Kavli Affiliate: Alexey Bondal
| Summary:
Given a relatively projective birational morphism $fcolon Xto Y$ of smooth algebraic spaces with dimension of fibers bounded by 1, we construct tilting relative (over $Y$) generators $T_X,f$ and $S_X,f$ in $mathcalD^b(X)$. We develop a piece of general theory of strict admissible lattice filtrations in triangulated categories and show that $mathcalD^b(X)$ has such a filtration $mathcalL$ where the lattice is the set of all birational decompositions $f colon X xrightarrowg Z xrightarrowh Y$ with smooth $Z$. The $t$-structures related to $T_X,f$ and $S_X,f$ are proved to be glued via filtrations left and right dual to $mathcalL$. We realise all such $Z$ as the fine moduli spaces of simple quotients of $mathcalO_X$ in the heart of the $t$-structure for which $S_X,g$ is a relative projective generator over $Y$. This implements the program of interpreting relevant smooth contractions of $X$ in terms of a suitable system of $t$-structures on $mathcalD^b(X)$.
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