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Wednesday, 4 November 2026: 3 new papers, 1 cross-list and 2 replacements in Symplectic Geometry (math.SG), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
We survey the recent development of spinal open books and nearly Lefschetz fibrations. The focus is on explaining past and new results regarding the detection of symplectic fillability and classification of symplectic fillings of contact 3-manifolds in light of the new machinery. We give many examples and a list of open questions. This can be considered as an additional chapter to the lecture notes by Ozbacgi and Wendl.
Let $W$ be a Weinstein manifold and let $f: W \to W$ be a compactly supported symplectic automorphism. Using Floer theories, one can define two different entropies of $f$, called "Floer-theoretic" entropy and "categorical" entropy. The current article compares these two entropies and show that the former bounds the latter from below. As an application of the comparison, using that these entropies are invariants up to the conjugate relation, we count conjugacy classes in symplectic mapping classes. Especially, we show an exponential abundance of conjugacy classes that are smoothly isotopic to the identity, when $W$ is a Weinstein manifold of dimension $4n \geq 6$, satisfying a local condition.
Let $Y$ be a smooth cubic 3-fold. We show that $S^2 \times Y$ admits an infinite sequence of cohomologous symplectic forms which are all deformation equivalent but pairwise non-isotopic; McDuff showed the corresponding result on $(S^2)^2 \times T^2$, but these appear to be the first simply-connected and monotone examples. The results are obtained from McDuff's bordism-flavoured Gromov--Witten invariant, re-interpreted cohomologically in terms of an evaluation map valued in the space of maps from the 2-sphere to a symplectic target. The application then relies on classical properties of the Fano space of lines on $Y$. We also discuss higher-dimensional generalisations.
We introduce a version of convex surface theory for immersed ribbon surfaces with Legendrian boundary. As an application, we establish a special case of K3: Problem 1.70 on sharpness in the slice-Bennequin inequality implying quasipositivity, and a special case of a conjecture of Cornwell, Ng, and Sivek on characterizing Lagrangian sliceness. Specifically, we prove that if $K$ is a knot that bounds a "star-shaped" ribbon disk and satisfies $\overline{\mathrm{sl}}(K) = -1$ (resp.\ $\overline{\mathrm{tb}}(K) = -1$), then $K$ is quasipositive (resp.\ Lagrangian slice). In fact, we prove these statements for any fixed transverse (resp.\ Legendrian) representative.