This question might be really easy (or stupid), but I have only vague (heard-about) knowledge of DG categories, so I don't know where to look for an answer.


Let $X$ be a smooth projective variety over a field $k$ (I am mostly interested in $k = \mathbb{C}$). Assume that I have a class $\alpha \in H^k(X,\mathcal{O}\_X)$, for some $0 \leq k \leq \dim X$. Then $\alpha$ gives a natural tansformation of triangulated functors:


$$ \alpha \otimes id : id\_X \rightarrow id\_X[k],$$


where $id\_X$ is the identity functor in the derived category $D^b(X)$ (seen as a triangulated category).


Let's now endow $D^b(X)$ with its DG enhancement. Is it possible to lift $\alpha \otimes id\_X$ to a natural transformation of DG functors:


$$\tilde{\alpha} : id\_X \rightarrow id\_X,$$


such that $\mathcal{H}^k(\tilde{\alpha}) = \alpha \otimes id$ and $\mathcal{H}^p(\tilde{\alpha}) = 0$ for all $p \neq k$?


Perhpas it is not possible in the general case, but are there conditions on $X$ which would guarantee that it is possible in some cases? 


Thanks a lot in advance!



DG natural transformation Serre functors
It is possible to lift $\alpha \otimes \mathrm{id}$ to a dg-enhancement, one way is the following. Just to fix notation, this natural transformation is induced from 
$$
\alpha \colon \mathcal{O}\_X \to \mathcal{O}\_X[k] \in \mathrm{Hom}(\mathcal{O}\_X,\mathcal{O}\_X[k]) = \mathrm{H}^k(X,\mathcal{O}\_X).
$$
by tensoring with $\mathrm{id} \colon A \to A$ for $A \in \mathcal{D}^b(X)$.


As a dg-enhancement, I choose $\mathrm{Inj}(X)$, the bounded complex of injective sheaves with coherent cohomology. Let $I^\bullet \in \mathrm{Inj}(X)$ be an injective resolution of $\mathcal{O}\_X$. Then $\alpha$ can be lifted to a morphism of complexes
$$
\tilde\alpha \colon I^\bullet \to I^\bullet[k].
$$
The next step should be again to tensor with $\mathrm{id}\colon J^\bullet \to J^\bullet$ for $J^\bullet \in \mathrm{Inj}(X)$ and obtain the desired natural transformation of $\mathrm{Inj}(X)$.


This is not possible directly, but can be done using enriched model category theory. As Olaf Schnürer worked out in details in [Six operations on dg enhancements of derived categories of sheaves](https://arxiv.org/abs/1507.08697), the whole six-functor-formalism can be set up for $\mathrm{Inj}(X)$, in particular there is a tensor product.


Actually, he establishes all these functors for the unbounded derived category of quasi-coherent sheaves $\mathcal{D}(X)$ using unbounded h-injective complexes as a dg-enhancement. So by passing to the unbounded categories, the variety $X$ does not even need to be smooth.


