The local motivic DT/PT correspondence

Area of Science:

• Algebraic Geometry
• Motivic Theory
• Enumerative Geometry

Background:

• The Quot scheme parameterizes quotients of sheaves, crucial in algebraic geometry.
• Previous work by Behrend, Bryan, and Szendrői established connections between virtual Euler characteristics and symmetric products.

Purpose of the Study:

• To demonstrate that the Quot scheme is a global critical locus.
• To compute the motivic partition function for this scheme.
• To define and compute a virtual motive for specific Quot schemes and explore related motivic phenomena.

Main Methods:

• Characterizing the Quot scheme as a global critical locus.
• Calculating motivic partition functions within the ring of relative motives.
• Utilizing techniques from Behrend-Bryan-Szendrői for virtual motive construction.
• Analyzing the pushforward of sheaves of vanishing cycles along the Hilbert-Chow map.

Main Results:

• The Quot scheme parameterizing length quotients is shown to be a global critical locus.
• The motivic partition function for this scheme is computed.
• A virtual motive is defined for Quot schemes of points on an ideal sheaf, with its motivic partition function calculated.
• Results align with motivic wall-crossing formulas and refine existing relations.

Conclusions:

• The 'relative' analysis provides new insights into the pushforward of vanishing cycles.
• Conjectures are proposed regarding the Hilbert-Chow map and its connection to cohomological Hall algebra representations.
• The study deepens the understanding of virtual motives and their applications in enumerative geometry.