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Let $X$ and $Y$ be smooth projective varieties. Let $\mathcal{P} \in \mathrm{D}^{b}(X \times Y)$ be Fourier-Mukai kernel. Define the support of $\mathcal{P}$ by $\mathrm{supp}(\mathcal{P}) = \bigcup_{i}\mathrm{supp}(\mathcal{H}^{i}(\mathcal{P})) \subset X \times Y$, union of the supports of all its cohomology sheaves.

I want to show that the fibres of the projection $\varphi:\mathrm{supp}(\mathcal{P}) \longrightarrow X$ are connected. I already proved that $\varphi$ is proper and surjective. Now assume for contradiction that there exists a point $x \in X$ over which the fibre is not connected.

I believe one can reduce to the case where $x$ is a closed point, but I am unsure why this reduction is justified.

Is it true that if a fibre over some (possibly non-closed) point is disconnected, then there exists a closed point with disconnected fibre? Is this a consequence of semicontinuity or Stein factorization?

I would appreciate clarification on this reduction step.

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If the fiber of $\mathrm{supp}(P)$ over a closed point $x$ is not connected then the image $F_x$ of $\mathcal{O}_x$ under the Fourier--Mukai functor has disconnected support, hence $$ F_x = \bigoplus F_i $$ (the sum with respect to the connected components of the support) and then $$ \mathrm{Hom}(F_x,F_x) = \bigoplus \mathrm{Hom}(F_i,F_i) $$ has dimension higher than 1. But it should be isomorphic to $\mathrm{Hom}(\mathcal{O}_x,\mathcal{O}_x)$, which is 1-dimensional.

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