Let $X\subset \mathbf{P}^n$ be a hypersurface such that the singular locus of $X$ consists of a single ordinary double point. I'm trying to find a reference to the "generalized" intermediate Jacobian of $X$.

When $n=2$, i.e. $X$ is a nodal curve, it is standard that the is a notion of generalized Jacobian of $X$, which is a $\mathbf{G}_m$-extension of the usual Jacobian of its normalization. See Ex II.6.9 in Hartshorne's book.

For higher dimensional cases, I have seen in several papers that experts use a notion of the generalized intermediate Jacobian of $X$, which is also a $\mathbf{G}_m$-extension of an abelian variety, without citing a reference.

So I'm wondering if there is a standard reference and definition for such generalized Jacobians. Thanks in advance.