LCK rank of locally conformally Kahler manifolds with potential

An LCK manifold with potential is a compact quotient of a Kahler manifold $X$ equipped with a positive Kahler potential $f$, such that the monodromy group acts on $X$ by holomorphic homotheties and multiplies $f$ by a character. The LCK rank is the rank of the image of this character, considered as a function from the monodromy group to real numbers. We prove that an LCK manifold with potential can have any rank between 1 and $b_1(M)$. Moreover, LCK manifolds with proper potential (ones with rank 1) are dense. Two errata to our previous work are given in the last Section.Comment: 14 pages. Supersedes arXiv:1512.00968. Contains errata to arXiv:math/030525

The ciliary machinery is repurposed for T cell immune synapse trafficking of LCK

Upon engagement of the T cell receptor with an antigen-presenting cell, LCK initiates TCR signaling by phosphorylating its activation motifs. However, the mechanism of LCK activation specifically at the immune synapse is a major question. We show that phosphorylation of the LCK activating Y394, despite modestly increasing its catalytic rate, dramatically focuses LCK localization to the immune synapse. We describe a trafficking mechanism whereby UNC119A extracts membrane-bound LCK by sequestering the hydrophobic myristoyl group, followed by release at the target membrane under the control of the ciliary ARL3/ARL13B. The UNC119A N terminus acts as a “regulatory arm” by binding the LCK kinase domain, an interaction inhibited by LCK Y394 phosphorylation, thus together with the ARL3/ARL13B machinery ensuring immune synapse focusing of active LCK. We propose that the ciliary machinery has been repurposed by T cells to generate and maintain polarized segregation of signals such as activated LCK at the immune synapse

Weighted Bott-Chern and Dolbeault cohomology for LCK-manifolds with potential

A locally conformally Kahler (LCK) manifold is a complex manifold with a Kahler structure on its covering and the deck transform group acting on it by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with a Kahler form taking values in a local system $L$, called the conformal weight bundle. The $L$-valued cohomology of $M$ is called Morse-Novikov cohomology. It was conjectured that (just as it happens for Kahler manifolds) the Morse-Novikov complex satisfies the $dd^c$-lemma. If true, it would have far-reaching consequences for the geometry of LCK manifolds. Counterexamples to the Morse-Novikov $dd^c$-lemma on Vaisman manifolds were found by R. Goto. We prove that $dd^c$-lemma is true with coefficients in a sufficiently general power $L^a$ of $L$ on any LCK manifold with potential (this includes Vaisman manifolds). We also prove vanishing of Dolbeault and Bott-Chern cohomology with coefficients in $L^a$. The same arguments are used to prove degeneration of the Dolbeault-Frohlicher spectral sequence with coefficients in any power of $L$.Comment: 18 pages, v. 2.0. Proof of Theorem 3.2 change