Let $X$ be a projective Fano 3-fold or 4-fold and let $D^b(X)$ be the bounded derived category of coherent sheaves on $X$. For what $X$ is it known a semi orthogonal decomposition into indecomposable subcategories of $D^b(X)$? I guess quite a lot of them...
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$\begingroup$ For any Fano variety in characteristic zero, $\mathcal{O}_{X}$ is exceptional and so $< \mathcal{O}_{X}, \mathcal{O}_{X}^{\perp}>$ forms a semi-orthogonal decomposition. The question then becomes whether $\mathcal{O}_{X}^{\perp}$ decomposes any further, and sometimes it does. $\endgroup$– Chris BravCommented Aug 29, 2012 at 14:26
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1$\begingroup$ So you might refine your question by asking not just for some semi-orthogonal decomposition, but a decomposition into indecomposable pieces. $\endgroup$– Chris BravCommented Aug 29, 2012 at 14:28
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$\begingroup$ Yes, you are right. I was assuming some non-triviality in order to exclude examples like yours. I will edit the question straight away. $\endgroup$– DannyBoyCommented Aug 29, 2012 at 14:35
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Idecomposability is not always easy to prove, but here is a list of Fano 3folds $X$ with $Pic(X) = {\mathbb{Z}}$ for which a decomposition with presumably indecomposable components is known:
1) index 4: $D(P^3) = < O,O(1),O(2),O(3)>$;
2) index 3: $D(Q^3) = < S,O,O(1),O(2)>$, $S$ is the spinor bundle;
3) index 2: one always have $D(X) = < B_X,O_X,O_X(1) >$; let $d = -K_X^3/8$; then
a) $d = 5$ then $B_X$ is generated by an exceptional pair;
b) $d = 4$ then $B_X = D(C_2)$, $C_2$ is a curve of genus $2$.
c) for $1 \le d \le 3$ the category $B_X$ is expected to be indecomosable.
4) index $1$, let $g_X = 1-K_X^3/2$; then such decomposition is known for $g_X \in \lbrace 12,10,9,8,7\rbrace$.
For those Fano 4folds which are unsections of Fano 3folds listed here a decomosition is also known.
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$\begingroup$ You may also take a look to recent papers by Bernardara-Bolognesi Categorical representability and intermediate Jacobians of Fano threefolds Derived categories and rationality of conic bundles they study very similar kind of questions. $\endgroup$– IMeasyCommented Aug 30, 2012 at 13:15