Quadratic Donaldson-Thomas invariants for $(\mathbb{P}^1)^3$ and some other smooth proper toric threefolds

TL;DR

This work refines Donaldson-Thomas theory by constructing and computing quadratic DT invariants valued in the Witt ring $W(k)$, focusing on $({\mathbb P}^1)^3$ and certain toric blow-ups. Central to the method is Levine’s motivic virtual localization, extended to an $N$-equivariant setting, which yields generating functions for the Witt-valued invariants. The main result shows the Witt-valued DT generating series for $({\mathbb P}^1)^3$ equals $M(q^2)^{-8}$, with the equivariant framework extending to $N_3$-oriented toric threefolds and their iterated blow-ups, giving a local-to-global mechanism via the equivariant vertex measure. A key outcome is the identification $ ilde{Z}^{N,\tau}(X,q)=M(q^2)^{\frac{1}{2}\deg_{\mathbb{R}}(c_3(T_X\otimes K_X))}$, linking the Witt-valued invariants to classical Chern data and exposing a real-structure refinement of DT theory with potential broader applications to refined enumerative geometry and spin-structure phenomena.

Abstract

Using virtual localization in Witt sheaf cohomology, we show that the generating series of quadratic Donaldson-Thomas invariants of $(\mathbb{P}^1)^3$, valued in the Witt ring of $\mathbb{R}$, $W(\mathbb{R})\cong \mathbb{Z}$, is equal to $M(q^2)^{-8}$, 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 $(\mathbb{P}^1)^3$ and other related smooth proper toric varieties.

Theorems & Definitions (95)

• Theorem 1
• Conjecture 2
• Theorem 3
• Remark 4
• Definition I.1
• Remark I.2
• Lemma I.1
• proof
• Remark I.3
• Definition I.4