# About a non-obvious (?) link between the jacobians of curves and differentials

To explain my problem, I must give a lemma:

Let $X$, $Y$, $Z$ be curves over $k$ (of characteristic 0) such that the genus of $Z$ is greater than 2, and $\pi : X \to Y$, $\phi : X \to Z$ two non-constant morphisms. If $\phi^\star(H^0(Z,\Omega))\subseteq\pi^\star(H^0(Y,\Omega))$, where $\Omega$ denotes the sheaf of regular 1-forms in each case, then there exists a non-constant morphism $u: Y \to Z$ such that $\phi = u \circ \pi$.

Now, in a proof, I saw the use of this lemma, except that the hypothesis was the inclusion $\mathrm{Image}(\mathrm{Jac}(Z) \to \mathrm{Jac}(X)) \subseteq \mathrm{Image}(\mathrm{Jac}(Y) \to \mathrm{Jac}(X))$, instead of $\phi^\star(H^0(Z,\Omega))\subseteq\pi^\star(H^0(Y,\Omega))$. I can guess it is equivalent, but why? Is it related to Grothendieck's duality? Did I miss something obvious?

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Since you are in char zero, you can assume the ground field is the complex numbers. The inclusion of jacobians follows from the inclusion of spaces of differentials via the description in terms of periods and calculus.

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I think the questioner wants to understand the deduction the other way, i.e. to go from the statement about Jacobians to the statement about differentials. –  Emerton Aug 30 '10 at 15:34
Perhaps writing the formulas $$Jac(X) = H^0(X,\Omega_X^1)^*/H_1(X,\mathbb{Z})$$ $$T^*_0Jac(X)= H^0(X,\Omega^1),$$ will give the questioner enough to work with. –  Donu Arapura Aug 30 '10 at 16:19
Then you take the derivative of the maps between the jacobians, again using the complex analytic description. $(\int_0^P \omega)' = \omega$ –  Felipe Voloch Aug 30 '10 at 16:21
What is the meaning of $T_0$? –  Bernikov Aug 31 '10 at 2:49
T_0 is the tangent space at zero. –  Felipe Voloch Aug 31 '10 at 2:54