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I am trying to learn perverse sheaves. They are complexes $M$ of sheaves with constructible cohomology (say we are working with algebraic varieties and the transcendental topology) such that the following holds: $$ \dim \mathrm{supp} \mathcal{H}^i(M) \leq -i, \qquad \dim \mathrm{supp} \mathcal{H}^i(\mathbb{D}M) \leq -i, $$ where $\mathbb{D}$ denotes the Verdier dual.

Here are two questions I am having trouble with:

(1) From the first inequality, it is obvious that $\mathcal{H}^i(M)=0$ if $i>0$. Why does the second one imply that $\mathcal{H}^i(M)=0$ for $i < -\dim X$?

(2) If X is a smooth curve, taking (1) for granted, a perverse sheaf has only $\mathcal{H}^0$ and $\mathcal{H}^{-1}0$ and the first one is supported at points. Where does the condition $\mathcal{H}^{-1}$ has no nonzero global sections supported at points come from?

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