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A regular semisimple Hessenberg variety $\mathrm{Hess}(S,h)$ is a smooth subvariety of the full flag variety $\mathrm{Fl}(\mathbb{C}^n)$ associated with a regular semisimple matrix $S$ of order $n$ and a function $h$ from $\{1,2,\dots,n\}$ to itself satisfying a certain condition. We show that when $\mathrm{Hess}(S,h)$ is connected and not the entire space $\mathrm{Fl}(\mathbb{C}^n)$, the reductive part of the identity component $\mathrm{Aut}^0(\mathrm{Hess}(S,h))$ of the automorphism group $\mathrm{Aut}(\mathrm{Hess}(S,h))$ of $\mathrm{Hess}(S,h)$ is an algebraic torus of dimension $n-1$ and $\mathrm{Aut}(\mathrm{Hess}(S,h))/\mathrm{Aut}^0(\mathrm{Hess}(S,h))$ is isomorphic to a subgroup of $\mathfrak{S}_n$ or $\mathfrak{S}_n\rtimes \{\pm 1\}$, where $\mathfrak{S}_n$ is the symmetric group of degree $n$. As a byproduct of our argument, we show that $\mathrm{Aut}(X)/\mathrm{Aut}^0(X)$ is a finite group for any projective GKM manifold $X$., Comment: 16 pages, 1 figure

In dimension 4, we extend the correspondence between compact nonsingular toric varieties and regular fans to a correspondence between almost complex torus manifolds and families of multi-fans in a geometric way, where an (almost) complex torus manifold is a $2n$-dimensional compact connected (almost) complex manifold equipped with an effective action of a real $n$-dimensional torus $T^n$ that has fixed points. Let $M$ be a 4-dimensional almost complex torus manifold. To $M$, we associate two equivalent combinatorial objects, a family $\Delta$ of multi-fans and a graph $\Gamma$, which encode the data on the fixed point set. We find a necessary and sufficient condition for each of $\Delta$ and $\Gamma$. Moreover, we provide a minimal model and operations for each of $\Delta$ and $\Gamma$. We introduce operations on a multi-fan and a graph that correspond to blow up and down of a manifold, and show that we can blow up and down $M$ to a minimal manifold $M'$ whose weights at the fixed points are unit vectors in $\mathbb{Z}^2$, $\Delta$ to a family of minimal multi-fans that has unit vectors only, and $\Gamma$ to a minimal graph whose edges all have unit vectors as labels. As an application, if $M$ is complex, $\Delta$ is a fan and determines $M$, $\Gamma$ encodes the equivariant cohomology of $M$, and $M'$ is $\mathbb{CP}^1 \times \mathbb{CP}^1$. This implies that any two 4-dimensional complex torus manifolds are obtained from each other by equivariant blow up and down.

In the present paper, we consider an action of the circle group on a compact oriented 4-manifold. We derive the Atiyah-Hirzebruch formula for the manifold, and associate a graph in terms of data on the fixed point set. We show in the case of isolated fixed points that if an abstract graph satisfies the Atiyah-Hirzebruch formula, then there exists a corresponding 4-dimensional oriented $S^1$-manifold., Comment: Improved results; we can reduce every weight to 1

An almost complex torus manifold is a $2n$-dimensional compact connected almost complex manifold equipped with an effective action of a real $n$-dimensional torus $T^n \simeq (S^1)^n$ that has fixed points. For an almost complex torus manifold, there is a labeled directed graph which contains information on weights at the fixed points and isotropy spheres. Let $M$ be a 6-dimensional almost complex torus manifold with Euler number 6. We show that two types of graphs occur for $M$, and for each type of graph we construct such a manifold $M$, proving the existence. Using the graphs, we determine the Chern numbers and the Hirzebruch $\chi_y$-genus of $M$.

It is known that the complex projective space $\mathbb{CP}^n$ admits a spin structure if and only if $n$ is odd. In this paper, we provide another proof that $\mathbb{CP}^{2m}$ does not admit a spin structure, by using a circle action.

In this paper, we classify the fixed point data (weights and signs at the fixed points), of a circle action on a 6-dimensional compact oriented manifold with 4 fixed points. We prove that it agrees with that of a disjoint union of rotations on two 6-spheres, or that of a linear action on $\mathbb{CP}^3$. The former case includes that of Petrie's exotic action on $\mathbb{CP}^3$.

Let a torus act on a compact oriented manifold $M$ with isolated fixed points, with an additional mild assumption that its isotropy submanifolds are orientable. We associate a signed labeled multigraph encoding the fixed point data (weights and signs at fixed points and isotropy submanifolds) of the manifold. We study operations on $M$ and its multigraph, (self) connected sum and blow up, etc. When the circle group acts on a 6-dimensional $M$, we classify such a multigraph by proving that we can convert it into the empty graph by successively applying two types of operations. In particular, this classifies the fixed point data of any such manifold. We prove this by showing that for any such manifold, we can successively take equivariant connected sums at fixed points with itself, $\mathbb{CP}^3$, and 6-dimensional analogue $Z_1$ and $Z_2$ of the Hirzebruch surfaces (and these with opposite orientations) to a fixed point free action on a compact oriented 6-manifold. We also classify a multigraph for a torus action on a 4-dimensional $M$., Comment: Added the assumption on the orientability of isotropy submanifolds. This paper supercedes arXiv:2108.07560; main results are new, while including all results of the previous one

Let a $k$-dimensional torus $T^k$ act on a $2n$-dimensional compact connected almost complex manifold $M$ with isolated fixed points. As for circle actions, we show that there exists a (directed labeled) multigraph that encodes weights at the fixed points of $M$. This includes the notion of a GKM graph as a special case that weights at each fixed point are pairwise linearly independent. If in addition $k=n$, i.e., $M$ is an almost complex torus manifold, the multigraph is a graph; it has no multiple edges. We show that the Hirzebruch $\chi_y$-genus $\chi_y(M)=\sum_{i=0}^n a_i(M) \cdot (-y)^i$ of an almost complex torus manifold $M$ satisfies $a_i(M) > 0$ for $0 \leq i \leq n$. In particular, the Todd genus of $M$ is positive and there are at least $n+1$ fixed points. Petrie's conjecture asserts that if a homotopy $\mathbb{CP}^n$ admits a non-trivial circle action, its Pontryagin class agrees with that of $\mathbb{CP}^n$. Petrie proved this conjecture if instead it admits a $T^n$-action. We prove that if a $2n$-dimensional almost complex torus manifold $M$ only shares the Euler number with the complex projective space $\mathbb{CP}^n$, an associated graph agrees with that of a linear $T^n$-action on $\mathbb{CP}^n$; consequently $M$ has the same weights at the fixed points, Chern numbers, equivariant cobordism class, Hirzebruch $\chi_y$-genus, Todd genus, and signature as $\mathbb{CP}^n$. If furthermore $M$ is equivariantly formal, the equivariant cohomology and the Chern classes of $M$ and $\mathbb{CP}^n$ also agree., Comment: Changed from almost complex toric manifolds to almost complex torus manifolds, to avoid confusion with toric manifolds that are complex. Minor revisions

Let the circle group act on a 6-dimensional compact oriented manifold $M$ with isolated fixed points. The fixed point data of $M$ is the collection of signs and weights at the fixed points. To classify such a manifold $M$, the fixed point data is an essential information. Under an assumption that each isotropy submanifold is orientable, we classify the fixed point data of $M$, by showing that we can convert it to the empty collection by performing a combination of a number of types of operations. We do so by proving that we can successively take equivariant connected sums of $M$ at fixed points with $S^6$, $\mathbb{CP}^3$, and 6-dimensional analogue $Z_1$ and $Z_2$ of the Hirzebruch surfaces (and these with opposite orientations), to a fixed point free action on a compact oriented 6-manifold., Comment: Added the assumption on the orientability of isotropy submanifolds

Let the circle group act on a compact oriented manifold $M$ with a non-empty discrete fixed point set. Then the dimension of $M$ is even. If $M$ has one fixed point, $M$ is the point. In any even dimension, such a manifold $M$ with two fixed points exists, a rotation of an even dimensional sphere. Suppose that $M$ has three fixed points. Then the dimension of $M$ is a multiple of 4. Under the assumption that each isotropy submanifold is orientable, we show that if $\dim M=8$, then the weights at the fixed points agree with those of an action on the quaternionic projective space $\mathbb{HP}^2$, and show that there is no such 12-dimensional manifold $M$., Comment: Major revision. Added the assumption on the orientability of isotropy submanifolds

In this paper, we prove various results for circle actions on compact unitary manifolds with discrete fixed point sets, generalizing results for almost complex manifolds. For a circle action on a compact unitary manifold with a discrete fixed point set, we prove relationships between the weights at the fixed points. As a consequence, we show that there is a multigraph that encodes the fixed point data (a collection of multisets of weights at the fixed points) of the manifold; this can be used to study unitary $S^1$-manifolds in terms of multigraphs. We derive results regarding the first equivariant Chern class, obtaining a lower bound on the number of fixed points under an assumption on a manifold. We determine the Hirzebruch $\chi_y$-genus of a compact unitary manifold admitting a semi-free $S^1$-action, and obtain a lower bound on the number of fixed points., Comment: To appear in Indiana University Mathematics Journal

In an earlier paper, the second author resolved a question of McDuff by constructing a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points. In this paper, we improve on this example by reducing the number of fixed points. More concretely, we construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly $2k$ fixed points for any $k \geq 5$.

Consider a circle action on an 8-dimensional compact almost complex manifold with 4 fixed points. To the author's knowledge, $S^2 \times S^6$ is the only known example of such a manifold. In this paper, we prove that if the circle acts on an 8-dimensional compact almost complex manifold $M$ with 4 fixed points, all the Chern numbers and the Hirzebruch $\chi_y$-genus of $M$ agree with those of $S^2 \times S^6$. In particular, $M$ is unitary cobordant to $S^2 \times S^6$.

We establish a necessary and sufficient condition for pairs of integers to arise as the weights at the fixed points of an effective circle action on a compact almost complex 4-manifold with a discrete fixed point set. As an application, we provide a necessary and sufficient condition for a pair of integers to arise as the Chern numbers of such an action, answering negatively a question by Sabatini whether $c_1^2[M] \leq 3 c_2[M]$ holds for any such manifold $M$. We achieve this by demonstrating that pairs of integers that arise as weights of a circle action, also arise as weights of a restriction of a $\mathbb{T}^2$-action. Furthermore, we discuss applications to circle actions on complex/symplectic 4-manifolds and semi-free circle actions with discrete fixed point sets., Comment: Major revision. Readability improved

The conjecture of Kosniowski asserts that if the circle acts on a compact unitary manifold $M$ with a non-empty fixed point set and $M$ does not bound a unitary manifold equivariantly, then the dimension of the manifold is bounded above by a linear function on the number of fixed points. We confirm the conjecture for almost complex manifolds under an assumption on weights. For instance, we confirm the conjecture if there is one type $\pm a$ of weights, or there are two types $\pm a$, $\pm b$ of weights.

Landweber and Stong prove that if a closed spin manifold $M$ admits a smooth $S^1$-action of odd type, then its signature $\mathrm{sign}(M)$ vanishes. In this paper, we extend the result to a torus action on a closed oriented manifold with generalized odd type.

In this paper, we study a circle action on a compact oriented manifold with a discrete fixed point set. The fixed point data consists of the weights of the $S^1$-representations at the fixed points. We prove various results and properties of the action, in terms of the fixed point data. We show that the manifold can be described by a multigraph associated to it. Specializing into the case of dimension 4, we classify the fixed point data. Moreover, we prove that there exist oriented $S^1$-manifolds with these fixed point data. Finally, we show that a certain multigraph behaves like a manifold.

Let the circle act on a compact almost complex manifold $M$. In this paper, we classify the fixed point data of the action if there are 4 fixed points and the dimension of the manifold is at most 6. First, if $\dim M=2$, then $M$ is a disjoint union of rotations on two 2-spheres. Second, if $\dim M=4$, we prove that the action alikes a circle action on a Hirzebruch surface. Finally, if $\dim M=6$, we prove that six types occur for the fixed point data; $\mathbb{CP}^3$ type, complex quadric in $\mathbb{CP}^4$ type, Fano 3-fold type, $S^6 \cup S^6$ type, and two unknown types that might possibly be realized as blow ups of a manifold like $S^6$. When $\dim M=6$, we recover the result by Ahara in which the fixed point data is determined if furthermore $\mathrm{Todd}(M)=1$ and $c_1^3(M)[M] \neq 0$, and the result by Tolman in which the fixed point data is determined if furthermore the base manifold admits a symplectic structure and the action is Hamiltonian.

The author proved that if the circle acts symplectically on a compact, connected symplectic manifold $M$ with three fixed points, then $M$ is equivariantly symplectomorphic to some standard action on $\mathbb{CP}^2$. In this paper, we extend the result to a circle action on an almost complex manifold; if the circle acts on a compact, connected almost complex manifold $M$ with exactly three fixed points, then $\dim M=4$. Moreover, we deal with the cases of one fixed point and two fixed points.

Consider a symplectic circle action on a closed symplectic manifold with non-empty isolated fixed points. Associated to each fixed point, there are well-defined non-zero integers, called weights. We prove that the action is Hamiltonian if the sum of an odd number of weights is never equal to zero (the weights may be taken at different fixed points). Moreover, we show that if $\dim M=6$, or if $\dim M=2n \leq 10$ and each fixed point has weights $\{\pm a_1, \cdots, \pm a_n\}$ for some positive integers $a_i$, it is enough to consider the sum of three weights. As applications, we recover the results for semi-free actions, and for certain circle actions on six-dimensional manifolds.

Consider a Hamiltonian circle action on a closed $8$-dimensional symplectic manifold $M$ with exactly five fixed points, which is the smallest possible fixed set. In their paper, L. Godinho and S. Sabatini show that if $M$ satisfies an extra "positivity condition" then the isotropy weights at the fixed points of $M$ agree with those of some linear action on $\mathbb{CP}^4$. Therefore, the (equivariant) cohomology rings and the (equivariant) Chern classes of $M$ and $\mathbb{CP}^4$ agree; in particular, $H^*(M;\mathbb{Z}) \simeq \mathbb{Z}[y]/y^5$ and $c(TM) = (1+y)^5$. In this paper, we prove that this positivity condition always holds for these manifolds. This completes the proof of the "symplectic Petrie conjecture" for Hamiltonian circle actions on on 8-dimensional closed symplectic manifolds with minimal fixed sets., Comment: To appear in Transformation Groups

Let the circle act symplectically on a compact, connected symplectic manifold $M$. If there are exactly three fixed points, $M$ is equivariantly symplectomorphic to $\mathbb{CP}^2$., Comment: arXiv admin note: text overlap with arXiv:1003.4787 by other authors