Ringel–Samokhin decomposition conjecture for moduli spaces of rank-two bundles

From papers

Let CC be a smooth projective curve of genus g2g\geq 2, let L{\mathcal L} be the fixed determinant, and let \moduliC(2,L){\moduli_C(2,\mathcal L)} be the moduli space of rank-two bundles with determinant L{\mathcal L}. Ringel–Samokhin decomposition conjecture. There exists a Ringel–Samokhin-type semiorthogonal decomposition

DbCoh(\moduliC(2,L))=DbCoh(pt),DbCoh(C),,DbCoh(Symg2C),DbCoh(Symg1C),DbCoh(Symg2C),,DbCoh(C),DbCoh(pt)D^b\operatorname{Coh}({\moduli_C(2,\mathcal L)})=\langle D^b\operatorname{Coh}(\mathrm{pt}),D^b\operatorname{Coh}(C),\ldots,D^b\operatorname{Coh}(\operatorname{Sym}^{g-2}C),D^b\operatorname{Coh}(\operatorname{Sym}^{g-1}C),D^b\operatorname{Coh}(\operatorname{Sym}^{g-2}C),\ldots,D^b\operatorname{Coh}(C),D^b\operatorname{Coh}(\mathrm{pt})\rangle

with anti-equivalence given by R ⁣Hom(,O\moduliC(2,L)(1))\mathbf{R}\!\operatorname{Hom}(-,{\mathcal O}_{{\moduli_C(2,\mathcal L)}}(1)). This is a proposed refinement imposing duality symmetry on the expected decomposition; the source gives no proof or resolution.

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Sources & referencesView supporting material

Primary source

Pieter Belmans, Sergey Galkin and Swarnava Mukhopadhyay, “Decompositions of moduli spaces of vector bundles and graph potentials”, arXiv:2009.05568 (2022).

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