Alexander Perry’s categorical reconstruction conjecture for smooth Fano varieties

Establish that, for any smooth complex Fano varieties X and X′, a ℂ-linear enhanced equivalence between the right orthogonal categories \(\mathcal C_X=\langle\mathcal O_X\rangle^\perp\) and \(\mathcal C_{X'}=\langle\mathcal O_{X'}\rangle^\perp\) implies \(X\cong X'\).

Background

For a smooth complex Fano variety X, the structure sheaf $\mathcal O_X$ is exceptional, so its right orthogonal $\mathcal C_X=\langle\mathcal O_X\rangle^\perp$ is an admissible subcategory of $D^b(X)$. Unlike the full derived category, this orthogonal category does not retain the position of $\mathcal O_X$ or the gluing data that embeds the orthogonal and the exceptional object into $D^b(X)$. Consequently, an equivalence of the orthogonal categories does not formally extend to an equivalence of the ambient derived categories.

The conjecture asserts that, despite this loss of information, the enhanced orthogonal category still determines the isomorphism class of the Fano variety. The paper proves this assertion only for smooth complex del Pezzo surfaces of degree at most four, leaving the general statement for smooth complex Fano varieties unresolved. The enhancement hypothesis is essential because the Hochschild–Serre algebra used in the reconstruction is defined through derived natural transformations and is preserved by enhanced Morita equivalences, but not necessarily by arbitrary triangulated equivalences.