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

TL;DR

This paper demonstrates that for closed 3-manifolds with reduced SL2 character schemes, the skein module at an odd root of unity forms a line bundle over the character variety, linking quantum invariants to algebraic geometry.

Contribution

It establishes that the skein module can be viewed as a line bundle over the character variety when the scheme is reduced, connecting quantum topology with algebraic geometry.

Findings

Skein module S(M) corresponds to global sections of a sheaf over the character scheme.

When the scheme is reduced, this sheaf is proven to be a line bundle.

The approach uses the Frobenius morphism to relate skein modules to geometric structures.

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.