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This is always true, it follows from the Künneth formula for coherent sheaves (see Kunneth formula for sheaf cohomology of varietiesKunneth formula for sheaf cohomology of varieties).

This is always true, it follows from the Künneth formula for coherent sheaves (see Kunneth formula for sheaf cohomology of varieties).

This is always true, it follows from the Künneth formula for coherent sheaves (see Kunneth formula for sheaf cohomology of varieties).

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Angelo
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This is always true, it follows from the Künneth formula for coherent sheaves (see Kunneth formula for sheaf cohomology of varieties).