by the $\mu_{k,p}^{M}$, and by
$\widetilde{\mathcal{KLK}}(M)\subset\mathcal{C}(M)$ the space spanned by the
$\Delta_{k,p}^{M}$ and the $B_{k,p}^{M}$.

Theorem 4.9 implies the following compatibility with restrictions.

######  Proposition 6.2.

Let $M\subset\tilde{M}$ be an isometric holomorphic embedding of Kähler
manifolds. Then

$\displaystyle\Delta_{k,p}^{\tilde{M}}|_{M}$ $\displaystyle=\Delta_{k,p}^{M}$
$\displaystyle B_{k,p}^{\tilde{M}}|_{M}$ $\displaystyle=B_{k,p}^{M}.$

We will see (Proposition 6.12) that
$[B_{k,p}^{M}]=[\Delta_{k,p}^{M}]=\mu_{k,p}^{M}$, and (Theorem 6.9) that
$\mathcal{KLK}(M)$ is closed with respect to Alesker multiplication. The
resulting subalgebra of $\mathcal{V}(M)$ is the Kähler-Lipschitz-Killing
algebra of $M$.

These curvature measures fit the previously developed schema from
[bernig_fu_hig, bernig_fu_solanes]:

######  Proposition 6.3.

Taking $M=\mathbb{C}^{n}$,

$\Delta_{k,p}^{\mathbb{C}^{n}}=\Delta_{k,p},\quad
B_{k,p}^{\mathbb{C}^{n}}=B_{k,p},$

where the right hand sides are the curvature measures of
$\operatorname{Curv}_{k}^{\operatorname{U}(n)}$ as defined in [bernig_fu_hig,
bernig_fu_solanes].

###### Proof.

We show first that $\Delta_{k,p}=\Delta_{k,p}^{\mathbb{C}^{n}}$. Consider in
complex projective space $\operatorname{\mathbb{C}P}_{\lambda}^{N}$ of
constant holomorphic curvature $4\lambda\neq 0$ the image of
$\Delta_{k,p}\in\operatorname{Curv}^{\operatorname{U}(n)}$ under the transfer
map, which we denote again by $\Delta_{k,p}$. By [fu_wannerer, Eqs. (75),
(82)] we have

$\displaystyle\Delta_{k,p}$
$\displaystyle=(-1)^{p}\left(\frac{\pi}{2}\right)^{\lceil\frac{k}{2}\rceil}\frac{1}{(k+1)!!}\sum_{j,l\geq
0}\binom{k+1}{2p+2l+1}\binom{p+l}{l}\binom{p+l}{j}(-\lambda)^{-j}C_{k,j}^{\lambda}$
$\displaystyle=\frac{\pi^{k}}{(k+1)!\omega_{k}}\sum_{j\geq
0}c_{k,p,j}(2\lambda)^{-j}C_{k,j}^{\lambda},$

where, under the identification (28),

$\displaystyle C_{k,j}^{\lambda}$
$\displaystyle=\left[0,\frac{\omega_{k}}{\pi^{k}(2n-k)!\omega_{2n-k}}\bm{\alpha}^{\vee}\wedge(2\lambda\operatorname{\mathsf{G}})^{j}\wedge\operatorname{\mathsf{g}}^{k-2j}\wedge\bm{\omega}^{2n-k-1}\right]$
$\displaystyle\mbox{for }k<2n,$ $\displaystyle C_{2n,j}^{\lambda}$
$\displaystyle=\left[\frac{\omega_{2n}}{\pi^{2n}}(2\lambda\operatorname{\mathsf{G}})^{j}\wedge\operatorname{\mathsf{g}}^{2n-j},0\right].$

Hence, for $k<2n$, and recalling (80),

(95)
$\Delta_{k,p}=\frac{1}{(k+1)!(2n-k)!\omega_{2n-k}}\left[0,\bm{\alpha}^{\vee}\wedge\bm{\phi}_{k,p}\wedge\bm{\omega}^{2n-k-1}\right],$

while

(96)
$\Delta_{2n,p}=\frac{1}{(2n+1)!}\left[\bm{\phi}_{2n,p},0\right]=\frac{\delta_{p,n}}{(2n)!}\left[\operatorname{\mathsf{g}}^{2n},0\right],$

by Proposition 5.12.

On the other hand, since
$\operatorname{\mathsf{R}}^{\operatorname{\mathbb{C}P}_{\lambda}^{n}}=\lambda\operatorname{\mathsf{G}}$,

$\displaystyle\bm{\kappa}_{k,p}$
$\displaystyle=\delta_{2p,k}{\bm{\kappa}}_{k}=\frac{\delta_{2p,k}\delta_{k,2n}}{(2n)!}\operatorname{\mathsf{g}}^{2n}+O(\lambda),$
$\displaystyle\bm{\omega}_{k,p}$
$\displaystyle=\frac{1}{(k+1)!(2n-k)!\omega_{2n-k}}\bm{\alpha}^{\vee}\wedge\bm{\phi}_{k,p}\wedge\bm{\omega}^{2n-k-1}+O(\lambda).$

It follows for all $k,p$ that

$\Delta_{k,p}^{\operatorname{\mathbb{C}P}_{\lambda}^{n}}=\Delta_{k,p}+O(\lambda).$

To prove that $\Delta_{k,p}=\Delta_{k,p}^{\mathbb{C}^{n}}$, it suffices to
prove this identity inside the unit ball of $\mathbb{C}^{n}$, where we may
approximate the flat metric by a family of metrics $g_{\lambda}$ compatible
with the complex structure and with constant holomorphic curvature $4\lambda$.
Letting $\lambda\to 0$ we have
$\Delta_{k,p}^{\operatorname{\mathbb{C}P}_{\lambda}^{n}}\to\Delta_{k,p}^{\mathbb{C}^{n}}$
which yields the claim.

Now we prove $B_{k,p}^{\mathbb{C}^{n}}=B_{k,p}$ for $2p<k<2n$. Consider on
$S\mathbb{C}^{n}=\mathbb{C}^{n}\times S^{2n-1}$ the horizontal vector field
$X$ defined by $X_{(x,\xi)}=(J\xi,0)$. Recall that
$B_{k,p},\Delta_{k,p}\in\mathrm{Curv}^{\operatorname{U}(n)}$ are given by (cf.
[bernig_fu_solanes, §3.1])

$\displaystyle B_{k,p}$ $\displaystyle=[0,\bm{\beta}_{k,p}]$
$\displaystyle\Delta_{k,p}$
$\displaystyle=\frac{1}{2n-k}(2(n-k+p)[0,\bm{\gamma}_{k,p}]+(k-2p)[0,\bm{\beta}_{k,p}]),$

where $\bm{\beta}_{k,p}$ is a multiple of $\bm{\beta}$ and
$\iota_{X}\bm{\gamma}_{k,p}\equiv 0$ modulo $\bm{\alpha}$. Comparing with (95)
it follows that, under the identification (28)

(97)
$\frac{1}{(k+1)!(2n-k-1)!\omega_{2n-k}}\bm{\alpha}^{\vee}\wedge\bm{\phi}_{k,p}\wedge\bm{\omega}^{2n-k-1}=2(n-k+p)\bm{\gamma}_{k,p}+(k-2p)\bm{\beta}_{k,p}+\bm{\rho},$

where
$\bm{\rho}\in\Omega^{2n-1}(S\mathbb{C}^{n})^{\overline{\operatorname{U}(n)}}$
is in the ideal $I$ generated by $\bm{\alpha},d\bm{\alpha}$. Since

$\iota_{X}\bm{\alpha}=0,\qquad\iota_{X}d\bm{\alpha}=-\omega_{1,0}=:-\bm{\gamma}$

we see that,

$\bm{\beta}\wedge(\iota_{X}\bm{\rho})\equiv\bm{\beta}\wedge\bm{\gamma}\wedge\bm{T}\quad\mod
I,$

where $\bm{T}$ belongs to the space of translation invariant
$\operatorname{U}(n)$-invariant elements of $\Omega^{2n-3}(S\mathbb{C}^{n})$.
Since this space is generated by $\bm{\alpha},\bm{\beta},\bm{\gamma}$ and four
forms of degree $2$, while $(2n-3)$ is odd, we deduce
$\bm{\beta}\wedge(\iota_{X}\bm{\rho})\in I$.

By (47) we have

$\iota_{X}\operatorname{\mathsf{g}}=\bm{\beta}^{\vee},\qquad\iota_{X}\operatorname{\mathsf{G}}\equiv\bm{\beta}^{\vee}\wedge\operatorname{\mathsf{g}}\qquad\iota_{X}\bm{\omega}=0,$

modulo $\bm{\alpha},\bm{\alpha}^{\vee}$. Hence,

$\displaystyle\iota_{X}\bm{\phi}_{k,p}$
$\displaystyle=\sum_{j}c_{k,p,j}(j\bm{\beta}^{\vee}\wedge\operatorname{\mathsf{G}}^{j-1}\wedge\operatorname{\mathsf{g}}^{k-2j+1}+(k-2j)\bm{\beta}^{\vee}\wedge\operatorname{\mathsf{G}}^{j}\wedge\operatorname{\mathsf{g}}^{k-2j-1})$
$\displaystyle=\sum_{j}((j+1)c_{k,p,j+1}+(k-2j)c_{k,p,j})\bm{\beta}^{\vee}\wedge\operatorname{\mathsf{G}}^{j}\wedge\operatorname{\mathsf{g}}^{k-2j-1}$
$\displaystyle=(k+1)\sum_{j}c_{k-1,p,j}\bm{\beta}^{\vee}\wedge\operatorname{\mathsf{G}}^{j}\wedge\operatorname{\mathsf{g}}^{k-2j-1}$
$\displaystyle=(k+1)\bm{\beta}^{\vee}\wedge\bm{\phi}_{k-1,p},$

where the third equality follows from (82).

Applying $\bm{\beta}\wedge\iota_{X}(\cdot)$ to both sides of (97), and using
the previous relation, yields $B_{k,p}=B_{k,p}^{M}$. ∎

###  6.2. Basic properties of the Kähler-Lipschitz-Killing curvature measures

The two localizations $\Delta_{k,p}^{M}$ and $B_{k,p}^{M}$ of the hermitian
intrinsic volumes have different, particularly desirable properties with
respect to the Alesker product. As will be proved in Proposition 6.11 below,
multiplication by the first intrinsic volume $t=\frac{2}{\pi}\mu_{1,0}^{M}$
stabilizes the family of $\Delta_{k,p}^{M}$, while multiplication by
$\sigma_{M}:=\frac{1}{2\pi}(\mu_{2,0}^{M}+2\mu_{2,1}^{M})$ stabilizes the
$B_{k,p}^{M}$.

Initially defined for all $2p\leq k\leq 2n$ and $2p<k<2n$, respectively, as a
consequence of the relations in the Gray algebra, some of the curvature
measures $\Delta^{M}_{k,p}$ and $B^{M}_{k,p}$ are actually trivial.

######  Proposition 6.4.

On an $n$-dimensional Kähler manifold $M$,

$\Delta_{k,p}^{M}=0\quad\text{and}\quad B_{k,p}^{M}=0$

whenever $k-p>n$.

###### Proof.

Since
$\bm{\phi}_{k,p}={g_{k,p}(\operatorname{\mathsf{G}},\operatorname{\mathsf{g}})}=0$
whenever $k-p>n$ by Proposition 5.11, the first assertion is clear from the
definition of $\Delta_{k,p}^{M}$. To prove the second assertion, observe that

$\bm{\alpha}\wedge\big{(}\bm{\alpha}^{\vee}\wedge\bm{\beta}\wedge\bm{\beta}^{\vee}\wedge\bm{\phi}_{k-1,p}\big{)}=(\bm{\alpha}\wedge\bm{\alpha}^{\vee})\wedge\bm{\beta}\wedge\bm{\beta}^{\vee}\wedge(\bm{\phi}_{k-1,p}|_{H})=0,$

where $H\subset T_{x}M$ denotes the complex hyperplane
$\ker\bm{\alpha}\cap\ker\bm{\beta}$. Thus
$\bm{\alpha}^{\vee}\wedge\bm{\beta}\wedge\bm{\beta}^{\vee}\wedge\bm{\phi}_{k-1,p}$
is divisible by the contact form $\bm{\alpha}$ and consequently
$B_{k,p}^{M}=0$. ∎

We may now clarify the relation between the $\Delta_{k,p}^{M}$ and the
Lipschitz-Killing curvature measures $\Lambda_{k}^{M}$ of Lemma 3.5.

######  Proposition 6.5.

$\sum_{p=0}^{\left\lfloor\frac{k}{2}\right\rfloor}\Delta^{M}_{k,p}=\Lambda_{k}^{M}.$

###### Proof.

By (91), for $k<2n$,

$\bm{\alpha}^{\vee}\wedge\sum_{p}\bm{\phi}_{k,p}\wedge\bm{\psi}_{2n-k-1}=(k+1)\bm{\alpha}^{\vee}\wedge\operatorname{\mathsf{g}}^{k}\wedge\bm{\psi}_{2n-k-1},$

which is the boundary term of $\Lambda_{k}^{M}$ from Lemma 3.5. Comparison of
the interior terms is trivial. ∎

###  6.3. Structure of $\mathcal{K}\mathcal{L}\mathcal{K}(M)$

In this section we prove our main result, Theorem 6.9. The following will be
helpful.

######  Lemma 6.6.

Given $x\in M$, let
$\tau_{x}:\mathcal{C}(M)\to\operatorname{Curv}(T_{x}M){\simeq\operatorname{Curv}(\mathbb{C}^{n})}$
denote the transfer map (16). Let
$\pi_{k}\colon\operatorname{Curv}\to\operatorname{Curv}_{k}$ be the projection
onto the $k$-th degree component. Then, for $0,k-n\leq p\leq\frac{k}{2}\leq
n$,

$\pi_{k}\circ\tau_{x}(\Delta_{k,p}^{M})=\Delta_{k,p},\qquad\pi_{k}\circ\tau_{x}(B_{k,p}^{M})=B_{k,p}.$

###### Proof.

Let us first treat the case $k=2n$. By (94), $\Delta^{M}_{2n,n}$ is the
riemannian volume measure on $M$, and thus
$\tau_{x}(\Delta_{2n,n})=\Delta_{2n,n}$. Since $B^{M}_{2n,n}=0$, this finishes
the case $k=2n$.

Let us now assume $k<2n$. In this case
$\tau_{x}(\Delta_{k,p})=[(\bm{\kappa}_{k,p})_{x},\bar{\tau}_{x}(\bm{\omega}_{k,p})]$,
where $\bar{\tau}_{x}$ is the composition of (50) with the identification
$\Omega^{\bullet,\bullet}(S\mathbb{C}^{n},\mathbb{C}^{n})|_{\\{0\\}\times
S^{2n-1}}\simeq\Omega^{\bullet,\bullet}(S\mathbb{C}^{n},\mathbb{C}^{n})^{\mathbb{C}^{n}}.$

By (51) we have that $\bar{\tau}_{x}(\bm{\omega})$ is the connection double
form on $\mathbb{C}^{n}$. Likewise,
$\bar{\tau}_{x}(\operatorname{\mathsf{g}}),\bar{\tau}_{x}(\operatorname{\mathsf{G}})$
are the metric and the Gray double forms on $\mathbb{C}^{n}$. By Proposition
6.3, it follows that $\Delta_{k,p}\in\operatorname{Curv}_{k}$ is represented
by
$d_{n,k,0}\bar{\tau}_{x}(\bm{\alpha}^{\vee}\wedge\bm{\phi}_{k,p}\wedge\bm{\omega}^{2n-k-1})$.
Therefore, since
$[0,\bar{\tau}_{x}(\bm{\alpha}^{\vee}\wedge\bm{\phi}_{k,p}\wedge\operatorname{\mathsf{R}}^{l}\wedge\bm{\omega}^{2n-2l-k-1})]\in\operatorname{Curv}_{k+l}$,
we have

$\displaystyle\pi_{k}\circ\tau_{x}(\Delta_{k,p}^{M})$
$\displaystyle=[0,d_{n,k,0}\bar{\tau}_{x}(\bm{\alpha}^{\vee}\wedge\bm{\phi}_{k,p}\wedge\bm{\omega}^{2n-k-1})]=\Delta_{k,p}.$

The proof of the second stated equality is completely analogous. ∎

In the following we consider the linear map

$r_{M}\colon\operatorname{Val}^{\operatorname{U}(n)}\to\mathcal{KLK}(M)$

given by $r_{M}(\mu_{k,p})=\mu_{k,p}^{M}$. In Theorem 6.9 we will see that
$r_{M}$ is an isomorphism of algebras. A first step is the following.

######  Proposition 6.7.

$r_{M}$ is a bijection.

###### Proof.

Surjectivity of $r_{M}$ follows from the definition of
$\mathcal{K}\mathcal{L}\mathcal{K}(M)$. To prove that $r_{M}$ is injective,
suppose that

$\sum_{k=l}^{2n}\sum_{q}a_{k,q}\mu_{k,q}^{M}=0$

with $l$ maximal and $0,k-n\leq q\leq\frac{k}{2}$. With respect to the
filtration
$\mathcal{W}_{0}\supset\mathcal{W}_{1}\supset\dots\supset\mathcal{W}_{2n}$ of
$\mathcal{V}(M)$ introduced in [alesker_val_man2], we have
$\mu_{k,p}^{M}\in\mathcal{W}_{k}$ by [alesker_val_man1, Proposition 5.2.5]. By
Lemma 6.6 and [solanes_wannerer, Corollary 3.6], the projection
$\mathcal{W}_{l}\to\mathcal{W}_{l}/\mathcal{W}_{l+1}=\operatorname{Val}_{l}(TM)$
maps $\mu_{l,q}^{M}\mapsto\mu_{l,q}$, while $\mu^{M}_{k,p}\mapsto 0$ for
$k>l$. We deduce that $\sum_{q}a_{l,q}\mu_{l,q}=0$ in
$\operatorname{Val}_{l}^{\operatorname{U}(n)}$, and thus all $a_{l,q}$ vanish,
contradicting the maximality of $l$. ∎

As a second step toward Theorem 6.9, we prove it for Kähler submanifolds of
$\mathbb{C}^{N}$.

######  Proposition 6.8 (Algebra structure for embedded Kähler manifolds).

Let $M\subset\mathbb{C}^{N}$ be a Kähler submanifold of complex dimension $n$.
Then $\mathcal{K}\mathcal{L}\mathcal{K}(M)\subset\mathcal{V}(M)$ is a
subalgebra, and the linear map
$r_{M}\colon\operatorname{Val}^{\operatorname{U}(n)}\to\mathcal{KLK}(M)$
defined by $r_{M}(\mu_{k,q})=\mu_{k,q}^{M}$, is an isomorphism of algebras.
Moreover, restriction of valuations on $\mathbb{C}^{N}$ to $M$ factors as

${\operatorname{Val}^{\operatorname{U}(N)}}$${\operatorname{Val}^{\operatorname{U}(n)}}$${\mathcal{KLK}(M)}$$|_{M}$$|_{\mathbb{C}^{n}}$$r_{M}$

###### Proof.

For $p\geq k-n$, we have $\mu_{k,p}|_{\mathbb{C}^{n}}=\mu_{k,p}$, and
$\mu_{k,p}|_{M}=\mu_{k,p}^{M}$ by Proposition 6.2. For $p<k-n$, both
$\mu_{k,p}|_{\mathbb{C}^{n}}$ and $\mu_{k,p}|_{M}=\mu_{k,p}^{M}$ vanish by
Proposition 6.4. Hence the previous diagram commutes.

Since restriction and Alesker product commute, the diagonal maps in the
previous diagram are algebra morphisms. Since the restriction
$\operatorname{Val}^{\operatorname{U}(N)}\to\operatorname{Val}^{\operatorname{U}(n)}$
is surjective, it follows that $r_{M}$ is also a morphism of algebras. Indeed,
given $\phi,\nu\in\operatorname{Val}^{\operatorname{U}(N)}$ we have

$\displaystyle r_{M}(\phi|_{\mathbb{C}^{n}}\cdot\nu|_{\mathbb{C}^{n}})$
$\displaystyle=r_{M}((\phi\cdot\nu)|_{\mathbb{C}^{n}})$
$\displaystyle=(\phi\cdot\nu)|_{M}$ $\displaystyle=\phi|_{M}\cdot\nu|_{M}$
$\displaystyle=r_{M}(\phi|_{\mathbb{C}^{n}})\cdot
r_{M}(\nu|_{\mathbb{C}^{n}}).$

Together with Proposition 6.7, this concludes the proof. ∎

The pointwise embedding lemma (Theorem 4.12) implies that this is enough to
prove Theorem 6.9 in full generality.

######  Theorem 6.9 (Structure of the Kähler-Lipschitz-Killing algebra).

Given any Kähler manifold $M$ of dimension $n$, the subspace
$\mathcal{K}\mathcal{L}\mathcal{K}(M)\subset\mathcal{V}(M)$ is a subalgebra,
and the map
$r_{M}\colon\operatorname{Val}^{\operatorname{U}(n)}\to\mathcal{K}\mathcal{L}\mathcal{K}(M)$,
given by

$r_{M}(\mu_{k,q})=\mu_{k,q}^{M},$

is an isomorphism of algebras.

###### Proof.

Let $\operatorname{\mathsf{R}}\in\mathcal{K}(\mathbb{C}^{n})$ be an embedded
curvature tensor, and take a complex submanifold $S\subset\mathbb{C}^{N}$ such
that $\operatorname{\mathsf{R}}=\sigma^{*}\operatorname{\mathsf{R}}^{S}_{x}$
for some $x\in S$ and some linear isometry $\sigma\colon\mathbb{C}^{n}\to
T_{x}S$.

By Proposition 6.8, if $a_{i}$ are constants such that

$\mu_{k,p}\cdot\mu_{l,q}=\sum_{i}a_{i}\mu_{k+l,i}$

in $\operatorname{Val}^{\operatorname{U}(n)}$, then also

$\displaystyle\mu_{k,p}^{S}\cdot\mu_{l,q}^{S}$
$\displaystyle=\sum_{i}a_{i}\mu_{k+l,i}^{S}.$

Recalling that $\Delta_{k,p}=\sum_{j}c_{k,p,j}\Gamma_{k,j}$, put

$\displaystyle Q_{k,p}=\sum_{j}c_{k,p,j}P_{k,p},\qquad
Q_{k,p,l,q}=\sum_{i,j}c_{k,p,i}c_{l,q,j}P_{k,i,l,j}$

where $P_{k,p}$ are the polynomials given in Proposition 4.6, and
$P_{k,p,l,q}$ are those from Proposition 4.7. By these propositions, at
$\xi=\sigma(e_{0})$, we have

$\displaystyle(\bar{\sigma}_{\xi}^{*})^{-1}Q_{k,p,l,q}(\sigma^{*}\operatorname{\mathsf{R}}_{x}^{S})$
$\displaystyle=\tau(\mu_{k,p}^{S}\cdot\mu_{l,q}^{S})_{\xi}$
$\displaystyle=\sum_{i}a_{i}\tau(\mu_{k+l}^{S})_{\xi}$
$\displaystyle=\sum_{i}a_{i}(\bar{\sigma}_{\xi}^{*})^{-1}Q_{k+l,i}(\sigma^{*}\operatorname{\mathsf{R}}_{x}^{S})|_{\xi},$

where $\tau$ is given by (5). Hence

$Q_{k,p,l,q}(\operatorname{\mathsf{R}})=\sum_{i}a_{i}\left.Q_{k+l,i}(\operatorname{\mathsf{R}})\right|_{e_{0}}.$

By Theorem 4.12, the previous relation holds for all
$\operatorname{\mathsf{R}}\in\mathcal{K}$. In particular, again by
Propositions 4.7 and 4.6, for every $n$-dimensional Kähler manifold $M$,

$\tau(\mu_{k,p}^{M}\cdot\mu_{l,q}^{M})=\sum_{i}a_{i}\tau(\mu_{k+l,i}^{M}).$

Since $\zeta(\mu^{M}_{k,p}\cdot\mu^{M}_{l,q})_{x}$ does not depend on
$\operatorname{\mathsf{R}}_{x}^{M}$ by (12), we have

$\zeta(\mu_{k,p}^{M}\cdot\mu_{l,q}^{M})=\sum_{i}a_{i}\zeta(\mu_{k+l,i}^{M}),$

and thus $\mu_{k,p}^{M}\cdot\mu_{l,q}^{M}=\sum_{i}a_{i}\mu_{k+l,i}^{M}.$ This
shows that $\mathcal{KLK}(M)$ is a subalgebra of $\mathcal{V}(M)$ and $r_{M}$
is an algebra morphism. By Proposition 6.7 it is an isomorphism. ∎

###  6.4. Structure of $\widetilde{\mathcal{KLK}}$ as a
$\mathcal{K}\mathcal{L}\mathcal{K}$-module

Recall that $\widetilde{\mathcal{KLK}}(M)\subset\mathcal{C}(M)$ is the vector
space spanned by the curvature measures $\Delta_{k,q}^{M},B_{k,q}^{M}$. In the
present section we show that $\widetilde{\mathcal{KLK}}(M)$ is a module over
$\mathcal{KLK}(M)$, canonically isomorphic to
$\operatorname{Curv}^{\operatorname{U}(n)}$ with respect to the isomorphism
$\operatorname{Val}^{\operatorname{U}(n)}\simeq\mathcal{K}\mathcal{L}\mathcal{K}(M)$.

######  Proposition 6.10.

The linear map
$R_{M}\colon\operatorname{Curv}^{\operatorname{U}(n)}\to\widetilde{\mathcal{KLK}}(M)$
defined by $R_{M}(\Delta_{k,p})=\Delta_{k,p}^{M}$ and
$R_{M}(B_{k,p})=B_{k,p}^{M}$ is a bijection.

###### Proof.

The surjectivity of $R_{M}$ is clear by the definition of
$\widetilde{\mathcal{KLK}}(M)$.

Let $\mathcal{C}_{2n}\subset\cdots\subset\mathcal{C}_{0}$ be the filtration of
$\mathcal{C}(M)$ introduced in [solanes_wannerer, §3]. By [solanes_wannerer,
Propositions 3.2 and 3.5] and Lemma 6.6, there is an isomorphism
$\mathcal{C}_{k}/\mathcal{C}_{k+1}\to\operatorname{Curv}_{k}$ which maps
$\Delta_{k,p}^{M}\mapsto\Delta_{k,p}$ and $B_{k,p}^{M}\mapsto B_{k,p}$. The
injectivity of $R_{M}$ follows easily, as in the proof of Proposition 6.8. ∎

######  Proposition 6.11.

There is a $\mathcal{KLK}(M)$-module structure on
$\widetilde{\mathcal{KLK}}(M)$ such that the following diagram commutes:

${\operatorname{Val}^{\operatorname{U}(n)}\otimes\operatorname{Curv}^{\operatorname{U}(n)}}$${\operatorname{Curv}^{\operatorname{U}(n)}}$${\mathcal{K}\mathcal{L}\mathcal{K}(M)\otimes\widetilde{\mathcal{K}\mathcal{L}\mathcal{K}}(M)}$${\widetilde{\mathcal{K}\mathcal{L}\mathcal{K}}(M)}$${\mathcal{V}(M)\otimes\mathcal{C}(M)}$${\mathcal{C}(M)}$$r_{M}\otimes
R_{M}$$R_{M}$

where the horizontal maps are the module structures. Explicitly,

(98) $R_{M}(\mu\cdot\Phi)=r_{M}(\mu)\cdot
R_{M}(\Phi),\qquad\forall\mu\in\operatorname{Val}^{\operatorname{U}(n)},\forall\Phi\in\operatorname{Curv}^{\operatorname{U}(n)}.$

The structure of $\widetilde{\mathcal{KLK}}(M)$ as a module over
$\mathcal{KLK}(M)$ is thus explicitly determined by Propositions 5.7 and 5.10
of [bernig_fu_solanes].

###### Proof.

For $M\subset\mathbb{C}^{N}$, the relation (98) follows from Propositions 6.3
and 6.2, and the compatibility of the module product with restrictions (cf.
[bernig_fu_solanes]*Propositions 2.3 and 2.4). The general case follows by
polynomiality, similarly to the proof of Theorem 6.9, using Proposition 2.1
and Theorem 4.12. ∎

We will denote by
$\operatorname{glob}_{M}\colon\mathcal{C}(M)\to\mathcal{V}(M)$ and
$\operatorname{glob}_{0}\colon\operatorname{Curv}^{\operatorname{U}(n)}\to\operatorname{Val}^{\operatorname{U}(n)}$
the respective globalization maps of $M$ and $\mathbb{C}^{n}$. The following
proposition shows that $R_{M}$ is an isomorphism of modules lifting the
algebra isomorphism $r_{M}$.

######  Proposition 6.12.

The following diagram commutes:

${\operatorname{Curv}^{\operatorname{U}(n)}}$${\widetilde{\mathcal{K}\mathcal{L}\mathcal{K}}(M)}$${\operatorname{Val}^{\operatorname{U}(n)}}$${\mathcal{K}\mathcal{L}\mathcal{K}(M)}$$R_{M}$$r_{M}$$\operatorname{glob}_{0}$$\operatorname{glob}_{M}$

In particular
$\operatorname{glob}_{M}(B_{k,q}^{M})=\operatorname{glob}_{M}(\Delta_{k,q}^{M})$.

###### Proof.

We have
$\operatorname{glob}_{M}(R_{M}(\Delta_{k,q}))=\mu_{k,q}^{M}=r_{M}(\operatorname{glob}_{0}\Delta_{k,q})$
by the definitions. For a general
$\Phi\in\operatorname{Curv}^{\operatorname{U}(n)}$ we have by
[bernig_fu_solanes, Theorem 5.13]

$\Phi=\mu\cdot\Delta_{0,0}+\varphi\cdot N_{1,0}$

for some $\mu,\varphi\in\operatorname{Val}^{\operatorname{U}(n)}$ and
$N_{1,0}=\Delta_{1,0}-B_{1,0}$. By (98) and the compatibility of module
product and globalization we get

$\operatorname{glob}_{M}(R_{M}(\Phi))=r_{M}(\mu)+r_{M}(\varphi)\cdot\operatorname{glob}_{M}(\Delta_{1,0}^{M}-B_{1,0}^{M})$

whereas

$r_{M}(\operatorname{glob}_{0}(\Phi))=r_{M}(\mu).$

The proof is completed by noting that
$\operatorname{glob}_{M}(\Delta_{1,0}^{M})=\operatorname{glob}_{M}(B_{1,0}^{M})$
by Proposition 4.8, since
$\Delta_{1,0}^{M}=2\Gamma_{1,0}^{M},B_{1,0}^{M}=2\tilde{\Gamma}_{1,0}^{M}$.

∎

##  7\. Integral geometry of complex space forms

In this section we specialize our results to the complex space forms
$M=\mathbb{C}P^{n}_{\lambda}$, viz. the complex projective space for
$\lambda>0$, the flat euclidean space for $\lambda=0$, and the complex
hyperbolic space when $\lambda<0$, always with a metric of constant
holomorphic curvature $4\lambda$. For $\lambda\neq 0$ we let $G_{\lambda}$ be
the isometry group of $\mathbb{C}P^{n}_{\lambda}$, while $G_{0}$ will be the
group generated by $\operatorname{U}(n)$ and the translations of
$\mathbb{C}^{n}$. We abbreviate

$r_{\lambda}:=r_{\mathbb{C}P^{n}_{\lambda}},\quad
R_{\lambda}:=R_{\mathbb{C}P^{n}_{\lambda}}.$

The algebra $\mathcal{KLK}(M)$ coincides with the algebra of
$G_{\lambda}$-invariant valuations which we denote by
$\mathcal{V}_{\lambda}^{n}$. Indeed, the former is clearly included in the
latter, and by Theorem 6.9 they have the same dimension. Similarly, the module
$\widetilde{\mathcal{KLK}}(M)$ equals the module of $G_{\lambda}$-invariant
curvature measures $\mathcal{C}_{\lambda}^{n}$, which can be identified with
$\operatorname{Curv}^{\operatorname{U}(n)}$ via the transfer map, as in
[bernig_fu_solanes].

###  7.1. The isomorphisms $r_{\lambda}$ and $R_{\lambda}$

The algebra $\mathcal{V}^{n}_{\lambda}$ is generated by two elements $t,s$
(see [bernig_fu_solanes, eq. (3.1)]), which we do not distinguish notationally
from the generators $t,s$ of $\operatorname{Val}^{\operatorname{U}(n)}$. Let
us put

$\sigma_{\lambda}:=r_{\lambda}(s).$

Our first goal is to describe the algebra isomorphism $r_{\lambda}$ in these
terms.

Let
$\mathrm{Beta}^{n}\subset\operatorname{Curv}^{\operatorname{U}(n)}\simeq\mathcal{C}_{\lambda}^{n}$
denote the span of the curvature measures $B_{k,p}$, and denote
$\mathrm{Beta}_{k}^{n}=\mathrm{Beta}^{n}\cap\operatorname{Curv}_{k}$.
Equivalently, $\mathrm{Beta}^{n}$ is the space of curvature measures
represented by invariant differential forms divisible by $\bm{\beta}$.

Recall the first variation map $\delta\colon\mathcal{V}(N)\to\mathcal{C}(N)$
defined for a riemannian manifold $N^{m}$ by
$\delta(\mu)=[0,\iota_{T}(\tau(\mu))]$, where $T$ is the Reeb vector field of
$SN$, and $\tau(\mu)$ is the $m$-form associated to $\mu$ by (5).

######  Lemma 7.1.

Let $\varphi\in\mathcal{V}_{\lambda}^{n}$. Then
$\delta\varphi\in\mathrm{Beta}^{n}$ if and only if $\varphi$ is a polynomial
in $s$.

###### Proof.

That $\delta\varphi\in\mathrm{Beta}^{n}$ if $\varphi$ is a polynomial in $s$
was proved in [abardia_gallego_solanes]*Corollary 4.5. To prove the converse,
we proceed as follows. By [bernig_fu_hig, Prop. 4.6] the composition

$p:\operatorname{Val}_{k}^{\operatorname{U}(n)}\stackrel{{\scriptstyle\delta}}{{\rightarrow}}\operatorname{Curv}_{k-1}^{\operatorname{U}(n)}\to\operatorname{Curv}_{k-1}^{\operatorname{U}(n)}/\mathrm{Beta}_{k-1}^{n}$

is surjective. Since

$\displaystyle\dim\operatorname{Val}_{k}^{\operatorname{U}(n)}$
$\displaystyle=\left\lfloor\frac{\min(k,2n-k)}{2}\right\rfloor+1,$
$\displaystyle\dim\operatorname{Curv}_{k-1}^{\operatorname{U}(n)}/\mathrm{Beta}_{k-1}^{n}$
$\displaystyle=\left\lfloor\frac{\min(k,2n-k)-1}{2}\right\rfloor+1,$

it follows that $p$ is injective for odd $k$ and has one-dimensional kernel if
$k$ is even. By [abardia_gallego_solanes]*Corollary 4.5 this kernel is spanned
by $s^{\frac{k}{2}}$.

Suppose $\varphi\in\mathcal{W}_{k}$ and let
$\varphi_{0}\in\operatorname{Val}^{\operatorname{U}(n)}$ be its image under
the projection
$\mathcal{W}_{k}\rightarrow\mathcal{W}_{k}/\mathcal{W}_{k+1}\cong\operatorname{Val}_{k}^{\operatorname{U}(n)}$.
Modulo $\oplus_{i\geq k}\operatorname{Curv}_{i}^{\operatorname{U}(n)}$ one has
$\delta\varphi\equiv\delta\varphi_{0}$, e.g., by [abardia_gallego_solanes,
Prop. 3.7]. Hence $\delta\varphi_{0}\in\mathrm{Beta}_{k-1}^{n}$, by the
assumption. It follows from the previous paragraph that
$\varphi_{0}=as^{\frac{k}{2}}$ for some $a\in\mathbb{R}$ if $k$ is even and
hence $\varphi-as^{\frac{k}{2}}\in\mathcal{W}_{k+1}$ in this case. If $k$ is
odd, then $\varphi_{0}=0$ and hence $\varphi\in\mathcal{W}_{k+1}$. In either
case we can proceed by induction. ∎

######  Proposition 7.2.

The algebra isomorphism
$r_{\lambda}\colon\operatorname{Val}^{\operatorname{U}(n)}\to\mathcal{V}_{\lambda}^{n}$
is given by

$\displaystyle r_{\lambda}(t)$ $\displaystyle=t,$ (99) $\displaystyle
r_{\lambda}(s)$ $\displaystyle=\sigma_{\lambda}=\frac{s}{1-\lambda
s}=\sum_{i=1}^{n}\lambda^{i-1}s^{i}.$

###### Proof.

By [bernig_fu_hig, Prop. 4.6], on $\mathbb{C}^{m}$ we have $\delta
s\in\mathrm{Beta}^{m}$. This means that
$\iota_{T}\tau(s)\equiv\bm{\beta}\wedge\eta$ modulo the ideal
$(\bm{\alpha},d\bm{\alpha})$ for some $\eta\in\Omega^{2m-2}(S\mathbb{C}^{m})$.
Therefore

$\tau(s)=\bm{\alpha}\wedge\iota_{T}\tau(s)=\bm{\alpha}\wedge\bm{\beta}\wedge\eta+\bm{\alpha}\wedge
d\bm{\alpha}\wedge\rho$

for some $\rho\in\Omega^{2m-3}(S\mathbb{C}^{m})$.

Let now $M\subset\mathbb{C}^{m}$ be a complex submanifold, and put
$\sigma_{M}=r_{M}(s)=s|_{M}$. By [alesker_intgeo, eq. (3.5.4)], and using the
notation of Subsection 4.5, we have

$\tau(\sigma_{M})=P_{*}F^{*}(\tau(s))=P_{*}F^{*}(\bm{\alpha}\wedge\bm{\beta}\wedge\eta+\bm{\alpha}\wedge
d\bm{\alpha}\wedge\rho).$

Since (cf. (69), (70))

$F^{*}\bm{\beta}=\cos(\theta)\bm{\beta},\quad
F^{*}\bm{\alpha}=\cos(\theta)\bm{\alpha},\quad
F^{*}d\bm{\alpha}=\cos(\theta)d\bm{\alpha}-\sin(\theta)d\theta\wedge\bm{\alpha},$

we deduce that $\tau(\sigma_{M})$ belongs to the ideal
$I=(\bm{\alpha}\wedge\bm{\beta},\bm{\alpha}\wedge d\bm{\alpha})$.

Let now $M$ be an arbitrary Kähler manifold $M$ and put again
$\sigma_{M}=r_{M}(s)=\frac{1}{2\pi}(\mu_{2,0}^{M}+2\mu_{2,1}^{M})$. By
Proposition 4.6, the form $\tau(\sigma_{M})$ depends polynomially on
$\operatorname{\mathsf{R}}^{M}$. By Theorem 4.12, and the previous paragraph,
this form vanishes modulo $I$ for all $M$. It follows that
$i_{T}(\tau(\sigma_{M}))$ is a multiple of $\bm{\beta}$ modulo the ideal
$(\bm{\alpha},d\bm{\alpha})$. Taking $M=\mathbb{C}P^{n}_{\lambda}$ it follows
that $\delta\sigma_{\lambda}\in\mathrm{Beta}^{n}$.

We can therefore apply the previous lemma to deduce that

(100) $\sigma_{\lambda}=\sum_{i=1}^{n}a_{i}s^{i},\qquad a_{i}\in\mathbb{R}.$

Since $\sigma_{\lambda}$ and $s^{i}$ are invariant under totally geodesic
embeddings $\mathbb{C}P^{n}_{\lambda}\to\mathbb{C}P^{n+d}_{\lambda}$, the
coefficients $a_{i}$ are independent of $n$.

Assume $\lambda>0$. Since $\mu_{2,0}$ has zero interior term, we have
$\mu_{2,0}^{M}(M)=0$. Recalling that
$\sigma_{\lambda}=\frac{1}{2\pi}(\mu_{2,0}^{M}+2\mu_{2,1}^{M})$ and
$t^{2}=\frac{2}{\pi}(\mu_{2,0}^{M}+\mu_{2,1}^{M})$ we deduce

$\displaystyle\sigma_{\lambda}(\operatorname{\mathbb{C}P}_{\lambda}^{n})$
$\displaystyle=\frac{1}{2}t^{2}(\operatorname{\mathbb{C}P}_{\lambda}^{n})=\frac{1}{\lambda}{n+1\choose
2},$

by Corollary 2.34 of [bernig_fu_solanes]. On the other hand we have (cf.
[bernig_fu_solanes, eq. (3.2)])

$s^{i}(\operatorname{\mathbb{C}P}_{\lambda}^{n})=\frac{n-i+1}{\lambda^{i}},\qquad
0\leq i\leq n.$

Plugging these values into (100) gives

$\frac{1}{\lambda}{n+1\choose
2}=\sum_{i=1}^{n}a_{i}\frac{n-i+1}{\lambda^{i}}.$

Induction on $n$ yields $a_{i}=\lambda^{i-1}$. This proves the statement for
$\lambda>0$.

Since the curvature tensor of $\mathbb{C}P^{n}_{\lambda}$ is
$R=\lambda\operatorname{\mathsf{G}}$, every curvature measure
$\Delta_{k,p}^{M}$ yields an element of
$\operatorname{Curv}^{\operatorname{U}(n)}$ depending polynomially on
$\lambda$. In particular
$\sigma_{\lambda}=\operatorname{glob}(\frac{1}{2\pi}(\Delta_{2,0}^{M}+2\Delta_{2,1}^{M}))$
is the globalization of a polynomial in $\lambda$ with values in
$\operatorname{Curv}^{\operatorname{U}(n)}$. By [bernig_fu_solanes, Lemma
5.4], also the right hand side of (99) is the globalization of a polynomial
family in $\operatorname{Curv}^{\operatorname{U}(n)}$. It follows from
[bernig_fu_solanes]*Lemma 2.7 that (99) holds for all $\lambda$. ∎

######  Remark 7.3.

It is amusing to compare Proposition 7.2 with a surprisingly similar
fundamental relation arising in spherical integral geometry. Let
$S^{n}_{\lambda}$ be the $n$-dimensional sphere of curvature $4\lambda$ and
define

$\phi=\frac{1}{\sqrt{\lambda}}\int_{\operatorname{SO}(n+1)}\chi(\bullet\cap
gS^{n-1}_{\lambda})dg,$

where $S^{n-1}_{\lambda}$ is a totally geodesic subsphere, and $dg$ is the
Haar probability measure. Then the algebra of
$\operatorname{SO}(n+1)$-invariant valuations on $S^{n}_{\lambda}$ is
generated by $\phi$. At the same time this algebra is generated by the first
Lipschitz-Killing valuation $t$. The relation between these two generators is

$t=\frac{\phi}{\sqrt{1-\lambda\phi^{2}}}.$

Next we express $r_{\lambda}$ in terms of the bases
$\tau_{k,p},\tau_{k,p}^{\lambda}$ of
$\operatorname{Val}^{\operatorname{U}(n)},\mathcal{V}_{\lambda}^{n}$
introduced in [bernig_fu_hig, bernig_fu_solanes] respectively.

######  Proposition 7.4.

Given $k$, put $r=\left\lfloor\frac{k}{2}\right\rfloor$, and set $\epsilon=0$
for $k$ even and $\epsilon=1$ for $k$ odd. Define the function

$g_{k,q}(\xi,\eta)=\frac{1}{r!}{r\choose
q}\left(\frac{\pi}{\lambda}\right)^{r}\xi^{r-q}(1-\xi)^{q-r-\epsilon-\frac{1}{2}}\eta^{q}(1-\eta)^{-q-\frac{1}{2}}.$

Then

(101)
$r_{\lambda}(\tau_{k,q})=\sum_{i,j}\left(\frac{\lambda}{\pi}\right)^{i+j}\left.\frac{\partial^{i+j}g_{k,q}}{\partial\xi^{i}\partial\eta^{j}}\right|_{\xi=\eta=0}\tau_{2i+2j+\epsilon,j}^{\lambda}.$

###### Proof.

Let

$\displaystyle v$ $\displaystyle=t^{2}(1-\lambda
s)=\frac{t^{2}}{1+\lambda\sigma_{\lambda}},$ $\displaystyle u$
$\displaystyle=4s-v=\frac{4\sigma_{\lambda}-t^{2}}{1+\lambda\sigma_{\lambda}}.$

By [bernig_fu_hig, Proposition 3.7], and reading $\sqrt{v}$ as
$t\sqrt{1-\lambda s}$,

(102) $\displaystyle r_{\lambda}(\tau_{k,q})$
$\displaystyle=\frac{\pi^{k}}{\omega_{k}(k-2q)!(2q)!}t^{k-2q}(4\sigma_{\lambda}-t^{2})^{q}$
(103)
$\displaystyle=\frac{\pi^{k}}{\omega_{k}(k-2q)!(2q)!}v^{\frac{k-2q}{2}}u^{q}(1+\lambda\sigma_{\lambda})^{\frac{k}{2}}$
(104) $\displaystyle=\frac{\pi^{k}}{\omega_{k}(k-2q)!(2q)!}(1-\lambda
s)^{-\frac{k}{2}}v^{\frac{k-2q}{2}}u^{q}.$

Consider now the operator
$\mathcal{O}\colon\mathbb{R}[[\xi,\eta]]\to\mathbb{R}[[v,u]]$ given by

$\mathcal{O}\left(\sum_{m,p}{c_{m,p}}\xi^{m}\eta^{p}\right)=\sum_{m,p}{c_{m,p}}\frac{{m+p\choose
m}}{{2m\choose m}{2p\choose p}}v^{m}u^{p}.$

By [bernig_fu_solanes, Lemma 3.16] we have

$\mathcal{O}(g_{2i,0})=\frac{i!}{(2i)!}\left(\frac{\pi}{\lambda}\right)^{i}v^{i}\left(1-\frac{v+u}{4}\right)^{-1-i}.$

Using

$g_{2r,q}=\left(\frac{4\pi}{\lambda}\right)^{q}\frac{1}{(2q)!}\eta^{q}\frac{\partial^{q}}{\partial\eta^{q}}g_{2r-2q,0}$

and

$\mathcal{O}\circ\left(\eta^{i}\frac{\partial^{i}}{\partial\eta^{i}}\right)=\left(u^{i}\frac{\partial^{i}}{\partial
u^{i}}\right)\circ\mathcal{O}$

yields

$\displaystyle\mathcal{O}(g_{2r,q})$
$\displaystyle=\frac{r!}{(2r-2q)!(2q)!}\left(\frac{\pi}{\lambda}\right)^{r}u^{q}v^{r-q}\left(1-\frac{u+v}{4}\right)^{-r-1}.$

By (19),

$\displaystyle\sum_{i,j}\left(\frac{\lambda}{\pi}\right)^{i+j}\left.\frac{\partial^{i+j}g_{2r,q}}{\partial\xi^{i}\partial\eta^{j}}\right|_{\xi=\eta=0}\tau_{2i+2j,j}^{\lambda}$
$\displaystyle=(1-\lambda
s)\sum_{i,j}\lambda^{i+j}\left.\frac{\partial^{i+j}g_{2r,q}}{\partial\xi^{i}\partial\eta^{j}}\right|_{\xi=\eta=0}\frac{(i+j)!}{(2i)!(2j)!}v^{i}u^{j}$
$\displaystyle=(1-\lambda
s)\sum_{i,j}\frac{1}{i!j!}\left.\frac{\partial^{i+j}g_{2r,q}}{\partial\xi^{i}\partial\eta^{j}}\right|_{\xi=\eta=0}\frac{{i+j\choose
i}}{{2i\choose i}{2j\choose j}}(\lambda v)^{i}(\lambda u)^{j}$
$\displaystyle=(1-\lambda s)\mathcal{O}(g_{2r,q})(\lambda v,\lambda u)$
$\displaystyle=\frac{r!}{(2r-2q)!(2q)!}\pi^{r}u^{q}v^{r-q}(1-\lambda s)^{-r},$

since $s=\frac{u+v}{4}$. Comparing with (104), this proves the proposition for
$k=2r$. For odd $k$, the proof is similar using the operator $\mathcal{P}$ of
[bernig_fu_solanes, Lemma 3.16]. ∎

######  Proposition 7.5.

Given $l$, put $r=\left\lfloor\frac{l}{2}\right\rfloor$, and set $\epsilon=0$
for $l$ even and $\epsilon=1$ for $l$ odd. Define the function

$f_{l,q}(\xi,\eta):=\frac{1}{r!}\binom{r}{q}\left(\frac{\pi}{\lambda}\right)^{r}\xi^{r-q}\eta^{q}(1-(\xi+\eta))^{-r-\epsilon-\frac{1}{2}+q}(1-\eta)^{-r-\frac{1}{2}}.$

Then

$r_{\lambda}(\mu_{l,p})=\sum_{i,j}\left(\frac{\lambda}{\pi}\right)^{i+j}\left.\frac{\partial^{i+j}f_{l,p}}{\partial\xi^{i}\partial\eta^{j}}\right|_{\xi=\eta=0}\mu_{2i+2j+\epsilon,j}^{\lambda}.$

###### Proof.

Recalling that $\mu_{k,p}=\sum_{q}(-1)^{q+p}{q\choose p}\tau_{k,q}$ (cf.
[bernig_fu_hig]*(39)) we consider

$\displaystyle\sum_{q}(-1)^{p+q}{q\choose p}g_{l,q}(\xi,\eta)$
$\displaystyle=\frac{1}{r!}\left(\frac{\pi}{\lambda}\right)^{r}\sum_{q}(-1)^{p+q}{q\choose
p}{r\choose
q}\xi^{r-q}(1-\xi)^{q-\epsilon-r-\frac{1}{2}}\eta^{q}(1-\eta)^{-q-\frac{1}{2}}$
$\displaystyle=\frac{1}{r!}\left(\frac{\pi}{\lambda}\right)^{r}\xi^{r}(1-\xi)^{-r-\epsilon-\frac{1}{2}}(1-\eta)^{-\frac{1}{2}}(-1)^{p}\sum_{q}{q\choose
p}{r\choose q}x^{q},$

where we have put $x=-\frac{\eta(1-\xi)}{\xi(1-\eta)}$. Using the identity

$\sum_{q}{q\choose p}{r\choose q}x^{q-p}={r\choose p}(1+x)^{r-p},$

which can be proved expanding $(1+x+y)^{r}$ in two different ways, and using
also $1+x=\frac{\xi-\eta}{\xi(1-\eta)}$, we get

$\displaystyle\sum_{q}(-1)^{p+q}{p\choose q}g_{l,q}(\xi,\eta)$
$\displaystyle=\frac{1}{r!}\left(\frac{\pi}{\lambda}\right)^{r}{r\choose
p}(1-\xi)^{p-r-\epsilon-\frac{1}{2}}\eta^{p}(1-\eta)^{-r-\frac{1}{2}}(\xi-\eta)^{r-p}$
$\displaystyle=f_{l,p}(\xi-\eta,\eta).$

The trivial identity

$\left.\frac{\partial^{i+j}}{\partial\xi^{i}\partial\eta^{j}}\right|_{\xi=\eta=0}f(\xi-\eta,\eta)=\sum_{r}(-1)^{r}\binom{j}{r}\left.\frac{\partial^{i+j}}{\partial\xi^{i+r}\partial\eta^{j-r}}\right|_{\xi=\eta=0}f(\xi,\eta).$

combined with (18) thus gives

$\displaystyle r_{\lambda}(\mu_{l,p})$
$\displaystyle=\sum_{i,j}\left(\frac{\lambda}{\pi}\right)^{i+j}\sum_{p}(-1)^{p+q}{p\choose
q}\left.\frac{\partial^{i+j}g_{l,q}}{\partial\xi^{i}\partial\eta^{j}}\right|_{\xi=\eta=0}\tau_{2i+2j{+\epsilon},j}^{\lambda}$
$\displaystyle=\sum_{i,j}\left(\frac{\lambda}{\pi}\right)^{i+j}\left.\frac{\partial^{i+j}f_{l,p}(\xi-\eta,\eta)}{\partial\xi^{i}\partial\eta^{j}}\right|_{\xi=\eta=0}\tau_{2i+2j{+\epsilon},j}^{\lambda}$
$\displaystyle=\sum_{i,j}\left(\frac{\lambda}{\pi}\right)^{i+j}\left.\frac{\partial^{i+j}f_{l,p}(\xi,\eta)}{\partial\xi^{i}\partial\eta^{j}}\right|_{\xi=\eta=0}\mu_{2i+2j{+\epsilon},j}^{\lambda},$

as stated. ∎

We next establish an analogue of Proposition 7.5 for the Kähler-Lipschitz-
Killing curvature measures.

######  Theorem 7.6.

Define the function

$h_{l,q}:=\frac{1}{1-\eta}f_{l,q}.$

Then

(105) $\displaystyle\Delta_{l,q}^{M}$ $\displaystyle=\sum_{k,p\geq
0}\left(\frac{\lambda}{\pi}\right)^{k+p}\left.\frac{\partial^{k+p}h_{l,q}}{\partial\xi^{k}\partial\eta^{p}}\right|_{\xi=\eta=0}\Delta_{2k+2p+\epsilon,p}$
(106) $\displaystyle N_{l,q}^{M}$ $\displaystyle=\sum_{l,p\geq
0}\left(\frac{\lambda}{\pi}\right)^{k+p}\left.\frac{\partial^{k+p}(\eta
h_{l,q})}{\partial\xi^{k}\partial\eta^{p}}\right|_{\xi=\eta=0}\Delta_{2k+2p+\epsilon,p}$
$\displaystyle\quad+\sum_{k,p\geq
0}\left(\frac{\lambda}{\pi}\right)^{k+p}\left.\frac{\partial^{k+p}((1-\eta)h_{l,q})}{\partial\xi^{k}\partial\eta^{p}}\right|_{\xi=\eta=0}N_{2k+2p+\epsilon,p}.$

###### Proof.

By Proposition 4.13,
$\Delta_{l,q}^{M}\in\mathrm{span}\\{\Delta_{k,p}\\}_{k,p}\subset\operatorname{Curv}^{\operatorname{U}(n)}$.
Let $\Phi_{l,q}$ be the right hand side of (105). By (17),

$\operatorname{glob}_{\lambda}(\Phi_{l,q})=\sum_{k,p\geq
0}\left(\frac{\lambda}{\pi}\right)^{k+p}\left.\frac{\partial^{k+p}}{\partial\xi^{k}\partial\eta^{p}}h_{l,q}(\xi,\eta)(1-\eta)\right|_{\xi=\eta=0}\mu_{2k+2p+\epsilon,p}^{\lambda}.$

Since $h_{l,q}(\xi,\eta)(1-\eta)=f_{l,q}(\xi,\eta)$, we deduce with
Proposition 7.5 that

$\operatorname{glob}_{\lambda}(\Phi_{l,q})=r_{\lambda}(\mu_{l,q})=\operatorname{glob}_{\lambda}(R_{\lambda}(\Delta_{l,q})).$

Since $\operatorname{glob}_{\lambda}$ is injective on
$\mathrm{span}\\{\Delta_{k,p}\\}_{k,p}\subset\operatorname{Curv}^{\operatorname{U}(n)}$,
the relation (105) follows.

Let $\Psi_{l,q}$ be the right hand side of (106). Since
$B_{k,p}=\Delta_{k,p}-N_{k,p}$ we have
$R_{\lambda}(B_{l,p})=R_{\lambda}(\Delta_{l,p}-N_{l,p})$ and

$\Phi_{l,q}-\Psi_{l,q}=\sum_{k,p\geq
0}\left(\frac{\lambda}{\pi}\right)^{k+p}\left.\frac{\partial^{k+p}((1-\eta)h_{l,q})}{\partial\xi^{k}\partial\eta^{p}}\right|_{\xi=\eta=0}B_{2k+2p+\epsilon,p}.$

Proposition 7.5 and (17) yield

$\operatorname{glob}_{\lambda}(\Phi_{l,q}-\Psi_{l,q})=r_{\lambda}(\mu_{l,q})=\operatorname{glob}_{\lambda}(R_{\lambda}(B_{l,p})).$

Since $R_{\lambda}(B_{l,p})\in\mathrm{Beta}^{n}$ and
$\operatorname{glob}_{\lambda}$ is injective on $\mathrm{Beta}^{n}$, it
follows that $R_{\lambda}(B_{l,p})=\Phi_{l,q}-\Psi_{l,q}$. We conclude that

$R_{\lambda}(N_{l,p})=R_{\lambda}(\Delta_{l,p})-R_{\lambda}(B_{l,p})=\Psi_{l,q},$

as stated. ∎

###  7.2. Isomorphisms intertwining the kinematic operators

Recall that the kinematic formulas in
$\operatorname{\mathbb{C}P}_{\lambda}^{n}$ are encoded by a co-algebra
structure
$k_{\lambda}\colon{\mathcal{V}_{\lambda}^{n}\to\mathcal{V}_{\lambda}^{n}\otimes\mathcal{V}_{\lambda}^{n}}$
(cf. [bernig_fu_solanes]*§2.3.3). Let
$\operatorname{pd}_{\lambda}\colon\mathcal{V}_{\lambda}^{n}\to(\mathcal{V}_{\lambda}^{n})^{*}$
be the Alesker-Poincaré duality (see [bernig_fu_solanes]*Definition 2.14). As
in [bernig_fu_solanes]*Eq. (3.45), $r_{\lambda}$ being an isomorphism implies
that

$J_{\lambda}=\operatorname{pd}_{\lambda}^{-1}\circ(r_{\lambda}^{-1})^{*}\circ\operatorname{pd}_{0}:\operatorname{Val}^{\operatorname{U}(n)}\to\mathcal{V}_{\lambda}^{n}$

is an isomorphism of co-algebras. This means in particular that $J_{\lambda}$
intertwines the kinematic operators of $\mathbb{C}^{n}$ and
$\operatorname{\mathbb{C}P}^{n}_{\lambda}$:

${\operatorname{Val}^{\operatorname{U}(n)}}$${\operatorname{Val}^{\operatorname{U}(n)}\otimes\operatorname{Val}^{\operatorname{U}(n)}}$${\mathcal{V}_{\lambda}^{n}}$${\mathcal{V}_{\lambda}^{n}\otimes\mathcal{V}_{\lambda}^{n}}$$k_{0}$$J_{\lambda}$$k_{\lambda}$$J_{\lambda}\otimes
J_{\lambda}$

Since $J_{\lambda}$ admits a simple explicit description, we obtain a
corresponding relation between between the flat and the curved kinematic
operator.

######  Proposition 7.7.

$J_{\lambda}=(1-\lambda s)^{n+1}r_{\lambda}.$
$k_{\lambda}(r_{\lambda}(\phi))=(r_{\lambda}\otimes r_{\lambda})\circ
k_{0}((1+\lambda s)^{-(n+1)}\phi).$

###### Proof.

As in [bernig_fu_solanes, Prop. 3.18], to completely determine $J_{\lambda}$
it is enough to show $J_{\lambda}(\chi)=(1-\lambda s)^{n+1}$. This boils down
to checking

(107) $\langle\operatorname{pd}_{\lambda}((1-\lambda
s)^{n+1}),\varphi\rangle=\langle\operatorname{pd}_{0}(\chi),r_{\lambda}^{-1}(\varphi)\rangle$

for all $\varphi=s^{j}t^{2i}$ with $i,j\geq 0$. Using

$r_{\lambda}^{-1}(\varphi)=\left(\frac{s}{1+\lambda
s}\right)^{j}t^{2i}=\sum_{l}(-\lambda)^{l}{j-1+l\choose l}s^{l+j}t^{2i}$

and [bernig_fu_solanes, (3.48)], the relation (107) translates into the
identity

$\sum_{k}(-1)^{k}{n+1\choose k}{n-j-k+1\choose i+1}=(-1)^{n-j-i}{n-i-1\choose
n-i-j},$

which follows from [graham_knuth_patashnik, (5.25)]. The second statement
follows immediately from [bernig_fu_solanes]*Theorem 2.19 and

(108) $(1-\lambda
s)^{n+1}r_{\lambda}(\phi)=(1+\lambda\sigma_{\lambda})^{-(n+1)}r_{\lambda}(\phi)=r_{\lambda}((1+\lambda
s)^{-(n+1)}\phi).$

∎

A similar relation between $k_{0},k_{\lambda}$ was given in
[bernig_fu_solanes]: For the linear isomorphism
$F_{\lambda}\colon\mathcal{V}^{n}_{0}\to\mathcal{V}^{n}_{\lambda}$,
$F_{\lambda}(\mu_{k,q})=\mu^{\lambda}_{k,q}$, equation (3.50) of
[bernig_fu_solanes] shows that

$k_{\lambda}(F_{\lambda}(\phi))=(F_{\lambda}\otimes F_{\lambda})\circ
k_{0}((1-\lambda s)\phi).$

Also the local kinematic operators of $\mathbb{C}^{n}$ and
$\operatorname{\mathbb{C}P}_{\lambda}^{n}$ are isomorphic; in fact, as was
shown in [bernig_fu_solanes]*Theorem 2.23, the transfer map
$\tau\colon\mathcal{C}_{\lambda}^{n}\to\operatorname{Curv}^{\operatorname{U}(n)}$
is an isomorphism of co-algebras. However, no relation between the semi-local
kinematic operators $\bar{k}_{0},\bar{k}_{\lambda}$ (see
[bernig_fu_solanes]*§2.3.3) was previously known. As a consequence of
Proposition 6.11, we obtain that also these kinematic operators also
isomorphic.

######  Theorem 7.8.

The following diagram commutes

${\operatorname{Curv}^{\operatorname{U}(n)}}$${\operatorname{Curv}^{\operatorname{U}(n)}\otimes\operatorname{Val}^{\operatorname{U}(n)}}$${\mathcal{C}_{\lambda}^{n}}$${\mathcal{C}_{\lambda}^{n}\otimes\mathcal{V}_{\lambda}^{n}}$$\bar{k}_{0}$$R_{\lambda}$$\bar{k}_{\lambda}$$R_{\lambda}\otimes
J_{\lambda}$

As a consequence, the semi-local kinematic operators are related through

$\bar{k}_{\lambda}(R_{\lambda}(\Phi))=(R_{\lambda}\otimes
r_{\lambda})\circ\bar{k}_{0}((1+\lambda s)^{-(n+1)}\Phi).$

###### Proof.

Let
$\bar{m}_{\lambda}\in\mathrm{Hom}(\mathcal{V}_{\lambda}^{n}\otimes\operatorname{Curv}^{\operatorname{U}(n)},\operatorname{Curv}^{\operatorname{U}(n)})=\mathrm{Hom}(\operatorname{Curv}^{\operatorname{U}(n)},\operatorname{Curv}^{\operatorname{U}(n)}\otimes(\mathcal{V}_{\lambda}^{n})^{*})$
correspond to the module structure. Given
$\varphi\in\mathcal{V}_{\lambda}^{n}$ and
$\Phi\in\operatorname{Curv}^{\operatorname{U}(n)}$ we have

$\displaystyle\langle\bar{m}_{\lambda}(R_{\lambda}\Phi),\varphi\rangle$
$\displaystyle=\varphi\cdot R_{\lambda}\Phi$
$\displaystyle=R_{\lambda}(r_{\lambda}^{-1}(\varphi)\cdot\Phi)$
$\displaystyle=\langle(R_{\lambda}\otimes\mathrm{id})\circ\bar{m}_{0}(\Phi),r_{\lambda}^{-1}(\varphi)\rangle$
$\displaystyle=\langle(R_{\lambda}\otimes(r_{\lambda}^{-1})^{*})\circ\bar{m}_{0}(\Phi),\varphi\rangle.$

Therefore

$\bar{m}_{\lambda}\circ
R_{\lambda}=(R_{\lambda}\otimes(r_{\lambda}^{-1})^{*})\circ\bar{m}_{0}.$

By [bernig_fu_solanes, Corollary 2.20], this gives

$\displaystyle\bar{k}_{\lambda}\circ R_{\lambda}$
$\displaystyle=(\mathrm{id}\otimes\operatorname{pd}_{\lambda}^{-1})\circ\bar{m}_{\lambda}\circ
R_{\lambda}$
$\displaystyle=(\mathrm{id}\otimes\operatorname{pd}_{\lambda}^{-1})\circ(R_{\lambda}\otimes(r_{\lambda}^{-1})^{*})\circ(\mathrm{id}\otimes\operatorname{pd}_{0})\circ\bar{k}_{0}$
$\displaystyle=(R_{\lambda}\otimes J_{\lambda})\circ\bar{k}_{0}.$

The second statement follows immediately from [bernig_fu_solanes]*Theorem 2.19
and (108). ∎

##  8\. Open problems

  1. (1)

Motivated by the results in [fu_wannerer], it seems a promising task to study
the space of valuations of the form $[[0,\omega]]$, where $\omega$ is a
differential form built from coframe, connection and curvature forms on a
Kähler manifold such that the result does not depend on the choice of frame
and that $d\omega$ is vertical. With respect to the Alesker product, they form
a canonical algebra associated to a Kähler manifold. We conjecture that it
equals the Kähler-Lipschitz-Killing algebra. Even for low-dimensional Kähler
manifolds, it is difficult to prove or disprove the conjecture, since the
space of invariant differential forms is of large dimension.

  2. (2)

It is easy to show that the intrinsic volumes on riemannian manifolds are (up
to linear combinations) the only valuations that satisfy the Weyl principle,
i.e. are invariant under pullback along isometric immersions. The same
statement holds true on pseudoriemannian manifolds
[bernig_faifman_solanes_part2]. Is it true that any valuation that can be
associated to Kähler manifolds and that satisfies the Weyl principle is a
linear combination of the $\mu_{k,p}$?

  3. (3)

Can the Weyl principle for Kähler manifolds be extended to a larger class of
manifolds including submanifolds of Kähler manifolds? Note that a submanifold
of a Kähler manifold is endowed with a metric and a closed $2$-form, which may
by degenerate.

  4. (4)

A natural question is whether the main results of this paper remain valid in
the quaternionic Kähler case. It turns out that the analogue of Theorem 4.12
is not true in the quaternionic Kähler case. If $M$ is a quaternionic Kähler
manifold of real dimension $4n$ which is embedded in some quaternionic vector
space $\mathbb{H}^{n+m}$, then each complex structure $I$ of $M$ is parallel.
It follows that $R^{M}_{p}(IX,IY,Z,W)=R^{M}_{p}(X,Y,Z,W)$ for all $X,Y,Z,W\in
V$. The curvature tensor of a general quaternionic Kähler manifold does not
have this symmetry, as can be seen in the case of quaternionic projective
space. Hence the set of embedded algebraic quaternionic Kähler curvature
tensors spans a subspace of positive codimension in the space of all algebraic
quaternionic Kähler curvature tensors. Nevertheless, it could still be true
that there is a canonical algebra of valuations associated to each
quaternionic Kähler manifold. This would imply that the algebras of smooth
isometry invariant valuations on quaternionic projective space
$\operatorname{\mathbb{H}P}^{n}$ and the algebra of continuous,
$\mathrm{Sp}(n)\cdot\mathrm{Sp}(1)$ and translation invariant valuations on
$\mathbb{H}^{n}$ are isomorphic. It is not known, even in small dimensions, if
this is true.

  5. (5)

More generally, can we associate a canonical algebra of valuations to each
manifold with a (sufficiently flat) $G$-structure? An additional hint in this
direction is provided by the recent results in [solanes_wannerer], where it is
shown that the algebras of valuations on the isotropic spaces $(S^{6},G_{2})$
and $(\mathbb{C}^{3},\overline{\mathrm{SU}(3)}))$ are isomorphic, as well as
on the isotropic spaces $(S^{7},G_{2})$ and
$(\mathbb{R}^{7},\overline{G_{2}})$.

## References