Suppose $Y$ is a smooth projective variety of dimension $2p-1$ over $\mathbb{C}$. I have a few questions about the $p^{~\text{ th}}$ intermediate Jacobian $J^p(Y)$ of $Y$.
Does it come from (i.e. is analytification of) an abelian variety? The natural map $H^{2p-1}(Y,\mathbb{R})\rightarrow\frac{H^{2p-1}(Y, \mathbb{C})}{F^pH^{2p-1}(Y,\mathbb{C})}~$ is an isomorphism of real vector spaces, so I tried to check if the Pioncare duality pairing $H^{2p-1}(Y,\mathbb{R})\otimes H^{2p-1}(Y,\mathbb{R})\rightarrow\mathbb{R}$ induces a Riemann form on $\frac{H^{2p-1}(Y, \mathbb{C})}{F^pH^{2p-1}(Y,\mathbb{C})}~$, but I'm not sure if it satisfies the positive-definiteness (of the corresponding Hermitian form) requirement.
Let $X$ be a smooth projective curve over $\mathbb{C}$. Does it matter at all for the situation in 1 if we assume $Y=X^{2p-1}$?
What I'm really interested in is $J^p\left(H^1(X)^{\otimes 2p-1}\right)\subset J^p(X^{2p-1})$, where the first $J^p$ is in the sense of Carlson. (For a mixed Hodge structure $H$, $J^p(H)$ is by definition $\frac{H_{\mathbb{C}}}{F^pH_{\mathbb{C}}+H_{\mathbb{Z}}}~$. I'm not sure if the definition was first given by Carlson, sorry if I should say in someone else's sense.) I wanted to know if $J^p\left(H^1(X)^{\otimes 2p-1}\right)$ comes from an abelian variety, and the reason I thought about 1 and 2 is that I was (maybe naively) hoping to get a Riemann form from the Poincare pairing on $H^{2p-1}(X^{2p-1},\mathbb{R})$. Any thought on whether or not $J^p\left(H^1(X)^{\otimes 2p-1}\right)$ comes from an abelian variety? Also, does this object naturally appear anywhere? Any thing interesting about it?