Let $\phi:S\rightarrow \mathbb{CP}^1$ be an elliptic fibration of a K3 surface. When is the Leray spectral sequence associated to the fibration $E_2$-degenerate? Are there any good criteria for the $E_2$-degeneration?
1 Answer
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It's always true with $\mathbb{Q}$ coefficients. It follows from a general result of Zucker, who proved Leray degenerates whenever you have a projective map to curve. But in this case, it's simpler to check it by hand. There is only one (two !) differentials to worry about
$$d_2:H^0(\mathbb{P}^1, R^1\phi_*\mathbb{Q})\to H^2(\mathbb{P}^1,\mathbb{Q})$$
But right side injects into $H^2(S)$, so this has to vanish, we also have
$$d_2:H^0(\mathbb{P}^1, R^2\phi_*\mathbb{Q})\to H^2(\mathbb{P}^1,R^1\phi_*\mathbb{Q})$$
You get vanishing of the second map if you observe that the edge map from $H^2(S)$ to the left side is also surjective (you can see this if you identify it with the sum of $H^2$ of the fibres).
answered May 18, 2014 at 17:04
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2$\begingroup$ There should be two differentials to worry about, no? $H^0(\mathbf P^1,R^2 \phi_\ast \mathbf Q) \to H^2(\mathbf P^1,R^1\phi_\ast \mathbf Q)$ and $H^0(\mathbf P^1, R^1 \phi_\ast \mathbf Q) \to H^2(\mathbf P^1,R^0\phi_\ast \mathbf Q)$. $\endgroup$ Commented May 18, 2014 at 17:31
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2$\begingroup$ Right Dan. I guess I didn't think about it carefully. I'll edit it when I get a chance. Anyway, it's true... $\endgroup$ Commented May 18, 2014 at 18:04
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$\begingroup$ Dear Donu, where does the proof use the $\mathbb Q$ coefficient? $\endgroup$ Commented Aug 24, 2023 at 19:39