# The Duistermaat-Heckman formula and
Chern-Schwartz-MacPherson classes

Allen Knutson<EMAIL_ADDRESS>For Victor Guillemin, my friend,
advisor, and inspiration

###### Abstract.

Let $M$ be a smooth complex projective variety, bearing a Kähler symplectic
form $\omega$ and a Hamiltonian action of a torus $T$, with finitely many
fixed points $M^{T}$. One standard form of the Duistermaat-Heckman theorem
gives a formula for $M$’s Duistermaat-Heckman measure $DH_{T}(M,\omega)$ as an
alternating sum of projections of cones, with overall direction determined by
a Morse decomposition of $M$.

Using Victor Ginzburg’s construction of Chern-Schwartz-MacPherson classes, we
show that these individual cone terms can themselves be interpreted as
Duistermaat-Heckman measures of cycles in $T^{*}M$. (This has a similar goal
to the symplectic cobordism approach of Viktor Ginzburg, Guillemin, and
Karshon.) Our approach also suggests extensions of the formula, including the
Brianchon-Gram theorem.

###### Contents

  1. 1 The Duistermaat-Heckman formula (aprés [GiLerS96])
    1. 1.1 Cone terms
    2. 1.2 Fourier equivalent measures
  2. 2 Chern-Schwartz-MacPherson classes
    1. 2.1 An exact sequence of ${\mathcal{D}}$-modules
    2. 2.2 Characteristic cycles
    3. 2.3 Chern-Schwartz-MacPherson classes and their additivity
    4. 2.4 Weber’s divisibility property
  3. 3 The main theorem: the geometry of cone terms
  4. 4 The Brianchon-Gram theorem and other extensions
    1. 4.1 A problematic decomposition

##  1\. The Duistermaat-Heckman formula (aprés [GiLerS96])

Except for a minor twist, the material in this section is by now completely
classical (no pun intended), and we include it largely to fix notation. The
minor twist will be the inclusion of a cycle in our manifold. For more
leisurely treatments we direct the reader to [GiLerS96, HKa12].

Given

  * •

a complex projective manifold $M\subseteq{\mathbb{P}}V$ where $V$ bears a
fixed Hermitian form $\langle,\rangle$, hence

  * •

a symplectic form $\omega$ on $M$, restricted from the Fubini-Study form on
${\mathbb{P}}V$,

  * •

an algebraic cycle $C\subseteq M$, i.e. a formal ${\mathbb{Z}}$-linear
combination $\sum_{i}n_{i}C_{i}$ of subvarieties $C_{i}\subseteq M$ of a fixed
dimension, and

  * •

an action $T\circlearrowright V$ of a compact torus $T$ preserving each of
$M$, $C$ and $\langle,\rangle$, hence

  * •

a homomorphism $\rho:T\to U(V)$ and a moment map $\Phi:M\to{\mathfrak{t}}^{*}$
made by composing
$M\mathrel{\hookrightarrow}{\mathbb{P}}V\mathrel{\hookrightarrow}{\mathfrak{u}}(V)^{*}\xrightarrow{\rho^{*}}{\mathfrak{t}}^{*}$,
where ${\mathfrak{u}}(V)^{*},{\mathfrak{t}}^{*}$ are the duals of the Lie
algebras,

we have three ways to think about the Duistermaat-Heckman measure
$DH_{T}(C\subseteq M,\omega)$:

  1. (1)

We can use $\omega$ to define a Liouville measure on the smooth part of each
$C_{i}$, push those measures forward with $\Phi_{*}$ to ${\mathfrak{t}}^{*}$,
and consider the sum of those measures, weighted by the coefficients
$(n_{i})$.

  2. (2)

We can take the Fourier transform of $\int_{M}\exp(\tilde{\omega})[C]$,
where $\tilde{\omega}=\omega-\Phi$ is the equivariant extension of the
symplectic form (as in [ABo84]). Note that if $C$ is a smooth subvariety
(formally given coefficient $1$),
then $\int_{M}\exp(\tilde{\omega})[C]=\int_{C}\exp(\tilde{\omega}|_{C})$ so
$DH_{T}(C\subseteq M,\omega)=DH_{T}(C\subseteq C,\omega|_{C})$.

  3. (3)

(Following [BrPro90] or [GiLerS96, §3.4]) We can compute the weight
multiplicity diagrams of $\oplus_{i}\Gamma(C_{i};\mathcal{O}(d))^{n_{i}}$ on
$T$’s weight lattice $T^{*}$, divide the multiplicities by $d^{\dim C}$ and
the lattice spacing by $d$, and consider the limit of the measure (now on
${\mathfrak{t}}^{*}$) as $d\to\infty$.

In [ABo84] the first two ways are related (in the case $C=M$, but our
generalization is an easy change) when $M^{T}$ is finite, computing the
integral

$\int_{M}\exp(\tilde{\omega})[C]=\sum_{f\in
M^{T}}\frac{\exp(\tilde{\omega})|_{f}\ [C]|_{f}}{\prod_{\lambda\in
wts(T_{f}M)}\lambda}=\sum_{f\in
M^{T}}\exp(-\Phi(f))\frac{[C]|_{f}}{\prod_{\lambda\in wts(T_{f}M)}\lambda}$

where $wts(T_{f}M)$ denotes the set of weights (with multiplicity) in the
isotropy action of $T$ on the tangent space $T_{f}M$, and $\alpha|_{f}$
denotes the pullback of a class $\alpha$ along the ($T$-equivariant) inclusion
of the point $f$.

For general $C$ of complex dimension $d$, we can (nonuniquely) write the
restriction $[C]|_{f}$ of its equivariant cohomology class $[C]\in
H^{*}_{T}(M)$ to the point $f$ as

$[C]|_{f}=\sum_{S\in{wts(T_{f}M)\choose d}}n_{f,S}\prod_{\lambda\in
wts(T_{f}M)\setminus S}\lambda,\qquad\text{for some }n_{f,S}\in{\mathbb{Z}}$

where $S$ runs over sub-multisets of $wts(T_{f}M)$ of size $d$. (Proof: pass
to the formal neighborhood of $f$ in $M$, then apply [KnM05, Lemma D], which
is stated there over polynomial rings but applies without change to power
series rings.)

It becomes now very tempting to Fourier transform the sum term by term. As is
well-understood, defining the transform of such singular terms requires some
choices in regularization, which we recapitulate in a moment. For now, we
choose $\vec{v}\in{\mathfrak{t}}$ not perpendicular to any of the (finitely
many) weights in $\\{T_{f}M\colon f\in M^{T}\\}$, use it to define
$\lambda_{+}:=\operatorname{sign}(\langle\vec{v},\lambda\rangle)\lambda$, and
flip the signs on some of our $(n_{f,S})$ so that

$\frac{[C]|_{f}}{\prod_{\lambda\in
wts(T_{f}M)}\lambda}=\sum_{S\in{wts(T_{f}M)\choose d}}n_{f,S}\
\frac{1}{\prod_{\lambda\in S}\lambda_{+}}$

and having now rewritten the “equivariant multiplicity” of $[C]$ like so, we
have

$\displaystyle\int_{M}\exp(\tilde{\omega})[C]$ $\displaystyle=$
$\displaystyle\sum_{f\in M^{T},\ S\in{wts(T_{f}M)\choose d}}n_{f,S}\
\frac{\exp(-\Phi(f))}{\prod_{\lambda\in S}\lambda_{+}}$

###  1.1. Cone terms

Define a cone term associated to a weight $\mu\in T^{*}$ and a multiset
$P=\\{\lambda\\}$ of weights in $T^{*}$ as any multiple of

$cone\left(\mu,\\{\lambda\\}\right):=\pi_{*}(\text{Lebesgue measure on
}{\mathbb{R}_{\geq 0}}^{P}),$

where

$\begin{array}[]{rrcl}\pi:&{\mathbb{R}_{\geq 0}}^{P}&\to&{\mathfrak{t}}^{*}\\\
&(x_{\lambda})_{\lambda\in P}&\mapsto&\mu+\sum_{\lambda\in
P}x_{\lambda}\lambda\end{array}$

This measure is only locally finite on ${\mathfrak{t}}^{*}$ if $\pi$ is
proper, or equivalently, if the vectors $\lambda\in P$ live in an open half-
space in $T^{*}$. This, we will arrange by requiring
$\langle\vec{v},\lambda\rangle>0$. At this point we define the Fourier
transform of ${\exp(-\Phi(f))}/{\prod_{\lambda\in S}\lambda_{+}}$ to be
$cone(\Phi(f),S)$. Summing these cone terms, we have arrived at Heckman’s
formula (as it is called in [GiLerS96, §3.3]) for the Duistermaat-Heckman
measure.

### Example: ${\mathbb{C}\mathbb{P}}^{2}$

Let $T^{2}\circlearrowright{\mathbb{C}}^{3}$ with weights
$P=\\{(0,0),(1,0),(0,1)\\}$, and let $\vec{v}=(1,2)$. Let $M$ and $C$ be the
projectivization ${\mathbb{C}\mathbb{P}}^{2}$. Then
$DH_{T}({\mathbb{C}\mathbb{P}}^{2},\omega)$ is Lebesgue measure on the
triangle with vertices $P$. The formula above computes it as pictured, where
cyan edges indicate flipped edges, i.e. those for which
$\lambda_{+}=-\lambda$. (The dotted lines, which indicate a certain
$3$-dimensionality of the picture, will be explained later.)

###  1.2. Fourier equivalent measures

There is a clear subtlety – if we change $\vec{v}$, it changes the individual
terms, so what precisely is guaranteeing that the final sum of measures is
independent of $\vec{v}$? (Other than the obvious: the measure was originally
defined as a pushforward not depending on this choice.)

In what is to come the only measures we need consider on ${\mathfrak{t}}^{*}$
will be linear combinations of cone terms. Call two cone terms $C\cdot
cone(\mu,{\lambda})$ and $D\cdot cone(\mu^{\prime},{\lambda^{\prime}})$
Fourier equivalent if $\mu=\mu^{\prime}$, the two multisets
$\\{\lambda\\},\\{\lambda^{\prime}\\}$ agree up to negating $k$ many weights,
and $C=(-1)^{k}D$. Call two measures Fourier equivalent if one can be obtained
from the other by replacing terms in one with Fourier equivalent terms to
obtain the other. We thank Terry Tao for pointing out the following lemma:

######  Lemma 1.

If $f,g$ are Fourier equivalent linear combinations of cone terms, and there
is a pointy cone $P$ such that $f,g$ are both supported with $P$, then $f=g$.

###### Proof.

Let $\partial_{\lambda}$ denote the differencing operation
$(\partial_{\lambda}f)(x)=f(x)-f(x+\lambda)$. If
$\partial_{\nu}f=\partial_{\nu}g$ with $\nu\neq\vec{0}$, then $f$ and $g$
differ by a function $h$ invariant under translation by $\nu$. This $h$ will
also be supported inside $P$, hence $h=0$. So we can simplify the equality to
be checked by applying such operators, without losing information.

If we take any single cone term $cone(\mu,\\{\lambda\\})$ and apply
$\prod_{\lambda}\partial_{\lambda}$, the result is the projection of a
parallelepiped. At this point the two measures we are comparing are compactly
supported, and are thus determined by their Fourier transforms. ∎

Heckman’s formula produces a measure supported in a translate of the cone
spanned by $\\{\lambda_{+}\colon\lambda$ a weight in some $T_{f}M\\}$, which
is a proper cone (i.e. pointy) by the assumption on $\vec{v}$. Since the
manifold $M$ is compact its DH measure is compactly supported, hence is
supported in some translate of any proper cone. At this point we apply the
lemma.

In what is to come we will need noncompact extensions of (1-3), as to be found
in [PraWu94, Th92]. Instead of compactness of $M$, one reduces to the case $M$
connected, then asks that $M^{T}$ be compact, and finally that some component
of $M^{T}$ be attractive, meaning that all of the isotropy weights in its
normal bundle lie in an open half-space of $T^{*}$. The moment map is then
proper, so (1) makes sense, and its image lies inside a proper cone, as was
needed in the argument above. One uses the AB/BV localization theorem to make
sense of (2), and although $\Gamma(C_{i};{\mathcal{O}}(d))$ is infinite-
dimensional its weight spaces are finite-dimensional, making sense of (3).

Finally, we will need to relax the nondegeneracy of the symplectic form, to a
closed $2$-form. This changes (1) in that the moment map is no longer
determined up to translation (unless the form is generically nondegenerate)
but is explicit extra data. It does not affect definition (2). For definition
(3) one still needs $[\omega]$ to be the first Chern class of a holomorphic
line bundle ${\mathcal{O}}(1)$, and the weight multiplicity diagram is now
made using the entire Euler characteristic of the sheaf cohomology of
${\mathcal{O}}(d)$, rather than just $H^{0}$ the space of sections. In the
rest of the paper we work primarily with definition (2), based off the AB/BV
localization formula.

##  2\. Chern-Schwartz-MacPherson classes

We follow [Gi86] for our treatment of CSM classes, as derived from
${\mathcal{D}}$-modules.

###  2.1. An exact sequence of ${\mathcal{D}}$-modules

Let $\beta:B\mathrel{\hookrightarrow}A$ be the closed inclusion of one smooth
complex manifold into another, in codimension $1$, defined by the vanishing of
a function $f$. Then there is a short exact sequence of
${\mathcal{D}}_{A}$-modules

$0\to{\mathcal{O}}_{A}\xrightarrow{\cdot f}{\mathcal{O}}_{A\setminus
B}\to\beta_{*}({\mathcal{O}}_{B})\to 0$

where ${\mathcal{D}}_{A}$ is the sheaf of differential operators on $A$, and
$\beta_{*}$ is the pushforward of ${\mathcal{D}}$-modules.

Our running example is very simple:
$\\{0\\}\mathrel{\hookrightarrow}{\mathbb{C}}$. Since ${\mathbb{C}}$ is
affine, instead of working with sheaves we can take global sections
$\Gamma({\mathcal{D}}_{\mathbb{C}})\mathrel{\cong}{\mathbb{C}}[\hat{x},\frac{d}{dx}]$.
Then the short exact sequence of ${\mathbb{C}}[\hat{x},\frac{d}{dx}]$-modules

$\begin{array}[]{rccccccl}0&\to&{\mathbb{C}}[\hat{x},\frac{d}{dx}]/\langle\frac{d}{dx}\rangle&\mathrel{\hookrightarrow}&{\mathbb{C}}[\hat{x},\frac{d}{dx}]/\langle\frac{d}{dx}\hat{x}\rangle&\mathrel{\twoheadrightarrow}&{\mathbb{C}}[\hat{x},\frac{d}{dx}]/\langle\hat{x}\rangle&\to
0\\\ \\\ &&\hfill 1&\mapsto&\hat{x}\hfill 1&\mapsto&1\hfill\\\ \\\
\text{define
ODEs}&&\frac{d}{dx}f=0&&\left(\frac{d}{dx}\hat{x}\right)f=0&&\hat{x}f=0\\\ \\\
\text{with solutions}&&1&&x^{-1}&&\delta&\text{($\delta$ = Dirac
delta)}\end{array}$

and those solutions can be identified with generators of our
${\mathcal{D}}_{\mathbb{C}}$-modules:

$\begin{array}[]{rcccccccc}0&\to&{\mathbb{C}}[x]&\mathrel{\hookrightarrow}&x^{-1}{\mathbb{C}}[x^{\pm}]&\mathrel{\twoheadrightarrow}&\delta{\mathbb{C}}[\delta]&\to&0\end{array}$

###  2.2. Characteristic cycles

The ${\mathcal{O}}_{A}$-algebra ${\mathcal{D}}_{A}$ is generated by vector
fields, sections of $TA$, which are used to build directional derivatives and
therefore define (noncommuting) operators on ${\mathcal{O}}_{A}$. If we
instead interpret sections of $TA$ as fiberwise linear functions on $T^{*}A$,
then they generate a different, commutative, algebra: the sheaf of
(polynomial, not just linear) functions on $T^{*}A$. One can make the relation
more precise: the degree of differential operators induces a filtration on
${\mathcal{D}}_{A}$, whose associated graded algebra $gr\ {\mathcal{D}}_{A}$
is ${\mathcal{O}}_{T^{*}A}$.

Given a ${\mathcal{D}}_{A}$-module ${\mathcal{F}}$, one could hope to filter
it as well, compatibly with the ${\mathcal{D}}_{A}$ filtration. At that point
${\mathcal{O}}_{T^{*}A}\circlearrowright gr\ {\mathcal{F}}$, and we can
consider its support cycle $supp(gr\
{\mathcal{F}})\subseteq{\mathcal{O}}_{T^{*}A}$ (which will typically have
multiplicities). It turns out (see e.g. [Bj13, definition 1.8.5]) that such
“good filtrations” exist, not uniquely enough to canonically define the sheaf
$gr\ {\mathcal{F}}$, but uniquely enough to well-define its support cycle. In
the running example from §2.1, we get the following short exact sequence of
modules over $gr\ {\mathcal{D}}_{\mathbb{C}}\mathrel{\cong}{\mathbb{C}}[x,y]$:

$\begin{array}[]{rcccccccl}0&\to&{\mathbb{C}}[x,y]/\langle
y\rangle&\mathrel{\hookrightarrow}&{\mathbb{C}}[x,y]/\langle
xy\rangle&\mathrel{\twoheadrightarrow}&{\mathbb{C}}[x,y]/\langle
y\rangle&\to&0\\\ &&\hfill 1&\mapsto&x\hfill 1&\mapsto&1\hfill&\end{array}$

The gradedness of $gr\ {\mathcal{F}}$ can be interpreted as its bearing a
circle action. This has the specific consequence that $supp(gr\
{\mathcal{F}})$ is a conical cycle inside $T^{*}A$, meaning, invariant under
the dilation action ${\mathbb{C}}^{\times}\circlearrowright T^{*}A$ that
scales the cotangent vectors. Consequently, we get a well-defined class

$[supp(gr\ {\mathcal{F}})]\ \in\
H^{*}_{{\mathbb{C}}^{\times}}(T^{*}A)\quad\mathrel{\cong}H^{*}_{{\mathbb{C}}^{\times}}(A)\mathrel{\cong}H^{*}(A){\otimes}H^{*}_{{\mathbb{C}}^{\times}}(pt)\mathrel{\cong}H^{*}(A)[\hbar]$

in the dilation-equivariant cohomology of $T^{*}A$, taking $\hbar$ as the
generator of $H^{*}_{{\mathbb{C}}^{\times}}(pt)$.

Perhaps the simplest example is ${\mathcal{F}}={\mathcal{O}}_{A}$. Then
$supp(gr\ {\mathcal{F}})$ is the zero section $A\subseteq T^{*}A$, and its
associated class is a $(-\hbar)$-homogenized version of the total Chern class
of the tangent bundle of $A$.

In this paper the only (complexes of) $D_{M}$-modules we will need consider
are of the form $R\iota_{*}({\mathcal{O}}_{A})$ for
$\iota:A\mathrel{\hookrightarrow}M$ the inclusion of a locally closed
submanifold. Hereafter we write

$cc(A\subseteq M):=\sum_{i}(-1)^{i}\ supp\left(gr\
R^{i}\iota_{*}({\mathcal{O}}_{A})\right)$

to denote the resulting “characteristic cycle”, a conical Lagrangian cycle
inside $T^{*}M$ defining an element $[cc(A\subseteq M)]\in
H^{*}_{{\mathbb{C}}^{\times}}(T^{*}M)\mathrel{\cong}H^{*}(M)[\hbar]$. In the
running example the derived pushforward $R^{i}\iota_{*}$ vanishes for $i>0$,
but in other examples such as ${\mathbb{C}}^{2}\setminus
0\mathrel{\hookrightarrow}{\mathbb{C}}^{2}$ (described in more detail later)
one must include some such higher pushforwards.

In general $cc(A\subseteq M)$ is very complicated, with many components with
various multiplicities. One component is the closure of the conormal bundle to
$A$, and each other component is the closure of the conormal bundle to some
locally closed submanifold $B\subseteq\overline{A}\setminus A$.

###  2.3. Chern-Schwartz-MacPherson classes and their additivity

Recall that a constructible function $f$ on $M$ is a finite linear combination
(with ${\mathbb{Z}}$-coefficients, say) of characteristic functions of closed
subvarieties. By splitting a subvariety into its regular and singular locus,
we can instead think of $f$ as a linear combination
$\sum_{A\in{\mathcal{A}}}n_{A}1_{A}$ of characteristic functions of locally
closed algebraic submanifolds $A\subseteq M$. This expansion is not unique,
however, so we need to treat it with care.

For example, write $1_{\mathbb{C}}=1_{{\mathbb{C}}^{\times}}+1_{0}$. To these
three subsets we can associate ${\mathcal{D}}_{\mathbb{C}}$-modules as
computed in our running example, and the characteristic cycles $\\{y=0\\}$,
$\\{xy=0\\}$, $\\{x=0\\}$ respectively. The exactness of the sequence from
§2.1 leads to a vanishing of its Euler characteristic, as an equation on
cycles:

$\begin{array}[]{rccccccc}&cc({\mathbb{C}}\subseteq{\mathbb{C}})&-&cc({\mathbb{C}}^{\times}\subset{\mathbb{C}})&+&cc(\\{0\\}\subset{\mathbb{C}})&=&0\\\
\\\
\text{Pictorially:}&\text{\huge--}&\text{minus}&\text{\huge+}&\text{plus}&\big{|}&=&0\end{array}$

This alternating-sum statement doesn’t quite match
$1_{\mathbb{C}}-1_{{\mathbb{C}}^{\times}}-1_{0}=0$. To fix this mismatch, for
$\iota_{*}:A\subseteq M$ a locally closed submanifold, we define its Chern-
Schwartz-MacPherson class $csm(1_{A})$ as

$csm(1_{A}):=(-1)^{\operatorname{codim}_{M}A}\ [cc(A\subseteq M)]\qquad\in
H^{*}_{{\mathbb{C}}^{\times}}(T^{*}M).$

With the signs integrated into the definition, it is then a theorem that this
definition on $\\{1_{A}\\}$ extends in a well-defined way to constructible
functions, at which point it is additive. (While it wasn’t important in the
running example, in bigger examples this additivity relies on $cc$ having been
defined using the derived pushforward.)

The traditional definition is slightly off from this – it lives in homology
rather than cohomology, and is dehomogenized by setting $-\hbar$ to $1$. We
take this opportunity to rant about the horrific unnaturality of considering
inhomogeneous elements of cohomology, insofar as cohomology should so very
often be understood as the associated graded to $K$-theory. In an associated
graded space, only homogeneous elements can properly be asked to possess
lifts. While this concludes the rant, we will retain the powers of $\hbar$
through the rest of this paper.

The Deligne-Grothendieck conjecture, proven by MacPherson [Ma74],
characterized these CSM classes by a recurrence relation (a functoriality
under proper pushforward) and a base case ($A=M$ smooth and proper). One
philosophical reason to prefer the description from [Gi86] recapitulated here
is its individual definition for each $A\subseteq M$, rather than reliance on
a recurrence relation.

###  2.4. Weber’s divisibility property

Taking $A\subseteq M$ as before, and a torus $T$ acting on $M$ preserving $A$,
then we can use the same definition to associate an equivariant CSM class
$csm(1_{A})\in H^{*}_{T\times{\mathbb{C}}^{\times}}(T^{*}M)$ (a slightly
different approach appears in [Oh06]).

######  Lemma 2.

[We12, theorem 20] If $p\in M^{T}$ is isolated, and $p\notin A$, then
$csm(1_{A})|_{p}\equiv 0\bmod\hbar$.

That requires a straightforward bit of translation from [We12], as CSM classes
there are inhomogeneous.

##  3\. The main theorem: the geometry of cone terms

We are now ready to give an algebro-geometric interpretation of the individual
cone terms in Heckman’s formula. This is in similar spirit to the approach of
[GuOKaGi02], where these cone terms are interpreted as components of the
(other end of the) boundary of a noncompact symplectic cobordism. It would be
interesting to connect the two approaches, perhaps through the algebraic
cobordism of [LevP09].

Our input is a complex projective symplectic manifold $(M,\omega)$ with an
algebraic action of a torus $T$, and a Białynicki-Birula decomposition defined
using a circle $S\mathrel{\hookrightarrow}T$ (or equivalently, a Morse
decomposition defined using a component of $T$’s moment map). We assume
$M^{T}$ finite, and $S$ generic enough that $M^{S}=M^{T}$, then write the
decomposition into attracting sets as $M=\coprod_{p\in M^{T}}M_{p}^{\circ}$.
Each inclusion $\iota^{p}:M_{p}^{\circ}\mathrel{\hookrightarrow}M$ defines a
Lagrangian cycle $cc(M_{p}^{\circ}\subseteq M)$ in $T^{*}M$, as in §2.

We give $T^{*}M$ the degenerate $2$-form $\omega_{+}:=\pi^{*}(\omega)$ where
$M\xhookrightarrow{\iota}T^{*}M\xrightarrow[]{\pi}\mathrel{\mkern-14.0mu}\rightarrow
M$ are the inclusion and projection. This choice is dictated by wanting
$\omega_{+}$ to be invariant, not just a weight vector, under the dilation
action ${\mathbb{C}}^{\times}\circlearrowright T^{*}M$ on the fibers, and
wanting $\iota^{*}(\omega_{+})$ to be $\omega$. (There is a familiar
nondegenerate $2$-form “$d\alpha$” available on $T^{*}M$, which one might be
tempted to add to $\omega_{+}$. As that form is exact it wouldn’t affect our
cohomology-based calculations, but it would spoil the dilation-invariance.)
Now we compute:

$\begin{array}[]{rcll}&&DH_{{\mathbb{C}}^{\times}\times T}(M\subseteq
M,\omega)\\\\[2.84544pt] &=&DH_{{\mathbb{C}}^{\times}\times T}(M\subseteq
T^{*}M,\omega_{+})&\text{since $\iota^{*}(\omega_{+})=\omega$}\\\\[5.69046pt]
&=&F.T.\int_{T^{*}M}[M]\exp(\tilde{\omega}_{+})\ =\
F.T.\int_{T^{*}M}csm(1_{M})\exp(\tilde{\omega}_{+})\\\\[2.84544pt]
&=&F.T.\int_{T^{*}M}csm\left(\sum_{p}1_{M_{p}^{\circ}}\right)\exp(\tilde{\omega}_{+})\\\\[5.69046pt]
&=&\sum_{p}F.T.\int_{T^{*}M}csm\left(1_{M_{p}^{\circ}}\right)\exp(\tilde{\omega}_{+})&\text{by
additivity of CSM classes}\\\\[5.69046pt]
&=&\sum_{p}F.T.\int_{T^{*}M}(-1)^{\operatorname{codim}_{M}M_{p}^{\circ}}[cc(M_{p}^{\circ}\subseteq
M)]\exp(\tilde{\omega}_{+})&\text{as in \S\ref{ssec:csm}}\\\\[5.69046pt]
&=&\sum_{p}(-1)^{\operatorname{codim}_{M}M_{p}^{\circ}}\
F.T.\int_{T^{*}M}[cc(M_{p}^{\circ}\subseteq
M)]\exp(\tilde{\omega}_{+})\\\\[5.69046pt]
&=&\sum_{p}(-1)^{\operatorname{codim}_{M}M_{p}^{\circ}}\
DH_{{\mathbb{C}}^{\times}\times T}(cc(M_{p}^{\circ}\subseteq M)\subseteq
T^{*}M,\omega_{+})\end{array}$

Before comparing (not equating!) our terms $DH_{{\mathbb{C}}^{\times}\times
T}(cc(M_{p}^{\circ}\subseteq M)\subseteq T^{*}M,\omega_{+})$ to the cone terms
in the Heckman formula, we consider the basic example
$M={\mathbb{C}\mathbb{P}}^{1}={\mathbb{C}}\coprod\\{\infty\\}$. Pictured below
are the moment images of $T^{*}M$,
$cc({\mathbb{C}}\subseteq{\mathbb{C}\mathbb{P}}^{1})$,
$cc(\\{0\\}\subseteq{\mathbb{C}\mathbb{P}}^{1})$ all with respect to the two-
torus ${\mathbb{C}}^{\times}\times T$, where $T$ is the maximal torus of
$PGL_{2}({\mathbb{C}})$ and ${\mathbb{C}}^{\times}$ acts by dilation on the
cotangent fibers.

Note that in this tiny example, $T^{*}M$ is toric w.r.t. our augmented torus
${\mathbb{C}}^{\times}\times T$, but this will never happen in larger
examples.

Even without the pictures, there is an obvious difference between the terms in
this alternating sum vs. the ones in the Heckman formula: in this sum, the
terms involve an extra ${\mathbb{C}}^{\times}$ action, dilating the fibers of
the cotangent bundle. (Note too that the moment image of
$cc({\mathbb{C}}\subseteq{\mathbb{C}}{\mathbb{P}}^{1})$ is not a polytope,
which can be blamed on the characteristic cycle being reducible.)

To drop that action, consider the inclusion
$T\mathrel{\hookrightarrow}S^{1}\times T$ inducing
$T^{*}\times{\mathbb{Z}}\mathrel{\twoheadrightarrow}T^{*}$, which

  * •

on the cohomology algebra level, amounts to setting $\hbar\to 0$, and

  * •

on the moment polytope level, amounts to pushing forward the measure along the
projection
${\mathfrak{t}}^{*}\times\mathbb{R}\mathrel{\twoheadrightarrow}{\mathfrak{t}}^{*}$.

In the $2$-dimensional pictures above, that amounts to projecting the measures
to the horizontal line, from which to obtain the usual formula for
$DH_{T}({\mathbb{C}\mathbb{P}}^{1},\omega)$:

.

One might consider those $2$-dimensional pictures above as “bent” versions of
the half-lines that we really want, but that we only obtain after the dilation
action is suppressed. Looking back at the ${\mathbb{C}\mathbb{P}}^{2}$ example
in §1, this extra action can be pictured by seeing the triangle as flat in the
page, and the other regions as coming out of the page. The dotted lines
indicate level sets in those $3$-dimensional pictures.

We have now arrived at the main theorem, giving geometric interpretation to
the individual terms in the Heckman formula: it is almost correct (and
conjecturally correct) to say they are themselves DH measures, not of $M$ but
of the characteristic cycles $cc(M_{p}^{\circ}\subseteq M)\subseteq T^{*}M$.

######  Theorem 1.

Let $S\mathrel{\hookrightarrow}T\circlearrowright(M,\omega)$, $M=\coprod_{p\in
M^{T}}M_{p}^{\circ}$ be as described at the beginning of §3. Then
$DH_{{\mathbb{C}}^{\times}\times T}(cc(M_{p}^{\circ}\subseteq M)\subseteq
T^{*}M,\omega_{+})$ is Fourier equivalent (and conjecturally equal) to a
measure whose projection to ${\mathfrak{t}}^{*}$ is proper, and that
projection is $p$’s cone term from the Heckman formula.

There are two subtleties in the theorem’s statement. What we are really after
could reasonably be called $DH_{T}(cc(M_{p}^{\circ}\subseteq M)\subseteq
T^{*}M,\omega_{+})$. One problem is that the $T$-moment map on $T^{*}M$ isn’t
proper for $\dim M>0$. We believe that its restriction to
$cc(M_{p}^{\circ}\subseteq M)$ is proper, but (a) this has been frustratingly
elusive and (b) that characteristic cycle is typically singular so we prefer
to keep our integration definition on the manifold $T^{*}M$. (In §4.1 we study
a slightly different situation where the $T$-moment map is not proper on the
characteristic cycle.)

###### Proof.

Let $C$ denote the cycle $cc(M^{\circ}_{p}\subseteq M)$. We recall that it
consists of the closure of the conormal bundle to ${M^{\circ}_{p}}$ union
various other conormal varieties living over
$\overline{M^{\circ}_{p}}\setminus M^{\circ}_{p}$. In particular

$[C]|_{p}=\prod_{\lambda\in T_{p}M}\begin{cases}\lambda&\text{if $\lambda$
defines a positive $S$-weight}\\\ \hbar-\lambda&\text{if $\lambda$ defines a
negative $S$-weight}\\\ \end{cases}$

(no $\lambda$ will define the $S$-weight $0$, by our choice of $S$).

The Fourier transform of $C$’s DH measure on
${\mathfrak{t}}^{*}\times\mathbb{R}$ is $\sum_{f\in
M^{T}}\exp(-\Phi(f))\frac{[C]|_{f}}{\prod_{\lambda\in
wts(T_{f}M)}\lambda(\hbar-\lambda)}$. To associate a measure to it (which
might only be Fourier equivalent to the actual DH measure), as in §1 we need
to flip some weights in denominators. We make that choice using the generator
of $S$’s Lie algebra. It is easy to see that the resulting measure has proper
projection along the composite
${\mathfrak{t}}^{*}\times\mathbb{R}\mathrel{\twoheadrightarrow}{\mathfrak{t}}^{*}\mathrel{\twoheadrightarrow}{\mathfrak{s}}^{*}$,
hence has proper projection to ${\mathfrak{t}}^{*}$.

To compute the projection to ${\mathfrak{t}}^{*}$, on the Fourier transform
side, amounts to setting $\hbar\to 0$. Now we use lemma 2 to note that
$[C]|_{f}\equiv 0\bmod\hbar$ for $f\neq p$. Hence our sum reduces to a single
term

$\left(\exp(-\Phi(p))\frac{[C]|_{p}}{\prod_{\lambda\in
wts(T_{p}M)}\lambda(\hbar-\lambda)}\right)\bigg{|}_{\hbar\to
0}=(-1)^{\operatorname{codim}_{M}M_{p}^{\circ}}\exp(-\Phi(p))\bigg{/}{\prod_{\lambda\in
wts(T_{p}M)}}\lambda_{+}$

which is exactly the term in the localization formula. ∎

It is worth spelling out the interconnectedness of the different points of
view in the case of the flag manifold, as in Heckman’s thesis, which gives the
asymptotic version of Kostant’s multiplicity formula (see [GiLerS96, §3] for
the connection). Our derivation is based on the ${\mathcal{D}}_{G/B}$-modules
associated to Bruhat cells; the global sections of these are the Verma
modules. The exact sequence given in §2.1 for a single divisor, when extended
to the full Bruhat decomposition, gives the BGG resolution involving those
Verma modules (see [Ke77]). The complexity we meet here, with the “bending” of
the individual cone terms, is closely related to the complexity (the non-
simplicity) of Verma modules. (It is not quite the same complexity, as even a
simple ${\mathcal{D}}$-module can have reducible characteristic cycle, a well-
known example being that of Kashiwara-Saito.)

##  4\. The Brianchon-Gram theorem and other extensions

Let $M$ be a smooth projective toric variety, with a moment polytope
$P\subset{\mathfrak{t}}^{*}$. Instead of using a Morse decomposition, we
consider the full decomposition $M=\bigcup_{F\subset P}M_{F}^{\circ}$ into
$T^{\mathbb{C}}$-orbits, one for each face of $P$. Then as in §2 we obtain

(1) $DH_{{\mathbb{C}}^{\times}\times T}(M\subseteq
M,\omega)=\sum_{F}(-1)^{\operatorname{codim}_{P}F}\
DH_{{\mathbb{C}}^{\times}\times T}(cc(M_{F}^{\circ}\subseteq M)\subseteq
T^{*}M,\ \omega_{+})$

(though the $\operatorname{codim}$ in the exponent is now the real
codimension). Then as in theorem 1, we project the measure from
$\mathbb{R}\times{\mathfrak{t}}^{*}$ to ${\mathfrak{t}}^{*}$.

######  Theorem 2.

Let $(M,\omega)$ is a symplectic toric manifold with moment polytope
$P\subseteq{\mathfrak{t}}^{*}$, let $M_{F}^{\circ}\subseteq M$ be the
$T$-orbit corresponding to a face $F\subseteq P$, and let $v$ be any vertex of
$F$. Let $T_{\mu}F$ denote the (real) tangent space to an interior point $\mu$
of $F$, and $W\subset T^{*}$ denote the primitive integer vectors along the
edges from $v$ out of $F$.

Then the pushforward to ${\mathfrak{t}}^{*}$ of the measure
$DH_{{\mathbb{C}}^{\times}\times T}(cc(M_{F}^{\circ}\subseteq M)\subseteq
T^{*}M,\ \omega_{+})$ is

$\pi_{*}(\text{Lebesgue measure on $T_{\mu}F\times{\mathbb{R}_{\geq
0}}^{W}$}),\quad\text{where}\quad\begin{array}[]{rrcl}\pi:&{T_{\mu}F\times\mathbb{R}_{\geq
0}}^{W}&\to&{\mathfrak{t}}^{*}\\\ &(\vec{v},(x_{\lambda})_{\lambda\in
W})&\mapsto&\mu+\left(\vec{v}-\sum_{\lambda\in
W}x_{\lambda}\lambda\right)\end{array}$

where we normalize the measure on $T_{\mu}F\leq{\mathfrak{t}}^{*}$ using its
intersection with the lattice $T^{*}$.

If we pushforward the LHS of equation (1) to ${\mathfrak{t}}^{*}$, we get
Lebesgue measure on $P$, and the pushforward of the RHS gives the “Brianchon-
Gram formula”: an alternating sum over all $F$ of the cone centered at $F$,
with lineality space $T_{\mu}F$, and generators $W$ as defined above.

The Brianchon-Gram formula was given a Heckman-like derivation also in
[HKa12], through a somewhat technical construction of a function with one
critical point in the interior of each face.

### Revisiting ${\mathbb{C}\mathbb{P}}^{2}$

Before getting into the proof of theorem 2, we look again at the
${\mathbb{C}\mathbb{P}}^{2}$ example from §1. There are seven orbits, where
the open orbit gives the entire plane, the $1$-dimensional orbits give half-
planes, and the fixed points give sectors. (As before, we have attempted to
indicate the $\hbar$ direction out of the plane, using dashed level sets. The
first characteristic cycle has seven components and the next three each have
three.) We exhort the reader to check that the choices of vertices $v\in F$
are immaterial.

###### Proof of theorem 2.

For this we use the decomposition of $M$ into its $T^{\mathbb{C}}$-orbits. For
each such orbit $E^{\circ}\subset M$, with closure we call $E$, observe that
$1_{E^{\circ}}=\sum_{F\subseteq E}(-1)^{\operatorname{codim}_{E}F}\ 1_{F}$
where the sum is over smaller $T^{\mathbb{C}}$-orbit closures. Then

$csm(1_{E^{\circ}})=\sum_{F\subseteq
E}(-1)^{\operatorname{codim}_{E}F}csm(1_{F})$

and since $F$ is smooth, and closed in $M$, its characteristic cycle is just
its conormal bundle $C_{M}F$. Consequently
$[cc(1_{E^{\circ}})]=\sum_{F\subseteq E}[C_{M}F]$ (where the sign we had from
inclusion-exclusion cancels with the one in Ginzburg’s formula for CSM
classes).

To understand the DH measure associated to this sum, consider the (non-
central) hyperplane arrangement defining the polytope $P$, and many other
regions in ${\mathfrak{t}}^{*}$. Not every region touches $P$ (unless $P$ is a
product of simplices – for a first cautionary example, consider a trapezoid),
so we work in a small open neighborhood $P_{+}$ of $P$ to avoid consideration
of those other regions. Each hyperplane divides space into an “inside” (where
$P$ is) and an “outside”. The moment polytopes of the individual $C_{M}F$ in
the sum, intersected with $P_{+}$, are the exactly the regions that touch $F$
and are on the outside of each of the hyperplanes through $F$. When we add
them, we get the Brianchon-Gram term associated to $F$. ∎

There is another theorem also called Brianchon-Gram, in which the cones point
inward rather than outward, but the total is $(-1)^{\dim P}$ times Lebesgue
measure on $P$. One can obtain that from this by scaling the symplectic form
on $M$ by $-1$, which turns $P$ inside out.

The additivity of CSM classes suggests that we wildly generalize to any
$T$-invariant decomposition of $M$ into locally closed submanifolds. We give
an example now to demonstrate the dangers.

###  4.1. A problematic decomposition

Let $T$ be one-dimensional this time, acting on $V$ with weights $0,1,2$, and
decompose ${\mathbb{P}}V$ into the projective point $[0,*,0]$ and the open
complement $A$. Then
$DH_{T\times{\mathbb{C}}^{\times}}({\mathbb{C}\mathbb{P}}^{2},\omega_{+})$ is
a piecewise-linear function times Lebesgue measure on the interval connecting
$(0,0)$ and $(2,0)$. The inclusion
$\iota:A\mathrel{\hookrightarrow}{\mathbb{C}\mathbb{P}}^{2}$ is perhaps
already worrisome in that $R^{1}\iota_{*}\neq 0$. Forging ahead, we calculate
$DH_{T\times{\mathbb{C}}^{\times}}(cc(S\subseteq M)\subseteq
T^{*}M,\omega_{+})$ for $S\in\\{\\{[0,*,0]\\},A\\}$ and obtain the following
picture:

In the shaded regions we have $\pm\frac{1}{2}$ Lebesgue measure (where the
“minus” comes from a contribution from the derived pushforward). There is now
a serious impropriety if we try to forget the ${\mathbb{C}}^{\times}$ action,
projecting out the vertical direction.

This could be fixed by replacing these measures with Fourier equivalent ones
pointing rightward. Our conjecture within theorem 1 is that in the case of BB
decompositions (which does not include this example), that replacement is
unnecessary.

### A non-Morse decomposition

Consider the decomposition of ${\mathbb{C}\mathbb{P}}^{2}=\\{[x,y,z]\\}$ into

$\\{xyz\neq 0\\}\coprod\\{x=0,y\neq 0\\}\coprod\\{y=0,z\neq
0\\}\coprod\\{z=0,x\neq 0\\}.$

If we follow the proof of theorem 2, but use this decomposition, we get the
following equality of measures

which of course one could obtain by partial cancelation of the Brianchon-Gram
formula we drew after theorem 2.

We didn’t here discuss nonabelian versions of our results, which hopefully
would allow for a similar geometric interpretation of [Pa99]. It is worth
pointing out that when
$T\mathrel{\hookrightarrow}G\circlearrowright(M,\omega)$, it is frequently
possible that $G$’s moment map is proper even though $T$’s isn’t (e.g.
$T^{1}\mathrel{\hookrightarrow}SU(2)\circlearrowright{\mathbb{C}}^{2}$), so
one can’t obviously derive the nonabelian from the abelian.

We end with a question, another conjecture, and an example.

> Q. Let $M$ be a smooth complex projective variety $M$ with a $T$-action, and
> $A\subseteq M$ a locally closed $T$-invariant smooth subvariety. What
> condition on $A$ guarantees that the projection of
> $supp(DH_{T\times{\mathbb{C}}^{\times}}(cc(A)\subseteq T^{*}M,\omega_{+}))$
> to ${\mathfrak{t}}^{*}$ is proper?

There are two issues to be wary of – higher cohomology involved in defining
$cc(A)$, and improperness of $A$ itself. The following is an attempt to deal
with each of those:

> Conjecture. Assume ${\overline{A}}\setminus A$ (the points added in the
> closure) supports an ample Cartier divisor in ${\overline{A}}$. Assume that
> there exists an open $T$-invariant subset $U\subseteq CA$ of the conormal
> bundle to $A$ such that $U/T$ is a proper scheme. Then the projection in the
> question above is proper.

We describe an example that would be covered by this conjecture. Consider the
flag manifold $GL(3)/B$, its divisor $X^{231}:=\overline{Br_{1}r_{2}B}/B$, and
its rotations $cX^{231},c^{2}X^{231}$ where $c=\begin{bmatrix}0&1&0\\\
0&0&1\\\ 1&0&0\end{bmatrix}$. When we intersect those divisors pairwise, we
get three ${\mathbb{P}}^{1}$s. Each stratum (the open stratum, three
${\mathbb{P}}^{1}\times{{\mathbb{G}}_{m}}$, three ${\mathbb{P}}^{1}$) has
trivial normal bundle. The analogue of theorem 2 for this example, computing
the usual piecewise-linear measure on the hexagon, looks as follows:

$\begin{matrix}+&\begin{matrix}\text{a multiple of}\\\ \text{Lebesgue
measure}\\\ \text{on the whole plane}\\\ \text{from open
stratum}\end{matrix}&-&\begin{matrix}\text{a piecewise}\\\ \text{linear
measure}\\\ \text{on a half-plane}\\\ \text{from a
divisor}\end{matrix}&-&\begin{matrix}\text{another}\end{matrix}&-&\begin{matrix}\text{another
}\end{matrix}+&\begin{matrix}\text{a piecewise}\\\ \text{linear measure}\\\
\text{on three half-planes}\\\ \text{from three
curves}\end{matrix}\end{matrix}$

This gives a manifestly $S_{3}$-invariant formula for the measure.

## References

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# Multiple closed geodesics on Finsler $3$-dimensional sphere

Huagui Duan, Zihao Qi

School of Mathematical Sciences and LPMC, Nankai University,
Tianjin 300071, P. R. China
Partially supported by National Key R&D Program of China (No. 2020YFA0713300)
and NNSFC (Nos. 12271268, 11671215 and 11790271)). E-mail:
<EMAIL_ADDRESS>supported by NNSFC (No. 12271268). E-mail:
<EMAIL_ADDRESS>
###### Abstract

In 1973, Katok constructed a non-degenerate (also called bumpy) Finsler metric
on $S^{3}$ with exactly four prime closed geodesics. And then Anosov
conjectured that four should be the optimal lower bound of the number of prime
closed geodesics on every Finsler $S^{3}$. In this paper, we proved this
conjecture for bumpy Finsler $S^{3}$ if the Morse index of any prime closed
geodesic is nonzero.

Key words: Closed geodesic, Finsler metric, $3$-dimensional sphere, Index
theory

2000 Mathematics Subject Classification: 53C22, 58E05, 58E10.

##  1 Introduction and main result

A closed curve on a Finsler manifold is a closed geodesic if it is locally the
shortest path connecting any two nearby points on this curve. As usual, on any
Finsler manifold $(M,F)$, a closed geodesic $c:S^{1}={\bf R}/{\bf Z}\to M$ is
prime if it is not a multiple covering (i.e., iteration) of any other closed
geodesics. Here the $m$-th iteration $c^{m}$ of $c$ is defined by
$c^{m}(t)=c(mt)$. The inverse curve $c^{-1}$ of $c$ is defined by
$c^{-1}(t)=c(1-t)$ for $t\in{\bf R}$. Note that unlike a Riemannian manifold,
the inverse curve $c^{-1}$ of a closed geodesic $c$ on a irreversible Finsler
manifold need not be a geodesic. We call two prime closed geodesics $c$ and
$d$ distinct if there is no ${\theta}\in(0,1)$ such that $c(t)=d(t+{\theta})$
for all $t\in{\bf R}$. On a reversible Finsler (or Riemannian) manifold, two
closed geodesics $c$ and $d$ are called geometrically distinct if
$c(S^{1})\neq d(S^{1})$, i.e., their image sets in $M$ are distinct. We shall
omit the word distinct when we talk about more than one prime closed geodesic.

For a closed geodesic $c$ on $n$-dimensional manifold $(M,\,F)$, denote by
$P_{c}$ the linearized Poincaré map of $c$. Then $P_{c}\in{\rm Sp}(2n-2)$ is
symplectic. For any $P\in{\rm Sp}(2k)$, we define the elliptic height  $e(P)$
of $P$ to be the total algebraic multiplicity of all eigenvalues of $P$ on the
unit circle ${\bf U}=\\{z\in{\bf C}|\;|z|=1\\}$ in the complex plane ${\bf
C}$. Since $P$ is symplectic, $e(P)$ is even and $0\leq e(P)\leq 2k$. A closed
geodesic $c$ is called elliptic if $e(P_{c})=2(n-1)$, i.e., all the
eigenvalues of $P_{c}$ locate on ${\bf U}$; non-degenerate if $1$ is not an
eigenvalue of $P_{c}$. A Finsler manifold $(M,\,F)$ is called bumpy if all the
closed geodesics (including iterations) on it are non-degenerate.

There is a famous conjecture in Riemannian geometry which claims there exist
infinitely many closed geodesics on any compact Riemannian manifold. This
conjecture has been proved except for CROSS’s (compact rank one symmetric
spaces). The results of Franks [Fra] in 1992, Hingston [Hin2] and Bangert
[Ban] in 1993 imply this conjecture is true for any Riemannian $S^{2}$.

For the Finsler metric, the closed geodesic problem is completely different.
It was quite surprising when Katok [Kat] in 1973 found some irreversible bumpy
Finsler metrics on CROSS’s with only finitely many closed geodesics and all
closed geodesics are non-degenerate and elliptic (cf. [Zil]). Based on Katok’s
examples, in 1974, Anosov [Ano] conjectured that the optimal lower bound of
the number of prime closed geodesics on any Finsler $S^{n}$ should be
$2[\frac{n+1}{2}]$. We refer readers to two excellent survey papers [Lon4] and
[BuK] for more interesting questions.

The index iteration theory of closed geodesics (cf.[Bot] and [Lon3]) has been
an important and powerful tool in studying the closed geodesic problem on
Finsler manifolds. For example, Bangert and Long in [BaL] (finished in 2005)
shows that there exist at least two prime closed geodesics on every Finsler
$S^{2}$, which solved the Anosov conjecture. And then a great number of
multiplicity and stability results about closed geodesics which are
established by index iteration theory have appeared (cf. [DuL1]-[DuL2], [DLW],
[Lon4], [Rad3], [Wan1]-[Wan3] and therein).

This paper is mainly devoted to study the Anosov conjecture of bumpy Finsler
$S^{3}$. In this direction, Duan and Long in [DuL1] and Rademacher in [Rad3]
proved the existence of at least two prime closed geodesics on bumpy Finsler
$S^{n}$, respectively. Wang in [Wan2] solved the Anosov conjecture for any
positively curved bumpy Finsler $S^{n}$. Recently, the Anosov conjecture on
bumpy Finsler CROSS’s (including spheres) has been proved by Duan, Long and
Wang in [DLW] under the much weaker curvature or index conditions, and
especially for $S^{3}$ the index condition is $i(c)\geq 2$ for any prime
closed geodesic $c$, which, to the authors’ knowledge, is the weakest
condition for Anosov conjecture of bumpy Finsler $S^{3}$. Actually we also
notice that in Katok’s emxample, the Morse index of every prime closed
geodesic on $S^{3}$ is equal to and greater than $2$.

Motivated by the Katok’s examples and above results, in this paper we
established the following result, which resolve the Anosov conjecture of bumpy
$S^{3}$ under much more relaxing restriction.

Theorem 1.1. Assume that $F$ is a bumpy Finsler metric on $S^{3}$, then there
exist at least four prime closed geodesics on $(S^{3},F)$ if the Morse index
of every prime closed geodesic is nonzero.

Under the assumption of Theorem 1.1, it follows from Remark 3.7 of [DuL2] and
Theorem 1.4 of [DLW] that there exist at least three prime closed geodesics on
bumpy Finsler $S^{3}$. Then, through assuming the existence of exactly three
prime closed geodesics, we will use a contradiction argument to complete the
proof of Theorem 1.1.

Since here we only assume the Morse index $i(c)\geq 1$ for any prime closed
geodesic $c$, generally the sequence $\\{i(c^{m})\\}$ will have no
monotonicity with respect to $m$, which will lead to the invalidness of
methods in [DLW] and [Wan2]. On the other hand, due to the assumption of the
existence of three prime closed geodesics, the methods in [DuL2] through some
complicated classification arguments and the mean index identity can not give
any valuable information in this situation.

To overcome the above difficulties, the main novelty in this paper is to
establish a variant of generalized common index jump theorem (see Theorem 3.8
below), which, compared with Theorem 3.6 of [DLLW] (cf. Theorem 3.6 below),
gives more information about some parameters including the common integer $N$
and iterations $m_{k}$s. Then we use the Morse theory, some precise analysis
and estimates of common iteration indices to derive a contradiction. Because
this method does not use the special classification of closed geodesics, it
may be hopeful to be applied to studying the corresponding problems on higher
Finsler manifolds.

When the assumption of Theorem 1.1 is not satisfied, i.e. such $(S^{3},F)$
possesses at least one prime closed geodesic with zero Morse index, one
suspects the existence of infinitely many prime closed geodesics. A related
result with this is the Corollary 2 of [BaK], which shows that for a compact
manifold $M$ with finite fundamental group, suppose there exists a closed
geodesic $c$ on $M$ such that $c^{m}$ is a local minimum of $E$ for infinitely
many $m\in{\bf N}$, then there exist infinitely many prime closed geodesics on
$M$. So we further propose the following conjecture.

Conjecure 1.2. Let $F$ be a bumpy Finsler metric on $n$-dimensional sphere
$S^{n}$. If there exist only finitely many prime closed geodesics on
$(S^{n},F)$, then none of the prime closed geodesics is a local minimal of the
energy functional $E$, which implies $i(c)\geq 1$ for any prime closed
geodesic $c$.

In this paper, let ${\bf N}$, ${\bf N}_{0}$, ${\bf Z}$, ${\bf Q}$, ${\bf R}$,
and ${\bf C}$ denote the sets of natural integers, non-negative integers,
integers, rational numbers, real numbers, and complex numbers respectively. We
use only singular homology modules with ${\bf Q}$-coefficients. For an
$S^{1}$-space $X$, we denote by $\overline{X}$ the quotient space $X/S^{1}$.
We define the functions

$\left\\{\matrix{[a]=\max\\{k\in{\bf Z}\,|\,k\leq a\\},&E(a)=\min\\{k\in{\bf
Z}\,|\,k\geq a\\},\cr\varphi(a)=E(a)-[a],&\\{a\\}=a-[a].\cr}\right.$ (1.1)

Especially, $\varphi(a)=0$ if $a\in{\bf Z}\,$, and $\varphi(a)=1$ if
$a\notin{\bf Z}\,$.

##  2 Critical modules of closed geodesics and Morse Theory

Let $M=(M,F)$ be a compact Finsler manifold $(M,F)$, the space
$\Lambda=\Lambda M$ of $H^{1}$-maps $\gamma:S^{1}\rightarrow M$ has a natural
structure of Riemannian Hilbert manifold on which the group $S^{1}={\bf
R}/{\bf Z}$ acts continuously by isometries, cf. [Kli], Chapters 1 and 2. This
action is defined by $(s\cdot\gamma)(t)=\gamma(t+s)$ for all
$\gamma\in{\Lambda}$ and $s,t\in S^{1}$. For any $\gamma\in\Lambda$, the
energy functional is defined by

$E(\gamma)=\frac{1}{2}\int_{S^{1}}F(\gamma(t),\dot{\gamma}(t))^{2}dt.$ (2.1)

It is $C^{1,1}$ and invariant under the $S^{1}$-action. The critical points of
$E$ of positive energies are precisely the closed geodesics $\gamma:S^{1}\to
M$. The index form of the functional $E$ is well defined along any closed
geodesic $c$ on $M$, which we denote by $E^{\prime\prime}(c)$. As usual, we
denote by $i(c)$ and $\nu(c)$ the Morse index and nullity of $E$ at $c$. In
the following, we denote by

${\Lambda}^{\kappa}=\\{d\in{\Lambda}\;|\;E(d)\leq\kappa\\},\quad{\Lambda}^{\kappa-}=\\{d\in{\Lambda}\;|\;E(d)<\kappa\\},\quad\forall\kappa\geq
0.$

For a closed geodesic $c$ we set ${\Lambda}(c)=\\{{\gamma}\in{\Lambda}\mid
E({\gamma})<E(c)\\}$.

For $m\in{\bf N}$ we denote the $m$-fold iteration map
$\phi_{m}:\Lambda\rightarrow\Lambda$ by $\phi_{m}({\gamma})(t)={\gamma}(mt)$,
for all $\,{\gamma}\in{\Lambda},t\in S^{1}$, as well as
${\gamma}^{m}=\phi_{m}(\gamma)$. If $\gamma\in\Lambda$ is not constant then
the multiplicity $m(\gamma)$ of $\gamma$ is the order of the isotropy group
$\\{s\in S^{1}\mid s\cdot\gamma=\gamma\\}$. For a closed geodesic $c$, the
mean index $\hat{i}(c)$ is defined as usual by
$\hat{i}(c)=\lim_{m\to\infty}i(c^{m})/m$. Using singular homology with
rational coefficients we consider the following critical ${\bf Q}$-module of a
closed geodesic $c\in\Lambda$:

$\overline{C}_{*}(E,c)=H_{*}\left(({\Lambda}(c)\cup S^{1}\cdot
c)/S^{1},{\Lambda}(c)/S^{1}\right).$ (2.2)

The following results of Rademacher will be used in our proofs below.

Proposition 2.1. (cf. Satz 6.11 of [Rad2] ) Let $c$ be a prime closed geodesic
on a bumpy Finsler manifold $(M,F)$. Then there holds

$\overline{C}_{q}(E,c^{m})=\left\\{\matrix{{\bf Q},&\quad{\it
if}\;\;i(c^{m})-i(c)\in 2{\bf Z}\;\;{\it and}\;\;q=i(c^{m}),\;\cr 0,&\quad{\it
otherwise}.\cr}\right.$

Definition 2.2. (cf. Definition 1.6 of [Rad1]) For a closed geodesic $c$, let
${\gamma}_{c}\in\\{\pm\frac{1}{2},\pm 1\\}$ be the invariant defined by
${\gamma}_{c}>0$ if and only if $i(c)$ is even, and $|{\gamma}_{c}|=1$ if and
only if $i(c^{2})-i(c)$ is even.

Let $(X,Y)$ be a space pair such that the Betti numbers $b_{i}=b_{i}(X,Y)=\dim
H_{i}(X,Y;{\bf Q})$ are finite for all $i\in{\bf Z}$. As usual the Poincaré
series of $(X,Y)$ is defined by the formal power series
$P(X,Y)=\sum_{i=0}^{\infty}b_{i}t^{i}$. We need the following well known
results on Betti numbers and the Morse inequality for
$\overline{{\Lambda}}\equiv\overline{{\Lambda}}S^{3}$ and
$\overline{{\Lambda}}^{0}=\overline{\Lambda}^{0}S^{3}=\\{{\rm
constant\;point\;curves\;in\;}S^{3}\\}\cong S^{3}$.

Proposition 2.3. (cf. Remark 2.5 of [Rad1] or [Hin1]) The Poincaré series is
given by

$\displaystyle P(\overline{{\Lambda}}S^{3},\overline{{\Lambda}}^{0}S^{3})(t)$
$\displaystyle=$ $\displaystyle
t^{2}\left(\frac{1}{1-t^{2}}+\frac{t^{2}}{1-t^{2}}\right)$ $\displaystyle=$
$\displaystyle
t^{2}(1+t^{2})(1+t^{2}+t^{4}+\cdots)=t^{2}+2t^{4}+2t^{6}+\cdots,$

which yields

${b}_{q}={b}_{q}(\overline{{\Lambda}}S^{3},\overline{{\Lambda}}^{0}S^{3})\;={\rm
rank}H_{q}(\overline{{\Lambda}}S^{3},\overline{{\Lambda}}^{0}S^{3})=\;\;\left\\{\matrix{1,&\quad{\it
if}\quad q=2,\cr 2,&\quad{\it if}\quad q=2k+2,\quad k\in{\bf N},\cr
0&\quad{\it otherwise}.\cr}\right.$ (2.3)

Proposition 2.4. (cf. Theorem I.4.3 of [Cha], Theorem 6.1 of [Rad2]) Suppose
that there exist only finitely many prime closed geodesics $\\{c_{j}\\}_{1\leq
j\leq k}$ on a Finsler $3$-sphere $(S^{3},F)$. Set

$M_{q}=\sum_{1\leq j\leq k,\;m\geq
1}\dim{\overline{C}}_{q}(E,c^{m}_{j}),\quad\forall q\in{\bf Z}.$

Then for every integer $q\geq 0$ there holds

$\displaystyle M_{q}-M_{q-1}+\cdots+(-1)^{q}M_{0}$ $\displaystyle\geq$
$\displaystyle b_{q}-b_{q-1}+\cdots+(-1)^{q}b_{0},$ (2.4) $\displaystyle
M_{q}$ $\displaystyle\geq$ $\displaystyle b_{q}.$ (2.5)

##  3 The common index jump theorem for symplectic paths

In [Lon1] of 1999, Y. Long established the basic normal form decomposition of
symplectic matrices. Based on this result he further established the precise
iteration formulae of indices of symplectic paths in [Lon2] of 2000. Since
every closed geodesic on a sphere is orientable, then, by Theorem 1.1 of
[Liu], the Morse index of a closed geodesic on $S^{n}$ coincides with the
Maslov-type index of a corresponding symplectic path.

As in [Lon2], the basic normal forms are denoted by

$\displaystyle N_{1}({\lambda},b)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ll}{\lambda}&b\\\
0&{\lambda}\end{array}\right),\qquad{\rm for\;}{\lambda}=\pm 1,\;b\in{\bf R},$
(3.3) $\displaystyle D({\lambda})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ll}{\lambda}&0\\\
0&{\lambda}^{-1}\end{array}\right),\qquad{\rm for\;}{\lambda}\in{\bf
R}\setminus\\{0,\pm 1\\},$ (3.6) $\displaystyle R({\theta})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ll}\cos{\theta}&-\sin{\theta}\\\
\sin{\theta}&\cos{\theta}\end{array}\right),\qquad{\rm
for\;}{\theta}\in(0,\pi)\cup(\pi,2\pi),$ (3.9) $\displaystyle
N_{2}(e^{\sqrt{-1}{\theta}},B)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ll}R({\theta})&B\\\
0&R({\theta})\end{array}\right),\qquad{\rm
for\;}{\theta}\in(0,\pi)\cup(\pi,2\pi)\;\;{\rm and}\;$ (3.15)
$\displaystyle\quad B=\left(\begin{array}[]{lll}b_{1}&b_{2}\\\
b_{3}&b_{4}\end{array}\right)\;{\rm with}\;b_{j}\in{\bf R},\;\;{\rm
and}\;\;b_{2}\not=b_{3}.$

As in [Lon2], the $\diamond$-sum (direct sum) of any two real matrices is
defined by

$\left(\begin{array}[]{lll}A_{1}&B_{1}\\\
C_{1}&D_{1}\end{array}\right)_{2i\times
2i}\diamond\left(\begin{array}[]{lll}A_{2}&B_{2}\\\
C_{2}&D_{2}\end{array}\right)_{2j\times
2j}=\left(\begin{array}[]{llll}A_{1}&0&B_{1}&0\\\ 0&A_{2}&0&B_{2}\\\
C_{1}&0&D_{1}&0\\\ 0&C_{2}&0&D_{2}\end{array}\right).$

For every $M\in{\rm Sp}(2n)$, the homotopy set $\Omega(M)$ of $M$ in ${\rm
Sp}(2n)$ is defined by

${\Omega}(M)=\\{N\in{\rm Sp}(2n)\,|\,{\sigma}(N)\cap{\bf
U}={\sigma}(M)\cap{\bf
U}\equiv\Gamma,\;\nu_{{\omega}}(N)=\nu_{{\omega}}(M),\,\forall{\omega}\in\Gamma\\},$

where ${\sigma}(M)$ denotes the spectrum of $M$,
$\nu_{{\omega}}(M)\equiv\dim_{{\bf C}}\ker_{{\bf C}}(M-{\omega}I)$ for
${\omega}\in{\bf U}$. The component ${\Omega}^{0}(M)$ of $P$ in ${\rm Sp}(2n)$
is defined by the path connected component of ${\Omega}(M)$ containing $M$.

Lemma 3.1. (cf. [Lon2], Lemma 9.1.5 and List 9.1.12 of [Lon3]) For $M\in{\rm
Sp}(2n)$ and ${\omega}\in{\bf U}$, the splitting number
$S_{M}^{\pm}({\omega})$ (cf. Definition 9.1.4 of [Lon3]) satisfies

$\displaystyle S_{M}^{\pm}({\omega})$ $\displaystyle=$ $\displaystyle
0,\qquad{\rm if}\;\;{\omega}\not\in{\sigma}(M).$ (3.16) $\displaystyle
S_{N_{1}(1,a)}^{+}(1)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lll}1,&\quad{\rm if}\;\;a\geq 0,\\\
0,&\quad{\rm if}\;\;a<0.\end{array}\right.$ (3.19)

For any $M_{i}\in{\rm Sp}(2n_{i})$ with $i=0$ and $1$, there holds

$S^{\pm}_{M_{0}\diamond
M_{1}}({\omega})=S^{\pm}_{M_{0}}({\omega})+S^{\pm}_{M_{1}}({\omega}),\qquad\forall\;{\omega}\in{\bf
U}.$ (3.20)

We have the following decomposition theorem

Theorem 3.2. (cf. [Lon2] and Theorem 1.8.10 of [Lon3]) For any $M\in{\rm
Sp}(2n)$, there is a path $f:[0,1]\to{\Omega}^{0}(M)$ such that $f(0)=M$ and

$f(1)=M_{1}\diamond\cdots\diamond M_{k},$ (3.21)

where each $M_{i}$ is a basic normal form listed in (3.3)-(3.15) for $1\leq
i\leq k$.

For every ${\gamma}\in\mathcal{P}_{\tau}(2n)\equiv\\{{\gamma}\in
C([0,\tau],Sp(2n))\ |\ {\gamma}(0)=I_{2n}\\}$, we extend ${\gamma}(t)$ to
$t\in[0,m\tau]$ for every $m\in{\bf N}$ by

$\displaystyle{\gamma}^{m}(t)={\gamma}(t-j\tau){\gamma}(\tau)^{j}\quad\forall\;j\tau\leq
t\leq(j+1)\tau\;\;{\rm and}\;\;j=0,1,\ldots,m-1,$ (3.22)

as in p.114 of [Lon1]. As in [LoZ] and [Lon3], we denote the Maslov-type
indices of ${\gamma}^{m}$ by $(i({\gamma},m),\nu({\gamma},m))$.

Then by Theorem 3.2 and some index computations of basic normal forms, the
following iteration formula from [LoZ] and [Lon3] can be obtained.

Theorem 3.3. (cf. Theorem 9.3.1 of [Lon3]) For any path
${\gamma}\in\mathcal{P}_{\tau}(2n)$, let $M={\gamma}(\tau)$ and
$C(M)=\sum_{0<{\theta}<2\pi}S_{M}^{-}(e^{\sqrt{-1}{\theta}})$. We extend
${\gamma}$ to $[0,+\infty)$ by its iterates. Then for any $m\in{\bf N}$ we
have

$\displaystyle i({\gamma},m)=m(i({\gamma},1)+S^{+}_{M}(1)-C(M))$
$\displaystyle\qquad+2\sum_{{\theta}\in(0,2\pi)}E\left(\frac{m{\theta}}{2\pi}\right)S^{-}_{M}(e^{\sqrt{-1}{\theta}})-(S_{M}^{+}(1)+C(M))$
(3.23)

and

$\hat{i}({\gamma},1)=i({\gamma},1)+S^{+}_{M}(1)-C(M)+\sum_{{\theta}\in(0,2\pi)}\frac{{\theta}}{\pi}S^{-}_{M}(e^{\sqrt{-1}{\theta}}).$
(3.24)

Theorem 3.4. (cf. Theorem 4.1 of [LoZ] and Theorem 3.4 of [DLLW]) Fix an
integer $q>0$. Let $\mu_{i}\geq 0$ and ${\beta}_{i}$ be integers for all
$i=1,\cdots,q$. Let ${\alpha}_{i,j}$ be positive numbers for
$j=1,\cdots,\mu_{i}$ and $i=1,\cdots,q$. Let ${\delta}\in(0,\frac{1}{2})$
satisfying ${\delta}\max\limits_{1\leq i\leq q}\mu_{i}<\frac{1}{2}$. Suppose
$D_{i}\equiv{\beta}_{i}+\sum\limits_{j=1}^{\mu_{i}}{\alpha}_{i,j}\neq 0$ for
$i=1,\cdots,q$. Then there exist infinitely many
$(N,m_{1},\cdots,m_{q})\in{\bf N}^{q+1}$ such that

$\displaystyle
m_{i}{\beta}_{i}+\sum_{j=1}^{\mu_{i}}E(m_{i}{\alpha}_{i,j})=\varrho_{i}N+\Delta_{i},\qquad\forall\
1\leq i\leq q.$ (3.25)
$\displaystyle\min\\{\\{m_{i}{\alpha}_{i,j}\\},1-\\{m_{i}{\alpha}_{i,j}\\}\\}<{\delta},\qquad\forall\
j=1,\cdots,\mu_{i},1\leq i\leq q,$ (3.26) $\displaystyle
m_{i}{\alpha}_{i,j}\in{\bf N},\ {\rm if}\ {\alpha}_{i,j}\in{\bf Q},$ (3.27)

where

$\displaystyle\varrho_{i}=\left\\{\begin{array}[]{cc}1,&{\rm if}\
D_{i}>0,\cr-1,&{\rm if}\
D_{i}<0,\end{array}\right.\quad\Delta_{i}=\sum_{0<\\{m_{i}{\alpha}_{i,j}\\}<{\delta}}1,\quad\forall\
1\leq i\leq q.$ (3.30)

Remark 3.5. (i) When $D_{i}>0$ for all $1\leq i\leq q$, this is precisely the
Theorem 4.1 of [LoZ] (also cf. Theorem 11.1.1 of [Lon3]). When $D_{i}\neq 0$
for all $1\leq i\leq q$, this has been proved in [DLLW] (cf. Theorem 3.4 of
[DLLW]).

(ii) According to Theorem 3.4 and its proof, it is easy to see that for any
two small enough integers ${\delta}_{1}$ and ${\delta}_{2}$ satisfying
${\delta}_{k}\in(0,\frac{1}{2})$ and ${\delta}_{k}\max\limits_{1\leq i\leq
q}\mu_{i}<\frac{1}{2}$ for $k=1,2$, Theorem 3.4 holds with

$\sum_{0<\\{m_{i}{\alpha}_{i,j}\\}<{\delta}_{1}}1=\sum_{0<\\{m_{i}{\alpha}_{i,j}\\}<{\delta}_{2}}1,\quad\forall\
1\leq i\leq q.$

In 2002, Y. Long and C. Zhu [LoZ] has established the common index jump
theorem for symplectic paths, which has become one of the main tools to study
the periodic orbit problem in Hamiltonian and symplectic dynamics. In [DLW] of
2016, H. Duan, Y. Long and W. Wang further improved this theorem to an
enhanced version which gives more precise index properties of
${\gamma}_{k}^{2m_{k}}$ and ${\gamma}_{k}^{2m_{k}\pm m}$ with $1\leq
m\leq\bar{m}$ for any fixed $\bar{m}$. Under the help of Theorem 3.4,
following the proofs of Theorem 3.5 in [DLW], this result has been further
generalized to the case of admitting the existence of symplectic paths with
negative mean indices.

Theorem 3.6. (Generalized common index jump theorem, cf. Theorem 3.6 of
[DLLW]) Let $\gamma_{i}\in\mathcal{P}_{\tau_{i}}(2n)$ for $i=1,\cdots,q$ be a
finite collection of symplectic paths with nonzero mean indices
$\hat{i}({\gamma}_{i},1)$. Let $M_{i}={\gamma}_{i}(\tau_{i})$. We extend
${\gamma}_{i}$ to $[0,+\infty)$ by (3.22) inductively.

Then for any fixed $\bar{m}\in{\bf N}$, there exist infinitely many
$(q+1)$-tuples $(N,m_{1},\cdots,m_{q})\in{\bf N}^{q+1}$ such that the
following hold for all $1\leq i\leq q$ and $1\leq m\leq\bar{m}$,

$\displaystyle\nu({\gamma}_{i},2m_{i}-m)$ $\displaystyle=$
$\displaystyle\nu({\gamma}_{i},2m_{i}+m)=\nu({\gamma}_{i},m),$ (3.31)
$\displaystyle i({\gamma}_{i},2m_{i}+m)$ $\displaystyle=$ $\displaystyle
2\varrho_{i}N+i({\gamma}_{i},m),$ (3.32) $\displaystyle
i({\gamma}_{i},2m_{i}-m)$ $\displaystyle=$ $\displaystyle
2\varrho_{i}N-i({\gamma}_{i},m)-2(S^{+}_{M_{i}}(1)+Q_{i}(m)),$ (3.33)
$\displaystyle i({\gamma}_{i},2m_{i})$ $\displaystyle=$ $\displaystyle
2\varrho_{i}N-(S^{+}_{M_{i}}(1)+C(M_{i})-2\Delta_{i}),$ (3.34)

where

$\displaystyle\varrho_{i}=\left\\{\begin{array}[]{cc}1,&{\rm if}\
\hat{i}({\gamma}_{i},1)>0,\cr-1,&{\rm if}\
\hat{i}({\gamma}_{i},1)<0,\end{array}\right.\qquad\Delta_{i}=\sum_{0<\\{m_{i}{\theta}/\pi\\}<\delta}S^{-}_{M_{i}}(e^{\sqrt{-1}{\theta}}),$
(3.37) $\displaystyle\
Q_{i}(m)=\sum_{e^{\sqrt{-1}{\theta}}\in{\sigma}(M_{i}),\atop\\{\frac{m_{i}{\theta}}{\pi}\\}=\\{\frac{m{\theta}}{2\pi}\\}=0}S^{-}_{M_{i}}(e^{\sqrt{-1}{\theta}}).$
(3.38)

Moreover we have

$\displaystyle\min\left\\{\\{\frac{m_{i}\theta}{\pi}\\},1-\\{\frac{m_{i}\theta}{\pi}\\}\\}\right\\}<{\delta},$
(3.39)

whenever $e^{\sqrt{-1}\theta}\in\sigma(M_{i})$ and ${\delta}$ can be chosen as
small as we want. More precisely, by (3.17) in [DLLW] and (4.40), (4.41) in
[LoZ] , we have

$\displaystyle
m_{i}=\left(\left[\frac{N}{M|\hat{i}(\gamma_{i},1)|}\right]+\chi_{i}\right)M,\quad\forall\
1\leq i\leq q,$ (3.40)

where $\chi_{i}=0$ or $1$ for $1\leq i\leq q$ and $\frac{M\theta}{\pi}\in{\bf
Z}$ whenever $e^{\sqrt{-1}\theta}\in\sigma(M_{i})$ and
$\frac{\theta}{\pi}\in{\bf Q}$ for some $1\leq i\leq q$. Furthermore, given
$M_{0}$, from the proof of Theorem 4.1 of [LoZ], we may further require $N$ to
be the mutiple of $M_{0}$, i.e., $M_{0}|N$.

Remark 3.7. In fact, let
$\mu_{i}=\sum_{0<{\theta}<2\pi}S_{M_{i}}^{-}(e^{\sqrt{-1}{\theta}})$,
$\alpha_{i,j}=\frac{{\theta}_{j}}{\pi}$ where
$e^{\sqrt{-1}{\theta}_{j}}\in\sigma(M_{i})$ for $1\leq j\leq\mu_{i}$ and
$1\leq i\leq q$. Let $l=q+\sum_{i=1}^{q}\mu_{i}$ and

$\displaystyle
v=\left(\frac{1}{M|\hat{i}(\gamma_{1},1)|},\cdots,\frac{1}{M|\hat{i}(\gamma_{1},1)|},\frac{{\alpha}_{1,1}}{|\hat{i}(\gamma_{1},1)|},\cdots,\frac{{\alpha}_{1,\mu_{1}}}{|\hat{i}(\gamma_{1},1)|},\cdots,\frac{{\alpha}_{q,1}}{|\hat{i}(\gamma_{q},1)|},\cdots,\frac{{\alpha}_{q,\mu_{q}}}{|\hat{i}(\gamma_{q},1)|}\right)\in{\bf
R}^{l}.$ (3.41)

Then Theorem 3.6 is equivalent to find a vertex

$\displaystyle\chi=(\chi_{1},\cdots,\chi_{q},\chi_{1,1},\cdots,\chi_{1,\mu_{1}},\cdots,\chi_{q,1},\cdots,\chi_{q,\mu_{q}})\in\\{0,1\\}^{l}$

of the cube $[0,1]^{l}$ and infinitely many $N\in{\bf N}$ such that for any
small enough $\epsilon\in(0,\frac{1}{2})$ there holds

$\displaystyle|\\{Nv\\}-\chi|<\epsilon.$ (3.42)

Next we use some multiple $\hat{N}$ of $N$ to replace $N$ in Theorem 3.6, then
we get the corresponding iterations $\hat{m}_{k}$s such that the equalities
similar to (3.31)-(3.34) hold. At the same time, we also get some relation
equalities about these integers, which will play a crucial role in the proof
of Theorem 1.1 in Section 4.

Theorem 3.8. (A variant of Theorem 3.6) Let
$\gamma_{i}\in\mathcal{P}_{\tau_{i}}(2n)$ for $i=1,\cdots,q$ be a finite
collection of symplectic paths with nonzero mean indices
$\hat{i}({\gamma}_{i},1)$. Let $M_{i}={\gamma}_{i}(\tau_{i})$. Fix a positive
integer $\hat{p}$, use $\frac{\delta}{\hat{p}}$ and $\frac{\epsilon}{\hat{p}}$
to substitute $\delta$ in (3.39) and $\epsilon$ in (3.42) respectively, and
then let $N$ be an integer satisfying (3.31)-(3.40).

Then for $\hat{N}=\hat{p}N$, there exists the corresponding $q$-tuples
$(\hat{m}_{1},\cdots,\hat{m}_{q})\in{\bf N}^{q}$ such that the following hold
for all $1\leq i\leq q$ and $1\leq m\leq\bar{m}$

$\displaystyle\nu({\gamma}_{i},2\hat{m}_{i}-m)$ $\displaystyle=$
$\displaystyle\nu({\gamma}_{i},2\hat{m}_{i}+m)=\nu({\gamma}_{i},m),$ (3.43)
$\displaystyle i({\gamma}_{i},2\hat{m}_{i}+m)$ $\displaystyle=$ $\displaystyle
2\varrho_{i}\hat{p}N+i({\gamma}_{i},m),$ (3.44) $\displaystyle
i({\gamma}_{i},2\hat{m}_{i}-m)$ $\displaystyle=$ $\displaystyle
2\varrho_{i}\hat{p}N-i({\gamma}_{i},m)-2(S^{+}_{M_{i}}(1)+Q_{i}(m)),$ (3.45)
$\displaystyle i({\gamma}_{i},2\hat{m}_{i})$ $\displaystyle=$ $\displaystyle
2\varrho_{i}\hat{p}N-(S^{+}_{M_{i}}(1)+C(M_{i})-2\hat{\Delta}_{i}),$ (3.46)

where $\varrho_{i}$ and $Q_{i}(m)$ are the same as those in (3.38) and

$\displaystyle\hat{m}_{i}=\left(\left[\frac{\hat{p}N}{M|\hat{i}(\gamma_{i},1)|}\right]+\hat{\chi}_{i}\right)M,\quad\hat{\Delta}_{i}=\sum_{0<\\{\hat{m}_{i}{\theta}/\pi\\}<\delta}S^{-}_{M_{i}}(e^{\sqrt{-1}{\theta}}),\quad\forall\
1\leq i\leq q.$ (3.47)

Furthermore, comparing with some corresponding integers in Theorem 3.6, there
holds

$\displaystyle\hat{\chi}_{i}=\chi_{i},\quad\hat{m}_{i}=\hat{p}m_{i},\quad\hat{\Delta}_{i}=\Delta_{i},\quad\forall\
1\leq i\leq q.$ (3.48)

Proof. Set $v_{i}=\frac{1}{|M\hat{i}(\gamma_{i},1)|},\forall 1\leq i\leq q$.
Firstly notice that

$\displaystyle\\{\hat{p}Nv_{i}\\}=\\{\hat{p}([Nv_{i}]+\\{Nv_{i}\\})\\}=\\{\hat{p}[Nv_{i}]+\hat{p}\\{Nv_{i}\\}\\}=\\{\hat{p}\\{Nv_{i}\\}\\}.$
(3.49)

When $\chi_{i}=0$, it follows from (3.42) with $\epsilon$ replaced by
$\frac{\epsilon}{\hat{p}}$ (i.e. $\\{Nv_{i}\\}<\frac{\epsilon}{\hat{p}}$) that
$|\\{\hat{p}Nv_{i}\\}-\chi_{i}|=\\{\hat{p}\\{Nv_{i}\\}\\}=\hat{p}\\{Nv_{i}\\}<\epsilon$.
When $\chi_{i}=1$, it follows from (3.42) that
$0<1-\\{Nv_{i}\\}<\frac{\epsilon}{\hat{p}}$. Therefore there holds
$0<\hat{p}-\hat{p}\\{Nv_{i}\\}<\epsilon$, which yields
$0<1-\\{\hat{p}\\{Nv_{i}\\}\\}=1-\\{\hat{p}Nv_{i}\\}<\epsilon$. In a word,
there always holds

$\displaystyle|\\{\hat{p}Nv_{i}\\}-\chi_{i}|=|\\{\hat{p}\\{Nv_{i}\\}\\}-\chi_{i}|<\epsilon,\qquad
1\leq i\leq q.$ (3.50)

Roughly speaking, when $\\{Nv_{i}\\}$ is close enough to $\chi_{i}$,
$\\{\hat{p}Nv_{i}\\}$ is also close enough to the same $\chi_{i}$. Therefore
$\hat{\chi}_{i}=\chi_{i},\forall\ 1\leq i\leq q$.

Now we prove the relation of $\hat{m}_{i}$ and $m_{i}$ for $1\leq i\leq q$. By
the similar arguments as those in proof of Theorem 3.6 (i.e. the proof of
similar (3.40)), we have

$\displaystyle\hat{m}_{i}$ $\displaystyle=$
$\displaystyle\left(\left[\frac{\hat{p}N}{M|\hat{i}(\gamma_{i},1)|}\right]+\hat{\chi}_{i}\right)M=\left(\left[\hat{p}Nv_{i}\right]+\chi_{i}\right)M$
(3.51) $\displaystyle=$
$\displaystyle\left(\left[\hat{p}([Nv_{i}]+\\{Nv_{i}\\})\right]+\chi_{i}\right)M=\left(\hat{p}[Nv_{i}]+\left[\hat{p}\\{Nv_{i}\\}\right]+\chi_{i}\right)M.$

When $\chi_{i}=0$, then $\\{Nv_{i}\\}<\frac{\epsilon}{\hat{p}}$, and thus
$\hat{p}\\{Nv_{i}\\}<\epsilon$. By (3.51) and (3.40) we obtain
$\hat{m}_{i}=\hat{p}[Nv_{i}]M=\hat{p}m_{i}$. When $\chi_{i}=1$, it follows
from (3.42) that $0<1-\\{Nv_{i}\\}<\frac{\epsilon}{\hat{p}}$. Therefore there
holds $\hat{p}-1<\hat{p}-\epsilon<\hat{p}\\{Nv_{i}\\}<\hat{p}$, which yields
$[\hat{p}\\{Nv_{i}\\}]=\hat{p}-1$. Again by (3.51) and (3.40) we obtain
$\hat{m}_{i}=(\hat{p}[Nv_{i}]+(\hat{p}-1)+1)M=\hat{p}([Nv_{i}]+1)M=\hat{p}([Nv_{i}]+\chi_{i})M=\hat{p}m_{i}$.
In a word, there always holds $\hat{m}_{i}=\hat{p}m_{i}$, $\forall 1\leq i\leq
q$.

Finally the proof of identities (3.43)-(3.28) are completely the same as those
of (3.31)-(3.34) in Theorem 3.6, so here we omit these details.

##  4 Proof of Theorem 1.1

In this section, let $F$ be any bumpy Finsler metric on $S^{3}$, i.e., all
closed geodesics (including iterations) on $(S^{3},F)$ are non-degenerate. To
prove theorem 1.1, firstly we assume that the Morse index $i(c)\geq 1$ for any
prime closed geodesic $c$ on $(S^{3},F)$.

On one hand, Theorem 1.4 in [DLW] showed that if every prime closed geodesic
$c$ on every bumpy $S^{3}$ satisfies $i(c)\geq 2$, then there exist at least
four prime closed geodesics. On the other hand, Remark 3.7 in [DuL2] showed
that if there exist exactly two prime closed geodesics $c_{1}$ and $c_{2}$ on
bumpy Finsler $S^{3}$, then only two possible cases happen: (i) $i(c_{1})=0$
and $i(c_{2})=1$; (ii) $i(c_{k})\geq 2$ for $k=1,2$.

Thus, it follows from Remark 3.7 in [DuL2] and the assumption of nonzero Morse
index that there exist at least 3 prime closed geodesics on $S^{3}$.
Therefore, in order to prove theorem 1.1, by the above assumption and Theorem
1.4 of [DLW], we only need to consider the case when there exist exactly 3
prime closed geodesics $c_{k}$ with $k=1,2,3$ on $(S^{3},F)$ and at least one
prime closed geodesic among them has Morse index $1$. Then we try to get a
contradiction.

Furthermore, by Propositions 2.1, 2.3 and 2.4, among three prime closed
geodesics there exists at least one prime closed geodesic with even Morse
index such that these closed geodesics can generate non-trivial Morse-type
numbers $M_{2j}\geq b_{2j}\geq 1$ for any $j\geq 1$. In fact, under the
assumption of existence of exactly three prime closed geodesics, it can be
further showed that there exist exactly two prime closed geodesics with even
Morse indices (cf. Lemma 4.2 and Assumption below).

It is well-known that the Morse index sequence $\\{i(c^{m})\\}$ either tends
to $+\infty$ asumptotically linearly or $i(c^{m})=0$ for all $m\geq 1$ (cf.
Proposition 1.3 of [Bot]). Therefore $\hat{i}(c)=0$ if and only if
$i(c^{m})=0$ for all $m\geq 1$. Since $i(c_{k})\geq 1$ for $k=1,2,3$ by the
above assumption, so the mean index satisfies $\hat{i}(c_{k})>0$. So
$\varrho_{k}=1$ in Theorem 3.6 by the definition (3.38), and
$i(c_{k}^{m})\rightarrow+\infty$ as $m\rightarrow+\infty$. So the positive
integer $\hat{m}$ defined by

$\displaystyle\overline{m}=\max\limits_{1\leq k\leq q}\min\\{m_{0}\in{\bf
N}\,|\,i(c_{k}^{m+m_{0}})\geq i(c_{k})+4,\ \forall\,m\geq 1\\}$ (4.1)

is well defined and finite.

Lemma 4.1. With the above $\overline{m}$, Theorem 3.6 yields $(q+1)$-tuples
$(N,m_{1},\cdots,m_{q})\in{\bf N}^{q+1}$ such that the following inequalities
hold

$\displaystyle i(c_{k}^{2m_{k}-m})$ $\displaystyle\leq$ $\displaystyle
2N-i(c_{k}),\quad\forall\ 1\leq m<2m_{k},$ (4.2) $\displaystyle
i(c_{k}^{2m_{k}+m})$ $\displaystyle\geq$ $\displaystyle
2N+i(c_{k}),\quad\forall\ m\geq 1.$ (4.3)

Proof. Note that there always holds $S^{+}_{M_{i}}(1)=Q_{i}(m)=0$ by the non-
degeneracy of $F$ and Lemma 3.1, and $i(c^{m})\geq i(c)\geq 1$ for any $m\geq
1$ by the Bott formulae (cf. Theorem 9.2.1 of [Lon3]). So it follows from
(3.32) and (3.33) of Theorem 3.6 that the above inequalities hold for
$m\leq\overline{m}$.

Now assume $m\geq\overline{m}+1$. Similar to the proof of Theorem 3.6 (cf.
equalities (3.33) and (3.36) in [DLW] for more details), by Theorem 3.3 and
(3.12), we have

$\displaystyle i(c_{k}^{2m_{k}-m})$ $\displaystyle=$
$\displaystyle-i(c_{k}^{m})+2N+2\Delta_{k}+2\sum\limits_{\theta}\left(E\left(\frac{2m_{k}-m}{2\pi}\theta\right)+\right.$
(4.4)
$\displaystyle\qquad\left.E\left(\frac{m}{2\pi}\theta\right)-E\left(\frac{m_{k}}{\pi}\theta\right)\right)S_{M_{k}}^{-}(e^{\sqrt{-1}\theta})-2C(M_{k}),$
$\displaystyle i(c_{k}^{2m_{k}+m})$ $\displaystyle=$ $\displaystyle
i(c_{k}^{m})+2N+2\Delta_{k}+2\sum\limits_{\theta}\left(E\left(\frac{2m_{k}+m}{2\pi}\theta\right)-\right.$
(4.5)
$\displaystyle\qquad\qquad\left.E\left(\frac{m}{2\pi}\theta\right)-E\left(\frac{m_{k}}{\pi}\theta\right)\right)S_{M_{k}}^{-}(e^{\sqrt{-1}\theta}).$

Note that $E(a)+E(b)-E(a+b)\leq 1$. Then by the definition of $\overline{m}$,
there holds

$i(c_{k}^{2m_{k}-m})\leq-i(c_{k}^{m})+2N+2\Delta_{k}\leq-i(c_{k})-4+2N+2\Delta_{k}\leq
2N-i(c_{k}),$ (4.6) $i(c_{k}^{2m_{k}+m})\geq
i(c_{k}^{m})+2N+2\Delta_{k}-2C(M_{k})\geq
i(c_{k})+4+2N+2\Delta_{k}-2C(M_{k})\geq 2N+i(c_{k}),$ (4.7)

where we use the fact $0\leq\Delta_{k}\leq C(M_{k})\leq 2$ in the case of
$S^{3}$.

Lemma 4.2. It is impossible that two of $c_{1},c_{2},c_{3}$ have odd indices
and one of them has an even index.

Proof. Without loss of generality, suppose that $c_{1},c_{2}$ have odd indices
and $c_{3}$ has an even index, then by Lemma 4.1, there exist positive
integers $N$ and $m_{3}$ such that

$\displaystyle i(c_{3}^{2m_{k}-m})$ $\displaystyle\leq$ $\displaystyle
2N-i(c_{3})\leq 2N-1,\quad\forall\ 1\leq m<2m_{3},$ (4.8) $\displaystyle
i(c_{3}^{2m_{k}+m})$ $\displaystyle\geq$ $\displaystyle 2N+i(c_{3})\geq
2N+1,\quad\forall\ m\geq 1.$ (4.9)

Then by Proposition 2.1, $c_{1}$ and $c_{2}$ have no contribution to $M_{2N}$
while $c_{3}$ contributes at most 1 to $M_{2N}$. Hence there holds $M_{2N}\leq
1<b_{2N}$, which contradicts to Proposition 2.4.

According to the above arguments and Lemma 4.2, we make the following
assumption and will get a contradiction to complete the proof of Theorem 1.1.

Assumption. There exist exactly 3 prime closed geodesics $c_{1},c_{2},c_{3}$
on bumpy Finsler 3-sphere $(S^{3},F)$ with $i(c_{1})=1,i(c_{2})\in 2{\bf
N},i(c_{3})\in 2{\bf N}$.

Now we use Lemma 4.1 to get the following two crucial inequalities.

Lemma 4.3. Under the assumption, with the above $\overline{m}$, Theorem 3.6
yields $(q+1)$-tuples $(N,m_{1},\cdots,m_{q})\in{\bf N}^{q+1}$ such that the
following inequalities hold

$\displaystyle\sum_{k=1}^{3}2m_{k}\gamma_{c_{k}}-(2N)_{+}^{e}+(2N)_{+}^{o}$
$\displaystyle\geq$ $\displaystyle 2N-1,$ (4.10)
$\displaystyle\sum_{k=1}^{3}2m_{k}\gamma_{c_{k}}-(2N-1)_{+}^{e}+(2N-1)_{+}^{o}$
$\displaystyle\leq$ $\displaystyle 2N-3,$ (4.11)

where the counting notations are defined by $n_{+}^{e}=\\#\\{k\ |\
i(c_{k}^{2m_{k}})>n,i(c_{k}^{2m_{k}})\equiv i(c_{k})\equiv 0\left(\rm
mod2\it\right)\\}$ and $n_{+}^{o}=\\#\\{k\ |\
i(c_{k}^{2m_{k}})>n,i(c_{k}^{2m_{k}})\equiv i(c_{k})\equiv 1\left(\rm
mod2\it\right)\\}$.

Proof. It follows from Proposition 2.1, Definition 2.2 and Theorem 3.3 that

$\displaystyle\sum_{m=1}^{2m_{k}}(-1)^{i(c_{k}^{m})}\dim\overline{C}_{i(c_{k}^{m})}(E,c_{k}^{m})$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{m_{k}-1}\sum_{m=2i+1}^{2i+2}(-1)^{i(c_{k}^{m})}\dim\overline{C}_{i(c_{k}^{m})}(E,c_{k}^{m})$
(4.12) $\displaystyle=$
$\displaystyle\sum_{i=0}^{m_{k}-1}\sum_{m=1}^{2}(-1)^{i(c_{k}^{m})}\dim\overline{C}_{i(c_{k}^{m})}(E,c_{k}^{m})$
$\displaystyle=$ $\displaystyle
m_{k}\sum_{m=1}^{2}(-1)^{i(c_{k}^{m})}\dim\overline{C}_{i(c_{k}^{m})}(E,c_{k}^{m})$
$\displaystyle=$ $\displaystyle 2m_{k}{\gamma}_{c_{k}},\qquad\forall\ 1\leq
k\leq 3,$

where the second equality follows from Proposition 2.1 and the fact
$i(c_{k}^{m+2})-i(c_{k}^{m})\in 2{\bf Z}$ for all $m\geq 1$ from Theorem 3.3,
and the last equality follows from Proposition 2.1 and Definition 2.2.

By Lemma 4.1 and Proposition 2.1, the sum of the left side of (4.12) with
respect to $k$ is

$\displaystyle\sum_{k=1}^{3}\sum_{m=1}^{2m_{k}}(-1)^{i(c_{k}^{m})}\dim\overline{C}_{i(c_{k}^{m})}(E,c_{k}^{m})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{2N}(-1)^{i}M_{i}+(2N)_{+}^{e}-(2N)_{+}^{o},$ (4.13)
$\displaystyle\sum_{k=1}^{3}\sum_{m=1}^{2m_{k}}(-1)^{i(c_{k}^{m})}\dim\overline{C}_{i(c_{k}^{m})}(E,c_{k}^{m})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{2N-1}(-1)^{i}M_{i}+(2N-1)_{+}^{e}-(2N-1)_{+}^{o}.$
(4.14)

On the other hand, Propositions 2.3 and 2.4 give the following inequalities

$\displaystyle\sum_{i=1}^{2N}(-1)^{i}M_{i}\geq\sum_{i=1}^{2N}(-1)^{i}b_{i}=2N-1,\qquad\sum_{i=1}^{2N-1}(-1)^{i}M_{i}\leq\sum_{i=1}^{2N-1}(-1)^{i}b_{i}=2N-3.$
(4.15)

Now the inequalities (4.10) and (4.11) follow from (4.12)-(4.15).

Again by Lemma 4.1 and Proposition 2.1, $c_{2}^{2m_{2}}$ and $c_{3}^{2m_{3}}$
contribute at most 1 to $M_{2N}$, respectively. Note that Propositions 2.3 and
2.4 yields $M_{2N}\geq b_{2N}=2$. So $c_{2}^{2m_{2}}$ and $c_{3}^{2m_{3}}$
must contribute exactly 1 to $M_{2N}$, respectively, that is, there must hold

$\displaystyle i(c_{2}^{2m_{2}})=2N,\qquad i(c_{3}^{2m_{3}})=2N.$ (4.16)

Hence, by the definitions of $n_{+}^{e}$ and $n_{+}^{o}$ we have
$(2N)_{+}^{e}=0$, $(2N-1)_{+}^{e}=2$, $(2N)_{+}^{o}\leq 1$ and
$0\leq(2N-1)_{+}^{o}$. And then it follows from (4.10) and (4.11) of Lemma 4.3
that

$\displaystyle 2N-2$ $\displaystyle\leq$ $\displaystyle 2N-1-(2N)_{+}^{o}$
(4.17) $\displaystyle\leq$ $\displaystyle\sum_{k=1}^{3}2m_{k}\gamma_{c_{k}}$
$\displaystyle\leq$ $\displaystyle 2N-1-(2N-1)_{+}^{o}$ $\displaystyle\leq$
$\displaystyle 2N-1.$

Fix a small enough $\epsilon$ in (3.24) and suppose $N$ satisfies (3.24) for
$\frac{\epsilon}{4}$ (i.e., taking $\hat{p}=4$ there), then by Theorem 3.8,
the integers $\hat{N}=4N$ and
$\hat{m}_{k}=\left(\left[\frac{\hat{N}}{M\hat{i}(c_{k})}\right]+\chi_{k}\right)M=4m_{k},k=1,\cdots,q$
yields the $(q+1)$-tuples $(\hat{N},\hat{m}_{1},\cdots,\hat{m}_{q})\in{\bf
N}^{q+1}$ such that Theorem 3.4 and (3.43)-(3.46) in Theorem 3.8 hold. Then by
the similar arguments above, it can be shown that Lemma 4.1, Lemma 4.3 and
(4.17) also hold with $N$ being replaced by $4N$. This is to say, we have

$\displaystyle
8N-2\leq\sum_{k=1}^{3}2\hat{m}_{k}\gamma_{c_{k}}=4\sum_{k=1}^{3}2m_{k}\gamma_{c_{k}}\leq
8N-1,$ (4.18)

where we use the fact $\hat{m}_{k}=4m_{k}$ in (3.48) of Theorem 3.8. Then the
integer in the middle is the multiple of $4$ and lies between $8N-2$ and
$8N-1$. Clearly this is a contradiction.

Therefore the Assumption is not true and the proof of Theorem 1.1 is finished.

Acknowledgements The first author would like to thank Professor Yiming Long
for his valuable communications and constant encouragement.

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