Intrinsic Donaldson-Thomas theory. I. Component lattices of stacks

TL;DR

This paper introduces the component lattice of algebraic stacks as a foundational tool for extending Donaldson-Thomas theory and enumerative geometry to non-linear stacks, establishing key structural theorems.

Contribution

It develops the concept of component lattices for stacks, proving foundational results like constancy, finiteness, and associativity theorems, enabling broader applications in enumerative geometry.

Findings

Established the constancy theorem for component lattices.

Proved the finiteness criterion for isomorphism types.

Generalized Hall algebra structures to non-linear stacks.

Abstract

This is the first paper in a series on intrinsic Donaldson-Thomas theory, where we develop a new framework for enumerative geometry that allows the generalization of constructions and results from linear moduli stacks to general non-linear algebraic stacks. In this paper, we introduce the component lattice of an algebraic stack. This is a key object in our theory, defined using the formalism of stacks of graded and filtered points. It provides the combinatorial data needed to formulate various results in enumerative geometry, such as decomposition-type theorems and wall-crossing formulae. Later papers in the series will focus on extending Donaldson-Thomas theory to the non-linear case, and we expect that our approach will be useful for extending many other flavours of enumerative invariants beyond the linear case as well. This paper proves several foundational results of our framework. The first is the constancy theorem, which states that the isomorphism types of connected components of the stacks of graded and filtered points stay constant within chambers in the component lattice. The second is the finiteness theorem, which provides a criterion for the finiteness of the number of possible isomorphism types of these components. The third is the associativity theorem, generalizing the structure of Hall algebras from linear stacks to general stacks. We also discuss some applications of these results outside Donaldson-Thomas theory, including a construction of stacks of real-weighted filtrations, and a generalization of the semistable reduction theorem to real-weighted filtrations.