MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 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 $D(C)$?

share|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.