arXiv subjects
Publications and source records attributed to Saket Shah.
Following ideas of Kapustka-Kapustka, we use Lagrangian fibrations to construct twisted hyperholomorphic sheaves on products of hyperk\"ahler manifolds of K3$^{[n]}$- and OG10-type. As applications, we prove the Lefschetz standard conjecture and the D-equivalence conjecture for hyperk\"ahler manifolds of OG10-type.
Kuznetsov showed that for a flat quadric fibration $\mathcal{Q}$ over a smooth base $S$, $\mathrm{D}^b(\mathcal{Q})$ admits a semiorthogonal decomposition where one of the components is the derived category of the sheaf of even parts of a Clifford algebra $\mathrm{D}^b(S,\mathcal{C}l_0)$. \par As progress towards a generalization, we show that for a quadric fibration with a fairly minor condition on the rank of the quadric fibers, the category $\mathrm{D}^b(S,\mathcal{C}l_0)$ embeds fully faithfully into the derived category of the relative orthogonal Grassmannian $\mathrm{D}^b(\mathrm{OGr}(k,\mathcal{Q}))$. When $k = 2$, we use this to produce a semiorthogonal decomposition of $\mathrm{D}^b(\mathrm{OGr}(2,\mathcal{Q}))$ up to a residual category; we compute this residual category in the smooth case and produce a conjecture for in the case of a pencil of quadrics with smooth base locus.
In this note, we generalize results of Donagi and Pantev on twisted derived equivalences between elliptically fibered surfaces to higher dimensions. First, we establish a twisted derived equivalence between torsors under abelian schemes satisfying a certain compatibility condition. Then, relying on the work of Arinkin on compactified Jacobians, we extend the equivalence to twisted compactified Jacobians associated to curves on K3 surfaces. This positively answers a question stated by Mattei and Meinsma. We then extend a result of Bottini and Huybrechts for Fano varieties of lines on cubic fourfolds to general moduli spaces of Bridgeland-stable objects on Kuznetsov components admitting rational Lagrangian fibrations.
For a cubic hypersurface $X$, work of Galkin--Shinder and Voisin shows the existence of a birational map relating the Hilbert scheme of two points $X^{[2]}$ with a certain projective bundle over $X$. Belmans--Fu--Raedschelders show that this is a standard flip, a particularly nice type of birational map inducing decompositions of derived categories. We show that this geometric construction extends to produce standard flips for Hilbert schemes of quadrics on various higher-dimensional del Pezzo varieties of degree at least 3, including cubics, intersections of two quadrics, and linear sections of $\mathrm{Gr}(2, 5)$. The resulting construction also generalizes results of Chung--Hong--Lee for quintic del Pezzo varieties. As an application, we produce a conjectural semiorthogonal decompositions for orthogonal Grassmannians of lines.
We construct stability conditions on crepant resolutions of certain quotients of product varieties, giving as a special case the first examples of stability conditions on strict Calabi-Yau varieties of arbitrary dimension. Along the way, we prove the crepant resolutions are derived equivalent to the corresponding quotient stacks, verifying an instance of a conjecture of Bondal and Orlov.
We propose conjectural semiorthogonal decompositions for Fano schemes of linear subspaces on intersections of two quadrics, in terms of symmetric powers of the associated hyperelliptic (resp. stacky) curve. When the intersection is odd-dimensional, we moreover conjecture an identity in the Grothendieck ring of varieties and other motivic contexts. The evidence for these conjectures is given by upgrading recent results of Chen-Vilonen-Xue, to obtain formulae for the Hodge numbers of these Fano schemes. This allows us to numerically verify the conjecture in the hyperelliptic case, and establish a combinatorial identity as evidence for the stacky case.