Is there a notion of projective dg category?

Question

Since the paper Smooth and proper noncommutative schemes and gluing of DG categories by Orlov, dg categories are considered the non-commutative counterpart of algebraic geometry. More specifically, we call a dg category a non-commutative scheme if it is an admissible dg subcategory of the dg category $\mathfrak{Perf}(X)$ for a smooth projective scheme $X$. Now, many properties of a scheme $X$ defined over a field $k$ can be translated into properties of the category $\mathfrak{Perf}(X)$ (a dg enhancement of $Perf(X)$), e.g.

(1) the scheme $X$ is proper over $k$ if and only if for any $E,F \in \mathfrak{Perf}(X)$ we have (here I am identifying $E$ and $F$ with their image in the homotopy category of $\mathfrak{Perf}(X)$, which by definition is $Perf(X)$) $$ \sum_{n \in \mathbb{Z}} \text{dim} \, \text{Hom}_{Perf(X)}(E,F[n]) < +\infty$$

(2) the scheme $X$ is smooth over $k$ if and only if the diagonal bimodule associated to $\mathfrak{Perf}(X)$ is perfect in the derived category of $\mathfrak{Perf}(X)-\mathfrak{Perf}(X)$ bimodules.

From the above, one can then generalise these notions to that and smooth and proper dg categories. My question is whether there exists a similar analogy for the notion of projective scheme, and therefore the notion of a "projective dg category".

Thanks in advance.

Answer 1

If $X$ is a smooth projective threefold with a flopping curve $C$ then typically the variety $Y$ obtained from $X$ by a flop in $C$ is not projective, but smooth, proper, and derived equivalent to $X$. This shows that projectivity is not invariant under derived equivalence, hence does not correspond to a property of the derived category.