arXiv subjects
Publications and source records attributed to Nicholas Lindsay.
Let $X$ be a smooth complex prime Fano fourfold having a semi-free action of $\mathbb{C}^*$, then $X$ is contained in one of the families $\mathbb{P}^4,Q^4,W_5, X^{m}_{8}$. All of the families contain members that have a semi-free $\mathbb{C}^*$-action.
This paper is dedicated to the question: Is the sequence of odd Betti numbers of a closed symplectic manifold with a non-trivial Hamiltonian torus action unimodal? Recently, there was some progress on the question for the sequence of even Betti numbers by Cho-Kim and the author. The results of this paper give positive evidence in the case of odd Betti numbers, in dimensions $6,8$ and $10$ under progressively stronger symmetry assumptions.
We investigate when finite-order Hamiltonian diffeomorphisms extend to Hamiltonian circle actions, probing the transition from discrete to continuous symmetry in symplectic topology. Focusing on irrational ruled symplectic $4$-manifolds, we show that homologically trivial symplectic cyclic actions of order $k>2$ always extend to Hamiltonian $S^1$-actions, possibly after modifying the symplectic form. In contrast, we construct explicit symplectic involutions that cannot be so extended, even on minimal irrational ruled surfaces. These examples reveal geometric obstructions to extending discrete symmetries and highlight new exotic symplectic actions not equivalent to holomorphic ones. Our results also apply to higher-dimensional and non-cyclic group actions, and we establish several structural results on the isomorphism types of finite groups that can act on irrational ruled symplectic $4$-manifolds.
To begin the paper we revisit a cohomological localization result of Jones-Rawnsley which was subsequently improved by Farber, further generalizing the result. We then proceed to improve a previous result of the author on complete intersections of dimension $8k$ with a Hamiltonian $S^1$-action in two directions. Firstly, in dimension $8$ we remove the assumption on the fixed point set. Secondly, in any dimension we prove the result under an analogous assumption on the fixed point set. We also give some applications towards the unimodality of Betti numbers of symplectic manifolds having a Hamiltonian $S^1$-action, and discuss the relation to symplectic rationality problems.
Motivated by a problem of Hirzebruch, we study $8$-dimensional, closed, symplectic manifolds having a Hamiltonian torus action with isolated fixed points and second Betti number equal to $1$. Such manifolds are automatically positive monotone. Our main result concerns those endowed with a Hamiltonian $T^2$-action and fourth Betti number equal to $2$. We classify their isotropy data, (equivariant) cohomology rings and (equivariant) Chern classes, and prove that they agree with those of certain explicit Fano $4$-folds with torus actions. Moreover, under more general assumptions, we prove several finiteness results concerning Betti and Chern numbers of $8$-dimensional, positive monotone symplectic manifolds with a Hamiltonian torus action.
Jang has proven a remarkable classification of $6$-dimensional manifolds having an almost complex circle action with $4$ fixed points. Jang classifies the weights and associated multigraph into six cases, leaving the existence of connected manifolds fitting into three of the cases unknown. We show that one of the unknown cases may be constructed by a surgery construction of Kustarev, and the underlying manifold is diffeomorphic to $S^4 \times S^2$. We show that the action is not equivariantly diffeomorphic to a linear one, thus giving a new exotic $S^1$-action of on a product of spheres that preserves an almost complex structure. We also prove a uniqueness statement for the almost complex structures produced by Kustarev's construction and prove some topological applications of Jang's classification.
We show that there exist symplectic structures on a $\mathbb CP^1$-bundle over $\mathbb CP^2$ that do not admit a compatible Kähler structure. These symplectic structures were originally constructed by Tolman and they have a Hamiltonian $\mathbb T^2$-symmetry. Tolman's manifold was shown to be diffeomorphic to a $\mathbb CP^1$-bundle over $\mathbb CP^{2}$ by Goertsches, Konstantis, and Zoller. The proof of our result relies on Mori theory, and on classical facts about holomorphic vector bundles over $\mathbb CP^{2}$.
We study $3$-folds with an action of a algebraic torus $T$ and finite fixed point set. In particular, assuming the torus action has (exactly) $6$ fixed points we show that aside from Mori fibre spaces, the topology of such spaces is strongly restricted. For $T= \mathbb{C}^{*}$ we prove that there are two explicit infinite families plus a finite number of exceptional cases. For $T = \mathbb{C}^{*} \times \mathbb{C}^{*}$ there are $2$ exceptional cases which are described explicitly.
We study the problem of determining which diffeomorphism classes of Kähler manifolds admit a Hamiltonian circle action. Our main result is the following: Let $M$ be a closed symplectic manifold, diffeomorphic to a complete intersection with complex dimension $4k$, having a Hamiltonian circle action such that each component of the fixed point set is an isolated fixed point or has dimension $2 \mod 4$. Then $M$ is diffeomorphic to $\mathbb{CP}^{4k}$, a quadric $Q \subset \mathbb{CP}^{4k+1}$ or an intersection of two quadrics $Q_1 \cap Q_2 \subset \mathbb{CP}^{4k+2}$.
We prove that any symplectic Fano $6$-manifold $M$ with a Hamiltonian $S^1$-action is simply connected and satisfies $c_1 c_2(M)=24$. This is done by showing that the fixed submanifold $M_{\min}\subseteq M$ on which the Hamiltonian attains its minimum is diffeomorphic to either a del Pezzo surface, a $2$-sphere or a point. In the case when $\dim(M_{\min})=4$, we use the fact that symplectic Fano $4$-manifolds are symplectomorphic to del Pezzo surfaces. The case when $\dim(M_{\min})=2$ involves a study of $6$-dimensional Hamiltonian $S^1$-manifolds with $M_{\min}$ diffeomorphic to a surface of positive genus. By exploiting an analogy with the algebro-geometric situation we construct in each such $6$-manifold an $S^1$-invariant symplectic hypersurface ${\cal F}(M)$ playing the role of a smooth fibre of a hypothetical Mori fibration over $M_{\min}$. This relies upon applying Seiberg-Witten theory to the resolution of symplectic $4$-orbifolds occurring as the reduced spaces of $M$.