Kefeng Liu

Let $(X,ω_0)$ be a compact Kähler manifold and $\mathcal X\to B$ its Kuranishi family, where the base $B$ may be singular with $\dim_{\C} B \ge 1$. Using explicit sections of Hodge bundles obtained from algebraic and geometric constructions, we define an intrinsic period map and a Hodge map that parametrizes nearby $(p,p)$-classes. For deformations over irreducible analytic bases, we introduce $\nabla^{1,1}$-flat extensions of Kähler cones and obtain explicit positive representatives, leading to an upper semicontinuity property for these extensions. Combined with the characterization of Kähler cones due to Demailly--Paun, this yields a complete local description of Kähler cones in terms of analytic cycles. We further show that this upper semicontinuity persists on a large region of the base determined by a uniform bound on the operator norm of the Beltrami differential. As further applications, we generalize Green's density criterion to strong algebraic approximation and to the approximation of real $(p,p)$-forms, and give an intrinsic analytic description of Hodge loci, leading to a Beltrami-differential criterion for the variational Hodge conjecture.

In this note, we compute the cyclotomic expansion formula for colored Jones polynomial of double twist knots with an odd number of half-twists $\mathcal{K}_{p,\frac{s}{2}}$ by using the Kauffman bracket skein theory. It answers a question proposed by Lovejoy and Osburn in 2019.

Motivated by the cubic forms and anomaly cancellation formulas of Witten-Freed-Hopkins, we give some new cubic forms on spin, spin$^c$, spin$^{w_2}$ and orientable 12-manifolds respectively. We relate them to $\eta$-invariants when the manifolds are with boundary, and mod 2 indices on 10 dimensional characteristic submanifolds when the manifolds are spin$^c$ or spin$^{w_2}$. Our method of producing these cubic forms is a combination of (generalized) Witten classes and the character of the basic representation of affine $E_8$.