Let $X$ be a smooth finite type variety over $\mathbb{C}$. Suppose that $\theta$ is a closed algebraic $1$-form whose cohomology class is weight $2$.
Can we always express $\theta$ as
$$\theta = \sum\_i c\_i \ d \log f\_i$$
where the $c\_i$ are constants and the $f\_i$ are nowhere vanishing algebraic functions?


As long as I'm at it, I'll ask for something more. Let $L(X)$ be the sub-$\mathbb{C}$-algebra of $H^{\ast}(X)$ consisting of forms of degree $p$ and weight $2p$. Is $L(X)$ always generated in degree $1$?  I just remembered that I know a counterexample to this. $\mathbb{P}^2 \setminus (\mbox{nodal cubic})$ has betti numbers $(1,0,1)$, and the $H^2$ is in weight $4$, but there is no $H^1$ and, for that matter, no units in the coordinate ring.


Motivation: In an [earlier question](https://mathoverflow.net/questions/130129/if-a-d-log-form-is-exact-is-it-zero), we showed that representations of cohomology classes as polynomials in $d \log$ forms are unique. This question is about which forms are of that type. It looks a little bit like a variant of the Hodge conjecture, but hopefully much easier!



Log forms and Tate classes
What is true is that the exponential map gives surjection
$$H^0(\mathcal{O}\_X^\*)\to Hom\_{MHS}(\mathbb{Z},H^1(X,\mathbb{Z}(1))$$
Classes on the right can be interpreted as weight two classes represented by $1$-forms logarthmic at infinity with integral periods; the map can be identified with $d\log$. I'm not sure if this exactly what you want, but it is at least close to it. The argument is not hard, but I'm too lazy to try to reproduce it here. So at the risk of self advertising, you can look at second page of my paper "Beilinson-Hodge cycles on semiabelian varieties" with Manish Kumar. Here's a link
<https://arxiv.org/abs/0808.2990>


