11.1: the proof is clumsy. Use $j_*j^*j_*\cal{F}=j_*\cal{O}\otimes j_*\cal{F}=j_*(j^*j_*\cal{O}\otimes F)$ and then apply (i) to prove the second equality in (ii). And for (iii), use $Hom(j_*\cal{O},j_*\cal{F})=j_*Hom(j^*j_*\cal{O},\cal{F})$ coupled with (i).
1. In Remark 10.8ii, it is mentioned that the corresponding fact for abelian varieties is deeper. What part of the argument of Cor 10.7 breaks down for abelian varieties? Is it integrality of Mukai vectors?
2. It's a bit odd that 10.3 says nothing about why M is a K3.
(It's derived equivalent to a K3, hence a K3.)
3. In the proof of 10.10, why does $\phi$ induce an isometry on $H^2$ if $\phi(0,0,1) = \pm(0,0,1)$?
4. Proposition 10.24 seems to say, in particular, that every (?) (-2) Mukai vector is the class of a spherical sheaf.
5. The proof of Lemma 10.6 went a bit quickly for me so it would be nice to talk briefly about it.
6. I found the first part of Section 10.3 to be a very readable explanation of coarse and fine moduli spaces. But I got lost in the second part, so I would like to have a discussion about it.
7. Why is the Kahler cone connected?
1. Trivial canonical bundle implies first Chern class is zero?
2. Can there be an isomorphism of K3 surfaces (as complex manifolds) which is not algebraic?
1. The digression on page 217 is pretty cool. I'd like to discuss it.
2.
1. Proof of 9.19: the paragraph at the top of page 202 is superfluous.
2. 9.20: as stated the closed point $a\in A$ plays no role. Of course one can reverse the roles of $A$ and $\hat A$, tenroing with line bundles on $\hat A$ and translating by points in $A$.
3. 9.26: Doesn't seem correct. The commutative square giving the fact that $f:B\to A$ is a homomorphism isn't cartesian, hence there is no reason to expect that the base-change map $f^*m_*(\cal{F}\boxtimes\cal{E})\to m_*(f^*\cal{F}\boxtimes f^*\cal{E})$ should be an isomorphism. I think it even fails for skypscraper sheaves.