FRG Workshop on Moduli Spaces and Stability - Shared screen with speaker view - Recording 1/3
Have we lost Sasha?
Is there some condition that could be imposed either on the collection or on X_H to guarantee that after restricting to X_H one still gets a full collection?
it can be read off from the geometry of the homological projective dual, but that's a rather tautological answer
the question is somewhat similar to applying the usual Lefschetz theorem: can one find conditions that guarantee that there is no new middle cohomology?
I'm not aware of such conditions in this classical setting, and also not in the exceptional collections setting :)
Ok, makes sense
But does the count work at least on the level of K groups of X_H and X (if say X_H) is smooth)?
I think it goes the other way? if you can compute something on X_H (say Hodge numbers, or rank of K_0) you know whether there's any chance of the Lefschetz collection for X restricting to X_H without additional components
I believe no, because take e.g. X=P^3, the ample line bundle O(2), so H=quadric=P^1 x P^1. Then restriction of the part of FEC from P^3 has length 2, but a FEC on P^1 x P^1 has length 4
Ok, that is what I thought =(
Sveta, sure, thanks