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This is a Quantitative Study study.

September 29, 2025Open Access

Reconstruction theorems in the supported case

Key Points

Equivalence preserves support dimension when the (anti)canonical bundle is ample, and irreducible supports are considered.
Skyscraper sheaves at closed points retain their structure under the equivalence mapping.
In ample cases, closed points of the support are recovered up to homeomorphism, enhancing previous understandings.
The findings apply to the bounded derived categories of coherent sheaves on smooth projective complex varieties.

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.

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Luigi Lombardi (Tue,) studied this question.

www.synapsesocial.com/papers/68da58c9c1728099cfd10aa8 — DOI: https://doi.org/10.48550/arxiv.2503.08419

Reconstruction theorems in the supported case

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