# Light propagation in Kerr spacetime

Pravin Kumar Dahal<EMAIL_ADDRESS>School of Mathematical &
Physical Sciences, Macquarie University

###### Abstract

We explicitly solve the equations for the propagation of an electromagnetic
wave up to the subleading order geometric optics expansion in the Kerr
spacetime. This is done in two nontrivial steps. We first construct a set of
parallel propagated null tetrad in Kerr spacetime. Two of the components of
such tetrad give the propagation and polarization of an electromagnetic wave
in geometric optics approximation. Then we use the parallel propagated tetrad
to solve the modified trajectory equation in Kerr spacetime. We obtain the
wavelength-dependent deviation of the trajectory of electromagnetic waves,
which gives the mathematical description of the gravitational spin Hall effect
in Kerr spacetime.

##  I Introduction

One way to study a black hole is from the light emitted by the matter in the
vicinity of the black hole. Such radiation has to propagate a long distance
through the strong gravitational fields before reaching the observer. So, one
needs to solve the equation for the propagation of electromagnetic waves in
curved spacetime in order to extract the information contained in
astrophysical observations.

Solution of the wave equations, in general, in curved spacetimes is a
cumbersome task. A well-known and widely employed approximation technique
developed long ago for this purpose is geometric optics approximation [1, 2].
In this approximation, valid in the limit of an infinite frequency, we reduce
the wave equation to the ray equation and transport equation along the ray.
However, the geometric optics approximation does not capture the wavelike
behaviour, which is essential for the wave of large but finite frequency
propagating in curved spacetime. For this purpose, geometric optics
approximation is generalized by assuming it as the leading order approximation
of some perturbative expansion of the trajectory and transport equation in
terms of the inverse frequency $1/\omega$ of the wave [3, 4, 5]. We call spin
optics to this approximation up to the subleading order in $1/\omega$, and it
is sufficient to describe physical phenomena originating from the interaction
of spin with the extrinsic orbital angular momentum of the wave [3, 6, 5]. An
example of such a phenomenon we discuss here is the gravitational spin Hall
effect [7]. This article is concerned only with extrinsic and not intrinsic
angular momentum.

The analogous phenomenon known as the spin Hall effect or optical Magnus
effect is observed in condensed matter physics [11, 12, 13]. This effect
results when light propagates in an inhomogeneous medium, where the
inhomogeneous medium imparts orbital angular momentum, thereby interacting
with the intrinsic spin angular momentum of light [16]. This effect is well
developed [17] and experimentally verified [14, 15, 12, 13], thus serving as a
basis for our calculation and comparison.

Modified geometric optics was initially developed for stationary spacetimes in
Refs [2, 3, 5]. Recently, it was devised for general spacetimes in Refs. [8,
6]. We have also developed the covariant formulation of spin optics in general
spacetime in Ref. [9, 10], which yield slightly different results. Using the
tools developed there, we demonstrate the gravitational spin Hall effect in
the Kerr background analytically and numerically. Previously, Ref. [18] has
used the subleading order geometric optics correction, devised for the
stationary spacetimes, to demonstrate the gravitational spin Hall effect in
the Kerr spacetime (see, also, Ref. [19]). Electromagnetic waves from
astrophysical sources might pass through the vicinity of massive gravitating
bodies, which act as gravitational lenses. The leading order geometric optics
expansion suffices if the characteristic wavelength of electromagnetic waves
is negligible compared to the length scale of the inhomogeneities in spacetime
curvature from such bodies. However, subleading order correction from the
geometric optics might be necessary when studying gravitational lensing of
electromagnetic waves with wavelengths larger than the Schwarzschild radius of
lensing objects [20, 21]. In such a situation, wave effects must be
appropriately taken into account. To perform this task, we proceed in two
steps. First, we construct the parallel propagated null tetrad, which is the
solution of the wave equation in the geometric optics approximation. Then, we
need to use that tetrad to find the subleading order correction in the
trajectory. Propagation and polarization relations in the subleading order are
calculated explicitly in terms of this tetrad.

This article is organized as follows. In Sec. II, we describe the general
procedure of solving the parallel transport equations for null geodesics in
Petrov type-D spacetimes admitting an additional integral of motion [22],
which is quadratic in particle momenta. We then apply this procedure to find
the parallel propagated null tetrad in Kerr spacetime. In Sec. III, we use
this parallel propagated tetrad to find the leading order deviation in the
null trajectory from the geometric optics approximation analytically and
numerically. We then discuss our results and conclude the article in Sec. IV.

We consider a spacetime manifold $M$ with the metric $g_{\mu\nu}$ of
Lorentzian signature $\left(-,+,+,+\right)$. The phase space is the cotangent
bundle $T^{*}M$, whose points are written as $(x,p)$. Similarly, $\tilde{m}$
represents the complex conjugate of $m$. We use the system of units with
$G=c=1$ and adopt the Einstein summation convention. A semicolon (;) denotes
the covariant derivative, $\lambda$ denotes the parameter of electromagnetic
wave curves and $\dot{x}=dx/d\lambda$. The curvature convention is adopted
from Ref. [1].

##  II Parallel propagated null tetrad

In general, it is not straightforward to explicitly solve the parallel
transport equations. Fortunately, the Kerr geometry has some remarkable
properties, including the separability of wave equations and the complete
integrability of geodesics. These special separability properties that permit
the explicit integration of the geodesic equations can also be used to provide
an explicit solution to the problem of constructing a parallel propagated
orthonormal tetrad. For timelike geodesics, this was done by Marck [23] in two
steps: 1) First, construct a locally defined orthonormal tetrad along a null
geodesic such that two of its components are already parallel propagated. 2)
Then obtain the entirely parallel transported tetrad by rotating along some
spatial hypersurface. The rotation angle would be the sum of two functions,
one of $r$ coordinates only and the other of $\theta$ coordinates expressed in
terms of elliptic integrals.

The procedure would be different for the null geodesic. Unlike in this
reference, we will not restrict ourselves to the static orthonormal frame and
will solve the equations of parallel transport in general. This parallel
propagated tetrad would be useful in studying tidal effects near the Kerr
black hole and the gravitational spin Hall effect. They are also useful while
investigating particles and fields with spins [24]. In quantum physics, the
point-splitting method is used to calculate the renormalized values of local
observables in curved spacetime, and it relies on the parallel transported
frame. It also plays a role in proving the peeling-off property of the
gravitational radiation [25, 26, 27].

Using the separation of variables on the Hamilton-Jacobi equation, Carter
found the existence of the fourth constant of the Kerr spacetime, making the
geodesic equation analytically soluble [28]. This constant of motion is
quadratic in particle momenta [29] and has a direct correspondence with the
rank two Killing tensor. A rank two Killing tensor $K_{\alpha\beta}$ is a
symmetric tensor with vanishing symmetrized covariant derivative,
$\nabla^{(\gamma}K^{\alpha\beta)}=0$. The spacetime symmetry associated with
the Killing tensor of rank two and higher is known as hidden symmetry.

Spacetime with some symmetries always has its counterpart in the phase space.
However, not all phase space symmetries are associated with the configuration
space. Symmetries that can be reduced to the configuration space are explicit
symmetries of the spacetime, and those which do not have its counterpart in
the configuration space are hidden symmetries. Explicit continuous symmetries
are described by the Killing vectors and hidden symmetries by the Killing
tensors. Killing tensors do not generate a spacetime diffeomorphism; hence,
they are not associated with the configuration space. However, the existence
of these tensors could be realized from the geodesic equations of motion.

Following Carter’s discovery, Penrose [30] and Floyd [31] showed that there
exists the Killing-Yano tensor $f_{\alpha\beta}$ in the Kerr geometry, which
resembles a square root of the Killing tensor and obeys
$\nabla_{(\gamma}f_{\alpha)\beta}=0$. Its Hodge dual
$h_{\alpha\beta}=*f_{\alpha\beta}$ is again a two-form that satisfies

$\nabla_{\gamma}h_{\alpha\beta}=g_{\gamma\alpha}\xi_{\beta}-g_{\gamma\beta}\xi_{\alpha};\quad\xi_{\alpha}=\frac{1}{D-1}\nabla_{\beta}h^{\beta}_{~{}~{}\alpha},$
(1)

where $\xi_{\alpha}$ is the Killing vector and $D$ is the spacetime dimension
(for our case, $D=4$). The object satisfying such an equation is a closed
conformal Killing-Yano two-form. The wedge product of these two-forms is also
a closed conformal Killing-Yano tensor [32]. This quantity is called a
principal tensor by Frolov et al. [33]. The principal tensor describes hidden
symmetry of the spacetime, whose existence ensures the complete integrability
of geodesic motion (consequences extend beyond this property; see Ref. [33]).
We will show below that the principal tensor $h_{\alpha\beta}$ also enables us
to construct a complete set of parallel transported frames along the
geodesics. In summary, the integrability of the geodesic equations is ensured
by the existence of Killing tensor $K_{\alpha\beta}$, whose square root
resembles the Killing-Yano two-form $f_{\alpha\beta}$. As a result, we could
also analytically solve the equations of parallel transport applied to an
orthonormal tetrad along a null geodesic congruence.

###  II.1 Parallel transport along null geodesics

We consider an affine parametrized null geodesics ${\cal C}$, with tangent
vector $l^{\alpha}$. Let $h^{\alpha\beta}$ be the principal tensor. Then,
defining

$v^{\alpha}=h^{\alpha\beta}l_{\beta}-A(r,\theta)l^{\alpha},$ (2)

we obtain

$\dot{v}^{\alpha}=l^{\alpha}\left(-\dot{A}+l^{\beta}\xi_{\beta}\right),$ (3)

where Eq. (1) has been used to find this. Thus, the requirement that
$v^{\alpha}$ be parallel transported gives

$\dot{A}=l^{\beta}\xi_{\beta}.$ (4)

Now, using $v^{\alpha}$ as a seed vector and defining the equation analogous
to Eq. (2) enables us to construct another parallel transported vector
immediately

$n^{\alpha}=h^{\alpha\beta}v_{\beta}-B(r,\theta)l^{\alpha},\qquad\dot{B}=v^{\beta}\xi_{\beta}.$
(5)

The vector $n^{\alpha}$ does not belong to the null plane of vectors
orthogonal to $l^{\alpha}$, because of which it cannot be used as a new seed
to construct equations like Eq. (2) to create new parallel transported
vectors.

We can define another parallel transported vector depending on particle
momentum $l^{\alpha}$ and position through the tensor $f_{\alpha\beta}$

$u^{\alpha}=f^{\alpha}_{~{}~{}\beta}l^{\beta},$ (6)

Then, this quantity satisfies the parallel transport equations if and only if
tensor $f_{\alpha\beta}$ satisfies the Killing tensor equation [34]

$\nabla_{(\alpha}f_{\beta\gamma)}=0.$ (7)

If the tensor is completely antisymmetric, it is called the Killing-Yano form
[35]

$\nabla_{(\alpha}f_{\beta)\gamma}=0.$ (8)

As $f_{\alpha\beta}=*h_{\alpha\beta}$ is the Hodge dual of $h_{\alpha\beta}$,
the above two equations indeed hold, and hence $f_{\alpha\beta}$ is parallel
transported along the geodesics. This is because the Hodge dual of a closed
conformal Killing-Yano tensor is a Killing-Yano tensor and vice versa.
Corresponding vector $u^{\alpha}$ is parallel propagated and also
perpendicular to the momentum of the particle

$u_{\alpha}l^{\alpha}=0.$ (9)

Conversely, any skew-symmetric vector $u^{\alpha}$ that is linear in momentum
and parallel propagated along and orthogonal to any geodesic describes the
Killing-Yano tensor $f_{\alpha\beta}$. Thus, by construction, vectors
$(l^{\alpha},n^{\alpha},v^{\alpha},u^{\alpha})$ are parallel transported along
the geodesic ${\cal C}$. This property has been used to explicitly construct
the parallel propagated frame in Kerr spacetime [23, 36] and higher dimensions
[37].

###  II.2 Application to Kerr spacetime

Let us take the Kerr metric in Boyer-Lindquist coordinates

$ds^{2}=-\left(1-\frac{2Mr}{\rho^{2}}\right)dt^{2}-\frac{4aMr}{\rho^{2}}\sin^{2}\theta
dtd\phi+\frac{\rho^{2}}{\Delta}dr^{2}+\\\ \rho^{2}d\theta^{2}+\frac{{\cal
A}}{\rho^{2}}\sin^{2}\theta d\phi^{2},$ (10)

where

$\displaystyle\rho^{2}=$ $\displaystyle
r^{2}+a^{2}\cos^{2}\theta,\quad\Delta=r^{2}+a^{2}-2Mr,$ $\displaystyle{\cal
A}=$ $\displaystyle(r^{2}+a^{2})^{2}-\Delta a^{2}\sin^{2}\theta.$ (11)

The equation of motions for null geodesics in Kerr spacetime is

$\displaystyle\dot{t}=$ $\displaystyle\frac{{\cal
A}-2aMr\zeta}{\Delta\rho^{2}},\quad\rho^{4}\dot{r}^{2}={\cal R}^{2},$
$\displaystyle\rho^{4}\dot{\theta}^{2}=$
$\displaystyle\Theta^{2},\quad\dot{\phi}=\frac{2aMr+(\rho^{2}-2Mr)\zeta\csc^{2}\theta}{\Delta\rho^{2}},$
(12)

where $\eta$, $\zeta$ are constants of motion for the null geodesic trajectory
and

$\displaystyle{\cal R}^{2}=$
$\displaystyle\left(a^{2}-a\zeta+r^{2}\right)^{2}-\Delta\left((a-\zeta)^{2}+\eta\right),$
$\displaystyle\Theta^{2}=$ $\displaystyle
a^{2}\cos^{2}\theta+\eta-\zeta^{2}\cot^{2}\theta.$ (13)

One of the obvious choices of the parallel propagated orthonormal tetrad would
be the unit vector tangent to the geodesic

$l_{0\mu}=\left(-1,\frac{{\cal R}}{\Delta},\Theta,\zeta\right).$ (14)

Now, to calculate three others, we take the principal tensor of Kerr spacetime
in coordinates $(t,r,\theta,\phi)$

$h_{\alpha\beta}=\left(\begin{array}[]{cccc}0&r&a^{2}\sin\theta\cos\theta&0\\\
-r&0&0&ar\sin^{2}\theta\\\
-a^{2}\sin\theta\cos\theta&0&0&a\left(a^{2}+r^{2}\right)\sin\theta\cos\theta\\\
0&-ar\sin^{2}\theta&-a\left(a^{2}+r^{2}\right)\sin\theta\cos\theta&0\\\
\end{array}\right),$ (15)

and the corresponding Killing vector $\xi^{\mu}=(1,0,0,0)$ related by Eq. (1).
This allows us to calculate one of the parallel transported vectors using Eq.
(2)

$v_{\alpha}=\bigg{(}\frac{A(r,\theta)\rho^{2}+r{\cal
R}+a^{2}\sin\theta\cos\theta\Theta}{\rho^{2}},-\frac{A(r,\theta){\cal
R}+a^{2}r-a\zeta
r+r^{3}}{\Delta},a(\zeta\cot\theta-a\sin\theta\cos\theta)-A(r,\theta)\Theta,\\\
-\frac{\zeta A(r,\theta)\rho^{2}+ar\sin^{2}\theta{\cal
R}+a\left(a^{2}+r^{2}\right)\sin\theta\cos\theta\Theta}{\rho^{2}}\bigg{)}.$
(16)

Moreover, using Eq. (4) for the derivative of $A(r,\theta)$, we could also
calculate $A$ as

$\dot{A}(r,\theta)=-1,\qquad\mathrm{or}\quad A(r,\theta)={\cal F}(r)+{\cal
G}(\theta),$ (17)

where

${\cal F}(r)=-\int\frac{r^{2}}{{\cal R}}dr;\quad{\cal
G}(\theta)=-\int\frac{a^{2}\cos^{2}\theta}{\Theta}d\theta.$ (18)

Now, we have all the ingredients that allow us to integrate $\dot{B}$
appearing in Eq. (5), from which one can express $B$ in terms of $A$ as

$B(r,\theta)=\frac{1}{2}\left(r^{2}-a^{2}\cos^{2}\theta-A^{2}(r,\theta)\right).$
(19)

Thus, using Eq. (5), we could calculate another parallel transported vector

$n_{0\alpha}=\bigg{(}\frac{-4A(r,\theta)\left(2r{\cal R}+a^{2}\sin
2\theta\Theta\right)-4A^{2}(r,\theta)\rho^{2}+2\rho^{2}\left(a^{2}\cos
2\theta-3a^{2}+4a\zeta-2r^{2}\right)}{8\rho^{2}},\\\
\frac{2A^{2}(r,\theta){\cal
R}+4r\left(a^{2}-a\zeta+r^{2}\right)A(r,\theta)+2\rho^{2}{\cal
R}}{4\Delta},\frac{1}{4}\left(2A^{2}(r,\theta)\Theta+2aA(r,\theta)(a\sin
2\theta-2\zeta\cot\theta)-2\rho^{2}\Theta\right),\\\ \frac{8a\sin\theta
A(r,\theta)\left(r\sin\theta{\cal
R}+\left(a^{2}+r^{2}\right)\cos\theta\Theta\right)+4\zeta
A^{2}(r,\theta)\rho^{2}-4\rho^{2}\left(\zeta
a^{2}\sin^{2}\theta+\left(a^{2}+r^{2}\right)\left(\zeta-2a^{2}\sin^{2}\theta\right)\right)}{8\rho^{2}}\bigg{)}.$
(20)

To calculate the last parallel transported vector, we need the Hodge dual of
$h_{\alpha\beta}$. It is given as

$*h_{\alpha\beta}=\left(\begin{array}[]{cccc}0&-a\cos\theta&ar\sin\theta&0\\\
a\cos\theta&0&0&-a^{2}\sin^{2}\theta\cos\theta\\\
-ar\sin\theta&0&0&r\left(a^{2}+r^{2}\right)\sin\theta\\\
0&a^{2}\sin^{2}\theta\cos\theta&-r\left(a^{2}+r^{2}\right)\sin\theta&0\\\
\end{array}\right).$ (21)

From Eq. (6), the final parallel transported vector is

$u_{\alpha}=\bigg{(}\frac{ar\sin\theta\Theta-a\cos\theta{\cal
R}}{\rho^{2}},\frac{a\cos\theta\left(a^{2}-a\zeta+r^{2}\right)}{\Delta},\zeta
r\csc\theta-ar\sin\theta,\\\
\frac{\sin\theta\left(a^{2}\sin\theta\cos\theta{\cal
R}-r\left(a^{2}+r^{2}\right)\Theta\right)}{\rho^{2}}\bigg{)}.$ (22)

These null trajectories $l_{0}^{\mu}$ and polarization vectors
$m_{0\mu}=(v_{\mu}+iu_{\mu})/2$ constitute the solutions of electromagnetic
wave equations in the geometric optics approximation (see Ref. [6]).

##  III Transverse deflection due to the spin Hall effect

After constructing the Fermi transported (parallel propagated) null tetrad, we
study the spin Hall effect on the Kerr spacetime. Let us consider the lensing
object described by the Kerr geometry with mass $M$ and angular momentum per
unit mass $a$. The geometric optics approximation is valid only when the
characteristic wavelength of the waves is much smaller than the horizon radius
($1/\omega\ll M+\sqrt{M^{2}-a^{2}}$) [38, 39]. To get the subleading order
correction of the null trajectory, we substitute the above relations into the
propagation equation (see Appendix B) [10]

$\frac{D^{2}x^{\alpha}}{D\lambda^{2}}=-\frac{i}{\omega}R^{\alpha}_{~{}\beta\mu\nu}l_{0}^{\beta}m_{0}^{\mu}\tilde{m}_{0}^{\nu}.$
(23)

The results are cumbersome in general, but a numerical solution is possible.
However, restricting ourselves near the equatorial plane considerably
simplifies calculations, and we will present our results on the transverse
deflection of the light ray trajectory there. At the equatorial plane
$\theta=\pi/2$, $\eta=0$ and

$\displaystyle\frac{D^{2}t}{D\lambda^{2}}=0,\quad\frac{D^{2}r}{D\lambda^{2}}=0,$
$\displaystyle\frac{D^{2}\theta}{D\lambda^{2}}=-\frac{3M(a-\zeta)A\left(r,\pi/2\right)}{\omega
r^{6}},\quad\frac{D^{2}\phi}{D\lambda^{2}}=0.$ (24)

where $A\left(r,\pi/2\right)={\cal F}(r)$. Integrating this equation gives

$\dot{\theta}=-\frac{3M(a-\zeta)}{\omega r}\int\frac{{\cal F}(r)}{r^{3}{\cal
R}}dr\bigg{|}_{\eta=0},$ (25)

where the integration constant is chosen to be zero as the term containing it
gives divergent results on integration. Thus, the deflection of light rays in
the $\theta$-direction as it passes close to the lensing object of mass $M$
and angular momentum $aM$ could be calculated using the relation

$d\theta=\frac{\dot{\theta}}{\dot{r}}dr=-\frac{3M(a-\zeta)r}{{\omega\cal
R}}\int\frac{{\cal F}(r)}{r^{3}{\cal R}}dr\bigg{|}_{\eta=0}dr.$ (26)

This result for the transverse deflection is very similar to that for the
Schwarzschild spacetime given in Ref [9]. However, the difference comes from
the appearance of the $a-\zeta$ term, instead of $\zeta$ only, which tends to
cancel the magnitude of the spin Hall effect depending on the angular momentum
of particle $\zeta$ and hole $a$. Explicitly, for the Schwarzschild case,
$a=0$ and $\eta=0$

${\cal
F}(r)=-\int\frac{r^{2}}{\sqrt{r^{4}-\left(1-\frac{2M}{r}\right)r^{2}\zeta^{2}}}dr.$
(27)

A photon starting from infinity and approaching the lensing object within the
closest distance of $R_{0}$ has $\zeta\approx R_{0}$, up to leading order in
$M/r$. We can now integrate, up to leading order in $M/r$, to obtain the total
deflection angle of electromagnetic waves in the $\theta$-direction while
passing near a lensing object of mass $M$

$\theta=2\left(\theta_{\infty}-\theta_{0}\right)=\frac{\pi M}{2\omega
R_{0}^{2}}.$ (28)

This transverse deflection agrees with the result reported in Ref. [40] (it is
also interesting to compare the magnitude of this transverse deflection with
the second-order parametrized post-Newtonian expansion of the total deflection
in the geometric optics approximation [41]). Note that this deflection is half
the deflection of gravitational waves due to the spin-orbit correction [9].

For comparison, we take the situation where the external orbital angular
momentum of the light is zero $\zeta=0$ and consider only the angular momentum
of the hole. $\zeta=0$ corresponds to principle null rays with $\eta=a^{2}$.
As we want to approximate up to the leading order expansion in $M/r$
(parameter $a$ should scale accordingly in the post-Newtonian approximation
[42]), we could take $\eta\approx 0$. Thus, we have an electromagnetic wave
propagating through an equatorial plane $\theta=\pi/2$ of a rotating lensing
object. In this circumstance, we have

${\cal F}(r)=-\int\frac{r^{2}}{\sqrt{\left(r^{2}+a^{2}\right)^{2}-\Delta
a^{2}}}dr.$ (29)

We could substitute this into Eq. (25) to obtain $\dot{\theta}$. We have

$\dot{\theta}=-\frac{aM}{\omega r^{4}}.$ (30)

This is the only nonvanishing subleading order contribution (which could be
confirmed by directly integrating Eqs. (24)). So, from Eq. (46), we could
write subleading order components of the propagation vector as

$l_{1\mu}-b_{\mu}=\left(0,0,-\frac{aM}{r^{2}},0\right).$ (31)

Now, let us write $l_{1\mu}-b_{\mu}=\nabla\Psi$ for some $\Psi$. Then on
integration, we obtain

$\Psi=-\int_{\pi/2}^{\theta}\frac{aM}{r^{2}}d\theta=-\frac{aM}{r^{2}}(\theta-\pi/2)\approx\frac{aM}{r^{2}}\cos\theta,$
(32)

where we have used the result that $-(\theta-\pi/2)$ is the leading order
expansion of the $\cos\theta$ near $\pi/2$. Note that the gradient of this
solution for $\Psi$ also contributes to the radial trajectory. However, this
contribution is not on the leading order; thus, the deflection along the
$\theta$ direction in the subleading order remains unchanged. As a result, the
magnitude of the transverse deflection due to the spin-orbit interaction would
not be affected. Thus

$\Psi=\frac{\mathbf{J}\cdot\mathbf{r}}{r^{3}},\quad\mathrm{for}\quad|\mathbf{J}|=aM,$
(33)

gives the gravitomagnetic scalar potential. In these settings, which are not
manifestly covariant, the dispersion relation given in Eq. (44) reduces to the
form

$\mathbf{l_{0}}\cdot\mathbf{l_{0}}+\frac{2}{\omega}\nabla\Psi\cdot\mathbf{l_{0}}=0,$
(34)

which matches the dispersion relation given in Ref. [43, 44]. Although the
magnitude of the transverse deflection matches, other results, like the time
delay, would differ from these calculations (see above Eq. (33)). Also, note
that deviation in trajectory caused by the angular momentum of hole $a$ is in
the opposite direction to the deviation by the particle’s angular momentum
$\zeta$.

###  III.1 Numerical results

Let us take the propagation Eq. (23)

$\frac{D^{2}x^{\alpha}}{D\lambda^{2}}=-\frac{i}{\omega}R^{\alpha}_{~{}\beta\mu\nu}l^{\beta}m^{\mu}\tilde{m}^{\nu}.$
(35)

These are coupled second-order differential equations, and we do numerical
integration and plot the results. This calculation up to the subleading order
approximation is facilitated by the fact that we could substitute the leading
order term for each tetrad component. Every quantity here can be computed in
terms of the parallel propagated null tetrad. We thus can solve these
equations using Mathematica, for which we use the default integration method,
precision and accuracy. We can show that these quantities
$(x(\lambda),l(\lambda))$ are gauge invariant [10], which thus makes our
result gauge invariant. Fig. 1 shows the result of our numerical simulation
depicting the gravitational spin Hall effect in the Kerr background. We have
exaggerated the values of the parameters to make the effect visible. The plots
are for the rays of opposite circular polarization $s=\pm 1$, along with the
ray of spin-zero for reference. The circularly polarized rays are not confined
on the plane even at the equator, even though the null geodesics can be
restricted on the plane in Kerr spacetime.

Figure 1:  Demonstrating the spin Hall effect in Kerr spacetime for the black
hole of mass $M=1$ and angular momentum per unit mass $a=0.9$. Here, the red
trajectory is for $s=+1$, the blue trajectory is for $s=-1$, and the green
trajectory is for $s=0$ (geodesic curve). To make the effect visible, we have
chosen an exaggerated value of the characteristic frequency $1/\omega=0.7$.
The light source is on the equator $\theta=\pi/2$ with an arbitrary initial
position $\phi=7\pi/4$ and $r=20$ and emits light at time $t=0$. The initial
wave vector of these ingoing light rays is given by Eq. (14) with a negative
sign of ${\cal R}$ and a positive sign of ${\Theta}$. This initial condition
corresponds to the parameter $\lambda=0$, and we continue the propagation up
to $\lambda=50$. We chose the constants of motion for the trajectory $\eta=0$
and $\zeta=50$.

It is also interesting to consider the principal null rays, for which
$\zeta=0$. Although such rays do not encounter the gravitational spin Hall
effect in Schwarzschild spacetime, this is not true for Kerr spacetime. All
null trajectories generally undergo the gravitational spin Hall effect in Kerr
spacetime. The deviation for the principal null trajectories is shown in Fig.
2.

Observation of Fig. 1 and 2 shows that the deviation in the trajectory caused
by the angular momentum of the particle $\zeta$ is in the opposite direction
to the deviation by the angular momentum of the hole $a$. Also, as the rays
are scattered at a finite angle, the separation between the rays increases
with increasing distance, there is no reintersection of rays. The
gravitational spin Hall effect is caused by the interaction between the spin
and orbital angular momentum of a particle. When the particle crosses the
distance of closest approach, neither its spin nor orbital angular momentum
change direction, which causes the spin-orbit interaction effect to only grow
(compare Fig. 1 with the figures in Ref. [8]). So, this effect is visible at
large distances from the lensing object.

Figure 2:  Demonstrating the spin Hall effect for the principal null rays
(corresponding to $\zeta=0$ and $\eta=a^{2}$) in Kerr spacetime. To make the
effect visible, we chose the angular momentum per unit mass $a=0.99$ and the
helicity $s=\pm 2$ (positive sign for the red and negative sign for the green
trajectory). We have shown in Ref. [9] that propagation Eq. (23) holds for the
gravitational waves also, which have the helicity of two after we take account
of the difference in the helicity. Other values of the parameters/coordinates
are precisely the same as in Fig. 1. Because of infinite gravitational
blueshift, geometric optics approximation holds near the horizon regardless of
the gravitational field strength. This assumption also plays an important role
in deriving the Hawking radiation [45].

##  IV Discussions and conclusion

We have presented our numerical work to obtain the precise prediction of the
gravitational spin Hall effect by solving the trajectory equation for the
electromagnetic waves propagating in the Kerr spacetime. The trajectory
equation was derived using WKB expansion, where both the phase and amplitude
were expanded in inverse powers of an expansion parameter $\omega$. This
expansion in both the phase and amplitude was necessary because the
requirement to use the parallel propagated null tetrad in the geometric optics
approximation limits the flexibility of its transformation. Generalizing to
the subleading order requires using the Fermi-transported null tetrad, which
also constrains its transformation properties, resulting in an observer/gauge
independent spin Hall effect. This result of observer-independence was proved
in Ref. [10]. Thus, the spin Hall effect is distinct from observer/emitter-
dependent effects (Wigner rotation and gravitational Faraday rotation) that
occur in the geometric optics regime, which could be mitigated by some
encoding schemes (see Ref. [46]).

Additionally, the gravitational spin Hall effect results from the interaction
between the spin and orbital angular momentum. Neither spin nor orbital
angular momentum changes direction as light rays pass the distance of the
closest approach, thus causing the spin-orbit interaction effect to grow, not
decrease. As a result, the initially divergent trajectories between the rays
of different frequencies do not reconverge, which is confirmed by numerical
investigations.

Although weak, it is possible to observe the gravitational spin Hall effect:
first, the angular deviation, although small, can be observed at sufficiently
large distances. Second, a weak quantum measurement technique could detect the
spin Hall effect even if the spatial separation between the two circularly
polarized light rays in opposite directions is smaller than their wavelength
[47].

###### Acknowledgements.

PKD is supported by an International Macquarie University Research Excellence
Scholarship.

## Data Availability Statement

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

##  Appendix A WKB approximation

When the typical wavelength of the wave, propagating on approximately flat
spacetime, is very small (but nonnegligible) with respect to the length scale
of its amplitude and wavelength variations and the inhomogeneity of the
spacetime curvature where it travels, then spin optics correction is required.
We could formulate spin optics by using the WKB expansion

$A^{\alpha}=a^{\alpha}e^{i\omega{\cal S}},$ (36)

where $a^{\alpha}$ is the complex amplitude, and $\omega{\cal S}$ is the real
phase. Here, $\omega$ is the characteristic frequency of the wave. We denote
the square amplitude by $a=\left(\tilde{a}^{\alpha}a_{\alpha}\right)^{1/2}$,
wave vector by $l_{\alpha}={\cal S}_{;\alpha}$ and polarization vector by
$m^{\alpha}=a^{\alpha}/a$. Now, let us expand both the wave and polarization
vectors as

$\displaystyle l^{\alpha}=$ $\displaystyle
l^{\alpha}_{0}+\frac{l^{\alpha}_{1}}{\omega}+\frac{l^{\alpha}_{2}}{\omega^{2}}+...,$
(37) $\displaystyle m^{\alpha}=$ $\displaystyle
m^{\alpha}_{0}+\frac{m^{\alpha}_{1}}{\omega}+\frac{m^{\alpha}_{2}}{\omega^{2}}+....$
(38)

The necessity of expanding both the wave vector and amplitude comes from the
fact that higher-order phase factors like ${\cal S}_{1}(\lambda)$ can not be
absorbed into the leading-order amplitude $m^{\alpha}_{0}$ by transformation
$m^{\alpha}\to e^{i{\cal S}_{1}(\lambda)/\omega}m^{\alpha}$. This is because
geometric optics require using a parallel propagated null tetrad, which
constrains its transformation freedom $m^{\alpha}\to e^{i{\cal
S}_{1}(\lambda)/\omega}m^{\alpha}$ providing an additional restriction

$\frac{d{\cal S}_{1}(\lambda)}{d\lambda}=0.$ (39)

Next, we substitute this vector potential onto the Lorentz gauge condition for
electromagnetic waves, from which we obtain

$l_{0}^{\alpha}m_{0\alpha}+\frac{1}{\omega}\left(l_{0}^{\alpha}m_{1\alpha}+l_{1}^{\alpha}m_{0\alpha}-i\left(\frac{a_{;\alpha}}{a}m_{0}^{\alpha}+m^{\alpha}_{0;\alpha}\right)\right)=0.$
(40)

Again, substituting the vector potential into the source-free electromagnetic
wave equation, we obtain

$j^{\alpha}=\vcentcolon
m_{0}^{\alpha}l_{0\beta}l_{0}^{\beta}+\frac{1}{\omega}\bigg{(}m_{1}^{\alpha}l_{0\beta}l_{0}^{\beta}+2m_{0}^{\alpha}l_{1\beta}l_{0}^{\beta}\\\
-i\left(m_{0}^{\alpha}l^{\beta}_{0;\beta}+2m^{\alpha}_{0;\beta}l_{0}^{\beta}+2\frac{a_{;\beta}}{a}m_{0}^{\alpha}l_{0}^{\beta}\right)\bigg{)}=0,$
(41)

up to the subleading order in $\omega$. Let us now calculate an identically
vanishing quantity,
$\tilde{m}_{0\alpha}j^{\alpha}+m_{0\alpha}\tilde{j}^{\alpha}$

$l_{0\beta}l_{0}^{\beta}+\frac{2}{\omega}\left(l_{1\beta}-b_{\beta}\right)l_{0}^{\beta}=0.$
(42)

This relation can be considered as the generalization of the dispersion
relation of geometric optics. For simplification, we have applied
$\tilde{m}_{0}^{\alpha}m_{0\alpha}=1$ and substituted

$\frac{i}{2}\left(\tilde{m}^{\alpha}m_{\alpha;\beta}-m^{\alpha}\tilde{m}_{\alpha;\beta}\right)=i\tilde{m}^{\alpha}m_{\alpha;\beta}\vcentcolon=b_{\beta}.$
(43)

##  Appendix B Equations of spin optics

To find the propagation equation of spin optics, let us start with the
generalized dispersion relation (42) and express it as

$\frac{1}{2}g_{\alpha\beta}l_{0}^{\alpha}l_{0}^{\beta}+\frac{1}{\omega}g_{\alpha\beta}\left(l_{1}^{\alpha}-b^{\alpha}\right)l_{0}^{\beta}=0.$
(44)

This relation can be viewed as the Hamilton-Jacobi equation for the subleading
order phase function ${\cal S}$, where ${\cal
S}_{;\alpha}=l_{0\alpha}+l_{1\alpha}/\omega$. The Hamiltonian function on the
cotangent bundle $T^{*}M$, associated with the Hamilton-Jacobi equation, is

$H(x,l)=\frac{1}{2}g^{\alpha\beta}l_{0\alpha}l_{0\beta}+\frac{1}{\omega}g^{\alpha\beta}\left(l_{1\alpha}-b_{\alpha}\right)l_{0\beta}\\\
=\frac{1}{2\omega^{2}}g^{\alpha\beta}\left(\omega
l_{0\alpha}+l_{1\alpha}-b_{\alpha}\right)\left(\omega
l_{0\beta}+l_{1\beta}-b_{\beta}\right).$ (45)

Hamilton’s equations of motion are

$\frac{dx^{\alpha}}{d\lambda}=\frac{\partial H}{\partial
l_{\alpha}}=g^{\alpha\beta}\left(l_{\beta}-\frac{b_{\beta}}{\omega}\right),$
(46)

and

$\frac{dl_{\alpha}}{d\lambda}=-\frac{\partial H}{\partial
x^{\alpha}}=\frac{1}{2}\dot{x}^{\mu}\dot{x}^{\nu}\frac{\partial
g_{\mu\nu}}{\partial
x^{\alpha}}+\frac{1}{\omega}g^{\mu\nu}\dot{x}_{\nu}\frac{\partial
b_{\mu}}{\partial x^{\alpha}},$ (47)

where we have used Eq. (46) in obtaining this. Now, we could write the action
whose solution from the variation principle corresponds to the solution of the
Hamilton-Jacobi equation (44) (see Ref. [10])

${\cal S}(x,l)=\int_{\lambda}(\dot{x}^{\alpha}l_{\alpha}-H(x,l))d\lambda\\\
=\frac{1}{2}\int\dot{x}^{\alpha}\dot{x}_{\alpha}d\lambda+\frac{1}{\omega}\int
b_{\alpha}\dot{x}^{\alpha}d\lambda,$ (48)

where Eqs. (45) and (46) are used for simplification. The first term is the
optical path length, and the second term resembles the Berry connection of
optics. Either the variational principle $\delta{\cal S}=0$ or simplification
of Hamilton’s equation of motion (47) yields the same equation [10]

$\frac{D^{2}x_{\mu}}{D\lambda^{2}}+\frac{1}{\omega}\left(b_{\mu;\nu}-b_{\nu;\mu}\right)\dot{x}^{\nu}=0.$
(49)

Further simplification of
$b_{\beta;\alpha}-b_{\alpha;\beta}\vcentcolon=k_{\alpha\beta}$ gives

$k_{\alpha\beta}=-iR_{\alpha\beta\mu\nu}m^{\mu}\tilde{m}^{\nu}+i\left(\tilde{m}^{\nu}_{;\alpha}m_{\nu;\beta}-\tilde{m}^{\nu}_{;\beta}m_{\nu;\alpha}\right).$
(50)

After substituting this back into Eq. (49), we obtain

$\frac{D^{2}x^{\alpha}}{D\lambda^{2}}=-\frac{i}{\omega}R^{\alpha}_{~{}\beta\mu\nu}m^{\mu}\tilde{m}^{\nu}l_{0}^{\beta}\approx-\frac{i}{\omega}R^{\alpha}_{~{}\beta\mu\nu}l_{0}^{\beta}m_{0}^{\mu}\tilde{m}_{0}^{\nu},$
(51)

which is the propagation Eq. (23).

## References

  * Misner et al. [1973] Misner, C. W., Thorne, K. S., & Wheeler, J. A., Gravitation, San Francisco: W.H. Freeman and Co., 1973.
  * Dolan [2017] Dolan, S. R. 2017, Int. J. Mod. Phys. D 27, 1843010.
  * Frolov & Shoom [2011] Frolov, V. P. & Shoom, A. A. 2011, Phys. Rev. D, 84, 044026.
  * Yamamoto [2018] Yamamoto, N. 2018, Phys. Rev. D, 98, 061701.
  * Yoo [2012] Yoo, C.-M. 2012, Phys. Rev. D, 86, 084005.
  * Frolov [2020] Frolov, V. P. 2020, Phys. Rev. D, 102, 084013.
  * Oancea et al. [2019] Oancea, M. A., Paganini, C. F., Joudioux, J., et al., arXiv:1904.09963 (2019).
  * Oancea et al. [2020] Oancea, M. A., Joudioux, J., Dodin, I. Y., et al. 2020, Phys. Rev. D, 102, 024075.
  * Dahal [2021] Dahal, P. K.,  Astronomy, 1(3), 271-287 (2022).
  * Dahal [2021] Dahal, P. K., Appl. Phys. B 129, 11 (2023).
  * Bliokh & Bliokh [2015] Bliokh, K. Y., Bliokh, Y. P., Jetp Lett. 79, 519–522 (2005).
  * Hosten & Kwiat [2008] Hosten, O. & Kwiat, P., Science, 319, 787 (2008).
  * Bliokh et al. [2008] Bliokh, K. Y., Niv, A., Kleiner, V., et al. 2008, Nature Photonics, 2, 748 (2008).
  * Wunderlich et al. [2005] Wunderlich, J., Kaestner, B., Sinova, J., et al. 2005, Phys. Rev. Lett. , 94, 047204.
  * Kato et al. [2004] Kato, Y. K., Myers, R. C., Gossard, A. C., et al. 2004, Science, 306, 1910.
  * Bliokh et al. [2015] Bliokh, K. Y., Rodríguez-Fortuño, F. J., Nori, F., et al., Nature Photonics, 9, 796 (2015).
  * Ling et al. [2017] Ling, X., Zhou, X., Huang, K., et al. 2017, Reports on Progress in Physics, 80, 066401.
  * Frolov & Shoom [2012] Frolov, V. P. & Shoom, A. A., Phys. Rev. D, 86, 024010 (2012).
  * Mashhoon [1993] Mashhoon, B., Physics Letters A, 173, 347 (1993).
  * Cusin & Lagos [2020] Cusin, G. & Lagos, M., Phys. Rev. D, 101, 044041 (2020).
  * Ezquiaga et al. [2021] Ezquiaga, J. M., Holz, D. E., Hu, W., et al., Phys. Rev. D, 103, 064047 (2021).
  * [22] Chandrasekhar, S., The Mathematical Theory of Black Holes (Oxford University Press, Oxford, 1992).
  * Marck [1983] Marck, J.-A., Proceedings of the Royal Society of London Series A, 385, 431 (1983).
  * [24] Christensen, S. M., Phys. Rev. D 17, 946 (1978).
  * [25] Sachs, R., Proceedings of the Royal Society of London Series A. 264, 309 (1962).
  * [26] Sachs, R., Phys. Rev. 128, 2851 (1962).
  * Newman & Penrose [1962] Newman, E. & Penrose, R., Journal of Mathematical Physics, 3, 566 (1962).
  * [28] Carter, B., Phys. Rev. 174, 1559 (1968).
  * [29] Walker, M., Penrose, R., Commun.Math. Phys. 18, 265–274 (1970).
  * [30] Penrose, R., Annals of the New York Academy of Sciences, 224: 125-134 (1973).
  * [31] Floyd, R., The dynamics of Kerr fields, PhD Thesis, London (1973).
  * Krtouš et al. [2007] Krtouš, P., Kubizňák, D., Page, D. N., et al. Phys. Rev. D, 76, 084034 (2007).
  * Frolov et al. [2017] Frolov, V. P., Krtouš, P., & Kubizňák, D. Living Reviews in Relativity, 20, 6 (2017).
  * [34] Collinson, C.D., Howarth, L., General Relativity and Gravitation 32, 1767–1776 (2000).
  * [35] Yano, K., Annals of Mathematics, 55(2), 328–347 (1952).
  * Marck [1983] Marck, J.-A., Physics Letters A, 97, 140 (1983).
  * Connell et al. [2008] Connell, P., Frolov, V. P., & Kubizňák, D., Phys. Rev. D, 78, 024042 (2008).
  * Takahashi [2017] Takahashi, R., Astrophys. J. , 835, 103 (2017).
  * Nakamura [1998] Nakamura, T. T., Phys. Rev. Lett. , 80, 1138 (1998).
  * Gosselin et al. [2007] Gosselin, P., Bérard, A., & Mohrbach, H., Phys. Rev. D, 75, 084035 (2007).
  * Edery & Godin [2006] Edery, A. & Godin, J., General Relativity and Gravitation, 38, 1715 (2006).
  * Will [1993] Will, C. M., Theory and Experiment in Gravitational Physics, Cambridge University Press, 1993, Ch. 4.
  * Mashhoon & Rahvar [2022] Mashhoon, B. & Rahvar, S., arXiv:2211.01691 (2022).
  * Ramos & Mashhoon [2006] Ramos, J. & Mashhoon, B., Phys. Rev. D, 73, 084003 (2006).
  * Hawking [1975] Hawking, S. W., Communications in Mathematical Physics, 43, 199 (1975).
  * Dahal & Terno [2021] Dahal, P. K. & Terno, D. R., Phys. Rev. A, 104, 042610 (2021).
  * Hosten & Kwiat [2008] Hosten, O. & Kwiat, P., Science, 319, 787 (2008).


$M=X_{c}(F_{\ast})\setminus Z\subset W$ given by the set of nonsingular points
of $X_{c}(F_{\ast})$. Let $f\colon M\rightarrow W$ denote the inclusion map.
Since $X,X_{c}(F_{\ast})\subset G$ are $\mathcal{X}$-transverse by assumption,
it follows that $X\cap W=X$ and $X_{c}(F_{\ast})\cap W=M$ are
$\mathcal{X}$-transverse in $W$. We apply axiom (4) of the Gysin coherent
characteristic class $c\ell$ to the pure-dimensional compact subvariety
$Y:=M\cap X=X_{c}(F_{\ast})\cap X\subset M$ with inclusion map $j\colon
Y\rightarrow M$ to obtain

(32) $c\ell^{\ast}(f)^{-1}\cap f^{!}c\ell_{\ast}(i)=c\ell_{\ast}(j)\qquad\in
H_{\ast}(M;\mathbb{Q}).$

As shown above, the class $c\ell_{\ast}(i)\in H_{\ast}(W;\mathbb{Q})$ is
concentrated in even degrees of the form $2d-2r$ for nonnegative integers $r$.
The Gysin map $f^{!}$ drops the degree in homology by
$2k^{\prime}m^{\prime\prime}$ because $M\subset W$ has complex codimension
$k^{\prime}m^{\prime\prime}$. Setting $l:=d-k^{\prime}m^{\prime\prime}$, we
conclude that $f^{!}c\ell_{\ast}(i)\in H_{\ast}(M;\mathbb{Q})$ is concentrated
in even degrees of the form $2d-2r-2k^{\prime}m^{\prime\prime}=2l-2r$ for
nonnegative integers $r$. Therefore, the part in $H_{0}(M;\mathbb{Q})$ of
equation (32) is

(33) $\sum_{r=0}^{l}(c\ell^{\ast}(f)^{-1})^{2l-2r}\cap
f^{!}c\ell_{2d-2r}(i)=c\ell_{0}(j)\quad\in H_{0}(M;\mathbb{Q}).$

By inserting equation (31) into equation (33), we obtain

$\sum_{r=0}^{l}\sum_{\begin{subarray}{c}r^{\prime},r^{\prime\prime}\geq 0,\\\
r^{\prime}+r^{\prime\prime}=r\end{subarray}}(c\ell^{\ast}(f)^{-1})^{2l-2r}\cap
f^{!}\Sigma_{\ast}^{W}(c\ell_{2d^{\prime}-2r^{\prime}}(i^{\prime})\times
c\ell_{2d^{\prime\prime}-2r^{\prime\prime}}(i^{\prime\prime}))=c\ell_{0}(j).$

Therefore, by applying the augmentation $\varepsilon_{\ast}\colon
H_{\ast}(M;\mathbb{Q})\rightarrow\mathbb{Q}$, we have

$\sum_{r=0}^{l}\sum_{\begin{subarray}{c}r^{\prime},r^{\prime\prime}\geq 0,\\\
r^{\prime}+r^{\prime\prime}=r\end{subarray}}\langle
c\ell^{\ast}(f)^{-1},f^{!}\Sigma_{\ast}^{W}(c\ell_{2d^{\prime}-2r^{\prime}}(i^{\prime})\times
c\ell_{2d^{\prime\prime}-2r^{\prime\prime}}(i^{\prime\prime}))\rangle=|c\ell_{\ast}|(j).$

Using (9) to write

$\displaystyle c\ell_{2d^{\prime}-2r^{\prime}}(i^{\prime})$
$\displaystyle=\sum_{\begin{subarray}{c}b^{\prime}\in\mathcal{P}(m^{\prime},k^{\prime}),\\\
|b^{\prime}|=d^{\prime}-r^{\prime}\end{subarray}}\lambda_{i^{\prime}}^{b^{\prime}}\cdot[X_{b^{\prime}}]_{G^{\prime}},$
$\displaystyle c\ell_{2d^{\prime\prime}-2r^{\prime\prime}}(i^{\prime\prime})$
$\displaystyle=\sum_{\begin{subarray}{c}b^{\prime\prime}\in\mathcal{P}(m^{\prime\prime},k^{\prime\prime}),\\\
|b^{\prime\prime}|=d^{\prime\prime}-r^{\prime\prime}\end{subarray}}\lambda_{i^{\prime\prime}}^{b^{\prime\prime}}\cdot[X_{b^{\prime\prime}}]_{G^{\prime\prime}},$

we obtain

$|c\ell_{\ast}|(j)=\sum_{r=0}^{l}\sum_{\begin{subarray}{c}r^{\prime},r^{\prime\prime}\geq
0,\\\
r^{\prime}+r^{\prime\prime}=r\end{subarray}}\sum_{\begin{subarray}{c}b^{\prime}\in\mathcal{P}(m^{\prime},k^{\prime}),\\\
|b^{\prime}|=d^{\prime}-r^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}b^{\prime\prime}\in\mathcal{P}(m^{\prime\prime},k^{\prime\prime}),\\\
|b^{\prime\prime}|=d^{\prime\prime}-r^{\prime\prime}\end{subarray}}\lambda_{i^{\prime}}^{b^{\prime}}\lambda_{i^{\prime\prime}}^{b^{\prime\prime}}\cdot\langle
c\ell^{\ast}(f)^{-1},f^{!}\Sigma_{\ast}^{W}([X_{b^{\prime}}]_{G^{\prime}}\times[X_{b^{\prime\prime}}]_{G^{\prime\prime}})\rangle.$

Hence, equation (29) follows in view of (15). Consequently, the expression
$|c\ell_{\ast}|(i^{\prime},i^{\prime\prime}):=|c\ell_{\ast}|(X\cap
X_{c}(F_{\ast})\subset G)$ is independent of all choices involved.

This completes the proof of 7.10. ∎

######  Remark 7.11.

The characteristic number $|c\ell_{\ast}|(i^{\prime},i^{\prime\prime})$
introduced in 7.10 is defined as the genus of the inclusion $X\cap
X_{c}(F_{\ast})\subset G$, which may be called a characteristic subvariety in
view of its role in 7.1. Note that if the spaces $X^{\prime}$ and
$X^{\prime\prime}$ are Schubert varieties, then it follows from 5.10 that the
characteristic subvariety is the intersection of three Schubert varieties in
general position. It is worthwhile mentioning that for a finite number of
given Gysin coherent characteristic classes with respect to $\mathcal{X}$, we
may assume that the same notion of $\mathcal{X}$-transversality applies in
axiom (4), so that we can simultaneously use the same collection of
characteristic subvarieties in their computation.

###  7.3. Proof of 7.1

We apply 7.10 to the inclusions $i^{\prime}\colon X^{\prime}\hookrightarrow
G^{\prime}$ and $i^{\prime\prime}\colon
X^{\prime\prime}:=X_{a^{\prime\prime}}(D_{\ast}^{\prime\prime})\hookrightarrow
G^{\prime\prime}$, which are both contained in $\mathcal{X}$, to obtain

(34)
$|c\ell_{\ast}|(i^{\prime},i^{\prime\prime})=\sum_{r=0}^{l}\sum_{\begin{subarray}{c}0\leq
r^{\prime}\leq l,\\\ 0\leq r^{\prime\prime}\leq l,\\\
r^{\prime}+r^{\prime\prime}=r\end{subarray}}\sum_{\begin{subarray}{c}b^{\prime}\in\mathcal{P}(m^{\prime},k^{\prime}),\\\
|b^{\prime}|=d^{\prime}-r^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}b^{\prime\prime}\in\mathcal{P}(m^{\prime\prime},k^{\prime\prime}),\\\
|b^{\prime\prime}|=d^{\prime\prime}-r^{\prime\prime}\end{subarray}}\lambda_{i^{\prime}}^{b^{\prime}}\lambda_{i^{\prime\prime}}^{b^{\prime\prime}}\cdot\langle
c\ell^{\ast}\rangle(b^{\prime},b^{\prime\prime}),$

where we note that $l=d^{\prime}+d^{\prime\prime}-k^{\prime}m^{\prime\prime}$
with
$d^{\prime\prime}:=\operatorname{dim}_{\mathbb{C}}(X^{\prime\prime})=|a^{\prime\prime}|$
because we have $|a^{\prime}|+|a^{\prime\prime}|=k^{\prime}m^{\prime\prime}$
by definition of $a^{\prime\prime}$ (see Figure 2). Hence, to show (10), it
remains to show the equality

$\lambda^{a^{\prime}}_{i^{\prime}}=\sum_{\begin{subarray}{c}b^{\prime}\in\mathcal{P}(m^{\prime},k^{\prime}),\\\
|b^{\prime}|=|a^{\prime}|\end{subarray}}\sum_{\begin{subarray}{c}b^{\prime\prime}\in\mathcal{P}(m^{\prime\prime},k^{\prime\prime}),\\\
|b^{\prime\prime}|=|a^{\prime\prime}|\end{subarray}}\lambda_{i^{\prime}}^{b^{\prime}}\lambda_{i^{\prime\prime}}^{b^{\prime\prime}}\cdot\langle
c\ell^{\ast}\rangle(b^{\prime},b^{\prime\prime}),$

whose right hand side equals the summand indexed by
$(r,r^{\prime},r^{\prime\prime})=(l,l,0)$ on the right hand side of (34).

By definition of the Gysin coherent characteristic class $c\ell$, we have
$c\ell_{2d^{\prime\prime}}(X^{\prime\prime})=[X^{\prime\prime}]_{G^{\prime\prime}}$
in $H_{2d^{\prime\prime}}(G^{\prime\prime};\mathbb{Q})$. In other words, we
have
$\lambda_{i^{\prime\prime}}^{b^{\prime\prime}}=\delta_{a^{\prime\prime}b^{\prime\prime}}$
for all $b^{\prime\prime}\in\mathcal{P}(m^{\prime\prime},k^{\prime\prime})$
with $|b^{\prime\prime}|=|a^{\prime\prime}|$. Furthermore, by 7.8, we have
$\langle
c\ell^{\ast}\rangle(b^{\prime},a^{\prime\prime})=\delta_{a^{\prime}b^{\prime}}$,
and the claim follows.

This completes the proof of 7.1.

##  8\. Proof of 6.4

Let $c\ell$ and $\widetilde{c\ell}$ be Gysin coherent characteristic classes
with respect to $\mathcal{X}$ such that
$c\ell^{\ast}=\widetilde{c\ell}^{\ast}$ and
$|c\ell_{\ast}|=|\widetilde{c\ell}_{\ast}|$ for the associated genera. We
prove the claim by induction on the complex dimension $g=km$ (where $m:=n-k$)
of the target Grassmannian $G=G_{k}(\mathbb{C}^{n})$ ($0\leq k\leq n$) of
inclusions $i\colon X\hookrightarrow G$ of irreducible closed algebraic
subvarieties $X$. As for the induction basis $g=0$, we note that
$G=\operatorname{pt}$ is a one point space, and we have $X=\operatorname{pt}$
since irreducibility of $X$ implies that $X\neq\varnothing$. Hence, we obtain
$c\ell_{\ast}(i)=[X]_{G}=[\operatorname{pt}]=\widetilde{c\ell}_{\ast}(i)$ in
$H_{\ast}(G;\mathbb{Q})=\mathbb{Q}\cdot[\operatorname{pt}]$, so that the
inclusion $i\colon X\hookrightarrow G$ satisfies
$c\ell_{\ast}(i)=\widetilde{c\ell}_{\ast}(i)$. As for the induction
hypothesis, we fix an integer $g>0$, and assume that

  * $(\ast)_{g}$

for every inclusion $i\colon X\hookrightarrow G$ in $\mathcal{X}$ of an
irreducible closed algebraic subvariety $X$ in a Grassmannian
$G=G_{k}(\mathbb{C}^{n})$ of complex dimension $<g$, we have
$c\ell_{\ast}(i)=\widetilde{c\ell}_{\ast}(i)$.

As for the induction step, we consider the inclusion $i^{\prime}\colon
X^{\prime}\hookrightarrow G^{\prime}$ in $\mathcal{X}$ of an irreducible
closed $d^{\prime}$-dimensional algebraic subvariety $X^{\prime}$ in a
$g$-dimensional Grassmannian
$G^{\prime}=G_{k^{\prime}}(\mathbb{C}^{n^{\prime}})$ (where we note that
$0<k^{\prime}<n^{\prime}$ since $g=k^{\prime}m^{\prime}>0$ with
$m^{\prime}:=n^{\prime}-k^{\prime}$), and have to show that
$c\ell_{\ast}(i^{\prime})=\widetilde{c\ell}_{\ast}(i^{\prime})$ in
$H_{\ast}(G^{\prime};\mathbb{Q})$. That is, if $\\{\lambda^{b}_{i}\\}_{i,b}$
and $\\{\widetilde{\lambda}^{b}_{i}\\}_{i,b}$ denote the families of
coefficients associated to $c\ell$ and $\widetilde{c\ell}$, respectively, as
introduced in (9), then we have to show that
$\lambda^{b^{\prime}}_{i^{\prime}}=\widetilde{\lambda}^{b^{\prime}}_{i^{\prime}}$
for all $b^{\prime}\in\mathcal{P}(m^{\prime},k^{\prime})$ with
$|b^{\prime}|\in\\{0,\dots,d^{\prime}\\}$, which we shall prove by induction
on $l:=d^{\prime}-|b^{\prime}|$ (see Figure 5). As for the induction basis
$l=0$, we note that
$c\ell_{2d^{\prime}}(i^{\prime})=[X^{\prime}]_{G^{\prime}}=\widetilde{c\ell}_{2d^{\prime}}(i^{\prime})$
in $H_{2d^{\prime}}(G^{\prime};\mathbb{Q})$. As for the induction hypothesis,
we suppose that $l\in\\{1,\dots,d^{\prime}\\}$ is given such that

  * $(\ast\ast)_{l}$

for every $b^{\prime}\in\mathcal{P}(m^{\prime},k^{\prime})$ with
$d^{\prime}-|b^{\prime}|\in\\{0,\dots,l-1\\}$, we have
$\lambda^{b^{\prime}}_{i^{\prime}}=\widetilde{\lambda}^{b^{\prime}}_{i^{\prime}}$.

$0$$g$$g=\operatorname{dim}G$$l$$0$$0$$=d^{\prime}$$d^{\prime}$$l=d^{\prime}-|b^{\prime}|$

Figure 5. Induction scheme for the proof of 6.4. The outer induction runs over
the dimension $g$ of the ambient Grassmannian $G$, starting with $g=0$, while
the inner induction runs for every fixed embedding $i^{\prime}\colon
X^{\prime}\rightarrow
G^{\prime}=G_{k^{\prime}}(\mathbb{C}^{m^{\prime}+k^{\prime}})$ in
$\mathcal{X}$ with $\operatorname{dim}G^{\prime}=g$ and
$d^{\prime}:=\operatorname{dim}X^{\prime}$ over the formal codimension
$l:=d^{\prime}-|b^{\prime}|$ of partitions
$b^{\prime}\in\mathcal{P}(m^{\prime},k^{\prime})$ in $X^{\prime}$, starting
with $l=0$.

As for the induction step, we fix
$a^{\prime}\in\mathcal{P}(m^{\prime},k^{\prime})$ with
$d^{\prime}-|a^{\prime}|=l$, and have to show that
$\lambda^{a^{\prime}}_{i^{\prime}}=\widetilde{\lambda}^{a^{\prime}}_{i^{\prime}}$.
By applying 7.1 to the inclusion $i^{\prime}\colon X^{\prime}\hookrightarrow
G^{\prime}$ in $\mathcal{X}$, we obtain an inclusion $i^{\prime\prime}\colon
X_{a^{\prime\prime}}(D_{\ast}^{\prime\prime})\hookrightarrow G^{\prime\prime}$
of a Schubert subvariety $X_{a^{\prime\prime}}(D_{\ast}^{\prime\prime})\subset
G^{\prime\prime}$ into the Grassmannian
$G^{\prime\prime}=G_{k^{\prime\prime}}(\mathbb{C}^{m^{\prime\prime}+k^{\prime\prime}})$
whose dimension $k^{\prime\prime}m^{\prime\prime}$ is strictly smaller than
the dimension $g=k^{\prime}m^{\prime}$ of $G^{\prime}$ because we have $0\leq
k^{\prime\prime}<k^{\prime}$ and $0\leq m^{\prime\prime}\leq m^{\prime}$ by
definition of $k^{\prime\prime}$ and $m^{\prime\prime}$. Consequently, we have
$\lambda^{b^{\prime\prime}}_{i^{\prime\prime}}=\widetilde{\lambda}^{b^{\prime\prime}}_{i^{\prime\prime}}$
for all $b^{\prime\prime}\in\mathcal{P}(m^{\prime\prime},k^{\prime\prime})$ by
induction hypothesis $(\ast)_{g}$. Furthermore, we have
$\lambda^{b^{\prime}}_{i^{\prime}}=\widetilde{\lambda}^{b^{\prime}}_{i^{\prime}}$
for all $b^{\prime}\in\mathcal{P}(m^{\prime},k^{\prime})$ with
$d^{\prime}-|b^{\prime}|\in\\{0,\dots,l-1\\}$ by induction hypothesis
$(\ast\ast)_{l}$. All in all, we conclude that
$\lambda^{a^{\prime}}_{i^{\prime}}=\widetilde{\lambda}^{a^{\prime}}_{i^{\prime}}$
by comparing the equations (10) induced by $c\ell$ and $\widetilde{c\ell}$,
respectively. Here, note that for the genera, we have

$|c\ell_{\ast}|(i^{\prime},i^{\prime\prime})=|c\ell_{\ast}|(X\cap
X_{c}(F_{\ast})\hookrightarrow G)=|\widetilde{c\ell}_{\ast}|(X\cap
X_{c}(F_{\ast})\hookrightarrow
G)=|\widetilde{c\ell}_{\ast}|(i^{\prime},i^{\prime\prime})$

because we have $|c\ell_{\ast}|=|\widetilde{c\ell}_{\ast}|$, and we may use
the same collection of characteristic subvarieties by Remark 7.11. For the
integrals, we have $\langle
c\ell^{\ast}\rangle(b^{\prime},b^{\prime\prime})=\langle\widetilde{c\ell}^{\ast}\rangle(b^{\prime},b^{\prime\prime})$
since $(c\ell^{\ast})^{-1}=(\widetilde{c\ell}^{\ast})^{-1}$, and both of these
cohomology classes are integrated over the same triple intersection of
Schubert varieties by Remark 7.9.

This completes the proof of 6.4.

##  9\. The Goresky-MacPherson $L$-Class

Following Siegel [33], an oriented PL pseudomanifold $X$ is called a Witt
space if for some (and hence, as shown in [22, Section 2.4], for any) PL
stratification of $X$, the middle degree, lower middle perversity rational
intersection homology of all even-dimensional links vanishes. For example, any
pure-dimensional complex algebraic variety (endowed with the complex topology)
is a Witt space. (In fact, such a space has a natural orientation, and can be
PL stratified without strata of odd codimension.) Closed Witt spaces $X$ have
homological $L$-classes $L_{i}(X)\in H_{i}(X;\mathbb{Q})$ that generalize the
Poincaré duals of the cohomological $L$-classes of Hirzebruch [25] defined for
smooth manifolds. These classes were first introduced by Goresky and
MacPherson [21] for PL pseudomanifolds without strata of odd codimension. The
construction of the Goresky-MacPherson-Siegel $L$-classes is based on a Thom-
Pontrjagin type approach that exploits transversality techniques and bordism
invariance of the signature $\sigma(X)$ of the Goresky-MacPherson-Siegel
intersection form on middle-perversity intersection homology of the Witt space
$X$. Cheeger [15] gave a local formula for $L$-classes in terms of the eta-
invariant of simplicial links. In [2] and [3], the first author extended
$L$-classes to oriented stratified pseudomanifolds that allow for Lagrangian
sheaves along strata of odd codimension. For the central role of $L$-classes
in the topological classification of singular spaces similar to that of
Hirzebruch’s $L$-classes in the classification theory of high-dimensional
manifolds, we refer to works of Weinberger, Cappell and Shaneson ([13], [14],
[35]).

The collection of $L$-classes $L_{\ast}(X)=L_{0}(X)+L_{1}(X)+\dots\in
H_{\ast}(X;\mathbb{Q})$ with $X$ ranging over all closed Witt spaces is
characterized by the following axioms (see e.g. [12], and Proposition 8.2.11
in [4], and Theorem 9.4.18 in [18]):

  1. (1)

_(Signature normalization)_ For all $X$, we have
$\epsilon_{\ast}L_{\ast}(X)=\sigma(X)$.

  2. (2)

_(Gysin restriction for normally nonsingular inclusions with trivial normal
bundle)_ If $g\colon Y\hookrightarrow X$ is a normally nonsingular inclusion
of closed Witt spaces with trivial normal bundle, then

$g^{!}L_{\ast}(X)=L_{\ast}(Y)$

under the Gysin homomorphism $g^{!}\colon H_{\ast}(X;\mathbb{Q})\rightarrow
H_{\ast}(Y;\mathbb{Q})$.

For concrete computations in the complex projective algebraic setting, axiom
(2) is often not applicable due to the failure of triviality of normal
bundles. For example, normal bundles of normally nonsingular embeddings that
arise from transverse intersections of singular projective varieties with
smooth varieties are frequently nontrivial. By using the machinery of Banagl-
Laures-McClure [7], the first author established the following Gysin
restriction formula for the Goresky-MacPherson-Siegel $L$-class for arbitrary
normally nonsingular inclusions of even-dimensional Witt spaces.

######  Theorem 9.1 (see Theorem 3.18 in [6]).

Let $g\colon Y\hookrightarrow X$ be a normally nonsingular inclusion of closed
oriented even-dimensional PL Witt pseudomanifolds. Let $\nu$ be the
topological normal bundle of $g$. Then

$g^{!}L_{\ast}(X)=L^{\ast}(\nu)\cap L_{\ast}(Y).$

A different axiomatization involving codimension $0$ restrictions was used by
Matsui in [28, p. 61] to investigate ambient intersection formulae for
Goresky-MacPherson $L$-classes. In [5], the first author analyzed the behavior
of $L$-classes under Gysin transfers associated to finite degree covers.

The Goresky-MacPherson-Siegel $L$-class of complex projective algebraic
varieties fits into the framework of Gysin coherent characteristic classes
(6.2), as we shall detail next.

######  Theorem 9.2.

The pair $\mathcal{L}=(\mathcal{L}^{\ast},\mathcal{L}_{\ast})$ defined by
$\mathcal{L}^{\ast}(f)=L^{\ast}(\nu_{f})$ for every inclusion $f\colon
M\rightarrow W$ of a smooth closed subvariety $M\subset W$ in a smooth variety
$W$ with normal bundle $\nu_{f}$, and by
$\mathcal{L}_{\ast}(i)=i_{\ast}L_{\ast}(X)$ for every inclusion $i\colon
X\rightarrow W$ of a compact possibly singular subvariety $X\subset W$ in a
smooth variety $W$ is a Gysin coherent characteristic class.

###### Proof.

(Here, we take $\mathcal{X}$ to be the family of all inclusions $i\colon
X\rightarrow W$ of compact subvarieties $X$ in smooth varieties $W$ such that
$X$ is irreducible.) By the properties of the cohomological Hirzebruch
$L$-class, the class $\mathcal{L}^{\ast}(f)=L^{\ast}(\nu_{f})\in
H^{\ast}(M;\mathbb{Q})$ is normalized for all $f$. Moreover, the highest
nontrivial homogeneous component of the Goresky-MacPherson $L$-class
$L_{\ast}(X)=L_{0}(X)+L_{1}(X)+\dots\in H_{\ast}(X;\mathbb{Q})$ is known to be
$L_{2d}(X)=[X]_{X}$, where $d$ denotes the complex dimension of $X$.
Consequently, the highest nontrivial homogeneous component of
$\mathcal{L}_{\ast}(i)=i_{\ast}L_{\ast}(X)\in H_{\ast}(W;\mathbb{Q})$ is the
ambient fundamental class $i_{\ast}L_{2d}(X)=i_{\ast}[X]_{X}=[X]_{W}$. We
proceed to check the axioms of Gysin coherent characteristic classes for the
pair $\mathcal{L}$. As for axiom (1), we have $L_{\ast}(X\times
X^{\prime})=L_{\ast}(X)\times L_{\ast}(X^{\prime})$ in $H_{\ast}(X\times
X^{\prime};\mathbb{Q})$ for all Witt spaces $X$ and $X^{\prime}$ by a result
of Woolf [36, Proposition 5.16]. (Alternatively, multiplicativity of the
Goreksy-MacPherson $L$-class under products follows from results in [7],
especially Section 11 there.) Hence, for every $i\colon X\rightarrow W$ and
$i^{\prime}\colon X^{\prime}\rightarrow W^{\prime}$, the claim follows by
applying $(i\times i^{\prime})_{\ast}$ and using naturality of the cross
product:

$\displaystyle\mathcal{L}_{\ast}(i\times i^{\prime})$ $\displaystyle=(i\times
i^{\prime})_{\ast}L_{\ast}(X\times X^{\prime})=(i\times
i^{\prime})_{\ast}(L_{\ast}(X)\times L_{\ast}(X^{\prime}))$
$\displaystyle=i_{\ast}L_{\ast}(X)\times
i_{\ast}^{\prime}L_{\ast}(X^{\prime})=\mathcal{L}_{\ast}(i)\times\mathcal{L}_{\ast}(i^{\prime}).$

Next, let us show that the pair $\mathcal{L}$ is compatible with ambient
isomorphisms as stated in axiom (2). As for $\mathcal{L}^{\ast}$, we consider
$f\colon M\rightarrow W$ and $f^{\prime}\colon M^{\prime}\rightarrow
W^{\prime}$, and an isomorphism
$W\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}W^{\prime}$ that restricts
to an isomorphism $\phi\colon
M\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}M^{\prime}$. Then, we have
$\phi^{\ast}\nu_{f^{\prime}}=\nu_{f}$, and thus

$\phi^{\ast}\mathcal{L}^{\ast}(f^{\prime})=\phi^{\ast}L^{\ast}(\nu_{f^{\prime}})=L^{\ast}(\phi^{\ast}\nu_{f^{\prime}})=L^{\ast}(\nu_{f})=\mathcal{L}^{\ast}(f).$

As for $\mathcal{L}_{\ast}$, we consider $i\colon X\rightarrow W$ and
$i^{\prime}\colon X^{\prime}\rightarrow W^{\prime}$, and an isomorphism
$\Phi\colon W\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}W^{\prime}$ that
restricts to an isomorphism
$X\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}X^{\prime}$. Since
$W^{\prime}$ is smooth and quasiprojective, it follows that the compact
subvariety $X^{\prime}\subset W^{\prime}$ can be Whitney stratified with only
even-codimensional strata. We equip $X\subset W$ with the Whitney
stratification induced from that of $X^{\prime}\subset W^{\prime}$ by the
isomorphism $\Phi$. Let $\Phi_{0}\colon
X\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}X^{\prime}$ denote the
restriction of $\Phi$. As the isomorphism $\Phi_{0}$ is the restriction of an
ambient diffeomorphism underlying $\Phi$, it follows directly from the
construction of the Goresky-MacPherson $L$-class of Whitney stratified
pseudomanifolds with only even-codimensional strata (see [21, Section 5.3])
that $\Phi_{0\ast}L_{\ast}(X)=L_{\ast}(X^{\prime})$. Hence, we obtain

$\Phi_{\ast}\mathcal{L}_{\ast}(i)=\Phi_{\ast}i_{\ast}L_{\ast}(X)=i_{\ast}^{\prime}\Phi_{0\ast}L_{\ast}(X)=i_{\ast}^{\prime}L_{\ast}(X^{\prime})=\mathcal{L}_{\ast}(i^{\prime}).$

To verify axiom (3), we consider $i\colon X\rightarrow W$ and $f\colon
M\rightarrow W$ such that $X\subset M$. Then, the inclusion $i^{M}:=i|\colon
X\rightarrow M$ satisfies $f\circ i^{M}=i$, and we obtain

$f_{\ast}\mathcal{L}_{\ast}(i^{M})=f_{\ast}i^{M}_{\ast}L_{\ast}(X)=i_{\ast}L_{\ast}(X)=\mathcal{L}_{\ast}(i).$

Finally, to show axiom (4), let us call closed irreducible subvarieties
$Z,Z^{\prime}\subset W$ of a smooth variety $W$ $\mathcal{X}$-transverse if
$Z$ and $Z^{\prime}$ are simultaneously Whitney transverse and generically
transverse in $W$. This notion of $\mathcal{X}$-transversality has indeed all
required properties. (In fact, properness of $\mathcal{X}$-transverse
intersections holds by 3.4. Moreover, Kleiman’s transversality theorem for the
action of $GL_{n}(\mathbb{C})$ on the Grassmannians $G=G_{k}(\mathbb{C}^{n})$
holds for Whitney transversality by 2.2, and for generic transversality by
3.5. Here, we also use that Zariski dense open subsets are also dense in the
complex topology by [30, Theorem 1, p. 58]. Finally, locality clearly holds
for our notion of $\mathcal{X}$-transversality.) Now, consider an inclusion
$i\colon X\rightarrow W$ in $\mathcal{X}$ and an inclusion $f\colon
M\rightarrow W$ such that $M$ is irreducible, and $M$ and $X$ are
$\mathcal{X}$-transverse in $W$. Let $j\colon Y\rightarrow M$ and $g\colon
Y\rightarrow X$ denote the inclusions of the pure-dimensional compact
subvariety $Y:=M\cap X$. Then, the inclusion $g\colon Y\hookrightarrow X$ is
normally nonsingular, with topological normal bundle $\nu=j^{\ast}\nu_{f}$
given by the restriction of the normal bundle $\nu_{f}$ of $M$ in $W$ (see
2.3). According to the first author’s Gysin restriction formula for the
Goresky-MacPherson $L$-class (see 9.1), we have

$g^{!}L_{\ast}(X)=L^{\ast}(\nu)\cap L_{\ast}(Y).$

Using that $f^{!}i_{\ast}=j_{\ast}g^{!}$ by 2.4, as well as
$L^{\ast}(\nu)=L^{\ast}(j^{\ast}\nu_{f})=j^{\ast}L^{\ast}(\nu_{f})$, we
conclude that

$\displaystyle f^{!}\mathcal{L}_{\ast}(i)$
$\displaystyle=f^{!}i_{\ast}L_{\ast}(X)=j_{\ast}g^{!}L_{\ast}(X)=j_{\ast}(L^{\ast}(\nu)\cap
L_{\ast}(Y))$ $\displaystyle=j_{\ast}(j^{\ast}L^{\ast}(\nu_{f})\cap
L_{\ast}(Y))=L^{\ast}(\nu_{f})\cap
j_{\ast}L_{\ast}(Y)=\mathcal{L}^{\ast}(f)\cap\mathcal{L}_{\ast}(j).$

This completes the proof of 9.2. ∎

##  10\. An Example: The $L$-Class of $X_{3,2,1}$

We set $X=X_{3,2,1}$. Note that $X$ is a singular Schubert variety of real
dimension $12$ that does not satisfy global Poincaré duality over the
rationals since, for example,

$H_{8}(X;\mathbb{Q})=\mathbb{Q}[X_{3,1}]_{X}\oplus\mathbb{Q}[X_{2,2}]_{X}\oplus\mathbb{Q}[X_{2,1,1}]_{X},$

is a $3$-dimensional rational vector space, whereas

$H_{4}(X;\mathbb{Q})=\mathbb{Q}[X_{2}]_{X}\oplus\mathbb{Q}[X_{1,1}]_{X},$

has dimension $2$. Let us compute the total Goresky-MacPherson $L$-class

$L_{\ast}(X_{3,2,1})=L_{12}(X)+L_{8}(X)+L_{4}(X)+L_{0}(X)\in
H_{\ast}(X_{3,2,1};\mathbb{Q})$

with $L_{j}(X)\in H_{j}(X;\mathbb{Q})$. The highest class $L_{12}(X)=[X]_{X}$
is the fundamental class, and $L_{0}(X)=\sigma(X)\cdot[\operatorname{pt}]_{X}$
is determined by the signature. In the following, we are concerned with the
computation of the $L$-classes $L_{8}(X)\in H_{8}(X;\mathbb{Q})$ and
$L_{4}(X)\in H_{4}(X;\mathbb{Q})$, which can be uniquely written as rational
linear combinations

(35)
$L_{8}(X)=\lambda_{3,1}[X_{3,1}]_{X}+\lambda_{2,2}[X_{2,2}]_{X}+\lambda_{2,1,1}[X_{2,1,1}]_{X},\qquad\lambda_{3,1},\lambda_{2,2},\lambda_{2,1,1}\in\mathbb{Q},$

and

(36)
$L_{4}(X)=\lambda_{2}[X_{2}]_{X}+\lambda_{1,1}[X_{1,1}]_{X},\qquad\lambda_{2},\lambda_{1,1}\in\mathbb{Q},$

respectively. Note that our computation of $L_{4}(X)$ goes beyond the scope of
the computations in [6, Section 4], which are limited to $L$-classes in real
codimension $4$. For our purpose, we will consider $X=X_{3,2,1}(F_{\ast})$ as
a Schubert subvariety of the Grassmannian $G=G_{3}(\mathbb{C}^{6})$ with
respect to some complete flag $F_{\ast}$ on $\mathbb{C}^{6}$.

Similarly to the normally nonsingular expansion of $L_{6}(X_{3,2})$ in [6,
Section 4], our method is to produce equations for the unknown coefficients in
(35) and (36) as follows. By intersecting $X\subset G$ transversely with a
nonsingular subvariety $M\subset G$ with topological normal bundle $\nu_{M}$,
we obtain an oriented normally nonsingular inclusion

(37) $g\colon Y=M\cap X\hookrightarrow X$

with normal bundle $\nu_{Y}=\nu_{M}|_{Y}$. We consider the associated Gysin
homomorphism

(38) $g^{!}\colon H_{\ast}(X;\mathbb{Q})\rightarrow
H_{\ast-2c}(Y;\mathbb{Q}),$

where $c$ denotes the complex codimension of $M$ in $G$. Then, we compute the
values of $g^{!}$ on the Schubert generators of $H_{\ast}(X;\mathbb{Q})$ by
using intersection theory of Schubert cycles. Provided that $L_{\ast}(Y)$ is
known, the Gysin restriction formula

(39) $g^{!}L_{\ast}(X)=L^{\ast}(\nu_{Y})\cap L_{\ast}(Y)$

of 9.1 then yields equations in the unknown coefficients in (35) and (36).
Finally, for appropriate choices of nonsingular subvarieties $M\subset G$, we
are able to derive normally nonsingular expansions for these coefficients.

Compared to the calculations in [6, Section 4], our method is more general in
the following two ways. First, we will choose $M$ to be the regular part of a
possibly singular Schubert subvariety of $G$, thus allowing noncompact $M$ for
which the intersection $Y=M\cap X$ is still compact, so that $g^{!}$ and
$L_{\ast}(Y)$ are still defined. Second, the Segre product of Grassmannians in
an ambient Grassmannian is a new ingredient that allows us to identify $Y$ in
some cases with a product of Schubert varieties with known $L$-classes.

Let us compute $\lambda_{3,1}$. Our computation is similar to that of the
coefficient $\mu$ of $L_{6}(X_{3,2})$ in [6, Section 4]. We choose a direct
sum decomposition $\mathbb{C}^{6}=V^{\prime}\oplus V^{\prime\prime}$ with
$\operatorname{dim}_{\mathbb{C}}(V^{\prime})=5$ and
$\operatorname{dim}_{\mathbb{C}}(V^{\prime\prime})=1$. Let
$(E_{\ast}^{\prime},F_{\ast}^{\prime})$ be a pair of transverse flags on
$V^{\prime}$, and let $(E_{\ast}^{\prime\prime},F_{\ast}^{\prime\prime})$ be a
pair of transverse flags on $V^{\prime\prime}$. Then,
$(E_{\ast},F_{\ast})=(E_{\ast}^{\prime}\oplus
E_{\ast}^{\prime\prime},F_{\ast}^{\prime\prime}\oplus F_{\ast}^{\prime})$ is a
pair of transverse flags on $\mathbb{C}^{6}$. Hence, by 5.5,
$M=X_{3,3}(E_{\ast})$ and $X=X_{3,2,1}(F_{\ast})$ are Whitney transverse in
$G$ with respect to suitable Whitney stratifications. By a result of
Lakshmibai-Weyman (see 4.3), the Schubert subvariety
$M=X_{3,3}(E_{\ast})\subset G$ is nonsingular. In order to compute the
transverse intersection $Y=M\cap X$, we consider the Segre product

$S\colon G_{2}(V^{\prime})\times G_{1}(V^{\prime\prime})\hookrightarrow
G_{3}(\mathbb{C}^{6}).$

Writing $(3,3,0)\in\mathcal{P}(3,3)$ as $(3,3,0)=(0)\sqcup(3,3)$ with
$(0)\in\mathcal{P}(0,1)$ and $(3,3)\in\mathcal{P}(3,2)$, and
$(3,2,1)\in\mathcal{P}(3,3)$ as $(3,2,1)=(2,1)\sqcup(0)$ with
$(2,1)\in\mathcal{P}(3,2)$ and $(0)\in\mathcal{P}(0,1)$, we conclude from 5.10
that

$\displaystyle Y$ $\displaystyle=M\cap X$ $\displaystyle=X_{3,3}(E_{\ast})\cap
X_{3,2,1}(F_{\ast})$ $\displaystyle=X_{(0)\sqcup(3,3)}(E_{\ast}^{\prime}\oplus
E_{\ast}^{\prime\prime})\cap X_{(2,1)\sqcup(0)}(F_{\ast}^{\prime\prime}\oplus
F_{\ast}^{\prime})$ $\displaystyle=S((X_{0}(E_{\ast}^{\prime})\cap
X_{0}(F_{\ast}^{\prime}))\times(X_{3,3}(E_{\ast}^{\prime\prime})\cap
X_{2,1}(F_{\ast}^{\prime\prime})))$
$\displaystyle=S((X_{0}(E_{\ast}^{\prime})\cap
X_{0}(F_{\ast}^{\prime}))\times(X_{2,1}(F_{\ast}^{\prime\prime})\cap
X_{3,3}(E_{\ast}^{\prime\prime})))$
$\displaystyle=X_{(0)\sqcup(2,1)}(E_{\ast}^{\prime}\oplus
F_{\ast}^{\prime\prime})\cap X_{(3,3)\sqcup(0)}(E_{\ast}^{\prime\prime}\oplus
F_{\ast}^{\prime})$ $\displaystyle=X_{2,1}(E_{\ast}^{\prime}\oplus
F_{\ast}^{\prime\prime})\cap X_{3,3,3}(E_{\ast}^{\prime\prime}\oplus
F_{\ast}^{\prime})$ $\displaystyle=X_{2,1}(E_{\ast}^{\prime}\oplus
F_{\ast}^{\prime\prime}).$

We set $X_{2,1}^{\prime}:=X_{2,1}(E_{\ast}^{\prime}\oplus
F_{\ast}^{\prime\prime})$ and $X_{1}^{\prime}:=X_{1}(E_{\ast}^{\prime}\oplus
F_{\ast}^{\prime\prime})$. The Gysin homomorphism $g^{!}\colon
H_{8}(X;\mathbb{Q})\rightarrow H_{2}(Y;\mathbb{Q})$ associated to the normally
nonsingular inclusion $g\colon Y\hookrightarrow X$ is of the form

$g^{!}\colon\mathbb{Q}[X_{3,1}]_{X}\oplus\mathbb{Q}[X_{2,2}]_{X}\oplus\mathbb{Q}[X_{2,1,1}]_{X}\rightarrow\mathbb{Q}[X_{1}^{\prime}]_{Y}.$

Since $(E_{\ast},F_{\ast})$ is a pair of transverse flags on $\mathbb{C}^{6}$,
it follows from 5.5 that $M\subset G$ is Whitney transverse to $X_{3,1}$,
$X_{2,2}$, and $X_{2,1,1}$ (all defined with respect to the flag $F_{\ast}$)
with respect to suitable Whitney stratifications. Hence, using 2.5, we have
$g^{!}[X_{3,1}]_{X}=[M\cap X_{3,1}]_{Y}$, $g^{!}[X_{2,2}]_{X}=[M\cap
X_{2,2}]_{Y}$, and $g^{!}[X_{2,1,1}]_{X}=[M\cap X_{2,1,1}]_{Y}$. By a
computation similar to the above, 5.10 implies that $M\cap
X_{3,1}=X_{1}^{\prime}$. Moreover, we have $M\cap X_{2,2}=\varnothing$ and
$M\cap X_{2,1,1}=\varnothing$ by 5.8. All in all, we obtain

$g^{!}L_{8}(X)=\lambda_{3,1}\cdot[X_{1}^{\prime}]_{Y}.$

On the other hand, (39) yields

$\displaystyle g^{!}L_{8}(X)$
$\displaystyle=((1+L^{1}(\nu_{Y})+L^{2}(\nu_{Y})+\dots)\cap(L_{6}(Y)+L_{2}(Y)))_{2}$
$\displaystyle=L_{2}(Y)+L^{1}(\nu_{Y})\cap[Y]_{Y}.$

The $L$-class $L_{\ast}(Y)=L_{\ast}(X_{2,1}^{\prime})=L_{\ast}(X_{2,1})$ was
computed in [6, Section 4] to be

(40)
$L_{\ast}(X_{2,1})=L_{6}(X_{2,1})+L_{2}(X_{2,1})=[X_{2,1}]_{X_{2,1}}+\frac{2}{3}[X_{1}]_{X_{2,1}}\in
H_{\ast}(X_{2,1};\mathbb{Q}).$

Altogether,

$\lambda_{3,1}\cdot[X_{1}^{\prime}]_{Y}=\frac{2}{3}[X_{1}^{\prime}]_{Y}+L^{1}(\nu_{Y})\cap[Y]_{Y}\in
H_{2}(Y;\mathbb{Q})=\mathbb{Q}[X_{1}^{\prime}]_{Y},\qquad Y=X_{2,1}^{\prime}.$

The coefficient $\lambda_{2,1,1}$ can be computed in a similar way, by taking
$M=X_{3,3}(E_{\ast})$ for a suitable flag $E_{\ast}$ on $\mathbb{C}^{6}$. It
turns out that $\lambda_{3,1}=\lambda_{2,1,1}$.

Let us compute $\lambda_{2,2}$. We choose a direct sum decomposition
$\mathbb{C}^{6}=V^{\prime}\oplus V^{\prime\prime}$ with
$\operatorname{dim}_{\mathbb{C}}(V^{\prime})=3$ and
$\operatorname{dim}_{\mathbb{C}}(V^{\prime\prime})=3$. Let
$(E_{\ast}^{\prime},F_{\ast}^{\prime})$ be a pair of transverse flags on
$V^{\prime}$, and let $(E_{\ast}^{\prime\prime},F_{\ast}^{\prime\prime})$ be a
pair of transverse flags on $V^{\prime\prime}$. Then,
$(E_{\ast},F_{\ast})=(E_{\ast}^{\prime}\oplus
E_{\ast}^{\prime\prime},F_{\ast}^{\prime\prime}\oplus F_{\ast}^{\prime})$ is a
pair of transverse flags on $\mathbb{C}^{6}$. Hence, by 5.5,
$X_{3,1,1}(E_{\ast})$ and $X=X_{3,2,1}(F_{\ast})$ are Whitney transverse in
$G$ with respect to suitable Whitney stratifications. By a result of
Lakshmibai-Weyman (see 4.3), the singular locus of the Schubert subvariety
$X_{3,1,1}(E_{\ast})\subset G$ is $X_{0}(E_{\ast})\subset G$, a single point.
Let $M$ denote the regular part of $X_{3,1,1}(E_{\ast})$, which is an open
subvariety of $X_{3,1,1}(E_{\ast})$. Note that $M\cap
X=X_{3,1,1}(E_{\ast})\cap X$ because $X_{0}(E_{\ast})\cap X=\varnothing$ by
5.8. In order to compute the transverse intersection $Y=M\cap X$, we consider
the Segre product

$S\colon G_{1}(V^{\prime})\times G_{2}(V^{\prime\prime})\hookrightarrow
G_{3}(\mathbb{C}^{6}).$

Writing $(3,1,1)\in\mathcal{P}(3,3)$ as $(3,1,1)=(1,1)\sqcup(2)$ with
$(1,1)\in\mathcal{P}(1,2)$ and $(2)\in\mathcal{P}(2,1)$, and
$(3,2,1)\in\mathcal{P}(3,3)$ as $(3,2,1)=(1)\sqcup(1,0)$ with
$(1)\in\mathcal{P}(2,1)$ and $(1,0)\in\mathcal{P}(1,2)$, we conclude from 5.10
that

$\displaystyle Y$ $\displaystyle=M\cap X$
$\displaystyle=X_{3,1,1}(E_{\ast})\cap X_{3,2,1}(F_{\ast})$
$\displaystyle=X_{(1,1)\sqcup(2)}(E_{\ast}^{\prime}\oplus
E_{\ast}^{\prime\prime})\cap X_{(1)\sqcup(1,0)}(F_{\ast}^{\prime\prime}\oplus
F_{\ast}^{\prime})$ $\displaystyle=S((X_{1,1}(E_{\ast}^{\prime})\cap
X_{1}(F_{\ast}^{\prime}))\times(X_{2}(E_{\ast}^{\prime\prime})\times
X_{1}(F_{\ast}^{\prime\prime})))$
$\displaystyle=S(X_{1}(F_{\ast}^{\prime})\times
X_{1}(F_{\ast}^{\prime\prime}))$
$\displaystyle\cong\mathbb{P}^{1}\times\mathbb{P}^{1}.$

The Gysin homomorphism $g^{!}\colon H_{8}(X;\mathbb{Q})\rightarrow
H_{0}(Y;\mathbb{Q})$ associated to the normally nonsingular inclusion $g\colon
Y\hookrightarrow X$ is of the form

$g^{!}\colon\mathbb{Q}[X_{3,1}]_{X}\oplus\mathbb{Q}[X_{2,2}]_{X}\oplus\mathbb{Q}[X_{2,1,1}]_{X}\rightarrow\mathbb{Q}[\operatorname{pt}]_{Y}.$

Since $(E_{\ast},F_{\ast})$ is a pair of transverse flags on $\mathbb{C}^{6}$,
it follows from 5.5 and Lemma 2.1 that $M$ is transverse to all Whitney strata
of $X_{3,1}$, $X_{2,2}$, and $X_{2,1,1}$ (all defined with respect to the flag
$F_{\ast}$) in $G$ with respect to suitable Whitney stratifications. Hence,
using 2.5, we have $g^{!}[X_{3,1}]_{X}=[M\cap X_{3,1}]_{Y}$,
$g^{!}[X_{2,2}]_{X}=[M\cap X_{2,2}]_{Y}$, and $g^{!}[X_{2,1,1}]_{X}=[M\cap
X_{2,1,1}]_{Y}$. By 5.8, $M\cap X_{2,2}=\operatorname{pt}$ is a single point,
whereas $M\cap X_{3,1}=\varnothing$ and $M\cap X_{2,1,1}=\varnothing$. All in
all, we obtain

$g^{!}L_{8}(X)=\lambda_{2,2}\cdot[\operatorname{pt}]_{Y}.$

On the other hand, (39) yields

$\displaystyle g^{!}L_{8}(X)$
$\displaystyle=((1+L^{1}(\nu_{Y})+L^{2}(\nu_{Y})+\dots)\cap(L_{4}(Y)+L_{0}(Y)))_{0}$
$\displaystyle=L_{0}(Y)+L^{1}(\nu_{Y})\cap[Y]_{Y}.$

Note that $L_{0}(Y)=\sigma(Y)\cdot[\operatorname{pt}]_{Y}=0$ since the
signature of $Y\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$ vanishes. Altogether,
we obtain

$\lambda_{2,2}=\langle
L^{1}(\nu_{Y}),[Y]_{Y}\rangle=\int_{Y}L^{1}(\nu_{M}),\qquad
Y=S(X_{1}(F_{\ast}^{\prime})\times X_{1}(F_{\ast}^{\prime\prime}))\subset
M=X_{3,1,1}(E_{\ast}).$

Let us compute $\lambda_{2}$. We choose a direct sum decomposition
$\mathbb{C}^{6}=V^{\prime}\oplus V^{\prime\prime}$ with
$\operatorname{dim}_{\mathbb{C}}(V^{\prime})=2$ and
$\operatorname{dim}_{\mathbb{C}}(V^{\prime\prime})=4$. Let
$(E_{\ast}^{\prime},F_{\ast}^{\prime})$ be a pair of transverse flags on
$V^{\prime}$, and let $(E_{\ast}^{\prime\prime},F_{\ast}^{\prime\prime})$ be a
pair of transverse flags on $V^{\prime\prime}$. Then,
$(E_{\ast},F_{\ast})=(E_{\ast}^{\prime}\oplus
E_{\ast}^{\prime\prime},F_{\ast}^{\prime\prime}\oplus F_{\ast}^{\prime})$ is a
pair of transverse flags on $\mathbb{C}^{6}$. Hence, by 5.5,
$X_{3,3,1}(E_{\ast})$ and $X=X_{3,2,1}(F_{\ast})$ are Whitney transverse in
$G$ with respect to suitable Whitney stratifications. By a result of
Lakshmibai-Weyman (see 4.3), the singular locus of the Schubert subvariety
$X_{3,3,1}(E_{\ast})\subset G$ is $X_{3}(E_{\ast})\subset G$. Let $M$ denote
the regular part of $X_{3,3,1}(E_{\ast})$, which is an open subvariety of
$X_{3,3,1}(E_{\ast})$. Note that $M\cap X=X_{3,3,1}(E_{\ast})\cap X$ because
$X_{3}(E_{\ast})\cap X=\varnothing$ by 5.8. In order to compute the transverse
intersection $Y=M\cap X$, we consider the Segre product

$S\colon G_{1}(V^{\prime})\times G_{2}(V^{\prime\prime})\hookrightarrow
G_{3}(\mathbb{C}^{6}).$

Writing $(3,3,1)\in\mathcal{P}(3,3)$ as $(3,3,1)=(1)\sqcup(2,2)$ with
$(1)\in\mathcal{P}(1,1)$ and $(2,2)\in\mathcal{P}(2,2)$, and
$(3,2,1)\in\mathcal{P}(3,3)$ as $(3,2,1)=(2,1)\sqcup(1)$ with
$(2,1)\in\mathcal{P}(2,2)$ and $(1)\in\mathcal{P}(1,1)$, we conclude from 5.10
that

$\displaystyle Y$ $\displaystyle=M\cap X$
$\displaystyle=X_{3,3,1}(E_{\ast})\cap X_{3,2,1}(F_{\ast})$
$\displaystyle=X_{(1)\sqcup(2,2)}(E_{\ast}^{\prime}\oplus
E_{\ast}^{\prime\prime})\cap X_{(2,1)\sqcup(1)}(F_{\ast}^{\prime\prime}\oplus
F_{\ast}^{\prime})$ $\displaystyle=S((X_{1}(E_{\ast}^{\prime})\cap
X_{1}(F_{\ast}^{\prime}))\times(X_{2,2}(E_{\ast}^{\prime\prime})\times
X_{2,1}(F_{\ast}^{\prime\prime})))$
$\displaystyle=S(X_{1}(F_{\ast}^{\prime})\times
X_{2,1}(F_{\ast}^{\prime\prime})).$

The Gysin homomorphism $g^{!}\colon H_{4}(X;\mathbb{Q})\rightarrow
H_{0}(Y;\mathbb{Q})$ associated to the normally nonsingular inclusion $g\colon
Y\hookrightarrow X$ is of the form

$g^{!}\colon\mathbb{Q}[X_{2}]_{X}\oplus\mathbb{Q}[X_{1,1}]_{X}\rightarrow\mathbb{Q}[\operatorname{pt}]_{Y}.$

Since $(E_{\ast},F_{\ast})$ is a pair of transverse flags on $\mathbb{C}^{6}$,
it follows from 5.5 and Lemma 2.1 that $M$ is transverse to all Whitney strata
of $X_{2}$ and $X_{1,1}$ (all defined with respect to the flag $F_{\ast}$) in
$G$ with respect to suitable Whitney stratifications. Hence, using 2.5, we
have $g^{!}[X_{2}]_{X}=[M\cap X_{2}]_{Y}$ and $g^{!}[X_{1,1}]_{X}=[M\cap
X_{1,1}]_{Y}$. By 5.8, $M\cap X_{2}=\operatorname{pt}$ is a single point,
whereas $M\cap X_{1,1}=\varnothing$. All in all, we obtain

$g^{!}L_{4}(X)=\lambda_{2}\cdot[\operatorname{pt}]_{Y}.$

On the other hand, (39) yields

$\displaystyle g^{!}L_{4}(X)$
$\displaystyle=((1+L^{1}(\nu_{Y})+L^{2}(\nu_{Y})+\dots)\cap(L_{8}(Y)+L_{4}(Y)+L_{0}(Y)))_{0}$
$\displaystyle=L_{0}(Y)+L^{1}(\nu_{Y})\cap
L_{4}(Y)+L^{2}(\nu_{Y})\cap[Y]_{Y}.$

Note that $L_{0}(Y)=\sigma(Y)\cdot[\operatorname{pt}]_{Y}=0$ since the
signature of $Y\cong\mathbb{P}^{1}\times X_{2,1}$ vanishes. Next, we compute

$L_{4}(Y)\in H_{4}(Y;\mathbb{Q})=\mathbb{Q}[Y_{1\times
1}]_{Y}\oplus\mathbb{Q}[Y_{0\times
2}]_{Y}\oplus\mathbb{Q}[Y_{0\times(1,1)}]_{Y},$

where $Y_{1\times 1}:=S(X_{1}(F_{\ast}^{\prime})\times
X_{1}(F_{\ast}^{\prime\prime}))$, $Y_{0\times
2}:=S(X_{0}(F_{\ast}^{\prime})\times X_{2}(F_{\ast}^{\prime\prime}))$, and
$Y_{0\times(1,1)}:=S(X_{0}(F_{\ast}^{\prime})\times
X_{1,1}(F_{\ast}^{\prime\prime}))$. The Segre product $S$ restricts to an
isomorphism $G_{1}(V^{\prime})\times G_{2}(V^{\prime\prime})\cong
S(G_{1}(V^{\prime})\times G_{2}(V^{\prime\prime}))$, which in turn restricts
to an isomorphism $\varphi\colon X_{1}(F_{\ast}^{\prime})\times
X_{2,1}(F_{\ast}^{\prime\prime})\cong Y$. Hence, writing
$X_{1}^{\prime}=X_{1}(F_{\ast}^{\prime})$,
$X_{2,1}^{\prime\prime}=X_{2,1}(F_{\ast}^{\prime\prime})$, and
$X_{1}^{\prime\prime}=X_{1}(F_{\ast}^{\prime\prime})$, we have

$\displaystyle L_{4}(Y)$
$\displaystyle=\varphi_{\ast}L_{4}(X_{1}^{\prime}\times
X_{2,1}^{\prime\prime})$
$\displaystyle=\varphi_{\ast}(L_{\ast}(X_{1}^{\prime})\times
L_{\ast}(X_{2,1}^{\prime\prime}))_{4}$
$\displaystyle=\varphi_{\ast}(L_{2}(X_{1}^{\prime})\times
L_{2}(X_{2,1}^{\prime\prime}))$ $\displaystyle\stackrel{{\scriptstyle(\ref{l
class of x 2
1})}}{{=}}\frac{2}{3}\cdot\varphi_{\ast}([X_{1}^{\prime}]_{X_{1}^{\prime}}\times[X_{1}^{\prime\prime}]_{X_{2,1}^{\prime\prime}})$
$\displaystyle=\frac{2}{3}\cdot\varphi_{\ast}([X_{1}^{\prime}\times
X_{1}^{\prime\prime}]_{X_{1}^{\prime}\times X_{2,1}^{\prime\prime}})$
$\displaystyle=\frac{2}{3}\cdot[S(X_{1}^{\prime}\times
X_{1}^{\prime\prime})]_{Y}$ $\displaystyle=\frac{2}{3}\cdot[Y_{1\times
1}]_{Y}.$

Altogether, we obtain

$\displaystyle\lambda_{2}=\frac{2}{3}\cdot\langle L^{1}(\nu_{Y}),[Y_{1\times
1}]_{Y}\rangle+\langle
L^{2}(\nu_{Y}),[Y]_{Y}\rangle=\frac{2}{3}\cdot\int_{Y_{1\times
1}}L^{1}(\nu_{M})+\int_{Y}L^{2}(\nu_{M}),$ $\displaystyle Y_{1\times
1}=S(X_{1}(F_{\ast}^{\prime})\times X_{1}(F_{\ast}^{\prime\prime}))\subset
Y=S(X_{1}(F_{\ast}^{\prime})\times X_{2,1}(F_{\ast}^{\prime\prime}))\subset
M=X_{3,3,1}(E_{\ast}).$

The coefficient $\lambda_{1,1}$ can be computed in a similar way, by taking
$M=X_{3,2,2}(E_{\ast})$ for a suitable flag $E_{\ast}$ on $\mathbb{C}^{6}$.

## References

  * [1] P. Aluffi, L. C. Mihalcea, Chern classes of Schubert Cells and Varieties, J. Algebraic Geom. 18 (2009), no. 1, 63–100.
  * [2] M. Banagl, Extending Intersection Homology Type characteristic classes to non-Witt Spaces, Memoirs Amer. Math. Soc. 160 (2002), no. 760, 1 – 83.
  * [3] M. Banagl, The L-Class of non-Witt Spaces, Annals of Math. 163 (2006), 1 – 24.
  * [4] M. Banagl, Topological Invariants of Stratified Spaces, Springer Monographs in Math., Springer Verlag, 2007.
  * [5] M. Banagl, Topological and Hodge L-Classes of Singular Covering Spaces and Varieties with Trivial Canonical Class, Geometriae Dedicata 199 (2019), no. 1, 189 – 224.
  * [6] M. Banagl, Gysin Restriction of Topological and Hodge-Theoretic Characteristic Classes for Singular Spaces, New York J. Math. 26 (2020), 1273 – 1337.
  * [7] M. Banagl, G. Laures, J. McClure, The L-Homology Fundamental Class for IP-Spaces and the Stratified Novikov Conjecture, Selecta Math. 25:7 (2019), pp. 1 - 104.
  * [8] S. Billey, I. Coskun, Singularities of Generalized Richardson Varieties, Communications in Algebra 40 (2012), 1466 – 1495.
  * [9] J.-P. Brasselet, J. Schürmann, S. Yokura, Hirzebruch Classes and Motivic Chern Classes for Singular Spaces, J. Topol. Analysis 2 (2010), 1 – 55.
  * [10] S. E. Cappell, L. G. Maxim, J. Schürmann, J. L. Shaneson, _Characteristic classes of complex hypersurfaces_ , Adv. Math. 225 (2010), no. 5, 2616 – 2647.
  * [11] S. E. Cappell, L. G. Maxim, J. Schürmann, J. L. Shaneson, _Equivariant characteristic classes of singular complex algebraic varieties_ , Comm. Pure Appl. Math. 65 (2012), no. 12, 1722 – 1769.
  * [12] S. E. Cappell, J. L. Shaneson, Stratifiable maps and topological invariants, J. Amer. Math. Soc. 4 (1991), no. 3, 521 – 551.
  * [13] S. Cappell, J. Shaneson, S. Weinberger, Classes topologiques caractéristiques pour les actions de groupes sur les espaces singuliers, C. R. Acad. Sci. Paris 313 (1991), 293 – 295.
  * [14] S. E. Cappell and S. Weinberger, _Classification de certaines espaces stratifiés_ , C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), 399 – 401.
  * [15] J. Cheeger, Spectral Geometry of Singular Riemannian Spaces, J. Differential Geom. 18 (1983), 575 – 657.
  * [16] D. Eisenbud, J. Harris, 3264 and All That – Intersection Theory in Algebraic Geometry, Cambridge University Press, 2016.
  * [17] J. Fernández de Bobadilla, I. Pallarés, _The Brasselet-Schürmann-Yokura conjecture on $L$-classes of singular varieties_. Preprint, 2021. https://arxiv.org/abs/2007.11537
  * [18] G. Friedman, Singular Intersection Homology, New Mathematical Monographs, vol. 33, Cambridge University Press, 2020. manuscript available at: http://faculty.tcu.edu/gfriedman/IHbook.pdf
  * [19] W. Fulton, Intersection Theory, 2nd ed., Springer-Verlag, 1998.
  * [20] W. Fulton, _Young tableaux with applications to representation theory and geometry_ , London Mathematical Society Student Texts 35, Cambridge University Press, 1997.
  * [21] M. Goresky, R. MacPherson, Intersection Homology Theory, Topology 19 (1980), 135 – 162.
  * [22] M. Goresky, R. MacPherson, Intersection Homology II, Invent. Math. 71 (1983), 77 – 129.
  * [23] M. Goresky, R. MacPherson, Stratified Morse Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 14, Springer-Verlag, 1988.
  * [24] P. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley Classics Library ed., Wiley, 1994.
  * [25] F. Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 9, Springer-Verlag, 1956.
  * [26] S. L. Kleiman, _The transversality of a general translate_ , Compositio Math. 28 (1974), 287 – 297.
  * [27] V. Lakshmibai, J. Weyman, Multiplicities of Points on a Schubert Variety in a Minuscule $G/P$, Advances in Math. 84 (1990), 179 – 208.
  * [28] A. Matsui, Hirzebruch $L$-homology classes and the intersection formula, Kodai Math. J. 12 (1989), 56 – 71.
  * [29] L. Maxim, J. Schürmann, Characteristic Classes of Singular Toric Varieties, Comm. Pure Appl. Math. 68 (2015), 2177 – 2236.
  * [30] D. Mumford, _The Red Book of Varieties and Schemes_ , (second, expanded edition), Lecture Notes in Mathematics 1358, Springer-Verlag Berlin Heidelberg, 1999.
  * [31] R. W. Richardson, _Intersections of double cosets in algebraic groups_ , Indag. Math., N.S., 3 (1992), 69–77.
  * [32] J. Schürmann, Characteristic Classes of Mixed Hodge Modules, in: Topology of Stratified Spaces (Eds. Friedman, Hunsicker, Libgober, Maxim), MSRI publications vol. 58, Cambridge Univ. Press, 2011, 419 – 470.
  * [33] P. H. Siegel, Witt Spaces: A Geometric Cycle Theory for $KO$-Homology at Odd Primes, Amer. J. of Math. 105 (1983), 1067 – 1105.
  * [34] S. J. Sierra, _A general homological Kleiman-Bertini theorem_ , Algebra & Number Theory 3 (2009), 597 – 609.
  * [35] S. Weinberger, The Topological Classification of Stratified Spaces, Chicago Lectures in Math., The University of Chicago Press, 1994.
  * [36] J. Woolf, Witt Groups of Sheaves on Topological Spaces, Commentarii Math. Helv. 83 (2008), no. 2, 289 – 326.