Quadratic Donaldson-Thomas invariants for (ℙ¹)³ and some other smooth proper toric threefolds

2025/03/18 by Levine, Marc, Viergever, Anna M.
#14F42 #14N35 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.14420

Abstract

Using virtual localization in Witt sheaf cohomology, we show that the generating series of quadratic Donaldson-Thomas invariants of (ℙ¹)³, valued in the Witt ring of ℝ, W(ℝ)≅ ℤ, is equal to M(q²)^-8, where M(q) is the MacMahon function. This confirms a modified version of a conjecture of Viergever. We also show that a localized version of this conjecture holds for certain iterated blow-ups of (ℙ¹)³ and other related smooth proper toric varieties.

Related

• Hyperbolic localization in Donaldson-Thomas theory
• Hyperbolic localization of the Donaldson-Thomas sheaf
• Low degree motivic Donaldson-Thomas invariants of the three-dimensional projective space
• Virtual invariants of Quot schemes of points on threefolds
• Virtual Localization in equivariant Witt cohomology