Let's consider a smooth curve $C$ over $\mathbb{C}$. We know that the Jacobian variety $Jac(C)$ of $C$ is the moduli space of the degree $0$ line bundles on $C$. $Jac(C)$ is an abelian variety of dimension $=g$ and we have a very concrete construction of it, via the Abel-Jacobi map, etc.

Now consider $D(C)$, the derived category of bounded complex of coherent sheaves on $C$.

When the genus $g$ of $C$ is $0$, the Jacobian is a point. When $g=1$, the Jacobian is the elliptic curve itself. Now if we consider the case when $g>2$. In this case the canonical sheaf of $C$ is ample and by the reconstruction theorem of Bondal and Orlov (see A. Caldararu's notes http://arxiv.org/abs/math/0501094 Prop. 4.9, Thm 4.7), the curve $C$ is uniquely determined by the derived category $D(C)$. So we can expect that the Jacobian of $C$ can be also constructed from $D(C)$.

One thing I know so far is that there is a categorical description of the class of shifts of line bundles in $D(C)$, see also Caldararu's above notes Prop. 4.9.

$\textbf{My question}$ is: Could we construct the Jacobian variety $Jac(C)$ from the derived category $D(C)$, when the genus $g>2$? Maybe a related question is: do we have a categorical description of shifts of degree $0$ line bundles in $D(C)$?


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Of course, degree $0$ part cannot be described in terms of the category, because there are autoequivalences (twists) which do not preserve it. However, if you fix one object $E_0$ which is a line bundle up to a shift then you can consider those line bundles $E$ up to a shift such that $\chi(E_0,E) = 1 - g$ and then take a connected component of their moduli space.


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