#  Superfluid stars and Q-balls in curved spacetime

Konstantin G. Zloshchastiev  https://bit.do/kgz Institute of Systems Science,
Durban University of Technology, Durban 4000, South Africa

(received: 26 May 2020 [APS], 3 September 2020 [FNT])

###### Abstract

Within the framework of the theory of strongly-interacting quantum Bose
liquids, we consider a general relativistic model of self-interacting complex
scalar fields with logarithmic nonlinearity taken from dense superfluid
models. We demonstrate the existence of gravitational equilibria in this
model, described by spherically symmetric nonsingular finite-mass
asymptotically-flat solutions. These equilibrium configurations can describe
both massive astronomical objects, such as bosonized superfluid stars or cores
of neutron stars, and finite-size particles and non-topological solitons, such
as Q-balls. We give an estimate for masses and sizes of such objects.

###### pacs:

67.10.-j; 11.10.-z; 95.30.Sf
Keywords: quantum Bose liquid, superfluidity in cold stars, Q-ball,
logarithmic scalar gravity

††preprint: Low Temp. Phys. 47, 89 (2021) [DOI: 10.1063/10.0003166]

##  I Introduction

In the hierarchy of superdense stars, various objects exist, which occupy an
intermediate place between neutron stars and black holes. To a distant
observer, such objects would look almost like black holes; but they have no
horizon, for which reason they are often aggregated under the name of compact
stars (CS) and black hole mimickers (BHM). Widely known examples of such
objects include geons, quark stars, gravastars; and the most popular of them,
boson stars, which have long since been studied fi48 ; ka68 ; rb69 ; ty84 ;
csw86 . In this paper, we propose another type of CS/BHM-type bosonic objects
– superfluid stars, which are modeled by scalar field with logarithmic
nonlinearity motivated by the theory of strongly-interacting dense
superfluids. Furthermore, we demonstrate that our model can also describe
composite particle-like objects, or Q-balls, which self-interact and curve
spacetime.

Because superfluids are macroscopic wave-mechanical objects, the stability of
superfluid stars against gravitational collapse is expected to be enhanced by
the uncertainty principle, similar to boson star models. Moreover,
superfluidity introduces an additional effect here. The inviscid flow, caused
by suppression of dissipative fluctuations, makes the fluid parcels and volume
elements more resistant to coming to a full stop and adhering to each other.
Therefore superfluid stars are expected to have a larger degree of resistance
to gravitational collapse than the conventional boson stars.

Logarithmic Bose liquids are nonlinear effective models, which have
successfully been used to describe laboratory superfluids, such as the helium
II phase z12eb ; sz19 ; z19ijmpb . Unlike models with quartic nonlinearity,
such as the Gross-Pitaevskii one, logarithmic models go well beyond the two-
body interaction approximation. In fact, logarithmic nonlinearity occurs in a
leading-order approximation of any strongly-interacting many-body systems;
which can be described in terms of collective degrees of freedom by a single
wavefunction, and for which typical interparticle interaction potentials are
much larger than kinetic energies z18zna . This nonlinearity also occurs in a
theory of superflow-induced spacetime and emergent gravity z10gc ; z11appb ;
z20un1 .

While the original logarithmic models are non-relativistic, their Lorentz-
symmetric analogues are straightforward to construct. Relativistic logarithmic
scalar fields are known to possess a dilatation symmetry ros69 , which makes
them universally usable over a large range of length and mass scales. This
feature, together with the superfluid-enhanced stability against gravitational
collapse, could result in the substantial increase of the maximal mass of
logarithmic superfluid stars, which is a conjecture to be checked, among other
things, in this paper.

##  II The model

Adopting the units $c=1$ and metric signature $(-+++)$, we write a classical
Lagrangian of the model:

${\cal L}=\frac{R}{16\pi
G}-\frac{1}{2}\nabla_{\mu}\phi^{*}\nabla^{\mu}\phi-V(\phi,\phi^{*}),$ (1)

where the potential of a minimally coupled complex scalar field $\phi$ is
given by

$V(\phi,\phi^{*})=-b\left|\phi\right|^{2}\left[\ln\left(\left|\phi\right|^{2}/a\right)-1\right],$
(2)

where $a$ and $b$ are constant parameters. These parameters have a different
physical meaning to parameters in other relativistic scalar field theories,
such as the $\phi^{4}$ model csw86 . Their values are not determined by the
physics of point particles, but by the properties of a macroscopic wave-
mechanical object with collective degrees of freedom; which Bose liquids and
condensates are z12eb .

Correspondingly, nonlinear coupling $b$ is a linear function of the wave-
mechanical temperature $T_{\Psi}$, which is defined, in a non-relativistic
regime, as a thermodynamic conjugate of the Everett-Hirschman’s information
entropy,

$S_{\Psi}=-\langle\Psi|\ln{(|\Psi|^{2}/\bar{\rho})}|\Psi\rangle=-\int|\Psi|^{2}\ln{(|\Psi|^{2}/\bar{\rho})}\,d^{3}\vec{x},$

where $\Psi=\Psi(\vec{x},t)$ is condensate wavefunction, constant $\bar{\rho}$
is the decoupling density (a non-relativistic analogue of $a$, which
compensates dimensionality of condensate wavefunction inside the logarithm),
and the integral is taken over the volume occupied by the liquid, further
details can be found in Ref. az11 .

Thus, $b$ is not an a priori assumed parameter of the model, but a value
related to the wave-mechanical thermodynamical properties of the system and
its environment

$b\sim T_{\Psi}-T_{\Psi}^{(0)},$ (3)

where $T_{\Psi}^{(0)}$ is a reference or critical value z18zna ; z18epl ;
z19ijmpb .

Since the potential (2) is defined up to an overall sign, we choose this sign
so that, under our signature and curvature conventions, the scalar field
equation would have a static Gaussian solution in the Minkowski spacetime
limit ros68 . This would correspond to selecting the non-topological soliton
sector of the model; a detailed discussion of topological structure can be
found in Refs. z11appb ; z18epl ; z19ijmpb . We assume that $b>0$ in what
follows.

Furthermore, field equations can be derived from the Lagrangian (1) in a
standard way:

$R^{\mu}_{\nu}-\frac{1}{2}\delta^{\mu}_{\nu}R=8\pi GT^{\mu}_{\nu},$ (4)

where the energy-momentum tensor is determined by complex scalar field:

$\displaystyle T^{\mu}_{\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}g^{\mu\sigma}\left(\nabla_{\sigma}\phi^{*}\nabla_{\nu}\phi+\nabla_{\sigma}\phi\nabla_{\nu}\phi^{*}\right)$
(5)
$\displaystyle\quad-\frac{1}{2}\delta^{\mu}_{\nu}\left(g^{\alpha\beta}\nabla_{\alpha}\phi^{*}\,\nabla_{\beta}\phi+2V(\phi,\phi^{*})\right).$

The complex scalar field equation can be extracted from these equations via
Bianchi identities, or by varying Lagrangian (1) with respect to scalar field,
and written in a form

$\nabla_{\mu}\nabla^{\mu}\phi=2\frac{\partial}{\partial\phi^{*}}V(\phi,\phi^{*})=-2b\ln{(|\phi|^{2}/a)}\,\phi.$
(6)

This equation is a Lorentz-covariant analogue of the logarithmic Schrödinger
equation, which was extensively studied: to mention only a few very recent
works aff19 ; wz19 ; bcs19 ; ct19 ; lzh19 ; ss20 . Relativistic wave equations
with logarithmic nonlinearity have also been extensively studied in the past,
assuming fixed Minkowski spacetime ros68 ; z11appb ; dz11 ; z10gc ; szm16 .

Let us consider now spherically symmetric and time independent solutions of
Einstein field equations (4). The line element can be written in static
coordinates

$ds^{2}=-B(r)dt^{2}+A(r)dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right),$ (7)

and, due to a symmetry of the problem, we assume our scalar to be spherically
symmetric and stationary

$\phi(r,t)=e^{-i\omega t}\Phi(r),$ (8)

where $\Phi(r)$ is a real-valued function. Correspondingly, field equations
(4) and (6) reduce to a system of three ordinary differential equations

$\frac{A^{\prime}}{A^{2}x}+\frac{A-1}{Ax^{2}}=2\left[\frac{\Omega^{2}}{B}+1-\ln(\sigma^{2}/k)\right]\sigma^{2}+\frac{(\sigma^{\prime})^{2}}{A},$
(9)
$\frac{B^{\prime}}{ABx}-\frac{A-1}{Ax^{2}}=2\left[\frac{\Omega^{2}}{B}-1+\ln(\sigma^{2}/k)\right]\sigma^{2}+\frac{(\sigma^{\prime})^{2}}{A},$
(10)
$\sigma^{\prime\prime}+\left(\frac{2}{x}-\frac{A^{\prime}}{2A}+\frac{B^{\prime}}{2B}\right)\sigma^{\prime}+2A\left[\frac{\Omega^{2}}{B}+\ln(\sigma^{2}/k)\right]\sigma=0,$
(11)

where $x=r/L$, $L=1/\sqrt{b}$, $\sigma=(4\pi G)^{1/2}\Phi$, $k=4\pi Ga$,
$\Omega=\omega/\sqrt{2b}$, and a prime denotes $d/dx$. If we define

$A(x)=\left[1-2{\cal M}(x)/x\right]^{-1},$ (12)

we can replace Eq. (9) with

${\cal
M}^{\prime}(x)=x^{2}\left\\{\left[\frac{\Omega^{2}}{B}+1-\ln(\sigma^{2}/k)\right]\sigma^{2}+\frac{(\sigma^{\prime})^{2}}{2A}\right\\},$
(13)

and deal with a dimensionless mass function ${\cal M}(x)$ in actual
computations.

(a) $k<\tilde{k}$ (b) $k>\tilde{k}$

Figure 1: Function $\sigma(x)$ for different values of $k$, at $\sigma_{0}=1$.

(a) $k<\tilde{k}$ (b) $k>\tilde{k}$

Figure 2:  Function ${\cal M}(x)$ for different values of $k$, at
$\sigma_{0}=1$.

##  III Numerical analysis

We numerically solve Eqs. (9)-(13) using the shooting method rb69 ; csw86 .
According to this method, values of parameters are chosen in such a way that
our scalar field vanishes at spatial infinity, and has no nodes and
singularities. Then, nonsingular finite-mass solutions of Eqs. (9)-(13) are
expected to describe the lowest-energy bound states. Thus, Eqs. (9)-(13),
together with the initial conditions $\sigma(0)=\sigma_{0}$,
$\sigma^{\prime}(0)=0$, ${\cal M}(0)=0$, and $B(0)=B_{0}$, are regarded as an
eigenvalue problem for $\Omega$ and $B_{0}$. Once these are computed, the
total mass can be derived from an asymptotic value of ${\cal M}(x)$: $M={\cal
M}(\infty)L/G={\cal M}(\infty)/(G\sqrt{b})$.

Computations indicate that: when parameter $k$ goes above a certain critical
value $\tilde{k}$, then a level splitting occurs at
$\sigma_{0}\geqslant\tilde{\sigma}_{0}$, where $\tilde{\sigma}_{0}$ is a
critical value of the central scalar field. We have numerically established
that

$\tilde{k}\approx 0.066,$ (14)

whereas the value $\tilde{\sigma}_{0}$ is different for each $k$. Therefore,
in our computations, we restrict ourselves to the region
$\sigma_{0}<\tilde{\sigma}_{0}$, which is sufficient for the purposes of this
paper.

A profile of the scalar field is shown in Fig. 1, assuming the central field
value to be equal to one. The field rapidly decays at spatial infinity (in the
flat-spacetime limit it would have an exact Gaussian shape), it has no nodes
and singular points.

Profiles of the dimensionless mass function ${\cal M}(x)$ and relativisticity
$2{\cal M}(x)/x$, are shown in Figs. 2 and 3, respectively. Unlike the scalar
field, their behavior pattern changes as $k$ goes across the critical value
$\tilde{k}$. However, in both cases it indicates that the solutions remain
nonsingular, finite-mass and horizon-free.

(a) $k<\tilde{k}$ (b) $k>\tilde{k}$

Figure 3:  Relativisticity function for different values of $k$, at
$\sigma_{0}=1$.

(a) $k<\tilde{k}$ (b) $k>\tilde{k}$

Figure 4:  Effective compactness versus central field $\sigma_{0}$, for
different values of $k$.

The effective compactness of our solution can be defined as the ratio ${\cal
M}_{99}/x_{99}$, where ${\cal M}_{99}=0.99{\cal M}(\infty)$, and $x_{99}$ is
the dimensionless radius containing ${\cal M}_{99}$. Its profile, shown in
Fig. 4, is similar to that of boson stars formed by strongly self-interacting
scalar fields abb10 . The figure suggests that maximum compactness is achieved
in the models in which $k$ approaches the $\tilde{k}$ value.

From these figures one can estimate that most of the field’s energy and star’s
mass are localized inside the radius

$R\approx\alpha L\sim b^{-1/2},$ (15)

where $\alpha$ is a number of order one (its exact value depends on $k$ and
$\sigma_{0}$). Therefore, in general relativity, logarithmic superfluid tends
to form lumps whose dimensions scale as $b^{-1/2}$. What about the bounds on
their mass?

(a) $k<\tilde{k}$ (b) $k>\tilde{k}$

Figure 5:  Asymptotic value ${\cal M}(\infty)$ versus $\sigma_{0}$, for
different values of $k$.

(a) $k<\tilde{k}$ (b) $k>\tilde{k}$

Figure 6:  Asymptotic value ${\cal M}(\infty)$ versus $\Omega^{2}/B(0)$, for
different values of $k$.  Figure 7: Profile of ${\cal
M}_{\text{max}}=\text{max}\left[{\cal M}(\infty)\right]$ versus $k$.

Profiles of the asymptotic values of the asymptotic ${\cal M}(\infty)$ as a
function of central field value and frequency are given, respectively, in
Figs. 5 and 6. Furthermore, Fig. 7 is a summarizing plot of the maximum values
of ${\cal M}(\infty)$. One can see that the absolute maximum for ${\cal
M}(\infty)$ is achieved at $k\to\tilde{k}$, which yields the relation

$M_{\text{max}}\equiv\text{max}[{\cal M}(\infty)]L/G\approx(4G\sqrt{b})^{-1}.$
(16)

Therefore, masses of equilibrium configurations of the logarithmic model scale
as an inverse square root of nonlinear coupling, similarly to the sizes, cf.
Eq. (15).

##  IV Conclusions

It is shown that in general relativity, self-interacting logarithmic scalar
fields can form equilibria, which are described by nonsingular, finite-mass
and horizon-free solutions. According to Eqs. (15) and (16), their
characteristic scales of mass and size are determined by the nonlinear
coupling: $M\sim b^{-1/2}$ and $R\sim b^{-1/2}$. Since this coupling has no
known bounds other than being positive-definite, see discussion in between
Eqs. (2) and (4), both mass and size of the equilibrium configurations have no
upper and lower bounds. Therefore, our model can be used to describe objects
with lengths and masses of a wide scale range, which agrees with the
dilatation symmetry mentioned above.

If $b$ is small, then our model refers to astronomical-scale CS/BHM objects,
such as superfluid stars and cores of neutron stars mi59 ; ls01 . For example,
let us assume that our superfluid star has the maximum mass which is equal to
the mass of Sun, $M_{\odot}\approx 2\times 10^{33}\,\text{g}$. Then from Eq.
(16), we obtain

$b_{\odot}\equiv b({M_{\text{max}}=M_{\odot}})=(4GM_{\odot})^{-2}\approx
2.8\times 10^{-12}\,\text{cm}^{-2}.$

Correspondingly, Eq. (15) gives us an estimate for the radius of such a star:
$R_{M_{\odot}}=\alpha/\sqrt{b_{\odot}}\approx\alpha(2.8\times
10^{-12})^{-1/2}\,\text{cm}\sim\alpha(6\times 10^{5})\,\text{cm}\lesssim
10\,\text{km}$.

When $b$ is large, then the model describes composite particlelike objects of
finite size, such as Q-balls, which self-interact and curve spacetime
(logarithmic Q-balls in fixed flat spacetime were studied in Refs. ros68 ;
z11appb ; dz11 ; mo20 ).

Furthermore, if we assume wave-mechanical temperature to be proportional to
the thermal one, $T_{\Psi}\propto T$, then Eqs. (3) and (16) suggest that mass
of logarithmic superfluid stars and Q-balls scales as an inverse square of
temperature, $M\sim(T-T_{c})^{-1/2}$, where $T_{c}\propto T_{\Psi}^{(0)}$. It
means that temperature of these objects should depend on their mass as

$T\sim M^{-2},$ (17)

up to an additive constant $T_{c}$. If this constant is close to absolute
zero, then massive superfluid stars alone must be cold objects with low
evaporation rate by thermal radiation, whereas logarithmic Q-balls have a
longer lifetime in thermally hot and dense environments.

###### Acknowledgements.

This research is supported by Department of Higher Education and Training of
South Africa and in part by National Research Foundation of South Africa.
Proofreading of the manuscript by P. Stannard is greatly appreciated.

## References

  * (1)
  * (2) I. Z. Fisher, Zh. Eksp. Teor. Fiz. 18, 636 (1948) [arXiv:gr-qc/9911008].
  * (3) D. J. Kaup, Phys. Rev. 172, 1331 (1968).
  * (4) R. Ruffini and S. Bonazzola, Phys. Rev. 187, 1767 (1969).
  * (5) E. Takasugi and M. Yoshimura, Z. Phys. C 26, 241 (1984).
  * (6) M. Colpi, S. L. Shapiro, and I. Wasserman, Phys. Rev. Lett. 57, 2485 (1986).
  * (7) K. G. Zloshchastiev, Eur. Phys. J. B 85, 273 (2012).
  * (8) T. C. Scott and K. G. Zloshchastiev, Low Temp. Phys. 45, 1231 (2019).
  * (9) K. G. Zloshchastiev, Int. J. Mod. Phys. B 33, 1950184 (2019).
  * (10) K. G. Zloshchastiev, Z. Naturforsch. A 73, 619 (2018).
  * (11) K. G. Zloshchastiev, Grav. Cosmol. 16, 288 (2010).
  * (12) K. G. Zloshchastiev, Acta Phys. Polon. 42, 261 (2011).
  * (13) K. G. Zloshchastiev, Universe 6, 180 (2020).
  * (14) G. Rosen, Phys. Rev. 183, 1186 (1969).
  * (15) A. V. Avdeenkov and K. G. Zloshchastiev, J. Phys. B: At. Mol. Opt. Phys. 44, 195303 (2011).
  * (16) K. G. Zloshchastiev, Europhys. Lett. (EPL) 122, 39001 (2018).
  * (17) G. Rosen, J. Math. Phys. 9, 996 (1968).
  * (18) C. O. Alves, D. C. de Morais Filho, and G. M. Figueiredo, Math. Meth. Appl. Sci. 42, 4862 (2019).
  * (19) Z.-Q. Wang and C. Zhang, Arch. Rational Mech. Anal. 231, 45 (2019).
  * (20) W. Bao, R. Carles, C. Su, and Q. Tang, Numer. Math. 143, 461 (2019).
  * (21) S. Chen and X. Tang, Acta Math. Hungar. 157, 27 (2019).
  * (22) H. Li, X. Zhao, and Y. Hu, Appl. Numer. Math. 140, 91 (2019).
  * (23) J. Shertzer and T. C. Scott, J. Phys. Commun. 4, 065004 (2020).
  * (24) T. C. Scott, X. Zhang, R. B. Mann, and G. J. Fee, Phys. Rev. D 93, 084017 (2016).
  * (25) P. Amaro-Seoane, J. Barranco, A. Bernal, and L. Rezzolla, J. Cosmol. Astropart. Phys. 11, 002 (2010).
  * (26) A. B. Migdal, Nucl. Phys. 13, 655 (1959).
  * (27) U. Lombardo and H.-J. Schulze, Lect. Notes Phys. 578, 30 (2001).
  * (28) V. Dzhunushaliev and K. G. Zloshchastiev, Central Eur. J. Phys. 11, 325 (2013).
  * (29) M. Mohammadi, Phys. Scr. 95, 045302 (2020).


# Symplectic torus actions on complete intersections

Nicholas Lindsay

###### Abstract

In [20] the author classified complete intersections of dimension $8k$
admitting a Hamiltonian circle action, under a certain assumption on the fixed
point set. In this paper, we improve this result 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 of our methods to the unimodality of
Betti numbers of symplectic manifolds having a Hamiltonian $S^{1}$-action, and
discuss the relation to symplectic rationality problems.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Hamiltonian circle actions
    2. 2.2 Applications of Duistermaat-Heckman
    3. 2.3 Complete Intersections
  3. 3 Proof of Theorem 1.1
    1. 3.1 Proving the JR equation for $4$-dimensional fixed components
  4. 4 Results in arbitrary dimensions
    1. 4.0.1 Symplectic circle actions
    2. 4.0.2 GKM actions
  5. 5 Unimodality of Betti numbers

##  1 Introduction

To state our results, we set the following notation for complete
intersections. Let $X_{m}(d_{1},\ldots,d_{k})$ denote a smooth complete
intersection in $\mathbb{CP}^{k+m}$ with multidegree $(d_{1},\ldots,d_{k})$.
Here, $m$ is equal to the complex dimension, and $k$ the complex codimension.

Since complete intersections of dimension at least $4$ are simply connected,
there is no distinction between Hamiltonian and symplectic circle actions. To
shorten the statement of results, we keep the understanding that a complete
intersection $(M,\omega)$ is a closed symplectic manifold diffeomorphic to a
complete intersection. For some results appearing later in the article, some
more restrictions on the symplectic form are needed and these are stated
explicitly.

The major motivating result of this paper is a theorem of Benoist which gave a
complete characterization of smooth complete intersections whose holomorphic
automorphism groups are infinite [1, Theorem 3.1]. The aim of this work is to
obtain a symplectic analogue of this theorem. A comprehensive smooth analogue
in dimension $6$ was given by Dessai and Wiemeler [8].

The first main result is to remove the technical assumption on the fixed point
set in our previous result on complete intersections [20, Theorem 1.1] in
dimension 8.

######  Theorem 1.1.

Suppose that $(M,\omega)$ is an $8$-dimensional complete intersection with a
symplectic circle action. Then, $M$ is diffeomorphic to either
$X_{4}(1),X_{4}(2)$ or $X_{4}(2,2)$.

The main issue to overcome in the proof is to deal with $4$-dimensional fixed
components. For this we apply a result of $4$-dimensional symplectic geometry
due to Ohta and Ono (building on fundamental work of Gromov, McDuff, and
Taubes), to deduce information about their intersection form, and deduce via
localisation formulas, information about the intersection form of $M$.

In the next part of the paper, we prove a classification result for symplectic
circle actions on complete intersections under a condition on the fixed point
set analogous to [20, Theorem 1.1], therefore completely removing the
assumption on the dimension.

######  Theorem 1.2.

Suppose that $(M,\omega)$ is a complete intersection admitting a symplectic
circle action with no fixed components $F$ satisfying $\dim(F)>0$ and
$\dim(M)=\dim(F)\mod 4$.

Then $M$ is diffeomorphic to $X_{m}(1),X_{m}(2),X_{2m}(2,2),$ or $X_{2}(3),$
$m\in\mathbb{N}$.

Theorem 1.2 has the following corollary.

######  Corollary 1.3.

Suppose that $(M,\omega)$ is a complete intersection admitting a symplectic
circle action having only isolated fixed points. Then $M$ is diffeomorphic to
$X_{m}(1),X_{m}(2),X_{2m}(2,2),$ or $X_{2}(3),$ $m\in\mathbb{N}$.

Projective spaces and quadrics of any dimension admit a Hamiltonian torus
action with isolated fixed points. The underlying smooth $4$-manifolds of the
complete intersections $X_{2}(2,2),$ or $X_{2}(3),$ admit a symplectic form
with a toric action, since both are diffeomorphic to the blow-up of
$\mathbb{CP}^{2}$ in a sequence of points.

Hence, the only unknown case remaining from Theorem 1.3 is $X_{2m}(2,2)$ for
$m\geq 2$. We are able to deal with this when the action is GKM, and the
symplectic form is homotopic to the Fubini-Study form, which we denote
$\omega_{FS}$. Thus obtaining in Theorem 4.2, a Benoist type “if and only if”
classification of complete intersections having such symplectic torus actions,
precisely when the holomorphic automorphism group, denoted $Aut(M)$ is
infinite. The proof uses results from [14], to evaluate the integral of
$c_{1}$ along a hypothetical toric $1$-skeleton Poincaré dual to $c_{n-1}$,
and thus obtaining a contradiction with the known Chern numbers of the
intersection of two quadrics.

######  Theorem 1.4.

Let $M$ be a complete intersection. Then $(M,\omega_{FS})$ has a GKM
symplectic torus action $\iff\;\;|Aut(M)|=\infty$.

In the final section of the paper, we prove some new results regarding the
unimodality of Betti numbers of a closed symplectic manifold admitting a
Hamiltonian circle action. Tolman has asked whether for a closed symplectic
manifold with a Hamiltonian circle action, the even Betti numbers are unimodal
(i.e. weakly increasing up to the middle degree) [15, Problem 4.3]. For Kähler
manifolds the property follows from the hard Lefschetz theorem. [15, Problem
4.3] also asked about the sequence of odd Betti numbers, but the results of
this paper are only relevant for the even case.

In [5] Cho and Kim gave an affirmative answer to [15, Problem 4.3 (1)] when
$M$ is $8$-dimensional and the action has isolated fixed points. In Theorem
5.1 we prove the unimodality of even Betti numbers for $8$-dimensional
manifolds having a Hamiltonian $S^{1}$-action with no extremal, 4-dimensional
fixed components.

######  Theorem 1.5.

Suppose that $M$ is a closed symplectic $8$-manifold having a Hamiltonian
$S^{1}$-action, and neither $M_{\min}$ or $M_{\max}$ are $4$-dimensional. Then
$b_{2}(M)\leq b_{4}(M)$.

In Theorem 5.1 we prove that for a closed symplectic $12$-manifold with
$b_{2}(M)=1$, satisfying a similar assumption on the fixed point set the even
Betti numbers are unimodal. To the author‘s knowledge this is new under the
assumption of isolated fixed points.

Furthermore, we give versions of the above results in general dimensions,
generalizing an inequality of Cho [3] on the Betti numbers of a manifold
admitting a Hamiltonian $S^{1}$-action with isolated fixed points , to more
general fixed point sets. Finally, we combine this result with the Rokhlin-
Ochanine theorem to obtain a final application.

Acknowledgments. I would like to thank Leonor Godinho and Silvia Sabatini for
helpful discussions, in particular explaining to me the results of [14], which
allowed me to deal with the intersection of two quadrics in the GKM case. I
presented some of the results about unimodality of Betti numbers in the
workshop “Perspectives in equivariant topology” in Cologne September 2022, I
would like the thank those present for interesting discussions, in particular
Liat Kessler and Susan Tolman. I would like to thank Dmitri Panov for
collaborations and discussions which influenced the overall approach of the
project, and for comments on a draft version.

The author was supported by SFB-TRR 191 grant Symplectic Structures in
Geometry, Algebra and Dynamics funded by the Deutsche Forschungsgemeinschaft.

##  2 Preliminaries

###  2.1 Hamiltonian circle actions

Firstly, we recall the definition of Hamiltonian torus action on a symplectic
manifold. Let $\mathfrak{t}$ denote the Lie algebra of the torus $T^{k}=T$,
and $\mathfrak{t}^{*}$ its dual. Recall that if a manifold $M$ has a smooth
$T$-action, then for every $t\in\mathfrak{t}$ we can naturally associate an
invariant vector field on $M$; $X_{t}$, tangent to the orbits of the action of
the one-parameter family associated to $t$.

######  Definition 2.1.

Let $(M,\omega)$ be a closed symplectic manifold, with a smooth $T^{k}$-action
by symplectomorphims. Then, the action is called Hamiltonian if there is a
smooth function $\mu:M\rightarrow\mathfrak{t}^{*}$ such that

$\omega(X_{t},\cdot)=-d(\mu(t)),$

for all $t\in\mathfrak{t}$.

$\mu$ is called the moment map of the action. The fixed point set of the
action, denoted $M^{T}$ is equal to the critical set of $\mu$, and is a
collection of symplectic submanifolds.

When $k=1$ the moment map is ussually referred to as the Hamiltonian and
denoted $H:M\rightarrow\mathbb{R}$. The Hamiltonian $H$ is a perfect Morse-
Bott function [17], and a perfect Morse function if and only if $M^{S^{1}}$ is
finite.

Let $G\subset S^{1}$ be a subgroup. The subset $M^{G}=\\{x\in
M|\;g.p=p\;\forall g\in G\\}$ is an invariant symplectic submanifold for which
the action is Hamiltonian, with Hamiltonian $H|_{M^{G}}$. This will be used in
the proof of Theorem 1.1. Connected components of $M^{G}$ are called isotropy
submanifolds.

For a fixed component $F$ denote by $\lambda_{F}$ half the Morse-Bott index of
$H$ along $F$. It holds that $\lambda_{F}$ is also equal to the the number of
strictly negative weights of the $S^{1}$-action along $F$ counted with
multiplicity. We will need the following localisation formula for the Betti
numbers.

######  Theorem 2.2.

[17] Let $(M,\omega)$ be a closed symplectic manifold having a Hamiltonian
circle action. Then for each $i$,

$b_{i}(M)=\sum_{F\subset M^{S^{1}}}b_{i-2\lambda_{F}(M)}(F),$

where the sum runs over connected components of the fixed point set.

The following localisation formula is a consequence of the equivariant version
of the Atiyah-Singer index theorem. The formula we will use is derived in [12,
Section 5.8]. We use $\sigma(M)$ to denote the signature of $M$, which is
equal to zero for manifolds of dimension not divisible by $4$ and equal to $1$
for a single point.

######  Theorem 2.3.

Let $(M,\omega)$ be a closed symplectic manifold having a Hamiltonian
$S^{1}$-action. Then

$\sigma(M)=\sum_{F\subset M^{S^{1}}}(-1)^{\lambda_{F}}\sigma(F).$

###### Proof.

This formula follows from the formula of [12, Section 5.8], expressing the
signature as a sum over the fixed point components.

∎

We will need the following result which was proved in [20].

######  Theorem 2.4.

[20, Proposition 3.1, Lemma 3.2] Suppose that $M$ is a closed symplectic
manifold of dimension $8k$. Suppose that $M$ has a Hamiltonian $S^{1}$-action
such that each component of the fixed point set is an isolated fixed point or
has dimension $2\mod 4$. Suppose also that $b_{i}(M)=b_{i}(\mathbb{CP}^{4k})$
for $i\neq 4k$. Then the intersection form on $H^{4k}(M,\mathbb{Z})$ is
positive definite.

###  2.2 Applications of Duistermaat-Heckman

We begin by recalling the Dusitermaat-Heckman theorem.

######  Theorem 2.5.

Suppose that $(M,\omega)$ is a closed symplectic manifold with a Hamiltonian
$S^{1}$-action, with Hamiltonian $H$. Let $c$ be a regular value of $H$. Then

$\frac{d}{dt}[\omega_{t}]=e(H^{-1}(M)).$

Here $H^{-1}(M)$ is considered as an (orbifold) principal $S^{1}$-bundle over
the reduced symplectic orbifold $M_{t}$. Finally, $e$ denotes the Euler class
of this bundle in $H^{2}(M_{t},\mathbb{R})$.

Here we denote by $M_{\min}$ and $M_{\max}$ the fixed point components on
which the Hamiltonian obtains its minimum and maximum respectively. Recall
that they are both connected, symplectic submanifolds.

We call a fixed point component $Y$ internal if
$H(M_{\min})<H(Y)<H(M_{\max})$.

######  Lemma 2.6.

Suppose that $M$ is a closed symplectic manifold with a Hamiltonian circle
action. Suppose that $M_{\min}$ consists of an isolated fixed. Suppose that
$Y$ is an internal fixed point component such that $H(Y^{\prime})\geq H(Y)$
for any other internal fixed component $Y^{\prime}$. Then, $\lambda_{Y}=1$.

###### Proof.

This is an immediate consequence of [29, Lemma 3.1]. ∎

We will need the following result which follows from results of Ono.

######  Lemma 2.7.

[14, Lemma 5.2] Suppose that $(M,\omega)$ is a closed symplectic manifold with
a Hamiltonian $S^{1}$-action and $b_{2}(M)=1$. Then after possibly rescaling
the symplectic form by a positive constant we may assume that
$c_{1}(M)=[\omega]$.

###### Proof.

Since $b_{2}(M)=1$, $c_{1}(M)=k[\omega]$ for some $k\in\mathbb{R}$. The result
then follows from [25, Theorem 1,Theorem 2].

∎

For Hamiltonian $S^{1}$-actions on Fano varieties, the following normalization
for the Hamiltonian goes back to Futaki [11, Theorem 1]. In the case of
monotone symplectic manifolds proofs may be found for the semi-free case in
[6] and in the general case [14]. In [21, Proposition 1.9] the formula was
proven for closed symplectic manifolds with Hamiltonian $S^{1}$-actions, such
that $c_{1}$ and $[\omega]$ have the same evaluation on classes in the image
of the Hurewicz map $\pi_{2}(M)\rightarrow H_{2}(M)$, this will not be needed
in this paper.

######  Theorem 2.8.

Suppose that $(M,\omega)$ is a closed symplectic manifold with
$c_{1}(M)=[\omega]$. Then, after possibly adding a constant to the
Hamiltonian, for each fixed component $F$,

$H(F)=-\sum w_{i},$

where $w_{i}$ are the weights along $F$.

There has been a lot of recent progress towards classifying monotone
symplectic manifolds with Hamiltonian symmetries, see [4, 2, 14, 21, 28] and
the reference contained therein.

The following theorem is due to Ohta and Ona, and builds on fundamental work
of Gromov, McDuff and Taubes.

######  Theorem 2.9.

[26] Let $(N,\omega)$ be a closed symplectic $4$-manifold such that
$[\omega]=kc_{1}(N)$ for some $k>0$. Then, $N$ is diffeomorphic to a del Pezzo
surface.

The theorem has the following immediate corollary.

######  Corollary 2.10.

Let $(N,\omega)$ be a closed symplectic $4$-manifold such that
$[\omega]=kc_{1}(N)$ for some $k>0$. Then, $b_{2}^{+}(N)=1$.

###### Proof.

By Theorem 2.9, $N$ is diffeomorphic to a smooth del Pezzo surface. By the
classification of del Pezzo surfaces, $N$ is either diffeomorphic to
$\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ or the blow up of $\mathbb{CP}^{2}$ in
up to $9$ points. One may verify by direct calculation that each of these
manifolds satisfies $b_{2}^{+}=1$. ∎

###  2.3 Complete Intersections

A smooth complete intersection $M=X^{n}\subset\mathbb{CP}^{n+k}$ is a
subvariety of projective space cut out by a collection of smooth hypersurfaces
in general position. By applying the Lefschetz hyperplane consecutively on the
image of the variety via appropriate Veronese embeddings, it follows that
complete intersections satisfy $b_{i}(X)=\mathbb{CP}^{n}$ for $i\neq n$. The
topological consequences of this fact have a different flavour when $n$ is odd
or even. In both cases, there is an important characterization of when a
complete intersection is“close to projective space”, which is due to Ewing and
Moolgavkar when $n$ is odd and Libgober and Wood when $n$ is even.

In the case when $n$ is odd, the middle degree homology $H^{n}(M,\mathbb{R})$
has the structure of a symplectic vector space given by the intersection
product and does not contain cup products of cohomology classes of other
degrees. Thus the “odd part” of the cohomology ring of $M$ is essentially
determined by $b_{n}(M)\in 2\mathbb{Z}$. A complete characterization of when
this part of the cohomology ring vanishes, as it does for projective space,
was given by Ewing and Moolgavkar [9]. Let $Q=X_{n}(2)$ denote the quadric
$n$-fold.

######  Theorem 2.11.

[9, Corollary 5] Suppose that $X$ is a complete intersection of odd complex
dimension $n$, then if $\chi_{top}(M)=n+1$ then $X\cong\mathbb{CP}^{n}$ or
$X\cong Q$.

In the case when $n$ is even, then the intersection form on
$H^{n}(M,\mathbb{R})$ is symmetric, and $H^{n}(M,\mathbb{R})$ plays a more
interesting role in determining the cohomology ring of $M$. The most
fundamental invariant we can extract from the intersection form is the
signature $\sigma(M)$. A property satisfied by projective space is that the
signature is a equal to the alternating sum of even Betti numbers, as was
shown by Libgober and Wood, this almost gives a characterization of projective
space amongst complete intersections.

######  Theorem 2.12.

[19] Let $X$ be a complete intersection with $n$ even. Then

$\sigma(X)=\sum_{k\in\mathbb{Z}}(b_{4k}(X)-b_{4k+2}(X))$

if and only if $X$ is equal to $X_{n}(1),X_{n}(2),X_{2}(3)$ or $X_{n}(2,2)$.

Combining the condition
$\sigma(X)=\sum_{k\in\mathbb{Z}}(b_{4k}(X)-b_{4k+2}(X))$ with the Lefschetz
hyperplane theorem shows that this is equivalent to the intersection form
being positve definite when $4|n$ and $b_{n}^{+}(X)=1$ otherwise.

A beautiful fact, proved by Deligne [7, Page 339] is that when $4|n$ the
intersection form of $X_{n}(2,2)$ is the Korkine-Zolotareff form
$\Gamma_{n+4}$. That is the lattice of $\mathbb{R}^{n+4}$ generated by
$e_{i}+e_{j}$ and $\frac{1}{2}(e_{1}+\ldots+e_{n+4}).$ In particular for $n=4$
the intersection form is isomorphic to the $E_{8}$ lattice. Deligne gives a
Dynkin diagram description of the intersection form for $n=2\mod 4$, and it is
clear from this description that $b^{+}_{n}(X_{n}(2,2))=1$.

##  3 Proof of Theorem 1.1

We now proceed to the proof of our main result Theorem 3.4. Before giving the
details of the proof, we describe the general approach. It is necessary to
make the following definition. We follow the convention that for manifolds
with real dimension not divisible by $4$ the signature is $0$, and a point has
signature $1$.

######  Definition 3.1.

Suppose that $X$ is a closed manifold. Define

$I_{JR}(X)=\sigma(X)-\sum_{i\in\mathbb{Z}}(b_{4i}(X)-b_{4i+2}(X)),$

we refer to $I_{JR}(X)=0$ to be the JR equation.

The idea of the proof of the following proposition was motivated by a theorem
of Jones and Rawnsley [16, Theorem 1.1], which states that if a closed
symplectic $(M,\omega)$ has a Hamiltonian circle action with isolated fixed
points then it satisfies the JR equation.

In the following Proposition 3.2 we show that the invariant $I_{JR}$ localizes
in a simple way over the fixed point set. A form of Proposition 3.2 with
additional conditions has been proven by Faber [10, Corollary 6]. Fabers
result states that for a manifold with a Hamiltonian circle action such that
$I_{JR}(F)=0$, $\forall F\subset M^{S^{1}}$ and all the odd Betti numbers of
each $F$ vanish, ones has $I_{JR}(M)=0$. Proposition 3.2 improves on [10,
Corollary 6] in three ways. Firstly, it removes the assumption on the odd
Betti numbers. Secondly, it shows that only fixed submanifolds of positive
dimension divisible by $4$ are relevant. Lastly, the equation of Proposition
3.2 can be useful for an inequality analysis when both sides are not
necessarily zero, this is key for the proof of Theorem 5.1.

######  Proposition 3.2.

Suppose that $(M,\omega)$ is a closed symplectic manifold admitting a
Hamiltonian circle action. Then,

$I_{JR}(M)=\sum_{\\{F\subset
M^{S^{1}}|\dim(F)=4k>0\\}}(-1)^{\lambda_{F}}I_{JR}(F).$

###### Proof.

By Theorem 2.2 and Theorem 2.3 the alternating sum of even Betti numbers and
the signature may be be expressed as sums over the fixed points set, in a
similar way to the proof of [20, Proposition 3.1]. By Theorem 2.2:

$\sum_{j\in\mathbb{Z}}b_{4j}(M)-\sum_{j\in\mathbb{Z}}b_{4j+2}(M)=\sum_{F\subset
M^{S^{1}}}(\sum_{j\in\mathbb{Z}}b_{4j-2\lambda_{F}}(F)-\sum_{j\in\mathbb{Z}}b_{4j+2-2\lambda_{F}}(F)).$

For each component $F$, considering how shifting index changes the sign of an
alternating sum gives the equation:

$(\sum_{j\in\mathbb{Z}}b_{4j-2\lambda_{F}}(F)-\sum_{j\in\mathbb{Z}}b_{4j+2-2\lambda_{F}}(F))=(-1)^{\lambda_{F}}(\sum_{j\in\mathbb{Z}}b_{4j}(F)-\sum_{j\in\mathbb{Z}}b_{4j+2}(F)).$

On the other hand:

$\sigma(M)=\sum_{F\subset M^{S^{1}}}(-1)^{\lambda_{F}}\sigma(F).$

Using these formulas to express $I_{JR}(M)$ as a sum over the fixed point set,
gives the equation.

$I_{JR}(M)=\sum_{F\subset M^{S^{1}}}(-1)^{\lambda_{F}}I_{JR}(F).$

It remains to show that when $F$ is an isolated fixed point or that
$\dim(F)=2\mod 4$, that $I_{JR}(F)=0$. For $F$ an isolated fixed point, the
statement is straightforward to prove, recalling that a point has signature
$1$.

Suppose $N$ has dimension $2\mod 4$, it follows that $\sigma(N)=0$. So to
prove $I_{JR}(N)=0$ it is sufficient to prove

$\sum_{i\in\mathbb{Z}}(b_{4i}(N)-b_{4i+2}(N))=0.$

This follows from Poincarè duality and the assumption that $\dim(N)=2\mod 4$.

∎

The following simple lemma simplifies $I_{JR}$ for $4$-dimensional components.

######  Lemma 3.3.

Suppose that $N$ is a closed manifold of dimension $4$. Then

$I_{JR}(N)=2b_{2}^{+}(N)-2.$

###### Proof.

Recall that $b_{2}(N)=b_{2}^{+}(N)+b_{2}^{-}(N)$ ,
$\sigma(N)=b_{2}^{+}(N)-b_{2}^{-}(N)$ and that $b_{0}(N)=b_{4}(N)=1$. So it
follows that

$I_{JR}(N)=(b_{2}^{+}(N)-b_{2}^{-}(N))-2+(b_{2}^{+}(N)+b_{2}^{-})=2b_{2}^{+}(N)-2.$

∎

###  3.1 Proving the JR equation for $4$-dimensional fixed components

In the below theorem, we complete the main step to classifying $8$-dimensional
complete intersections with a symplectic circle action. The most important
result which motivates Theorem 3.4 is a result of Tolman [29], which
classified Hamiltonian $S^{1}$-actions on $6$-manifolds with $b_{2}=1$. In
[13], building on results of [14], substantial progress towards the problem of
classifying $8$-dimensional symplectic manifolds having a Hamiltonian torus
action with isolated fixed points and $b_{2}=1$ was completed. A sharp
classification result into four explicit cases was obtained when the torus is
two dimensional and $b_{4}=2$ or equivalently the action has exactly $6$ fixed
points [13, Main Theorem]. All of the examples subcircle actions with various
fixed components of positive dimension occurring.

######  Theorem 3.4.

Suppose that $(M,\omega)$ is a closed symplectic $8$-manifold admitting a
Hamiltonian circle action, and that $b_{2}(M)=1$. Then, the intersection form
of $M$ is positive definite.

###### Proof.

Firstly we note that the conclusion of the theorem is equivalent to showing
that $I_{JR}(M)=0$. By Theorem 3.2 and Lemma 3.3 it is sufficient to prove for
each $4$-dimensional fixed point component $N$, $b_{2}^{+}(N)=1$.

Let $N$ be a $4$-dimensional fixed component, since the downward flowing
normal bundle of $N$ has rank at most 2 (over $\mathbb{C}$), it follows that
$0\leq\lambda_{N}\leq 2$. We claim that if $\lambda_{N}$ is $0$ or $2$, then
by Theorem 2.2 $b_{2}(N)=1$. Since $N$ is a symplectic submanifold
$b_{2}(N)>0$. In the localisation sum for the Betti numbers Theorem 2.2,
$b_{2}(N)$ either contributes to $b_{2}(M)$ or $b_{6}(M)$ (when
$\lambda_{N}=0$ or $2$ respectively) via the localisation formula, but by the
assumption of Theorem 3.4 and Poincaré duality $b_{2}(M)=b_{6}(M)=1$. It
follows that $b_{2}(N)=1$ in each case. Finally, since $b_{2}(N)=1$ and $N$ is
a symplectic submanifold $\int_{N}[\omega_{N}]^{2}>0$ implying that
$b_{2}^{+}(N)=1$, proving Theorem 3.4 in this case.

Hence we may assume that $\lambda_{N}=1$, this case is more involved. Firstly
this assumption implies by Theorem 2.2 that $M_{\min}$ and $M_{\max}$ are
isolated fixed points.

By Theorem 2.2 every other fixed component is an isolated fixed point $p$ with
$\lambda_{p}=2$. We claim that such fixed points dont occur. If $H(p)\leq
H(N)$ then this follows from Lemma 2.6. If $H(p)\geq H(N)$ then this follows
from Lemma 2.6, the case $H(p)\leq H(N)$ is proved similarly.

Hence, there are three fixed components $M_{\min}$, $M_{\max}$ and $N$. By
Lemma 2.7 we may assume that $c_{1}(M)=[\omega]$. Since $\dim(N)=4$ and
$\lambda_{N}=1$ the normal bundle of $N$ splits as a sum of two line bundles,
one upward flowing with respect to the Hamiltonian, and one downward flowing.
We denote the splitting by $L_{1}\oplus L_{2}$. Let $w_{1}$ and $w_{2}$ denote
the weights of the circle action along $L_{1}$ and $L_{2}$ respectively.

If either of $w_{1}$ or $w_{2}$ has modulus greater than $1$, there is an
isotropy $6$-manifold $Q$ such that $Q^{S^{1}}$ has two fixed components.
Suppose without loss of generality that $w_{1}<-1$, so that $Q_{\max}=N$,
$Q_{\min}=M_{\min}$. Then applying Theorem 2.2 to the Hamiltonian
$S^{1}$-action on $Q$ gives $b_{2}(Q)=b_{0}(N)=1$. Then, by Poincaré duality
$b_{4}(Q)=1$, then applying Theorem 2.2 again $b_{4}(Q)=b_{2}(N)=1$. Then
since the symplectic form of $N$ squares positively we have that
$b_{2}^{+}(N)=1$ as required.

Hence, we may assume that $w_{1}=-1$ and $w_{2}=1$. In particular, $H(N)=0$ by
Theorem 2.8. To ease notation we set $H(M_{\max})=H_{max}$ and
$H_{\min}=H(M_{\min}).$ Then since $M_{\min}$ and $M_{\max}$ are isolated
points, all of the four weights along them are non-zero and of the same sign.
This implies by Theorem 2.8 that $H(N)-H_{\min}=-H_{\min}\leq-4$ and
$H_{\max}-H(N)=H_{\max}\geq 4$.

Considering how the tangent bundle of $M$ splits over $N$ shows that

$[\omega]|_{N}=c_{1}(M)|_{N}=c_{1}(N)+c_{1}(L_{1})+c_{1}(L_{2}).$ (1)

Let $\bar{N}$ be the union of gradient sphere with maximum in $N$. We consider
the cycle $N_{c}\subset M_{c}$ which is obtained as the projection of
$\bar{N}$ by the quotient map $N_{c}$. Considering levels of the Hamiltonian
slightly below $0=H(N)$, and applying the equivariant symplectic neighborhood
theorem gives that:

$e(H^{-1}(c))|_{N_{c}}=c_{1}(N_{1})|_{N_{c}}\in H^{2}(N_{c},\mathbb{Z}).$

Note that $N_{H_{\min}}$ is a point since $M_{min}$ is isolated, by the
equivariant Darboux threorem, the limit of the class of the reduced symplectic
form is zero as the Hamiltonian approaches $H_{\min}$. Applying Duistermaat-
Heckman Theorem 2.5 and the above observations yields the equation:

$[\omega]|_{N}=(H(N)-H_{\min})c_{1}(L_{1})=|H_{\min}|c_{1}(L_{1})$

Applying similar consinderations to the upward flowing manifold of $N$ gives
the following equation:

$[\omega]|_{N}=(H_{\max}-H(N))c_{1}(L_{2})=|H_{\max}|c_{1}(L_{2})$

.

Substituting these equations back into equation (1) gives

$[\omega]|_{N}=c_{1}(M)|_{N}=c_{1}(N)+(\frac{1}{|H_{\max}|}+\frac{1}{|H_{\min}|})[\omega]|_{N},$

By, Corollary 2.10 $b_{2}^{+}(N)=1$, proving the theorem.

∎

The following simple example may help the reader to get an overview on the
last part of the proof.

######  Example 3.5.

Consider $\mathbb{CP}^{4}$ with the Hamiltonian $S^{1}$-action.

$z.[x_{0}:x_{1}:x_{2}:x_{3}:x_{4}]=[x_{0}:zx_{1}:zx_{2}:zx_{3}:z^{2}x_{4}].$

Then $M_{\min}$ and $M_{\max}$ are isolated and are mapped to $-5$ and $5$
respectively by the moment map of Theorem 2.8. The only other fixed component
is $N=\mathbb{CP}^{2}$ with $H(N)=0$ and normal bundle
$\mathcal{O}(1)\oplus\mathcal{O}(1)$. Therefore
$c_{1}(M)|N=c_{1}(T\mathbb{CP}^{2})+2c_{1}(\mathcal{O}(1))=5H$, showing that
$c_{1}(M)|N=|H_{\max}|c_{1}(\mathcal{O}(1))=|H_{\min}|c_{1}(\mathcal{O}(1))$.

We also present the following corollaries which may be of independent
interest.

######  Corollary 3.6.

Suppose that $M$ is an $8$-dimensional closed symplectic manifold with a
Hamiltonian circle action and with $b_{2}(M)=1$. Then,

$\int_{M}c_{2}(M)^{2}\geq 0.$

##  4 Results in arbitrary dimensions

####  4.0.1 Symplectic circle actions

In the following theorem, we give a classification of the multidegrees of
complete intersections admitting a Hamiltonian $S^{1}$-action whose fixed
point sets satisfy an analogous assumption to [20, Theorem 1.1].

######  Theorem 4.1.

Suppose that $(M,\omega)$ is a complete intersection admitting a symplectic
circle action with no fixed components $F$ satisfying $\dim(F)>0$ and
$\dim(M)=\dim(F)\mod 4$.

Then $M$ is diffeomorphic to $X_{m}(1),X_{m}(2),X_{2m}(2,2),$ or $X_{2}(3),$
$m\in\mathbb{N}$.

###### Proof.

Firstly, suppose that $\dim(M)=2n$, with $n$ odd. By applying the Lefschetz
hyperplane theorem and Poincaré duality to $M$, all of the even Betti numbers
of $M$ are equal to $1$ and all of the odd Betti numbers are $0$, except
possibly $b_{n}(M)$.

Let $F$ be a fixed submanifold, then by Theorem 2.2, $b_{i}(F)\neq 0$ for at
most one odd $i$. Since, by assumption $\dim(F)$ is divisible by $4$ this
implies by applying Poincaré duality to $F$ that all of the odd Betti numbers
of $F$ vanish, because every odd Betti number appears in at least two degrees.
Therefore by Theorem 2.2 the odd Betti numbers of $M$ vanish. Since all of the
even Betti numbers are equal to one, it follows that $\chi(M)=n+1$. Hence, by
Theorem 2.11 the multidegree of $M$ is (1) or (2).

In the case that $\dim(M)$ is divisible by $4$, then by our assumptions all
terms on the right hand side of the equation given in Theorem 3.2 vanish. It
follows that

$\sigma(M)=\sum_{k\in\mathbb{Z}}(b_{4k}(M)-b_{4k+2}(M)).$

By Theorem 2.12 the multidegree of $M$ is $(1),(2)$ or $(2,2)$, or $M$ is a
cubic surface. ∎

####  4.0.2 GKM actions

In this section, we strengthen the above results under the additional
assumption that the action is GKM and preserves a symplectic form homotopic to
the Fubini-Study form. We are able to give a complete characterization as in
the classification of infinite automorphism groups due to Benoist [1, Theorem
3.1].

######  Theorem 4.2.

Let $X$ be a smooth complete intersection of complex dimension greater than
$2$, and let $\omega_{0}$ be the restriction of the Fubini-Study form to $X$.
Then $X$ admits a GKM torus action preserving a symplectic form that is
homotopic to $\omega_{0}$ $\iff$ $|Aut(X)|=\infty$.

###### Proof.

By Theorem 1.2 and the theorem of Benoist [1, Theorem 3.1], it is sufficient
to exclude the existence of such actions on a complete intersection of the
form $X_{2m}(2,2)$. Suppose that there is such a GKM action. Set $n=2m$ to be
the complex dimension of $X_{2m}(2,2)$. Since $X_{2m}(2,2))$ is Fano it is
possible to rescale the hypothetical invariant symplectic form so that
$c_{1}=[\omega]$.

A calculation with the adjunction formula and Bezout’s theorem gives that the
the Chern number $c_{1}c_{n-1}$ of $X_{n}(2,2)$ is given by

$\frac{n}{2}(n+2)(n-1).$

Moreover, the topological Euler characteristic is

$2(n+2).$

It follows that the the number of edges of the GKM graph is

$n(n+2).$

The collection of spheres corresponding to the edges is Poincaré dual to
$c_{n-1}$ [14] and by the adjunction formula $c_{1}$ is $(n-1)$ times the
primitive generator of $H^{2}$. Therefore, using the condition
$c_{1}=[\omega]$ the integral of $c_{1}$ on each of the spheres is a positive
integer multiple of $(n-1)$. Therefore implying that the Chern number
$c_{1}c_{n-1}$ is divisible by $n(n+2)(n-1)$, which is a contradiction.

∎

We note finally, that it is possible to obtain a statement covering complex
dimension $2$, at the price of requiring that the symplectic form preserved by
the GKM action is monotone. Since both $X_{2}(2,2)$ and $X_{2}(3)$ admit
Kähler forms preserved by a toric action, this is the strongest possible
statement which can cover complex dimension $2$. Since the proof is identical
to the proof of Theorem 4.2, it is omitted.

######  Theorem 4.3.

Let $X$ be a smooth complete intersection and let $\omega_{FS}$ be the
restriction of the Fubini-Study form to $X$. Then $X$ admits a GKM torus
action preserving a monotone symplectic form that is homotopic to
$\omega_{FS}$ $\iff$ $|Aut(X)|=\infty$.

Another, perhaps more elegant but weaker possibility, is to require that the
symplectic structure is symplectomorphic to one given by the Fubini-Study
symplectic form. This was stated in Theorem 1.4 of the introduction.

##  5 Unimodality of Betti numbers

Given the generality of Proposition 3.2, it is natural ask whether
$I_{JR}(M)=0$ for all closed symplectic manifolds admitting a Hamiltonian
circle action. The following example shows that this is not the case.

Consider the standard Hamiltonian $S^{1}$-action on $\mathbb{CP}^{4}$ fixing a
point and a hyperplane $H$. Let $S\subset H$ be a smooth degree $4$
hypersurface. Recall that $S$ is a K3 surface with signature $16$ and
$b_{2}(S)=22$. Let $M$ be the $S^{1}$-equivariant blow up of $\mathbb{CP}^{4}$
along $S$. Then by Proposition 3.2

$I_{JR}(M)=-I_{JR}(N)=-36.$

In this section, we will present some applications regarding the unimodality
of Betti numbers of manifolds admitting a Hamiltonian circle action. Tolman
has asked whether for a closed symplectic manifold with a Hamiltonian circle
action, the even Betti numbers are unimodal (i.e. weakly increasing up to the
middle degree) [15, Problem 4.3]. For Kähler manifolds the property holds from
the hard Lefschetz theorem.

In [5] Cho and Kim proved that $8$-dimensional closed symplectic manifolds
having a Hamiltonian $S^{1}$-action with isolated fixed points, the even Betti
numbers are unimodal. Here we prove the same result under the weaker
assumption that neither of the extremal submanifolds are $4$-dimensional.

######  Theorem 5.1.

Suppose that $(M,\omega)$ is a closed symplectic $8$-manifold admitting a
Hamiltonian circle action such that, $M_{\min},M_{\max}$ are not
$4$-dimensional. Then the even Betti numbers of $M$ are unimodal.

###### Proof.

By Theorem 3.2 $M$ satisfies:

$I_{JR}(M)=\sum_{\\{F\subset
M^{S^{1}}|\dim(F)=4k>0\\}}(-1)^{\lambda_{F}}I_{JR}(F).$

Since by assumption $M_{\min}$ and $M_{\max}$ are not $4$-dimensional, every
$4$-dimensional fixed component $F$, satisfies $\lambda_{F}=1$.

Therefore the above equation becomes:

$I_{JR}(M)=\sum_{\\{F\subset M^{S^{1}}|\dim(F)=4\\}}-I_{JR}(F).$

By Lemma 3.3 $I_{JR}(F)=2b^{+}(F)-2$, since $F$ is a symplectic $4$-manifold,
this gives $I_{JR}(F)>0$. Therefore giving $I_{JR}(M)<0$.

Or, the inequality may me restated as

$\sigma(M)-2+2b_{2}(M)-b_{4}(M)<0.$

Then, since $M$ is symplectic $-b_{4}(M)<\sigma(M)$, substituting this and
rearranging gives $1+b_{2}(M)<b_{4}(M)$. Then, since both $b_{2}(M),b_{4}(M)$
are integers this gives the desired inequality.

∎

Motivated by Theorem 5.1, we state a conjecture. Following [18] a closed
symplectic manifold $(M,\omega)$ is called symplectically rational if there is
a connected symplectic manifold with a semi-free Hamiltonian $S^{1}$-action
such that the reduced spaces at two regular levels are symplectomorphic to
$(M,\omega)$ and $(\mathbb{CP}^{n},\omega_{FS})$ respectively.

######  Conjecture 5.2.

Suppose that $M$ is a symplectically rational $8$-manifold admitting a
Hamiltonian $S^{1}$-action. Then, the even Betti numbers of $M$ are unimodal.

If a complex rational $4$-fold $M$ has such an algebraic torus action having
an algebraic surface $N$ as an extremal component, then $M$ is birational to a
weighted projective space bundle over $N$ via a standard GIT argument.
Therefore $N$ is stably rational and hence rational [30]. Therefore $N$
contributes $0$ to the equation of Proposition 3.2 and the proof of Theorem
5.1 goes through.

We also obtain a similar result for closed $12$-manifolds with $b_{2}=1$. To
the authors knowledge this result is new, under the weaker assumption of
isolated fixed points.

######  Proposition 5.3.

Suppose that $(M,\omega)$ is a closed symplectic $12$-manifold admitting a
Hamiltonian circle action with no $4$ or $8$ dimensional fixed components, and
$b_{2}(M)=1$. Then the even Betti numbers of $M$ are unimodal.

###### Proof.

By Theorem 3.2 $M$ satisfies the JR equation. Since the modulus of the
signature of a symmetric bilinear form is bounded above by the rank of the
middle homology group, we have that

$\sigma(M)\leq b_{6}(M).$

Then substituting the JR equation for $\sigma(M)$ gives

$2b_{4}(M)-b_{6}(M)\leq b_{6}(M).$

Which implies that $b_{4}(M)\leq b_{6}(M)$, as required. ∎

Next, we prove two inequalities on the Betti numbers of a closed symplectic
manifold admitting a Hamiltonian $S^{1}$-action, under the assumption that
there is no fixed components of positive dimension divisible by $4$. Under the
assumption of isolated points, the inequalities were proved by Cho [3, Theorem
1.2]. Both inequalities may be considered as evidence for the unimodality of
even Betti numbers of manifolds admitting a Hamiltonian $S^{1}$-action, in the
sense that they would be implied by the unimodality. In particular, they imply
that the possibilities for any particular Betti number of the form $b_{k}$
with $k=2n\mod 4$ are bounded, if all of the other Betti numbers are fixed,
provided $k\neq n$.

######  Proposition 5.4.

Suppose that $(M,\omega)$ is a closed symplectic manifold of dimension $2n$
having a Hamiltonian $S^{1}$-action, such that there is no fixed component of
positive dimension divisible by $4$. Then,

  1. 1.

If $\dim(M)=0\mod 8$ then:

$0<1-b_{2}(M)+b_{4}(M)-\dots+b_{n}(M).$

  2. 2.

If $\dim(M)=4\mod 8$ then:

$1-b_{2}(M)+b_{4}(M)-\dots-b_{n}(M)\leq 0.$

###### Proof.

By Proposition 3.2 the signature of $M$ is equal to the alternating sum of its
even degree Betti numbers.

The first inequality, follows $-b_{n}(M)<\sigma(M)$, after putting all terms
to the left hand side, applying Poincaré duality and dividing by $2$. In this
case the inequality is strict because the equality case is equivalent to the
symplectic form being negative definite, and the top power of the symplectic
form is positive.

The second inequality then follows from the inequality $\sigma(M)\leq
b_{n}(M)$, where $n=\frac{\dim(M)}{2}$, after putting all terms to the right
hand side, applying Poincaré duality and dividing by $2$.

∎

We finally state a proposition regarding the Betti numbers of spin manifolds
admitting a Hamiltonian circle action.

######  Proposition 5.5.

Let $(M,\omega)$ be a closed symplectic manifold of dimension $2n$ with a
Hamiltonian $S^{1}$-action having no fixed component of positive dimension
divisible by $4$. Suppose that $n=2\mod 4$ and $M$ is spin, and $b_{2i}(M)=1$
for $2i\neq n$. Then $b_{n}(M)=2\mod 16$.

###### Proof.

By Proposition 3.2 the signature of $M$ is equal to the alternating sum of its
even degree Betti numbers, by the assumptions this simplifies as $2-b_{n}(M)$.
By the Rokhlin-Ochanine theorem [27, 24], this quantity is divisible by $16$,
the result follows. ∎

The proposition has the following corollary.

######  Corollary 5.6.

There is no spin fake projective space of dimension $8k+4$ having a
Hamiltonian $S^{1}$-action with isolated fixed points.

## References

  * [1] O. Benoist, Separation et propriete de Deligne-Mumford des champs de modules d’intersections completes lisse., J. Lond. Math. Soc., II. Ser. 87 (2013), no. 1, 138–156.
  * [2] I. Charton and L. Kessler. Monotone symplectic six-manifolds that admit a hamiltonian gkm action are diffeomorphic to smooth fano threefolds. arXiv, 2308.10541v2, 2023
  * [3] Y. Cho. Remark on the Betti numbers for Hamiltonian circle actions. Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 113-117.
  * [4] Y. Cho. Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions I. Internat. J. Math. 30 (2019), No. 6, 1950032 (71 pages).
  * [5] Y. Cho, M. Kim. Unimodality of the Betti numbers for Hamiltonian circle actions with isolated fixed points. Math. Res. Lett. 21 (2014), No.4. 691-696.
  * [6] Y. Cho, M. Kim. Hamiltonian circle action with self-indexing moment map. MATHEMATICAL RESEARCH LETTERS, v.23, no.3, pp.719 - 732.
  * [7] P. Deligne, Le theoreme de Noether, Groupes de Monodromie en Geometrie Algebrique, Lecture Notes in Math., Vol. 340, Springer-Verlag, Berlin and New York, 1973, pp. 328-340.
  * [8] A. Dessai and M. Wiemeler. Complete Intersections with $S^{1}$-action, Transformation Groups, Vol. 22 (2017) pp. 295-320
  * [9] J. Ewing, S. Moolgavkar. Euler Characteristic of Complete Intersections. Proceedings of the American Mathematical Society, Volume 56 (1976).
  * [10] M. Farber. Signature in terms of Novikov numbers. Russ. Math. Surv. 54 209.
  * [11] A. Futaki. The Ricci Curvature of symplectic quoutients of Fano varieties. Tohoku Math. Journ. 39 (1987), 329-339.
  * [12] F. Hirzebruch, T. Berger, R. Jung. Manifolds and Modular Forms. Aspects of Mathematics book series (ASMA, volume E 20).
  * [13] L. Godinho, N. Lindsay, S. Sabatini. On a symplectic generalization of a Hirzebruch problem. arXiv:2403.00949.
  * [14] L. Godinho, F. v. Heymann, S. Sabatini, 12, 24 and Beyond. Advances in Mathematics, 319 (2017), 472 - 521.
  * [15] L. Jeffrey, T. Holm, Y. Karshon, E. Lerman, and E. Meinrenken, Moment maps in various geometries, available at http://www.birs.ca/workshops/2005/05w5072/report05w5072.pdf.
  * [16] J.D.S. Jones , J.H. Rawnsley. Hamiltonian circle actions and the Signature. Journal of Geometry and Physics Volume 23, Issues 3–4, November 1997, Pages 301-307
  * [17] F. C. Kirwan, The cohomology of quotients in symplectic and algebraic geometry, Princeton University Press, 1984.
  * [18] T.J.Li, Y. Ruan. Symplectic Birational Geometry. arXiv:0906.3265 (2009).
  * [19] A. S. Libgober and J. W. Wood, On the topological structure of even-dimensional complete intersections, Trans. Amer. Math. Soc. 267 (1981), no.2, 637–660
  * [20] N. Lindsay. Hamiltonian circle actions on complete intersections. Bulletin of the London Mathematical Society. Volume 54, Issue 1. (2022) Pages 206-212 .
  * [21] N. Lindsay, D. Panov. Invariant symplectic hypersurfaces in dimension $6$ and the Fano condition. Journal of Topology. Volume12, Issue1 (2019).Pages 221-285.
  * [22] N. Lindsay, D. Panov. Symplectic and Kähler structures on $\mathbb{CP}^{1}$-bundles over $\mathbb{CP}^{2}$. Selecta Mathematica volume 27, (2021) Article number: 93
  * [23] D. McDuff, D. Salamon. Introduction to Symplectic Topology, Second Edition. Oxford Mathematical Monographs.
  * [24] S. Ochanine. Signature modulo 16, invariants de Kervaire généralisés et nombres caractéristiques dans la K-théorie réelle, Mém. Soc. Math. France 1980/81, no. 5.
  * [25] K. Ono. Some remarks on group actions in symplectic geometry. J. Fac. Soc. Uni. Tokyo. Sect. 1A. Math 35 (1998), 431-417.
  * [26] H. Ohta and K. Ono. Notes on symplectic 4-manifolds with b+ = 1, II. Internat. J. Math. 7(6) 755–770, (1996).
  * [27] V. A. Rokhlin. New results in the theory of four-dimensional manifolds, Doklady Acad. Nauk. SSSR (N.S.) 84 (1952) 221–224.
  * [28] S. Sabatini, D. Sepe, On topological properties of positive complexity one spaces, Transformation Groups, 10.1007/s00031-020-09588.
  * [29] S. Tolman. On a symplectic generalization of petrie’s conjecture. Trans. Amer. Math. Soc., 362(8):3963–3996, 2010.
  * [30] O. Zariski.: On Castelnuovo’s criterion of rationality $P_{a}=P_{2}=0$ of an algebraic surface. Illinois J. Math.2, 303–315 (1958).

Department Mathematik/Informatik, Universitat zu Köln, Weyertal 86-90, 50931
Köln, Germany.

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