Reconstruction theorems in the supported case
Luigi Lombardi
TL;DR
The paper investigates when the dimension of the support is preserved under exact equivalences of bounded derived categories with support in a closed algebraic subset. It develops a reconstruction framework using the Serre functor for supported derived categories, point-like objects, and a Gabriel-type reconstruction with support to relate closed-point sets. Under the positivity of the restricted (anti)canonical bundle or under skyscraper-preserving equivalences, it proves homeomorphism of the sets of closed points and, in the one-dimensional case, full reconstruction of the supports. The approach yields both a partial and, in favorable cases, complete recovery of the underlying geometric supports from derived-category data, highlighting the strength of categorical methods in algebraic geometry. The results deepen the connection between derived-equivalences and the geometry of supports, with potential implications for birational and topological classifications under restricted positivity regimes.
Abstract
We show that any equivalence of bounded derived categories of coherent sheaves on a smooth projective complex variety supported in a closed algebraic subset preserves the dimension of the support in two cases: (i) the restriction of the (anti)canonical bundle to the support is ample; (ii) the supports are irreducible and the equivalence sends a skyscraper sheaf of a closed point to a skyscraper sheaf of a closed point. Moreover, in the first case the equivalence recovers the set of closed points of the support up to homeomorphism.