Rank $r$ DT theory from rank $1$

Abstract

Fix a Calabi-Yau 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as the quintic 3-fold. We express Joyce's generalised DT invariants counting Gieseker semistable sheaves of any rank $r$ on $X$ in terms of those counting sheaves of rank 1. By the MNOP conjecture they are therefore determined by the Gromov-Witten invariants of $X$.

arXiv commentReferee's corrections, added one figure. To appear in JAMS. 33 pages, 4 figures

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• Invariants of nested Hilbert and Quot schemes on surfaces