Skein modules of closed 3 manifolds define line bundles over character varieties

TL;DR

The paper links skein modules at an odd root of unity to geometric objects on SL$_2$ character varieties by using the Frobenius morphism to view skein modules as coherent sheaves over $X_{SL_2}(M)$. It establishes that, when $X_{SL_2}(M)$ is reduced, the associated sheaf $\mathscr{L}^M$ is a line bundle, extending prior results from irreducible and diagonal loci to encompass central representations via marked manifolds and Reynolds-type arguments. The authors develop the framework of stated skein modules, Frobenius-induced sheaves, and Azumaya loci, and they leverage Heegaard splittings to reduce the global problem to fiberwise dimension analyses, showing all fibers are 1-dimensional. Consequently, the geometric object given by $\mathscr{L}^M$ provides a canonical line bundle over the character scheme, bridging quantum skein theory, moduli spaces, and TQFT considerations with potential implications for computations and invariants in 3-manifold topology.

Abstract

Let M be a closed 3-manifold and S(M) the skein module of M at some odd root of unity. Using the Frobenius morphism, we can see S(M) as the space of global sections of a coherent sheaf over the SL2 character scheme of M. We prove that when the character scheme is reduced, this sheaf is a line bundle.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 2.4
• Theorem 2.5
• Corollary 2.6
• Proposition 2.7
• proof : Proof of Proposition \ref{['prop_injection']}