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Let $X$ be a smooth, projective variety, and $E_1$ and $E_2$ be two smooth divisors on it meeting each other normally. Suppose that $D^b_{E_1}(X) \subset D^b(X)$ and $D^b_{E_2}(X) \subset D^b(X)$ are admissible subcategories, so that there are semiorthogonal decompositions $$ D^b(X)=\langle D^b_{E_1}(X)^{\perp}, D^b_{E_1}(X) \rangle $$ and $$ D^b(X)=\langle D^b_{E_2}(X)^{\perp}, D^b_{E_2}(X) \rangle $$.

Is then $D^b_{E_1\cap E_2}(X) \subset D^b_{E_i}(X)$ admissible?

Or is there any further condition that can guarantee this?

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  • $\begingroup$ What is $D^b_{E_1}(X)$? $\endgroup$
    – Sasha
    Apr 19, 2017 at 16:06
  • $\begingroup$ I meant $D^b_{E_i}(X)$ to be the full subcategory of $D^b(X)$ of complexes of sheaves whose cohomology sheaves have their support contained in $E_i$. $\endgroup$
    – Adam Gyenge
    Apr 19, 2017 at 17:51
  • $\begingroup$ It is never admissible. $\endgroup$
    – Sasha
    Apr 19, 2017 at 18:50
  • $\begingroup$ 1. Can you explain this a bit to me? 2. What if I replace $D^b_{Z}(X)$ with $D^b(Z)$ everywhere in the question? $\endgroup$
    – Adam Gyenge
    Apr 19, 2017 at 19:32
  • $\begingroup$ It is also not admissible. $\endgroup$
    – Sasha
    Apr 19, 2017 at 19:52

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