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Consider a complex manifold $X$, and let $V$ be a holomorphic vector bundle on $X$ given by a Whitney sum of $n$ holomorphic line bundles $L_i$, i.e. let $V=\oplus_{i=1}^n L_i$. Further, if it is useful let us assume that this complex manifold has trivial canonical bundle.

The bundle cohomologies of the vector bundle $V$ are given by the sum of the bundle cohomologies of the line bundles $L_i$, so in particular for the cohomology dimensions we have, $$ h^i(V,X) = \sum_{i=1}^n h^i(L_i,X) \,.$$ My question is as follows. Consider now the tensor product of line bundles, e.g. $\tilde{L}=L_1 \otimes L_2$. I gather that there is no simple statement analogous to the above for the cohomologies of $\tilde{L}$ in terms of those of $L_1$ and $L_2$. However, is there anything useful that can be said in general in this case, or alternatively are there any special cases for which we can make is a neat statement? There are some similar questions on here, but I wonder if perhaps more can be said in this case of only tensor products of holomorphic line bundles on a complex manifold with trivial canonical bundle.

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I do not know of any results in the generality you are looking for. In very special cases, you do get a nice description:

Künneth formula (cf. MathOverflow). Suppose $X = Y \times_k Z$ is the product of two separated schemes over a field $k$, and let $\mathscr{F},\mathscr{G}$ be quasi-coherent sheaves on $Y$ and $Z$, respectively. Then, $$H^m\bigl(X,p_1^{-1}(\mathscr{F}) \otimes_{\mathcal{O}_X} p_2^{-1}(\mathscr{G})\bigr) \cong \bigoplus_{p+q=m} H^p(Y,\mathscr{F}) \otimes_k H^q(Z,\mathscr{G}).$$

Note that the requirements are very restrictive! I think one of the best results "in general" is the following result, which roughly says that the cohomology of a tensor product of tensor powers of line bundles is a polynomial in the powers on those line bundles:

The polynomial theorem of Snapper [Kleiman, §1]. Let $\mathscr{F}$ be a coherent sheaf on a complete scheme $X$ defined over an algebraically closed field $k$. Let $s = \dim \operatorname{Supp} \mathscr{F}$. Let $\mathscr{L}_1,\ldots,\mathscr{L}_t$ be $t$ line bundles on $X$. Then, the Euler–Poincaré characteristic $$\chi\bigl(\mathscr{F} \otimes \mathscr{L}_1^{\otimes n_1} \otimes \cdots \otimes \mathscr{L}_t^{\otimes n_t}\bigr) = \sum_{i=0}^{\dim X} (-1)^i \, h^i\bigl(X,\mathscr{F} \otimes \mathscr{L}_1^{\otimes n_1} \otimes \cdots \otimes \mathscr{L}_t^{\otimes n_t}\bigr)$$ is a numerical polynomial in $n_1,\ldots,n_t$ of total degree $s$.

The fact that the Euler–Poincaré characteristic is a polynomial can be useful with additional positivity assumptions on the $\mathscr{L}_i$; see [Kollár–Mori, §3.4; Lazarsfeld, Prop. 9.4.23], for example.

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