Y)$.

Does there exist an algebraic morphism $f: Y \to X$, such that the derived pushforward functor $Rf_*: \text{Perf}(X) \to \text{Perf}(Y)$ is isomorphic to $F$ in the homotopy category of dg-functors between $\text{Perf}(X)$ and $\text{Perf}(Y)$? Furthermore, if such an $f$ exists, is it unique up to isomorphism?

Any relevant papers would be appreciated.

Answer 1

The answer to your first question is no. Indeed let $X$ and $Y$ be smooth projective varieties and let $F: \operatorname{Perf}_{dg}X \to \operatorname{Perf}_{dg}Y$ be a dg-functor inducing an equivalence on the homotopy category (=the triangulated category of perfect complexes). Then the induced functor (which I will also call $F$) is of Fourier-Mukai type. There is a priori no reason for this functor to be induced from an actual isomorphism $f:X \to Y$. For an example, there exist K3 surfaces $X$ and $Y$ which are derived equivalent but not isomorphic ($Y$ is a fine moduli space of sheaves on $X$) see the book of Huybrechts on K3 surfaces for a proof and further references.

If we make the further assumption that the functor $F$ sends the skyscraper sheaves of closed points on $X$ to shifts of the skyscraper sheaves of closed points on $Y$, then in fact $F$ is isomorphic to $Rf_*(- \otimes L)[k]$ where $L$ is a line bundle and $f:X \to Y$ is an isomorphism of varieties. In general both $L$ and $[k]$ are nontrivial so this functor is not isomorphic to $Rf_*$, however I am not sure if the isomorphism $f$ appearing above is unique.