Moduli spaces of generalised tautological bundles on Hilbert schemes
Andreas Krug, Fabian Reede, Ziyu Zhang
TL;DR
The paper develops a general framework for constructing stable vector bundles on the Hilbert scheme $S^{[n]}$ by generalising tautological bundles through partitions $ abla$ of $n$ and irreducible representations $ extcircled{W}$, transported via the Bridgeland–King–Reid–Haiman equivalence. It proves stability of these generalized tautological bundles under suitable polarizations and, under natural moduli-space conditions, identifies components of the moduli space $M_{S^{[n]}}$ that arise from products of surface-moduli $M_j$, via an isomorphism given by $F_ abla^ extcircled{W}(E_1,dots,E_k)$; this extends known tautological constructions and connects to modular and hyperkähler geometry. A key technical achievement is a closed formula for the first Chern class $c_1(F_ abla^ extcircled{W}(E_1,dots,E_k))$, expressed as $c_1= B_ abla^ extcircled{W}- R_ abla^ extcircled{W}\delta$, with $oldsymbol{ extdelta}$ the boundary divisor, enabling both explicit computations and applications to stability and modular phenomena. The results unify and extend prior work on tautological bundles, O'Grady's modular sheaves, and the structure of $M_{S^{[n]}}$, while offering concrete tools to study new components and their geometric properties.
Abstract
We construct new stable vector bundles on Hilbert schemes of points on algebraic surfaces, which are parametrised by connected components of their moduli spaces. This work generalises aspects of our previous work on tautological bundles and of recent work of O'Grady.