Conjecture on a split limit mixed Hodge structure at an ideal point

Let MM be the relevant hyperbolic three-manifold, let X=X~(M)0smX=\tilde{X}(M)_0^{\mathrm{sm}}, and let PP be an ideal point of XX. Choose a local uniformizer zz at PP with m~(0)=1\tilde{m}(0)=1, and let a1,a2,b1,b2,u(z),v(z),cs(z)a_1,a_2,b_1,b_2,u(z),v(z),cs(z) and s(z)s(z) be the quantities occurring in the stated limit mixed Hodge structure. Split-limit conjecture. There exists an ideal point PP and such a uniformizer zz for which

lim⁡z→0((2πi)2cs(z)+(u(z)−2a1log⁡z)v(z))∈Q⋅(2πi)2,\lim_{z\rightarrow 0}\left((2\pi i)^2 cs(z)+(u(z)-2a_1\log z)v(z)\right)\in\mathbb{Q}\cdot(2\pi i)^2,

equivalently,

lim⁡z→0s(z)=lim⁡z→0((2πi)2cs(z)+(a2b1−a1b2)log⁡z−a1b1(log⁡z)2)∈Q⋅(2πi)2.\lim_{z\rightarrow 0}s(z)=\lim_{z\rightarrow 0}\left((2\pi i)^2 cs(z)+(a_2b_1-a_1b_2)\log z-a_1b_1(\log z)^2\right)\in\mathbb{Q}\cdot(2\pi i)^2.

In other words, for a suitable parameter, the limit Q\mathbb{Q}-mixed Hodge structure of the CS-VMHS at the tangent vector ∂∂z\frac{\partial}{\partial z} is split:

Q⊕Q(1)⊕Q(2).\mathbb{Q}\oplus\mathbb{Q}(1)\oplus\mathbb{Q}(2).

This predicts a rationality and splitting property for the limiting Chern–Simons variation at an ideal point of the character variety; the supplied text does not state whether the conjecture is known or open.

References

Primary source

Dong Uk Lee, “Chern-Simons invariants of hyperbolic three-manifolds, mixed Tate motives, and motivic path torsor of augmented character varieties”, arXiv:2502.11950 (2025).

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