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Let $C$ be a smooth, projective curve (can assume to be rational) and $X:=C \times C$. Denote by $p:X \to C$ one of the two natural projections. Let $E$ be a vector bundle on $X$. Is it true that,

$$H^1(E)=H^1(p_*E) \oplus H^0(R^1p_*E)?$$

i.e., the corresponding Leray spectral sequence degrenerates at $E_2$?

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    $\begingroup$ Yes: $E^{pq}_2=0$ if $p>1$ or $q>1$ and the differential has degree $(2, -1)$. Right? $\endgroup$
    – Piotr Achinger
    Commented Oct 21, 2018 at 10:51
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    $\begingroup$ Two remarks: 1) That would be true for any morphism of any variety $X$ onto a curve $C$. 2) Note however that the splitting is not canonical. $\endgroup$
    – abx
    Commented Oct 21, 2018 at 12:49
  • $\begingroup$ @abx and Piotr: Thanks for the answer. $\endgroup$
    – Jana
    Commented Oct 21, 2018 at 13:46

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