A birationality result for character varieties

Abstract

Let M be an orientable, cusped hyperbolic 3-manifold of finite volume. We show that the restriction map r : X 0 → X(∂M ) from a Dehn surgery component in the P SL 2 (C)-character variety of M to the character variety of the boundary of M is a birational isomorphism onto its image. This generalises a result by Nathan Dunfield. A key step in our proof is the exactness of Craig Hodgson's volume differential on the eigenvalue variety.

Volume of Representations and Birationality of Peripheral Holonomy

“…Under this assumption, we prove theorem 2 stating that Hol periph is a birational isomorphism between X n and its image. The first step of the proof is already proven -and crucial -when n = 2 in [7] and [14]: PGL(n, C)) to R such that for any [ρ] in a Zariski-open subset of X n , Vol([ρ]) = V (Hol periph ([ρ])).…”

“…We discuss in this paper a possible generalization of this result to the case of target group PGL(n, C) for n ≥ 2 and multicusped M. The generalization for n = 2 and M multicusped is proven by Klaff-Tillmann [14].…”

On the quantum Teichmüller invariants of fibred cusped 3-manifolds

“…Then, use it to definew 2 by the same formula. Replacing all uses of the relation (13) by (12) in the other computations definesw 0 = y 3 0 /y 2 0 andw 1 = y 1 1 /y 2 1 , and eventually proves the first two identities in (38). For instance, in the case of X = X 4 the result becomes…”

“…In fact, F w determines also some r wequivariant set {ξ Γ } Γ∈Π as above, so that eventually the map w → r w can be lifted to a regular "holonomy" map hol : G(T,b) → X(M φ ). We have (see Proposition 4.6 of [5] when M φ has a single cusp, and Remark 1.4 of [6] for the general case): In particular, every point w ∈ A encodes an augmented character of M φ ; the algebraic closure of hol(A) is the eigenvalue subvariety E(M φ ) of X 0 (M φ ), the irreducible component of X(M φ ) containing the discrete faithful holonomy r h (see [12]). If M φ has a single cusp, then E(M φ ) = X 0 (M φ ); in general E(M φ ) contains r h and has complex dimension equal to the number of cusps of M φ .…”

“…Hence Ψ w (U ) solves the equation (12). Conversely, U N = −Id implies that U is diagonalizable with eigenvalues in the set {−q 2i , i = 0, .…”