$\mathfrak{t}(\omega).\mathfrak{t}(\omega^{\prime})=\mathfrak{t}(\omega+\omega^{\prime}).$

One deduces that $\mu$ induces an isomorphism of bigraded
$\mathbb{E}^{**}(X)$-modules:

$\mathbb{E}^{**}(\operatorname{Th}(\omega))\otimes_{\mathbb{E}^{**}(X)}\mathbb{E}^{*}(\operatorname{Th}(\omega^{\prime}))\xrightarrow{\sim}\mathbb{E}^{**}(\operatorname{Th}(\omega+\omega^{\prime})).$

Finally the virtual Thom class extends the Thom class associated with a
symplectic vector bundle. Given a symplectic bundle $(V,\psi)$, one associates
a virtual symplectic bundle $[V,\psi]\in\operatorname{\underline{KSp}}(X)$
such that $f_{\mathrm{Sp}}([V,\psi])=[V]$ the stable vector bundles assocaited
with $V$. One further gets an identification
$\operatorname{Th}([V,\psi])=\operatorname{Th}([V])=\operatorname{Th}(V)$, by
construction of the Thom space of the virtual bundle $[V]$. Given this
identification, one has in the group $\mathbb{E}^{2r,r}(\operatorname{Th}(V))$
with $r=\operatorname{rk}(V)$:

$\mathfrak{t}([V,\psi])=\mathfrak{t}(V,\psi).$

###  3.4. Higher Borel classes and $\operatorname{GW}$-linearity

######  3.4.1.

Borel classes. Recall that to a symplectic orientation $b$ of $\mathbb{E}$ is
associated by Panin and Walter a theory of Borel classes
$b_{n}(\mathcal{U})\in\mathbb{E}^{4n,2n}(X)$, $\mathcal{U}/X$ a symplectic
vector bundle over a smooth $S$-scheme $X$. Besides, the Euler class
introduced in Paragraph 3.2.3 satisfies the relation:
$e(\mathcal{U})=b_{r}(\mathcal{U})$ when $\mathcal{U}$ has rank $2r$. We refer
the reader to [DF21, Section 2.2]. Following the classical scheme, one defines
the total Borel class associated with $\mathcal{U}$ as the following
polynomial expression in $t$:

(3.4.1.a) $b_{t}(\mathcal{U})=\sum_{n=0}^{r}b_{n}(\mathcal{U}).t^{r}.$

Note moreover that, using relation (3.2.5.a) and the construction of Borel
classes, one deduces further that for a morphism $\varphi$ as in 3.2.5, we
have the following relation in $\mathbb{F}^{4n,2n}(X)$:

(3.4.1.b)
$\varphi_{*}(b_{n}^{\mathbb{E}}(\mathcal{U}))=b_{n}^{\mathbb{F}}(\mathcal{U}).$

######  3.4.2.

The symplectic splitting principle. As for Chern classes, one has a splitting
principle for Borel classes. Indeed, given any symplectic bundle $\mathcal{V}$
over a scheme $X$, there exists an affine morphism $p:X^{\prime}\rightarrow X$
inducing a monomorphism
$p^{*}:\mathbb{E}^{**}(X)\rightarrow\mathbb{E}^{**}(X^{\prime})$ with the
property that $p^{-1}(\mathcal{V})$ splits as a direct sum of rank $2$
symplectic bundles:
$p^{-1}(\mathcal{V})=\mathcal{X}_{1}\perp\ldots\perp\mathcal{X}_{n}$.

One defines the Borel roots of $\mathcal{V}$ as the Borel classes
$\xi_{i}=b_{1}(\mathcal{X}_{i})$ so that by the Whitney sum formula [PW22,
Theorem 10.5] the Borel classes of $\mathcal{V}$ are the elementary symmetric
polynomials in the variables $\xi_{i}$.

As in the classical case, one can compute universal formulas involving Borel
classes of $\mathcal{V}$ by introducing Borel roots $\xi_{i}$, which reduces
to rank $2$ symplectic bundles, compute as if $\mathcal{V}$ was completely
split, and then express the resulting formula in terms of the elementary
symmetric polynomials in the $\xi_{i}$.

######  3.4.3.

We now give a complement to the theory of Borel classes which was known
previously only under the stronger assumption of the existence of an
$\mathrm{SL}$-orientation (see [DF21, 2.2.8]). To start with, let us recall
the following definition.

######  Definition 3.4.4.

Let $\lambda\in\mathcal{O}(S)^{\times}$. We denote by
$\langle\lambda\rangle\in\mathrm{End}_{\mathcal{SH}(S)}(\mathbf{1})$ the
endomorphism induced by the pointed morphism
$\lambda:\mathbb{P}^{1}_{S}\to\mathbb{P}^{1}_{S}$ given by
$[x:y]\mapsto[\lambda x:y]$.

As a result, $\lambda$ acts on the set $[X,Y]_{\mathcal{SH}(S)}$ for any
objects $X,Y$ via

$f\mapsto f\wedge\langle\lambda\rangle:X\wedge 1\to Y\wedge 1.$

By abuse of notation, we still denote by $\langle\lambda\rangle$ the
endomorphism $\mathrm{Id}_{X}\wedge\langle\lambda\rangle$. The aim of this
section is to prove the following theorem.

######  Theorem 3.4.5.

Let $(\mathbb{E},b)$ be an $\mathrm{Sp}$-oriented ring spectrum, and thom
classes $\mathfrak{t}$. Then for any unit
$\lambda\in\mathcal{O}_{S}(X)^{\times}$, and any $\mathrm{Sp}$-vector bundle
$(V,\psi)$ over $X$, the following relation holds:

$\mathfrak{t}(V,\lambda\psi)=\langle\lambda\rangle\mathfrak{t}(V,\psi).$

As a corollary, we obtain the following result whose proof is the same as
[DF21, 2.2.8].

######  Corollary 3.4.6.

Let $(\mathbb{E},b)$ be an $\mathrm{Sp}$-oriented ring spectrum and let
$\lambda\in\mathcal{O}_{X}(X)^{\times}$ be a unit. Then

$b_{i}(V,\lambda\psi)=\langle\lambda^{i}\rangle b_{i}(V,\psi)$

for any $\mathrm{Sp}$-vector bundle $(V,\psi)$ over $X$ and any
$i\in\mathbb{N}$.

The proof of the theorem will occupy the remaining of this section. We first
reduces to the case $\mathbb{E}=\mathbf{MSp}_{S}$ (using 3.2.4) and to the
case where of the tautological symplectic bundle $(U,\varphi)$ on
$\mathrm{BSp}_{2}$ seen over the base scheme $S$. Given
$\lambda\in\mathcal{O}_{S}(S)^{\times}$, the relation to prove is now:

$\mathrm{th}(U,\lambda\varphi)=\langle\lambda\rangle\mathrm{th}(U,\varphi).$

The proof of the theorem will require some elaborate computations with the
hyperbolic Grassmannians. Recall that
$\mathrm{H}\mathbb{P}^{n}_{S}=\mathrm{HGr}_{S}(2,\omega_{2n+2})$ and that
there are embeddings

$i_{n}:\mathrm{H}\mathbb{P}^{n}_{S}\to\mathrm{H}\mathbb{P}^{n+1}_{S}$

for any $n\in\mathbb{N}$. In particular, $\mathrm{H}\mathbb{P}^{0}_{S}=S$ and
we will consider $\mathrm{H}\mathbb{P}^{n}_{S}$ as pointed by the image of
$\mathrm{H}\mathbb{P}^{0}_{S}=S$ under the above map. In the sequel, we will
suppress the subscript $S$ from the notation to slightly lighten the
expressions.

There is an interesting cell structure on $\mathrm{H}\mathbb{P}^{n}$ for any
$n\in\mathbb{N}$, as explained by Panin-Walter in [PW22]. There is a
decomposition

$\mathrm{H}\mathbb{P}^{n}=X_{0}\sqcup X_{2}\sqcup\ldots\sqcup X_{2n}$

where each $X_{i}$ is of codimension $2i$, smooth and quasi-affine. The
closure $\overline{X}_{2i}$ of $X_{2i}$ is of the form
$\overline{X}_{2i}=X_{2i}\sqcup X_{2i+2}\sqcup\ldots\sqcup X_{2n}$ and there
is a projection

$q_{i}:\overline{X}_{2i}\to\mathrm{H}\mathbb{P}^{n-i}$

identifying $\overline{X}_{2i}$ with the total space of the universal bundle
$\mathcal{U}_{2}$ over $\mathrm{H}\mathbb{P}^{n-i}$. The composite of the zero
section $\mathrm{H}\mathbb{P}^{n-i}\to\overline{X}_{2i}$ with
$\overline{X}_{2i}\to\mathrm{H}\mathbb{P}^{n}$ is the composite of embeddings
$i_{n-1}\circ\ldots\circ i_{n-i}$ described above. In more detail, write
$e_{1},\ldots,e_{2n+2}$ for the usual basis of $\mathcal{O}_{S}^{2n+2}$.
Observe that $E_{i}=\mathrm{Vect}(e_{2j-1},j=1,\ldots,i)$ is a totally
isotropic subspace, in the sense that $E_{i}\subset E_{i}^{\perp}$, and that
we have a filtration of $\mathcal{O}_{S}^{2n+2}$ of the form

$0:=E_{0}\subset E_{1}\subset E_{n+1}=E_{n+1}^{\perp}\subset\ldots\subset
E_{1}^{\perp}\subset 0^{\perp}=\mathcal{O}_{S}^{2n+2}$

We have for any $i=0,\ldots,n$ an embedding
$\mathrm{Gr}(2,E_{i}^{\perp})\subset\mathrm{Gr}(2,\mathcal{O}_{S}^{2n+2})$ and
we set
$\overline{X}_{2i}:=\mathrm{Gr}(2,E_{i}^{\perp})\cap\mathrm{H}\mathbb{P}^{n+1}$.
In terms of the universal sequence, we can interpret these closed subschemes
as follows. Consider the projection
$p_{2}:\mathcal{O}_{S}^{2n+2}\to\mathcal{O}_{S}$ on the second factor. As
$E_{1}^{\perp}=\langle e_{1},e_{3},e_{4},\ldots,e_{2n+2}\rangle$ we see that
$\overline{X}_{2}$ is the closed subscheme where the section

$\mathcal{U}\xrightarrow{i}\mathcal{O}^{2n+2}\xrightarrow{p_{2}}\mathcal{O}$

vanishes. In particular, the normal bundle to $\overline{X}_{2}$ in
$\mathrm{H}\mathbb{P}^{n}$ is $(\mathcal{U}^{\vee})_{|\overline{X}_{2}}$.
Besides, if $p_{1}:\mathcal{O}_{S}^{2n+2}\to\mathcal{O}_{S}$ is the projection
to the first factor, it is easy to check that if the above composite vanishes
(i.e. we are on $\overline{X}_{2}$) then the composite

$\mathcal{U}\xrightarrow{i}\mathcal{O}^{2n+2}\xrightarrow{p_{1}}\mathcal{O}$

is arbitrary (i.e. the symplectic form on $\mathcal{U}$ is automatically
nondegenerate). Observing that the restriction of $\mathcal{U}$ to
$\mathrm{H}\mathbb{P}^{n-1}$ is the universal bundle, this realizes
$\overline{X}_{2}$ as the total space of $\mathcal{U}^{\vee}$ over
$\mathrm{H}\mathbb{P}^{n-1}$. We also deduce from this that the normal bundle
to $\mathrm{H}\mathbb{P}^{n-1}$ in $\mathrm{H}\mathbb{P}^{n}$ is
$\mathcal{U}^{\vee}\oplus\mathcal{U}^{\vee}$. To conclude, let us say that the
filtration $\mathrm{H}\mathbb{P}^{n-1}=X^{\prime}_{0}\sqcup\ldots\sqcup
X^{\prime}_{2n-2}$ pulls back to the filtration $X_{2}\sqcup\ldots\sqcup
X_{2n}$ under the projection
$q_{n-1}:\overline{X}_{2}\to\mathrm{H}\mathbb{P}^{n-1}$.

Let us now concentrate on the geometry of $X_{2i}$ for $i=0,\ldots,n$,
starting with $X_{0}$. By definition, $X_{0}$ is the open subscheme over which
the composite

$\pi:\mathcal{U}\xrightarrow{i}\mathcal{O}^{2n+2}\xrightarrow{p_{2}}\mathcal{O}$

is non trivial. This yields a nowhere vanishing section of $\mathcal{U}$ with
kernel $\mathcal{O}$ (as $\mathcal{U}$ has trivial determinant). We set
$Y=\\{f\in U|\pi(y)=1\\}$. This is a $\ker(\pi)$-torsor over $X_{0}$, i.e. an
$\mathbb{A}^{1}$-torsor over $X_{0}$ of the form $\mu:Y\to X_{0}$. We now work
over $Y$. It is easy to see that there exists a unique global section $e$ of
$\mu^{*}\mathcal{U}$ which has the property that the symplectic form on
$\mathcal{O}^{2}$ induced by the isomorphism
$j:\mathcal{O}^{2}\simeq\mathcal{U}$ obtained via $f,e$ is the usual
hyperbolic one.

We may then write $i:\mathcal{U}\to\mathcal{O}^{2n+2}$ as a matrix

$\begin{pmatrix}1&0\\\ a&b\\\ v&w\end{pmatrix}$

where $a,b\in\mathbb{A}^{1}$ and $v,w\in\mathbb{A}^{2n}$. The restriction of
$\sigma_{2n+2}$ under this map is the matrix

$\begin{pmatrix}0&b+v^{t}\sigma_{2n}w\\\ -b+w^{t}\sigma_{2n}v&0\end{pmatrix}$

and it follows that $b+v^{t}\sigma_{2n}w=1$. Now, the action of
$\mathbb{G}_{a}$ on $(a,v,w)$ is of the form
$\mathbb{G}_{a}\times\mathbb{A}^{1}\times\mathbb{A}^{2n}\times\mathbb{A}^{2n}\to\mathbb{A}^{1}\times\mathbb{A}^{2n}\times\mathbb{A}^{2n}$
given by

$\lambda\cdot\begin{pmatrix}1&0\\\ a&b\\\
v&w\end{pmatrix}=\begin{pmatrix}1&0\\\ a&b\\\
v&w\end{pmatrix}\begin{pmatrix}1&0\\\ \lambda&1\end{pmatrix},$

i.e. $\lambda\cdot(a,v,w)=(a+b\lambda,v+\lambda
w,w)=(a+\lambda(1-v^{t}\sigma_{2n}w),v+\lambda w,w)$. We may extend this
description to the other open cells $X_{2i}$ (or use inductively the fact that
the pull-back of $X^{\prime}_{0}\sqcup\ldots\sqcup X^{\prime}_{2n-2}$ is
$X_{2}\sqcup\ldots\sqcup X_{2n}$) to obtain the following result ([PW22,
Theorem 3.4]).

######  Lemma 3.4.7.

The cell $X_{2i}$ is the quotient of
$\mathbb{A}^{1}\times\mathbb{A}^{i}\times\mathbb{A}^{2n-2i}\times\mathbb{A}^{i}\times\mathbb{A}^{2n-2i}$
under the free action of $\mathbb{G}_{a}$ given by

$\lambda\cdot(a,v_{1},v_{2},w_{1},w_{2})\mapsto(a+\lambda(1-v_{2}^{t}\sigma_{2n-2i}w_{2}),v_{1}+\lambda
w_{1},v_{2}+\lambda w_{2},w_{1},w_{2}).$

In particular, the cells $X_{2i}$ are $\mathbb{A}^{1}$-contractible for
$i=0,\ldots,n$.

As a consequence, we obtain the following lemma.

######  Lemma 3.4.8.

For $n\geq 1$, there is a canonical weak-equivalence

$c_{n}:\mathrm{H}\mathbb{P}^{n}\simeq\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n-1}}).$

###### Proof.

Consider the open immersion $X_{0}\subset\mathrm{H}\mathbb{P}^{n}$, whose
closed complement is $\overline{X}_{2}$. The normal bundle to
$\overline{X}_{2}$ in $\mathrm{H}\mathbb{P}^{n}$ being $\mathcal{U}^{\vee}$,
we obtain by purity a cofiber sequence

$X_{0}\to\mathrm{H}\mathbb{P}^{n}\to\mathrm{Th}(U^{\vee}_{|\overline{X}_{2}})$

and therefore, as $X_{0}$ is contractible, a weak-equivalence
$\mathrm{H}\mathbb{P}^{n}\to\mathrm{Th}(U^{\vee}_{\overline{X}_{2}})$. The
inclusion $\mathrm{H}\mathbb{P}^{n-1}\subset\overline{X}_{2}$ is a weak-
equivalence (being the zero section of a vector bundle) and the pull back of
$U^{\vee}_{|\overline{X}_{2}}$ under this map is just
$U^{\vee}_{\mathrm{H}\mathbb{P}^{n-1}}$, which is isomorphic to its dual. We
then obtain a string of canonical weak-equivalences
$\mathrm{Th}(U^{\vee}_{|\overline{X}_{2}})\simeq\mathrm{Th}(U^{\vee}_{|\mathrm{H}\mathbb{P}^{n-1}})\simeq\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n-1}})$.
∎

For $n\geq 0$, the universal subbundle $\mathcal{U}$ on
$\mathrm{H}\mathbb{P}^{n+1}$ restricts to the canonical bundle on
$\mathrm{H}\mathbb{P}^{n}$ and we obtain an induced map

$j_{n}:\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})\to\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n+1}}).$

The following lemma will prove useful in the sequel.

######  Lemma 3.4.9.

Let $n\geq 1$ and let
$\tau_{n}:\mathrm{H}\mathbb{P}^{n}\to\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})$
be the canonical morphism. Then, the following diagram commutes in
$\mathcal{H}(S)$

$\textstyle{\mathrm{H}\mathbb{P}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{n}}$$\scriptstyle{i_{n}}$$\scriptstyle{\tau_{n}}$$\textstyle{\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n-1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{n-1}}$$\textstyle{\mathrm{H}\mathbb{P}^{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{n+1}}$$\textstyle{\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}}).}$

###### Proof.

We first prove that the bottom triangle commutes. Consider the composite of
the zero section $\overline{X}_{2}\to U_{|\overline{X}_{2}}$ with the
projection $U_{|\overline{X}_{2}}\to\mathrm{Th}(U_{|\overline{X}_{2}})$. The
deformation to the normal cone shows that the following diagram commutes in
$\mathcal{H}(S)$

$\textstyle{\overline{X}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U_{|\overline{X}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}\mathbb{P}^{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Th}(U_{|\overline{X}_{2}}).}$

On the other hand, we have a commutative diagram

$\textstyle{\mathrm{H}\mathbb{P}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U_{|\mathrm{H}\mathbb{P}^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{X}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U_{|\overline{X}_{2}}}$

inducing the weak-equivalence
$\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})\to\mathrm{Th}(U_{|\overline{X}_{2}})$.
The claim follows from the fact that the composite

$\mathrm{H}\mathbb{P}^{n}\to\overline{X}_{2}\to\mathrm{H}\mathbb{P}^{n+1}$

is $i_{n}$. We prove next that the other triangle commutes. For this, recall
that $X_{0}^{\prime}\subset\mathrm{H}\mathbb{P}^{n}$ is defined as the
complement of the vanishing locus of a section $s:\mathcal{U}\to\mathcal{O}$.
We then obtain a commutative diagram

$\textstyle{X_{0}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}\mathbb{P}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{\mathrm{Th}(U_{|\overline{X}_{2}^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U\setminus
0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Th}(U).}$

In $\mathcal{H}(S)$, $s$ and the zero section are equal, and we conclude
observing that the composite

$\textstyle{\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n-1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Th}(U_{|\overline{X}_{2}^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Th}(U)}$

is just $j_{n}$. ∎

We now pass to the definition of an action of $\mathbb{G}_{m}$ on
$\mathrm{H}\mathbb{P}^{n}$ for any $n\in\mathbb{N}$. Let
$\lambda\in\mathcal{O}(S)^{\times}$ and let
$m_{\lambda}:\mathcal{O}_{S}^{2}\to\mathcal{O}_{S}^{2}$ be given by the matrix

$\begin{pmatrix}\lambda&0\\\ 0&1\end{pmatrix}.$

A direct computation shows that
$m_{\lambda}^{\vee}\sigma_{2}m_{\lambda}=\lambda\cdot\sigma_{2}$ and it
follows that $m_{\lambda}$ induces a map

$\mathrm{Sp}_{2}(S)\to\mathrm{Sp}_{2}(S)$

given by $A=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\mapsto
m_{\lambda}^{-1}\cdot A\cdot m_{\lambda}=\begin{pmatrix}a&\lambda^{-1}b\\\
\lambda c&d\end{pmatrix}$. This induces an action of $\mathbb{G}_{m}$ on
$\mathrm{Sp}_{2}$, and consequently an action of $\mathbb{G}_{m}$ on
$\mathrm{BSp}_{2}$. This could be defined at the level of
$\mathrm{H}\mathbb{P}^{n}$ for $n\in\mathbb{N}$ as follows: Set

$\mathrm{Gr}(2,\mathcal{O}_{S}^{2n+2})\to\mathrm{Gr}(2,\mathcal{O}_{S}^{2n+2})$

by considering the composite

$\mathcal{U}\xrightarrow{i}\mathcal{O}^{2n+2}\xrightarrow{m_{\lambda}^{\oplus
n+1}}\mathcal{O}^{2n+2}.$

If $i^{\vee}\sigma_{2n+2}i$ is non degenerate, then so is the composite

$i^{\vee}(m_{\lambda}^{\oplus n+1})^{\vee}\sigma_{2n+2}(m_{\lambda}^{\oplus
n+1}i)=\lambda\cdot(i^{\vee}\sigma_{2n+2}i).$

Consequently, the above action of $\mathbb{G}_{m}$ on
$\mathrm{Gr}(2,\mathcal{O}_{S}^{2n+2})$ restricts to an action of
$\mathbb{G}_{m}$ on
$\mathrm{HGr}(2,\mathcal{O}_{S}^{2n+2})=\mathrm{H}\mathbb{P}^{n}$. Let us
denote by

$\mu:\mathbb{G}_{m}\times\mathrm{H}\mathbb{P}^{n}\to\mathrm{H}\mathbb{P}^{n}$

this action, and
$\pi:\mathbb{G}_{m}\times\mathrm{H}\mathbb{P}^{n}\to\mathrm{H}\mathbb{P}^{n}$
the projection to the second factor.

######  Lemma 3.4.10.

There is a canonical isomorphism $f:\pi^{*}\mathcal{U}\to\mu^{*}\mathcal{U}$
on $\mathbb{G}_{m}\times\mathrm{H}\mathbb{P}^{n}$.

###### Proof.

We start by exhibiting a $\mathbb{G}_{m}$-equivariant open covering
$\cup_{i=1}^{m}V_{i}$ of $\mathrm{H}\mathbb{P}^{n}$ which has the following
properties

  1. (1)

$V_{i}$ is affine for any $i=1,\ldots,m$.

  2. (2)

The restriction of $\mathcal{U}$ to $V_{i}$ is free for any $i=1,\ldots,m$.

To obtain this, we consider the composite morphism

$\mathrm{H}\mathbb{P}^{n}\to\mathrm{Gr}(2,\mathcal{O}_{S}^{2n+2})\to\mathbb{P}_{S}^{\frac{(2n+2)(2n+1)}{2}}$

where the second map is the Plücker embedding. We note that the
$\mathbb{G}_{m}$-action on $\mathrm{H}\mathbb{P}^{n}$ is the restriction of a
$\mathbb{G}_{m}$-action on $\mathrm{Gr}(2,\mathcal{O}_{S}^{2n+2})$. The latter
can be extended (taking exterior powers) to an action on
$\mathbb{P}_{S}^{\frac{(2n+2)(2n+1)}{2}}$ which is easily seen to be diagonal.
It follows that the usual covering of the latter by affine spaces (complement
of the zero locus of a given coordinate) is $\mathbb{G}_{m}$-equivariant. We
define the open subschemes $V_{i}\subset\mathrm{H}\mathbb{P}^{n}$ as the
intersections of this covering with $\mathrm{H}\mathbb{P}^{n}$. The latter
being affine, it follows that (1) above is satisfied. For (2), we observe that
the condition on the minors is equivalent to the composite

$\mathcal{U}\xrightarrow{i}\mathcal{O}^{2n+2}\to\mathcal{O}^{2}$

being an isomorphism (the projection corresponds to the identification of some
minor).

This covering being obtained, we now choose over each $V_{i}$ a symplectic
basis of $\mathcal{U}_{|V_{i}}$ and we may recover $\mathcal{U}$ from the open
covering $V_{i}$ and transition functions $V_{i}\cap V_{j}\to\mathrm{Sp}_{2}$.
A direct computations shows that $\mu^{*}\mathcal{U}$ is then obtained via the
transition functions

$V_{i}\cap V_{j}\to\mathrm{Sp}_{2}\to\mathrm{Sp}_{2}$

where the second map is given by

$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\mapsto\begin{pmatrix}a&tb\\\
t^{-1}c&d\end{pmatrix}=\begin{pmatrix}t&0\\\
0&1\end{pmatrix}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}t^{-1}&0\\\ 0&1\end{pmatrix}$

where $t$ is a parameter of $\mathbb{G}_{m}$. We now define
$\pi^{*}\mathcal{U}_{|V_{i}}\to\mu^{*}\mathcal{U}_{|V_{i}}$ using the matrix
$\begin{pmatrix}t&0\\\ 0&1\end{pmatrix}$. A direct computation shows that this
behaves well with the transition functions, yielding the required isomorphism
$f:\pi^{*}\mathcal{U}\to\mu^{*}\mathcal{U}$. ∎

We now fix $\lambda\in\mathcal{O}(S)^{\times}$, obtaining a multiplication by
$\lambda$ morphism

$\mu_{\lambda}:\mathrm{H}\mathbb{P}^{n}\to\mathrm{H}\mathbb{P}^{n}.$

Using the above lemma, we obtain a commutative diagram

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mu_{\lambda}^{*}U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}\mathbb{P}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\lambda}}$$\textstyle{\mathrm{H}\mathbb{P}^{n}}$

and then a morphism of Thom spaces
$\mu_{\lambda}^{f}:\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})\to\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})$
making the diagram

$\textstyle{\mathrm{H}\mathbb{P}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\lambda}}$$\textstyle{\mathrm{H}\mathbb{P}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\lambda}^{f}}$$\textstyle{\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})}$

commutative.

We note here that the action of $\mathbb{G}_{m}$ on $\mathrm{H}\mathbb{P}^{n}$
stabilizes the cellular filtration

$\mathrm{H}\mathbb{P}^{n}=X_{0}\sqcup\ldots\sqcup X_{2n}$

in the sense that the subschemes $X_{i}$ (and their closure) are
$\mathbb{G}_{m}$-equivariant. As a result, we see that the morphisms in Lemma
3.4.9 all commute with the relevant morphisms $\mu_{\lambda}$ and
$\mu_{\lambda}^{f}$.

######  Lemma 3.4.11.

The endomorphism
$\mu_{\lambda}:\mathrm{H}\mathbb{P}^{1}\to\mathrm{H}\mathbb{P}^{1}$ is equal
to $\langle\lambda\rangle$ in
$\mathrm{End}_{\mathcal{SH}(S)}(\mathrm{H}\mathbb{P}^{1})$.

###### Proof.

By the above remark, we have a commutative diagram

$\textstyle{\mathrm{H}\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{1}}$$\scriptstyle{\mu_{\lambda}}$$\textstyle{\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\lambda}^{f}}$$\textstyle{\mathrm{H}\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{1}}$$\textstyle{\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{0}})}$

Now, $U_{|\mathrm{H}\mathbb{P}^{0}}$ is trivial and $f$ is just the
multiplication by $\begin{pmatrix}\lambda&0\\\ 0&1\end{pmatrix}$. The claim
follows easily. ∎

By construction, there is a tautological Thom class

$\mathrm{th}(U_{|\mathrm{H}\mathbb{P}^{n}}):\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})\to\mathbf{MSp}(2)[4]$

which has the property that the following diagram commutes for any
$n\in\mathbb{N}$:

$\textstyle{\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{th}(U_{|\mathrm{H}\mathbb{P}^{n}})}$$\scriptstyle{j_{n}}$$\textstyle{\mathbf{MSp}(2)[4]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n+1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{th}(U_{|\mathrm{H}\mathbb{P}^{n+1}})}$$\textstyle{\mathbf{MSp}(2)[4]}$

######  Proposition 3.4.12.

For any $n\geq 0$, the composite

$\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})\xrightarrow{\mu_{\lambda}^{f}}\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})\xrightarrow{\mathrm{th}(U_{|\mathrm{H}\mathbb{P}^{n}})}\mathbf{MSp}(2)[4]$

is equal to $\langle\lambda\rangle\mathrm{th}(U_{|\mathrm{H}\mathbb{P}^{n}})$
in
$[\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}}),\mathbf{MSp}(2)[4]]_{\mathcal{SH}(R)}$.

###### Proof.

We work by induction on $n$. If $n=0$, this is a direct consequence of (the
proof of) Lemma 3.4.11. Let us then suppose that the result is true for some
$n-1\geq 0$ and consider the composite

$\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})\xrightarrow{\mu_{\lambda}^{f}}\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})\xrightarrow{\mathrm{th}(U_{|\mathrm{H}\mathbb{P}^{n}})}\mathbf{MSp}(2)[4].$

We have a commutative diagram

$\textstyle{\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n-1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\lambda}^{f}}$$\scriptstyle{j_{n-1}}$$\textstyle{\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n-1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{th}(U_{|\mathrm{H}\mathbb{P}^{n-1}})}$$\scriptstyle{j_{n-1}}$$\textstyle{\mathbf{MSp}(2)[4]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\lambda}^{f}}$$\textstyle{\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{th}(U_{|\mathrm{H}\mathbb{P}^{n}})}$$\textstyle{\mathbf{MSp}(2)[4]}$

By induction, we know that

$(\mathrm{th}(U_{|\mathrm{H}\mathbb{P}^{n}})\circ\mu_{\lambda}^{f})\circ
j_{n-1}=(\langle\lambda\rangle\mathrm{th}(U_{|\mathrm{H}\mathbb{P}^{n}}))\circ
j_{n-1}$

and it suffices to prove that

$j_{n-1}^{*}:[\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}}),\mathbf{MSp}(2)[4]]\to[\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n-1}}),\mathbf{MSp}(2)[4]]$

is injective for $n\geq 1$. Now, we may use Lemma 3.4.9 to obtain

$\tau_{n}^{*}=c_{n}^{*}\circ j_{n-1}^{*}$

and it suffices to show that $\tau_{n}^{*}$ is injective. However, it follows
from [PW22] that the cofiber sequence

$U\setminus
0\to\mathrm{H}\mathbb{P}^{n}\xrightarrow{\tau_{n}}\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}})$

induces a short split exact sequence

$0\to[\mathrm{Th}(U_{|\mathrm{H}\mathbb{P}^{n}}),\mathbf{MSp}(2)[4]]\xrightarrow{\tau_{n}^{*}}[\mathrm{H}\mathbb{P}^{n},\mathbf{MSp}(2)[4]]\to[U\setminus
0,\mathbf{MSp}(2)[4]]\to 0$

and in particular $\tau_{n}^{*}$ is injective. ∎

###### Proof of 3.4.5.

Taking colimits, it is sufficient to prove the result for the Thom classes of
$\mathcal{U}_{|\mathrm{H}\mathbb{P}^{n}}$ for any $n\in\mathbb{N}$. It follows
from the above proposition that we have

$\langle\lambda\rangle\mathrm{th}(U_{|\mathrm{H}\mathbb{P}^{n}})=\mathrm{th}(U_{|\mathrm{H}\mathbb{P}^{n}})\circ\mu_{\lambda}^{f}.$

Now, the property of Thom classes reads as
$\mathrm{th}(U_{|\mathrm{H}\mathbb{P}^{n}})\circ\mu_{\lambda}^{f}=\mathrm{th}(f^{*}\mu_{\lambda}^{*}U_{|\mathrm{H}\mathbb{P}^{n}})$
and a direct computation shows that
$f^{*}\mu_{\lambda}^{*}(U_{|\mathrm{H}\mathbb{P}^{n}},\varphi)=(U,\langle\lambda\rangle\varphi)$.
∎

###  3.5. The FTL associated to an $\mathrm{Sp}$-oriented spectrum

######  3.5.1.

Let $(\mathbb{E},b)$ be an $\mathrm{Sp}$-oriented ring spectrum over $S$. Its
ring of coefficients $\mathbb{E}^{**}:=\mathbb{E}^{**}(S)$ is a bigraded ring.
Moreover, the action of $\epsilon=-\langle-1\rangle$ on $\operatorname{SH}(S)$
allows to equip it with a bigraded $\mathbb{Z}_{\epsilon}$-algebra structure.
When dealing with symplectic orientations, and the forthcoming associated FTL,
it is more natural to consider the subring $\mathbb{E}^{\langle
2\rangle*}:=(\mathbb{E}^{4n,2n}(S),n\in\mathbb{Z})$, which is now a graded
$\mathbb{Z}_{\epsilon}$-algebra.171717Indeed, Borel classes live in degree
$\langle 2\rangle*$.

In [DF21, Def. 2.3.2], we proved that one can associate to $(\mathbb{E},b)$ a
$(3,4)$-series that we will denote by $F_{t}^{b}(x,y,z)$ with coefficients in
$\mathbb{E}^{\langle 2\rangle*}$. The method is directly imported from the
$\mathrm{GL}$-oriented case (compare [Dég18b, 2.1.19]): thanks to the
symplectic projective bundle theorem ([DF21, 2.2.3]), one gets a canonical
isomorphism

$\mathbb{E}^{\langle
2\rangle*}(\mathrm{H}\mathbb{P}^{\infty}_{S}\times_{S}\mathrm{H}\mathbb{P}^{\infty}_{S}\times_{S}\mathrm{H}\mathbb{P}^{\infty}_{S})\simeq\mathbb{E}^{\langle
2\rangle*}(S)[[x,y,z]]$

where $x$ (resp. $y$, $z$) are formal variables which corresponds to the Borel
class of the tautological symplectic bundle $\mathcal{U}_{1}$ (resp.
$\mathcal{U}_{2}$, $\mathcal{U}_{3}$) on $\mathrm{H}\mathbb{P}^{\infty}_{S}$
over the first (resp. second, third) coordinates. The triple tensor product
$\mathcal{U}_{1}\otimes\mathcal{U}_{2}\otimes\mathcal{U}_{3}$ is naturally a
rank $8$ symplectic bundle. It follows that one can write the total Borel
class (3.4.1.a):

(3.5.1.a)
$F_{t}^{b}(x,y,z):=b_{t}(\mathcal{U}_{1}\otimes\mathcal{U}_{2}\otimes\mathcal{U}_{3})$

as a polynomial in $t$ of degree $4$, with coefficients in
$\mathbb{E}^{**}[[x,y,z]]$. In other words, the expression $F_{t}^{b}(x,y,z)$
defines a $(3,4)$-series in the formal variables $x,y,z$ as required. It
satisfies the symmetry and associativity axioms of 3.1.2 according to the
analogous properties of the tensor product of symplectic bundles.

In _op. cit._ , the Neutral and Semi-neutral element axioms were only proved
under the additional assumption that $\mathbb{E}$ is $\mathrm{SL}$-oriented.
Thanks to 3.4.6, we can now avoid the latter assumption on $\mathbb{E}$. In
fact, the mentioned corollary readily implies that the FTL $F_{t}^{b}(x,y,z)$
satisfies the $\epsilon$-linearity axiom. Once this property is known, one
deduces the Neutral and Semi-neutral element properties as in the proof of
[DF21, Prop. 2.3.4].

The above procedure allows us to deduce the following theorem (which enhances
[DF21, §2.3] thanks to the $\epsilon$-linearity axiom).

######  Proposition 3.5.2.

The association $(\mathbb{E},b)\mapsto\big{(}\mathbb{E}^{\langle
2\rangle*},F_{t}^{b}(x,y,z)\big{)}$ described above defines a canonical
pseudo-functor from the scheme-fibred category of $\mathrm{Sp}$-oriented ring
spectra to the ring-fibred category of formal ternary laws $\mathcal{FTL}$.

###### Proof.

The association on objects was described in the paragraph preceding the
statement. The functoriality assertion follows from Formulas (3.5.1.a),
(3.4.1.b) and the symplectic splitting principle [DF21, Par. 2.2.5]. ∎

######  Remark 3.5.3.

To be more explicit about the pseudo-functoriality assertion (see also the
forthcoming proposition), we mean in particular that for any morphism of
scheme $f:T\rightarrow S$ and any s$\mathrm{Sp}$-oriented ring spectrum
$(\mathbb{E},b)$ over $S$, if we let $\mathbb{E}_{T}=f^{*}(\mathbb{E})$ with
the $\mathrm{Sp}$-orientation $b_{T}$ obtained by pullback (from the
$\mathbf{MSp}$-algebra structure), then one gets a canonical morphism of FTL:

$\big{(}\mathbb{E}^{\langle
2\rangle*}(S),F_{t}^{b}\big{)}\rightarrow\big{(}\mathbb{E}^{\langle
2\rangle*}(T),F_{t}^{b_{T}}\big{)}$

which can be decomposed as an extension of scalars along the ring extension
$\mathbb{E}^{\langle 2\rangle*}(T)/\mathbb{E}^{\langle 2\rangle*}(S)$ and then
an isomorphism.

######  Definition 3.5.4.

Let $S$ be a scheme. We say that a formal ternary law $(R,F_{t})$ is
_representable over $S$_ if there exists an $\mathrm{Sp}$-oriented ring
spectrum $(\mathbb{E},b)$ over $S$ such that
$(R,F_{t})\simeq(\mathbb{E}^{**}(S),F_{t}^{b})$. We will say that $(R,F_{t})$
is _weakly representable over $S$_ if there exists an $\mathrm{Sp}$-oriented
ring spectrum $(\mathbb{E},b)$ and a (graded) ring homomorphism
$\varphi:\mathbb{E}^{*,*}(S)\to R$ such that
$(R,F_{t})=(\mathbb{E}^{**},\varphi_{*}F_{t}^{b})$.

Representable (resp. weakly) will mean representable (resp. weakly) over some
scheme.

######  Example 3.5.5.

Recall 3.1.5 for additive and multiplicative FTL.

  1. (1)

The additive FTL with coefficients in $\mathbb{Z}_{\epsilon}$ (resp.
$\mathbb{Q}_{\epsilon}$) is representable by Milnor-Witt motivic cohomology
spectrum $\mathbf{H}_{\mathrm{MW}}\mathbb{Z}$ over $\mathbb{Q}$ (resp.
$\mathbb{Z}$) as proved in [DF21, Th. 3.3.2]) (resp. [DF21, Cor. 3.3.5]). More
generally, over any $\mathbb{Z}[1/6]$-scheme (any scheme) $S$, the
$\mathrm{Sp}$-oriented ring spectrum $\mathbf{H}_{\mathrm{MW}}\mathbb{Z}_{S}$
(resp. $\mathbf{H}_{\mathrm{MW}}\mathbb{Q}_{S}$) admits the additive formal
group law (see _loc. cit._ and use base change 3.5.3).

  2. (2)

The multiplicative FTL with coefficients in $\mathbb{Z}_{\epsilon}^{mul}$ is
representable by the (higher) Grothendieck-Witt (aka hermitian K-theory)
$\mathbf{GW}_{\mathbb{R}}$ ring spectrum over $\mathbb{R}$, according to
[FH23, Th. 6.6] (see also [CDFH21, §3.2]). More generally, using again base
change, over any $\mathbb{Z}[1/2]$-scheme $S$, the $\mathrm{Sp}$-oriented ring
spectrum $\mathbf{GW}_{S}$ admits the multiplicative formal group law.

######  Remark 3.5.6.

There is an obvious obstruction for an FTL to be representable: its ring of
coefficients must be isomorphic to the $\langle 2\rangle*$-graded part of the
ring of coefficients of some (motivic) ring spectrum. In particular, it must
be graded! Note on the other hand that the grading on the ring of coefficients
of a representable FTL is compatible with the notion of degree introduced in
3.1.2.

Apart from the obvious restriction on grading, the problem of determining
which FTLs are representable is a mystery. Note that it is already an open
question to determine which formal group laws are representable by a classical
complex-oriented ring spectrum, or even a $\mathrm{GL}$-oriented (motivic)
ring spectrum.

######  Proposition 3.5.7.

Let $(\mathbb{E},b)$ be an $\mathrm{Sp}$-oriented ring spectrum with
associated formal ternary law $F_{t}^{b}$.

Then there exists a one-to-one correspondence between the strict isomorphisms
of $FTL$ (3.1.7) of the form $(Id,\phi):(\mathbb{E}^{\langle
2\rangle*},F_{t}^{b})\rightarrow(\mathbb{E}^{\langle 2\rangle*},F_{t})$ and
the $\mathrm{Sp}$-orientations $b^{\prime}$ of $\mathbb{E}$.

###### Proof.

Indeed, according to 3.2.10, the $\mathrm{Sp}$-orientation $b^{\prime}$
corresponds to the power series $\phi$ with coefficients in $\mathbb{E}^{**}$
such that $b^{\prime}=\phi(b)$. The fact that $(Id,\phi)$ is an isomorphism
from $F_{t}^{b}$ to $F_{t}^{b^{\prime}}$ follows from the symplectic splitting
principle and Formula (3.2.10.a).

Given now a strict isomorphism of the form
$(Id,\phi):(\mathbb{E}^{**},F_{t}^{b})\rightarrow(\mathbb{E}^{**},F_{t})$, we
get that $b^{\prime}=\phi(b)$ is an $\mathrm{Sp}$-orientation and the
isomorphism $\phi$ gives an identification between $F_{t}$ and $F_{t}^{b}$. ∎

As in the classical case, one gets the following notable corollary.

######  Corollary 3.5.8.

Given an $\mathrm{Sp}$-orientable ring spectrum $\mathbb{E}$, there exists a
unique formal ternary law $F_{t}^{\mathbb{E}}$ up to strict isomorphism
associated with an $\mathrm{Sp}$-orientation on $\mathbb{E}$.

Moreover, given any ring spectrum $\mathbb{F}$, if $\mathbb{F}$ admits an
$\mathbb{E}$-algebra structure, then $\mathbb{F}$ is $\mathrm{Sp}$-orientable
and the FTL associated with any $\mathrm{Sp}$-orientation on $\mathbb{F}$ is
uniquely strictly isomorphism to $F_{t}^{\mathbb{E}}$ (after extension of
scalars along $\mathbb{F}^{\langle 2\rangle*}/\mathbb{E}^{\langle
2\rangle*}$).

We close this section with the following theorem.

######  Theorem 3.5.9.

Any representable formal ternary law $F_{t}$ with coefficients in a
$\mathbb{Q}$-algebra is uniquely strictly isomorphic to the additive FTL.

In other words, there exists a unique strict isomorphism of FTLs

$\mathrm{log}_{F_{t}}:(R,F_{t})\rightarrow\big{(}R,F_{t}^{add}\big{)}$

called the _logarithm_ of $F_{t}$.

###### Proof.

Let $(\mathbb{E},b)$ be a rational $\mathrm{Sp}$-oriented ring spectrum over
some scheme $S$ whose associated FTL is $F_{t}$. The rationalization
$L_{\mathbb{Q}}\mathbb{E}$ still represents $F_{t}$, so we can assume that $E$
is rational. According to [DFJK21, Cor. 6.2], the rational sphere spectrum
over $S$ is isomorphic to the rational Milnor-Witt motivic cohomology spectrum
$\mathbf{H}_{\mathrm{MW}}\mathbb{Q}_{S}$. In particular $\mathbb{E}$ admits a
canonical $\mathbf{H}_{\mathrm{MW}}\mathbb{Q}_{S}$-algebra structure. Then the
theorem follows from 3.5.5(1) and the preceding corollary. ∎

###  3.6. Todd classes and FTL

######  3.6.1.

As in 2.2.1, we consider $\mathrm{Sp}$-oriented ring spectra $(\mathbb{E},b)$
and $(\mathbb{F},b^{\prime})$ and a morphism of ring spectra
$\psi:\mathbb{E}\rightarrow\mathbb{F}$. For any scheme $S$, we consider the
induced morphism of bigraded rings:

$\psi_{\mathrm{H}\mathbb{P}^{\infty}_{S}}:\mathbb{E}^{**}(\mathrm{H}\mathbb{P}^{\infty}_{S})\rightarrow\mathbb{F}^{**}(\mathrm{H}\mathbb{P}^{\infty}_{S}).$

According to the symplectic projective bundle theorem ([DF21, Th. 2.2.3]),
$\psi_{\mathrm{H}\mathbb{P}^{\infty}_{S}}$ corresponds to a morphism of
bigraded rings:

$\mathbb{E}^{**}(S)[[u]]\rightarrow\mathbb{F}^{**}(S)[[t]]$

where $u$ and $t$ are indeterminates of bidegree $(4,2)$. Let us denote by
$\phi(t)$ the power series image of $u$ under this latter map. By definition,
$\phi(t)$ is uniquely determined by the relation in
$\mathbb{F}^{**}(\mathrm{H}\mathbb{P}^{\infty}_{S})$:

(3.6.1.a) $\psi_{\mathrm{H}\mathbb{P}^{\infty}_{S}}(b)=\phi(b^{\prime}).$

Moreover, $\phi(b^{\prime})$ is an $\mathrm{Sp}$-orientation of $\mathbb{F}$
over $S$ and it follows from 3.2.10 that it has the form:

$\phi(t)=t+\sum_{i>1}a_{i}.t^{i}$

where $a_{i}\in\mathbb{F}^{4-4i,2-2i}(S)$. In particular, the following power
series is well defined:

$\tilde{\Theta}(t):=\frac{t}{\Theta(t)}=1-a_{2}.t+(a_{2}^{2}-a_{3}).t^{2}+(-a_{4}+2a_{2}a_{3}-a^{3}_{2}).t^{4}+\cdots$

The next result is a direct analog of the $\mathrm{GL}$-oriented case (see
[Dég18b, Prop. 4.1.2]).

######  Proposition 3.6.2.

The Todd class
$\operatorname{td}_{\psi}:\mathbf{KSp}_{0}\rightarrow\mathbb{F}^{00\times}$
associated with the morphism $\psi$ (2.2.2) is uniquely characterized by the
relation

$\operatorname{td}_{\psi}([\mathcal{U}])=\left.\frac{t}{\Theta(t)}\right|_{t=b_{1}(\mathcal{U})}$

for a rank $2$ symplectic bundle $\mathcal{U}$ over $X$.

The case of an arbitrary symplectic bundle $\mathcal{U}$ of rank $2n$ is
determined by the symplectic splitting principle 3.4.2. One introduces Borel
roots $b_{1}^{(1)}=b_{1}(\mathcal{U}_{1})$, …,
$b_{1}^{(n)}=b_{1}(\mathcal{U}_{n})$ such that the Borel classes of
$\mathcal{U}$ satisfies the relations given by
$b_{t}(\mathcal{U})=\prod_{i}(1+b_{1}^{(i)}.t)$. Then we obtain the expression

$\operatorname{td}_{\psi}([\mathcal{U}])=\prod_{i=1}^{n}\tilde{\Theta}\big{(}b_{1}^{(i)}\big{)}$

which gives a computation of the left hand-side in terms of the Borel classes
$b_{i}(\mathcal{U})$.

##  4\. Todd classes and GRR formulas in the symplectic case

In this section, we compute the Todd classes associated to some morphisms of
ring spectra. We focus on two morphisms, both involving ring spectra that are
$\mathrm{Sp}$-oriented but not $\mathrm{GL}$-oriented.

###  4.1. Adams operations

Let $S=\operatorname{Spec}(\mathbb{Z}[\frac{1}{2}])$, and let $\mathbf{KO}$ be
the ring spectrum representing Hermitian $K$-theory (aka higher Grothendieck-
Witt groups) over $S$. For any integer $n\in\mathbb{N}$, let further
$n^{*}\in\mathbf{KO}^{0,0}(S)$ be defined by

$n^{*}=\begin{cases}n&\text{if $n$ is odd.}\\\
\frac{n}{2}(1-\epsilon)&\text{if $n$ is even.}\end{cases}$

For any $n\geq 1$, we have a morphism of ring spectra (constructed either in
[FH23] or [BH20b])

$\Psi^{n}:\mathbf{KO}\to\mathbf{KO}\left[\frac{1}{n^{*}}\right]$

called the _$n$ -th Adams operation_. These morphisms satisfy the same
properties as the classical Adams operations on $K$-theory. For instance, we
have $\Psi^{m}[\frac{1}{n^{*}}]\circ\Psi^{n}=\Psi^{mn}$ for any $m,n\geq 1$.
Moreover, denoting by $f:\mathbf{KO}\to\mathbf{KGL}$ the forgetful map, we
obtain a commutative diagram for any $n\geq 1$

$\textstyle{\mathbf{KO}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi^{n}}$$\scriptstyle{f}$$\textstyle{\mathbf{KO}[\frac{1}{n^{*}}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathbf{KGL}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi^{n}}$$\textstyle{\mathbf{KGL}[\frac{1}{n}]}$

where the bottom map is the classical Adams operation. Note that $n^{*}$ is
hyperbolic when $n$ is even. The Adams operations on $\mathbf{KO}$ are
obtained via unstable operations

$\psi^{n}:\operatorname{GW}^{2i}\to\operatorname{GW}^{2in}$

for any $i\in\mathbb{Z}$ and any $n\in\mathbb{N}$ ([FH23, §5]). Here
$\operatorname{GW}^{2i}$ represents symplectic $K$-theory if $i$ is odd and
orthogonal $K$-theory if $i$ is even. There is a periodicity element
$\gamma\in\operatorname{GW}^{4}(\mathbb{Z})$ having the property that

$\operatorname{GW}^{2i}(X)\xrightarrow{\cdot\gamma}\operatorname{GW}^{2i+4}(X)$

is an isomorphism for any $i\in\mathbb{Z}$ and any $X$. In that setting, the
unstable operation $\psi^{n}$ can be inductively computed using the formula
([FH23, (5.1.c)])

(4.1.0.a) $\psi^{n}(y)=y\psi^{n-1}(y)-\gamma\psi^{n-2}(y)$

where $y$ is the class in $\operatorname{GW}^{2}(X)$ of a rank $2$ symplectic
bundle.

Even though the spectrum we consider is a priori not defined over
$\mathbb{Z}$, it is still convenient to consider the graded
$\mathbb{Z}_{\epsilon}$-algebra

$\mathbb{Z}_{\epsilon}[\tau,\gamma^{\pm
1}]/\langle\tau^{2}-2h\gamma,(1+\epsilon)\tau\rangle$

in which $\tau$ is of degree $2$ and $\gamma$ of degree $4$. There is an
isomorphism of graded
$\mathbb{Z}_{\epsilon}=\operatorname{GW}^{0}(\mathbb{Z})$-algebras

$\mathbb{Z}_{\epsilon}[\tau,\gamma^{\pm
1}]/\langle\tau^{2}-2h\gamma,(1+\epsilon)\tau\rangle\simeq\bigoplus_{i\in\mathbb{Z}}\operatorname{GW}^{2i}(\mathbb{Z})$

under which $\tau$ is mapped to the class of the hyperbolic plane in
$\operatorname{GW}^{2}(\mathbb{Z})$. It is clear that for any $X$ as above,
$\mathbf{KO}^{*,*}(X)$ is an algebra over this ring.

If $E$ is a symplectic bundle of rank $2$ on a scheme $X$, its (first) Borel
class $b_{1}(E)$ is the class of $[E]-\tau$ in
$\mathbf{KO}^{4,2}(X)=\operatorname{GW}^{2}(X)$. In particular, we have seen
that $X=\mathrm{H}\mathbb{P}^{\infty}_{\mathbb{Z}[\frac{1}{2}]}$ (which can be
seen as an ind-object in the category of smooth schemes) has a universal
symplectic bundle $U$ with associated class $u\in\mathbf{KO}^{4,2}(X)$, and we
set $x:=b_{1}(U)=u-\tau$.

######  Lemma 4.1.1.

For any $n\geq 1$, there exists a unique _monic_ polynomial

$p_{n}(x)\in\mathbb{Z}_{\epsilon}[\tau,\gamma^{\pm
1},x]/\langle\tau^{2}-2h\gamma,(1+\epsilon)\tau\rangle$

of degree $n-1$ such that

$\psi^{n}(x)=xp_{n}(x).$

It satisfies the inductive formula $p_{1}(x)=1$, $p_{2}(x)=x+2\tau$ and

$p_{n}(x)=(x+\tau)p_{n-1}(x)-\gamma p_{n-2}(x)+\psi^{n-1}(\tau).$

###### Proof.

For $u$, we have $\psi^{0}(u)=2$, $\psi^{1}(u)=u$ and for $n\geq 2$

$\psi^{n}(u)=u\psi^{n-1}(u)-\gamma\psi^{n-2}(u)$

in view of formula (4.1.0.a). The same holds for $\tau$ and we we obtain

$\displaystyle\psi^{n}(u)-\psi^{n}(\tau)$ $\displaystyle=$ $\displaystyle
u\psi^{n-1}(u)-\gamma\psi^{n-2}(u)-\tau\psi^{n-1}(\tau)+\gamma\psi^{n-2}(\tau)$
$\displaystyle=$
$\displaystyle(u-\tau)\psi^{n-1}(u)+\tau(\psi^{n-1}(u)-\psi^{n-1}(\tau))-\gamma(\psi^{n-2}(u)-\psi^{n-2}(\tau))$
$\displaystyle=$
$\displaystyle(x+\tau)\psi^{n-1}(x)+x\psi^{n-1}(\tau)-\gamma\psi^{n-2}(x).$

This yields

$\displaystyle xp_{n}(x)$ $\displaystyle=$ $\displaystyle\psi^{n}(x)$
$\displaystyle=$
$\displaystyle(x+\tau)\psi^{n-1}(x)+x\psi^{n-1}(\tau)-\gamma\psi^{n-2}(x)$
$\displaystyle=$ $\displaystyle(x+\tau)xp_{n-1}(x)+x\psi^{n-1}(\tau)-\gamma
xp_{n-2}(x).$

As $x$ is a not a zero divisor, we conclude. ∎

As noted in the proof, the polynomials $p_{n}$ actually compute the quotients
$\psi^{n}(x)/x$. In view of (2.2.2.b), the Todd class of $u$ associated to the
operation $\Psi^{n}$ can actually be computed as

$\operatorname{td}_{\Psi^{n}}^{-1}(u)=\Psi^{n}(x)/x$

since $x=u-\tau=b_{1}(u)=e(U,\mathbf{KO})$ (in principle, we would have to
choose a stable orientation of $U$, but here we can take the “canonical one”
since $U$ is symplectic). Now, the stable operations $\Psi^{n}$ are obtained
out of the unstable ones by inverting a convenient element $\omega(n)$ defined
as follows ([FH23, Lemma 5.3.3]):

$\omega(n)=\begin{cases}0&\text{ if $n=0$.}\\\ 1&\text{ if $n=1$.}\\\
\tau\omega(n-1)-\gamma\omega(n-2)+\psi^{n-1}(\tau)&\text{ if $n\geq
2$.}\end{cases}$

One can then explicitly compute $\omega(n)$ as follows ([FH23, Proposition
5.3.4]):

$\omega(n)=\begin{cases}\displaystyle{n\Big{(}\frac{n-1}{2}(1-\epsilon)+\langle-1\rangle^{\frac{n-1}{2}}\Big{)}\gamma^{\frac{n-1}{2}}}&\text{if
$n$ is odd},\\\
\displaystyle{\frac{n^{2}}{2}\tau\gamma^{\frac{n-2}{2}}}&\text{if $n$ is
even}\end{cases}$

and set

(4.1.1.a) $\Psi^{n}=\frac{1}{\omega(n)}\psi^{n}$

for any $n\in\mathbb{N}$ (one observes that $\omega(n)$ is indeed invertible
whenever $n^{*}$ is).

######  Lemma 4.1.2.

For any $n\geq 2$, we have and $p_{n}(x)=x^{n-1}+n\tau x^{n-2}+r_{n-2}(x)$ for
some polynomial $r_{n-2}(x)$ of degree $\leq n-3$. Further, we have
$p_{n}(0)=r_{n-2}(0)=\omega(n)$.

###### Proof.

For the first assertion, we already know that $p_{n}$ is monic. To compute the
coefficient of $x^{n-2}$, it suffices work by induction using

$p_{n}(x)=(x+\tau)p_{n-1}(x)-\gamma p_{n-2}(x)+\psi^{n-1}(\tau)$

and $p_{2}=x+2\tau$. For the second statement, we observe that $p_{n}(0)$ and
$\omega(n)$ satisfy the same inductive definition. It suffices then to prove
that $\omega(2)=p_{2}(0)$ to conclude. The left-hand side is
$\tau+\psi^{1}(\tau)=2\tau$, which is precisely $p_{2}(0)$. ∎

######  Remark 4.1.3.

We were not able to give a closed formula for $p_{n}$. For small values of
$n$, they can be inductively computed, but it would be interesting to be
actually able to write them for all integers $n$.

As $p_{n}=\psi^{n}(x)/x$ and $\Psi^{n}=\frac{1}{\omega(n)}\psi^{n}$, this
immediately yields the following proposition.

######  Proposition 4.1.4.

For any $n\geq 1$, there exists unique polynomials

$q_{n}(x)\in\mathbb{Z}_{\epsilon}[\tau,\gamma^{\pm
1},\omega(n)^{-1},x]/\langle\tau^{2}-2h\gamma,(1+\epsilon)\tau\rangle$

of degree $n-1$ such that $q_{n}(0)=1$,
$q_{n}(x)=\frac{1}{\omega(n)}(x^{n-1}+n\tau x^{n-2})+r_{n-2}(x)$ with
$r_{n-2}$ of degree $\leq n-3$. They satisfy

$\operatorname{td}_{\Psi^{n}}^{-1}(u)=q_{n}(u-\tau).$

This proposition allows to compute the Todd class
$\operatorname{td}_{\Psi^{n}}$ associated to any symplectic bundle of rank
$2$, using the splitting principle and the fact that $U$ is the universal rank
$2$ bundle.

###  4.2. The Borel character

We begin by recalling that the Borel character is a morphism of ring spectra

$\mathrm{bo}_{t}:\mathbf{KO}\xrightarrow{\
(\mathrm{bo}_{2n})_{n\in\mathbb{Z}}\ }\bigoplus_{n\text{
even}}\mathbf{H}_{\mathrm{MW}}\mathbb{Q}(2n)[4n]\oplus\bigoplus_{n\text{
odd}}\mathbf{H}_{\mathrm{M}}\mathbb{Q}(2n)[4n].$

Let $p:\mathbf{H}_{\mathrm{MW}}\mathbb{Q}\to\mathbf{H}_{\mathrm{M}}\mathbb{Q}$
be the projection from MW-motivic cohomology to ordinary motivic cohomology
and let $f:\mathbf{KO}\to\mathbf{KGL}$ be the forgetful map considered in the
previous section (both are morphism of ring spectra over $S$). We will
consider $\mathbf{H}_{\mathrm{M}}\mathbb{Q}$ (resp. $\mathbf{KGL}$) as a
$\mathbf{H}_{\mathrm{MW}}\mathbb{Q}$-algebra using the morphism $p$ (resp. a
$\mathbf{KO}$-algebra using the morphism $f$). More explicitly, if $X$ is a
(ind-)smooth $S$-scheme, we will set $\alpha\cdot\beta:=p(\alpha)\cdot\beta$
for any $\alpha\in\mathbf{H}_{\mathrm{MW}}\mathbb{Q}(i)[j](X)$ and any
$\beta\in\mathbf{H}_{\mathrm{M}}\mathbb{Q}(a)[b](X)$.

We have a commutative diagram of spectra

(4.2.0.a)
$\textstyle{\mathbf{KO}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{bo}_{t}}$$\scriptstyle{f}$$\textstyle{\bigoplus_{n\text{
even}}\mathbf{H}_{\mathrm{MW}}\mathbb{Q}(2n)[4n]\oplus\bigoplus_{n\text{
odd}}\mathbf{H}_{\mathrm{M}}\mathbb{Q}(2n)[4n]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p\oplus\mathrm{Id}}$$\textstyle{\mathbf{KGL}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ch}_{t}}$$\textstyle{\bigoplus_{n\text{
even}}\mathbf{H}_{\mathrm{M}}\mathbb{Q}(2n)[4n]\oplus\bigoplus_{n\text{
odd}}\mathbf{H}_{\mathrm{M}}\mathbb{Q}(2n)[4n]}$

where $\mathrm{ch}_{t}$ is the usual Chern character.

Let further $X$ be a (ind-)smooth $S$-scheme and $E$ be a symplectic bundle of
rank $2d$ on $X$. We may see $E$ as the class of a morphism
$\Sigma_{T}^{\infty}X\to\mathbf{KO}_{S}(2)[4]$ whose image under the Borel
character is by definition of the form

(4.2.0.b)
$\mathrm{bo}_{t}(E)=2d+\tilde{\chi}_{2}(E)+\frac{1}{4!}\chi_{4}(f(E))+\frac{1}{\psi_{6}!}\tilde{\chi}_{6}(E)+\ldots$

in $\bigoplus_{n\text{
odd}}\mathbf{H}_{\mathrm{MW}}\mathbb{Q}(2n)[4n]\oplus\bigoplus_{n\text{
even}}\mathbf{H}_{\mathrm{M}}\mathbb{Q}(2n)[4n]$ where

$\tilde{\chi}_{2n}:\operatorname{GW}^{2}\to\mathbf{H}_{\mathrm{MW}}(2n)[4n]$

is defined inductively by setting $\tilde{\chi}_{2}=b_{1}$ (the first Borel
class in $\mathbf{H}_{\mathrm{MW}}(2)[4]$) and using

(4.2.0.c)
$\tilde{\chi}_{2n}-b_{1}\tilde{\chi}_{2n-2}+b_{2}\tilde{\chi}_{2n-4}+\ldots+(-1)^{n-1}b_{n-1}\tilde{\chi}_{2}+(-1)^{n}nb_{n}=0.$

The operation

$\chi_{2n}\circ f:\operatorname{GW}^{2}\to\mathbf{H}_{\mathrm{M}}(2n)[4n]$

is defined in a similar fashion, using the forgetful functor
$f:\operatorname{GW}^{2}\to\mathrm{K}_{0}$ and the operation
$\chi_{2n}:\mathrm{K}_{0}\to\mathbf{H}_{\mathrm{M}}(2n)[4n]$ worked out by
Riou ([Rio10, Remark 6.2.2.3]). The symmetric bilinear forms $\psi_{2n}$ are
defined by

$\psi_{2n}=\begin{cases}1&\text{ if $n=0,1$}\\\ n(2n-1)(2n-2)(2n-3)h&\text{if
$n\geq 2$ is even,}\\\ 2n(2n-2)\big{(}(2n^{2}-4n+1)h-\epsilon\big{)}&\text{if
$n\geq 3$ is odd,}\\\ \end{cases}$

and we set

$\psi_{2n}!=\psi_{2}\cdot\psi_{6}\cdot\ldots\cdot\psi_{2n}.$

One checks that their inverses are indeed well defined in
$\mathbf{H}_{\mathrm{MW}}\mathbb{Q}(0)[0](S)$, at least if
$\frac{1}{2}\in\mathcal{O}(S)$ ([DF21, Theorem 4.3.2]).

If $U$ is the universal bundle on $\mathrm{H}\mathbb{P}^{\infty}_{S}$ and $u$
is its class in $\operatorname{GW}^{2}(\mathrm{H}\mathbb{P}^{\infty}_{S})$, we
have $\tilde{\chi}_{2+4n}(E)=b_{1}(U)^{2n+1}$ according to [DF21, §4.1.7],
while $\chi_{4n}(f(U))=2c_{2}(U)^{2n}$ [DF21, Lemma 4.3.4]. If
$x:=b_{1}(u)=b_{1}(u-\tau)$, then its image $p(x)$ under the morphism
$p:\mathbf{H}_{\mathrm{MW}}\mathbb{Q}(2)[4]\to\mathbf{H}_{\mathrm{M}}\mathbb{Q}(2)[4]$
is $-c_{2}(U)$ and the $\mathbf{H}_{\mathrm{MW}}$-algebra structure on
$\mathbf{H}_{\mathrm{M}}$ reads in this case as

$p(x)^{2n}=x\cdot p(x)^{2n-1}$

for any $n\geq 1$. As $b_{1}(\tau)$ and $c_{2}(\tau)$ are both trivial, we
obtain an expression (slightly reordering terms) of $\mathrm{bo}_{t}(u-\tau)$
of the form

$\left(\sum_{n\geq 1}\frac{2}{(4n)!}p(x)^{2n}\right)+\left(x+\sum_{n\geq
1}\frac{1}{\psi_{2+4n}!}x^{2n+1}\right)=x\cdot\left(\sum_{n\geq
1}\frac{2}{(4n)!}p(x)^{2n-1}+1+\sum_{n\geq
1}\frac{1}{\psi_{2+4n}!}t^{2n}\right).$

As $x=u-\tau$ is the first Borel class of $U$ in $\mathrm{KGL}(2)[4]$, we then
see that

$\operatorname{td}_{\mathrm{bo}_{t}}^{-1}(u)=\left(1+\sum_{n\geq
1}\frac{2}{4n!}p(x)^{2n-1}+\sum_{n\geq 1}\frac{1}{\psi_{2+4n}!}x^{2n}\right).$

We now make use of the fact that rationally the ring spectrum representing MW-
motivic cohomology decomposes as a product (this could even by taken as a
definition [DFJK21, §5, §6])

$\mathbf{H}_{\mathrm{MW}}\mathbb{Q}\simeq\mathbf{H}_{\mathrm{M}}\mathbb{Q}\times\mathbf{H}_{\mathrm{W}}\mathbb{Q}$

where $\mathbf{H}_{\mathrm{W}}\mathbb{Q}$ is the spectrum representing the
cohomology of the rational Witt sheaf. Here the composite
$\mathbf{H}_{\mathrm{MW}}\mathbb{Q}\simeq\mathbf{H}_{\mathrm{M}}\mathbb{Q}\times\mathbf{H}_{\mathrm{W}}\mathbb{Q}\to\mathbf{H}_{\mathrm{M}}\mathbb{Q}$
is just the morphism we denoted by $p$, and we denote by
$p^{\prime}:\mathbf{H}_{\mathrm{MW}}\mathbb{Q}\simeq\mathbf{H}_{\mathrm{M}}\mathbb{Q}\times\mathbf{H}_{\mathrm{W}}\mathbb{Q}\to\mathbf{H}_{\mathrm{W}}\mathbb{Q}$
the composite of the decomposition and the second projection. This equivalence
yields in particular in degree $(0,0)$ an isomorphism

$\mathbf{H}^{0,0}_{\mathrm{MW}}\mathbb{Q}(S)\simeq\mathbf{K}^{\mathrm{MW}}_{0}(S)\simeq\underline{\mathbb{Q}}(S)\times\mathbf{W}(S)$

where $\underline{\mathbb{Q}}$ is the sheaf associated to the constant
presheaf $\mathbb{Q}$. In view of this decomposition and the commutative
diagram (4.2.0.a), we obtain

$p(\operatorname{td}_{\mathrm{bo}_{t}}^{-1}(u))=\operatorname{td}_{\mathrm{ch}_{t}}^{-1}(f(u))$

and we see that it suffices to compute the latter and
$p^{\prime}(\operatorname{td}_{\mathrm{bo}_{t}}(u))$ to obtain
$\operatorname{td}_{\mathrm{bo}_{t}}(u)$.

######  Proposition 4.2.1.

If $U$ is a rank $2$ symplectic bundle on a smooth $S$-scheme $X$ and
$u\in\operatorname{GW}^{2}(X)$ is its associated class, we have

$\operatorname{td}_{\mathrm{ch}_{t}}^{-1}(f(u))=2\sum_{i\geq
1}\frac{(-c_{2}(U))^{i-1}}{(2i)!}.$

###### Proof.

By universality, it suffices to consider the situation where
$X=\mathrm{H}\mathbb{P}^{\infty}_{S}$ and $U$ is the universal rank two
symplectic bundle. We may consider the projective bundle
$\pi:Y:={\mathbb{P}}(U)\to X$ and replace $X$ by $Y$ in view of the splitting
principle, as both $K$-theory and motivic cohomology are
$\mathrm{GL}$-oriented. Over $Y$, $U$ splits as an extension of a line bundle
$L$ with its dual, and we have

$\mathrm{ch}_{t}(f(u))=e^{c_{1}(L)}+e^{c_{1}(L^{\vee})}=e^{c_{1}(L)}+e^{-c_{1}(L)}=2\cosh(c_{1}(L)).$

This yields $\mathrm{ch}_{t}(f(u-\tau))=2\cosh(c_{1}(L))-2$. Using
$c_{1}(L)^{2}=-c_{1}(L)c_{1}(L^{\vee})=-c_{2}(U)$, the expression reads as

$\mathrm{ch}_{t}(f(u-\tau))=2\sum_{i\geq 1}\frac{(-c_{2}(U))^{i}}{(2i)!}$

As the first Borel class of $U$ in $\mathrm{CH}^{2}(X)$ is $-c_{2}(U)$, we
finally obtain

$\operatorname{td}_{\mathrm{ch}_{t}}^{-1}(f(u))=2\sum_{i\geq
1}\frac{(-c_{2}(U))^{i-1}}{(2i)!}.$

∎

We now start the computation of
$p^{\prime}(\operatorname{td}_{\mathrm{bo}_{t}}(u))$ which is obtained by
omitting the terms of the form $\chi_{2n}$ (including $2d$) in the expression
(4.2.0.b), replacing the operations $\tilde{\chi}_{2+4n}$ by the operations
$\tilde{\chi}_{2+4n}^{\mathrm{W}}:=p^{\prime}\circ\tilde{\chi}_{2+4n}$ and
taking the images of the symmetric bilinear forms $\psi_{2n}$ in the group
$\mathbf{W}(S)\otimes\mathbb{Q}$. Since $\epsilon=1$ and $h=0$ in
$\mathbf{W}(S)\otimes\mathbb{Q}$, we obtain for $n\geq 3$ odd

$\psi_{2n}=-2n(2n-2),$

so that $\psi_{4n+2}=-4n(4n+2)=-8n(2n+1)=-4(2n+1)2n$ for any $n\geq 1$ and
$\psi_{2}=1$. The following lemma is easily proved by induction.

######  Lemma 4.2.2.

For $n\geq 0$, we have $\psi_{4n+2}!=(-1)^{n}2^{2n}(2n+1)!$.

We then obtain an expression of the form

$p^{\prime}(\operatorname{td}_{\mathrm{bo}_{t}}^{-1}(u))=x+\sum_{n\geq
1}\frac{1}{\psi_{2+4n}!}x^{2n+1}=x+\sum_{n\geq
1}\frac{(-1)^{n}}{2^{2n}(2n+1)!}x^{2n+1}$

where $x=e(U,\mathbf{H}_{\mathrm{W}}\mathbb{Q})$ is the Euler class of $U$
(which coincides with the first Borel class). Dividing by $x$, we finally
obtain the following result.

######  Proposition 4.2.3.

The Todd class associated to the morphism of ring spectra

$\mathbf{KO}\xrightarrow{p^{\prime}\circ\mathrm{bo}_{t}}\bigoplus_{n\text{
even}}\mathbf{H}_{\mathrm{W}}\mathbb{Q}(2n)[4n]$

is of the form

$\operatorname{td}_{p^{\prime}\circ\mathrm{bo}_{t}}^{-1}(u)=1+\sum_{n\geq
1}\frac{(-1)^{n}}{(2n+1)!}\left(\frac{x}{2}\right)^{2n}$

where $x=e(U,\mathbf{H}_{\mathrm{W}}\mathbb{Q})$ is the Euler class of the
universal rank $2$ bundle $U$ on $\mathrm{H}\mathbb{P}^{\infty}$.

######  Remark 4.2.4.

The power series in the expression of the inverse of the Todd class of $u$ is
in fact the power series $\frac{\sin(\frac{x}{2})}{\frac{x}{2}}$.

If now $(E,\varphi)$ is a symplectic bundle of rank $2n$ over a smooth scheme
$X$, we may now use the splitting principle to suppose (up to passing to a
smooth scheme $Y/X$) that $E$ splits as a sum of rank $2$ symplectic bundles,
the so-called Borel roots $\alpha_{1},\ldots,\alpha_{n}$ of $E$. The Todd
class of $E$ is then of the form

$\operatorname{td}_{p^{\prime}\circ\mathrm{bo}_{t}}(E)=\prod_{i=1}^{n}\frac{\frac{\alpha_{i}}{2}}{\sin(\frac{\alpha_{i}}{2})}$

where we have somewhat abusively denoted by $\alpha_{i}$ the first Borel class
of the respective Borel roots. For the Todd class
$\operatorname{td}_{\mathrm{ch}_{t}}$, we refer to [Ful98, Example 3.2.4].

###  4.3. The quadratic Hirzebruch-Riemann-Roch theorem

######  4.3.1.

In this section, we illustrate Theorem 2.2.4 in the case of the Borel
character discussed in the previous section.

We consider a smooth proper scheme $X$ over a field $k$ of even dimension
$d=2n$, with projection $p\colon X\to\operatorname{Spec}(k)$ and canonical
bundle $\omega_{X/k}=\det(\Omega_{X/k})$. We let $f\colon L^{\otimes
2}\to\omega_{X/k}$ be an orientation of $X/k$ (see 2.1.2). Using this
isomorphism, we obtain an isomorphism (corresponding to the fact
$\operatorname{GW}$ is $\mathrm{SL}^{c}$-oriented)

$\operatorname{GW}^{2}(X)\simeq\operatorname{GW}^{2}(X,L^{\otimes
2})\simeq\operatorname{GW}^{2}(X,\omega_{X/k})$

mapping the (isometry class of the) symplectic bundle $(V,\varphi)$ to
$(V\otimes L,f\circ(\varphi\otimes\mathrm{Id}))$. The same formula applies for
$\operatorname{GW}^{0}(X)$.

The isomorphism $f\circ(\varphi\otimes\mathrm{Id})$ induces for any
$i\in\mathbb{N}$ an isomorphism

$[f\circ(\varphi\otimes\mathrm{Id})]_{i}\colon\operatorname{H}^{i}(X,V\otimes
L)\to\operatorname{H}^{i}(X,(V\otimes L)^{\vee}\otimes\omega_{X/k}).$

Serre duality yields a canonical isomorphism
$\mathrm{can}:\operatorname{H}^{i}(X,(V\otimes
L)^{\vee}\otimes\omega_{X/k})\simeq\operatorname{H}^{2n-i}(X,V\otimes L)^{*}$
and we finally obtain isomorphisms

$\varphi_{i}\colon\operatorname{H}^{i}(X,V\otimes
L)\to\operatorname{H}^{2n-i}(X,V\otimes L)^{*}.$

In particular, for $i=n$, the isomorphism

$\varphi_{n}\colon\operatorname{H}^{n}(X,V\otimes
L)\to\operatorname{H}^{n}(X,V\otimes L)^{*}$

yields either a symmetric or symplectic form on the vector space
$\operatorname{H}^{n}(X,V\otimes L)$ in the following way: If
$[V,\varphi]\in\operatorname{GW}^{2m}(X)$, then
$[\operatorname{H}^{n}(X,V\otimes
L),\varphi_{n}]\in\operatorname{GW}^{2m-2n}(k)$. For instance, if $\varphi$ is
symplectic and $n$ is odd then $\varphi_{n}$ is symmetric. The following
proposition can be seen as a computation of the tranfer map in Grothendieck-
Witt theory.

######  Proposition 4.3.2.

Let $[V,\varphi]\in\operatorname{GW}^{2m}(X)$. Then

$p_{*}([V,\varphi])=(-1)^{m+n}[\operatorname{H}^{n}(X,V\otimes
L),\varphi_{n}]+(\sum_{i=0}^{n-1}(-1)^{(m+i)}\mathrm{dim}_{k}\operatorname{H}^{i}(X,V\otimes
L))\cdot(1-\epsilon)$

in $\operatorname{GW}^{2m-2n}(k)$.

######  Remark 4.3.3.

The bundle $V$ should be seen as a complex concentrated in degree $m$ in the
statement. Moreover, one observes that if $m-n$ is odd then $\varphi_{n}$ is
symplectic, and then hyperbolic. Thus, the interesting information arises only
when $m$ and $n$ have the same parity, and is then encoded by $\varphi_{n}$.

In view of this computation, we may set the following definition.

######  Definition 4.3.4.

Let $X$ be a smooth proper scheme of even dimension $d=2n$ and let
$(V,\varphi)$ be either a symplectic bundle if $d=2\pmod{4}$ or a symmetric
bundle if $d=0\pmod{4}$. We define the quadratic Euler characteristic of
$(V,\varphi)$ under the orientation $f:L^{\otimes 2}\simeq\omega_{X/k}$ as the
following expression

$[\operatorname{H}^{n}(X,V\otimes
L),\varphi_{n}]+\sum_{i=0}^{n-1}(-1)^{(m+i)}\mathrm{dim}_{k}\operatorname{H}^{i}(X,V\otimes
L)\cdot(1-\epsilon)$

where $m=1$ if $\varphi$ is symplectic and $m=0$ else. We denote by
$\tilde{\chi}(X,f;V,\varphi)$ the class of this symmetric bilinear form.

The Grothendieck-Riemann-Roch formula for the Borel character
$\operatorname{bo}_{t}$ then reads as follows.

######  Theorem 4.3.5.

Let $X$ be a smooth proper scheme of even dimension $d$ such that its tangent
bundle $T_{X/k}$ admits a stable $\mathrm{Sp}$-orientation $\tilde{\tau}_{X}$.
We let $f$ be the orientation of $X/k$ associated with $\tilde{\tau}_{X}$.

Let $(V,\varphi)$ be either a symplectic bundle if $d=2\pmod{4}$ or a
symmetric bundle if $d=0\pmod{4}$. Then the following equality holds in
$\operatorname{GW}(k)$

$\tilde{\chi}(X,f;V,\varphi)=\tilde{\operatorname{deg}}_{\tilde{\tau}_{X}}\left(\operatorname{td}_{\operatorname{bo}_{t}}(\tilde{\tau}_{X}).\operatorname{bo}_{t}(V,\varphi)\right)$

where
$\tilde{\operatorname{deg}}_{\tilde{\tau}_{X}}\colon\widetilde{\operatorname{CH}}^{d}(X)\to\operatorname{GW}(k)$
is the quadratic degree map (pushforwar) associated with the symplectic
orientation $\tilde{\tau}_{X}$ of $X/K$.

######  Example 4.3.6.

Suppose that $d=2$ and that $X$ is a K3 surface. Then, a symplectic
orientation $\tilde{\tau}_{X}$ of $\Omega_{X/k}$ (or $T_{X/k}$) is just
obtained by choosing an isomorphism $f:\mathcal{O}_{X}\to\omega_{X/k}$.
Indeed, one just considers the composite
$\Omega_{X/k}\otimes\Omega_{X/k}\to\omega_{X/k}\xrightarrow{f^{-1}}\mathcal{O}_{X}$.
The Todd class can be easily computed using 4.2.1 and 4.2.3. We have

$\operatorname{td}_{\mathrm{ch}_{t}}^{-1}(\Omega_{X/k})=1-\frac{c_{2}(\Omega_{X/k})}{12},\operatorname{td}_{p^{\prime}\circ\mathrm{ch}_{t}}^{-1}(\Omega_{X/k})=1$

and it follows easily from the fact that $X$ is a surface that
$\operatorname{td}_{\operatorname{bo}_{t}}(\Omega_{X/k})=1+\frac{c_{2}(\Omega_{X/k})}{24}(1-\epsilon)$.
Note further that there is no need to tensor with $\mathbb{Q}$ as the Borel
character is defined integrally in case of surfaces. If $(V,\varphi)$ is a
symplectic bundle of rank $2r$ over $X$, we finally obtain

$\displaystyle\tilde{\chi}(X,f;V,\varphi)$ $\displaystyle=$
$\displaystyle\tilde{\operatorname{deg}}_{\tilde{\tau}_{X}}\left((2r+e(V,\varphi))\cdot(1+\frac{c_{2}(\Omega_{X/k})}{24}(1-\epsilon))\right)$
$\displaystyle=$
$\displaystyle\tilde{\operatorname{deg}}_{\tilde{\tau}_{X}}\left(r\cdot\frac{c_{2}(\Omega_{X/k})}{12}(1-\epsilon)+e(V,\varphi)\right)$
$\displaystyle=$
$\displaystyle\mathrm{deg}\left(r\cdot\frac{c_{2}(\Omega_{X/k})}{12}\right)\cdot(1-\epsilon)+\tilde{\operatorname{deg}}_{\tilde{\tau}_{X}}(e(V,\varphi)).$

Using the fact that $\mathrm{deg}(c_{2}(\Omega_{X/k}))=24$ (e.g. [Huy16]), we
finally obtain the formula:

$\tilde{\chi}(X,f;V,\varphi)=2r(1-\epsilon)+\tilde{\operatorname{deg}}_{\tilde{\tau}_{X}}(e(V,\varphi)).$

## References

  * [AGV73] M. Artin, A. Grothendieck, and J.-L. Verdier, _Théorie des topos et cohomologie étale des schémas_ , Lecture Notes in Mathematics, vol. 269, 270, 305, Springer-Verlag, 1972–1973, Séminaire de Géométrie Algébrique du Bois–Marie 1963–64 (SGA 4).
  * [AHW18] A. Asok, M. Hoyois, and M. Wendt, _Affine representability results in $\mathbb{A}^{1}$-homotopy theory, II: Principal bundles and homogeneous spaces_, Geom. Topol. 22 (2018), no. 2, 1181–1225.
  * [BB20] T. Barthel and A. Beaudry, _Chromatic structures in stable homotopy theory_ , Handbook of homotopy theory, CRC Press/Chapman Hall Handb. Math. Ser., CRC Press, Boca Raton, FL, [2020] ©2020, pp. 163–220.
  * [BD17] M. Bondarko and F. Déglise, _Dimensional homotopy t-structures in motivic homotopy theory_ , Adv. Math. 311 (2017), 91–189.
  * [Bea83] A. Beauville, _Variétés Kähleriennes dont la première classe de Chern est nulle_ , J. Differ. Geom. 18 (1983), 755–782.
  * [BH20a] T. Bachmann and M. J. Hopkins, _$\eta$ -periodic motivic stable homotopy theory over fields_, arXiv: 2005.06778, 2020.
  * [BH20b] by same author, _$\eta$ -periodic motivic stable homotopy theory over fields_, arXiv:2005.06778, 2020.
  * [BO74] S. Bloch and A. Ogus, _Gersten’s conjecture and the homology of schemes_ , Ann. Sci. École Norm. Sup. (4) 7 (1974).
  * [CD12] D.-C. Cisinski and F. Déglise, _Mixed weil cohomologies_ , Adv. in Math. 230 (2012), no. 1, 55–130.
  * [CD19] D.-C. Cisinski and F. Déglise, _Triangulated categories of mixed motives_ , Springer Monographs in Mathematics, Springer, Cham, 2019.
  * [CDFH21] D. Coulette, F. Déglise, J. Fasel, and J. Hornbostel, _Formal ternary laws and Buchstaber’s 2-groups_ , arXiv: 2112.03646, 2021.
  * [CHN24] B. Calmès, Y. Harpaz, and D. Nardin, _A motivic spectrum representing Hermitian $K$-theory_, arXiv:2402.15136, 2024.
  * [Dég18a] F. Déglise, _Bivariant theories in motivic stable homotopy_ , Doc. Math. 23 (2018), 997–1076.
  * [Dég18b] by same author, _Orientation theory in arithmetic geometry_ , $K$-Theory—Proceedings of the International Colloquium, Mumbai, 2016, Hindustan Book Agency, New Delhi, 2018, pp. 239–347.
  * [Dég23] by same author, _Notes on Milnor-Witt K-theory_ , arXiv: 2305.18609v1, 2023.
  * [Del87] P. Deligne, _Le déterminant de la cohomologie_ , Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 93–177.
  * [DF21] F. Déglise and J. Fasel, _The Borel character_ , Journal of the IMJ (2021), 1–51.
  * [DFJK21] F. Déglise, J. Fasel, F. Jin, and A. A. Khan, _On the rational motivic homotopy category_ , Journal de l’École polytechnique — Mathématiques 8 (2021), 533–583 (en).
  * [DJK21] F. Déglise, F. Jin, and A. Khan, _Fundamental classes in motivic homotopy theory_ , J. Eur. Math. Soc. (JEMS) 23 (2021), no. 12, 3935–3993.
  * [Fas08] J. Fasel, _Groupes de Chow-Witt_ , Mém. Soc. Math. Fr. (N.S.) 113 (2008), viii+197. MR MR2542148
  * [Fel20] N. Feld, _Milnor-Witt cycle modules_ , J. Pure Appl. Algebra 224 (2020), no. 7, 106298, 44.
  * [Fel21] by same author, _Morel homotopy modules and Milnor-Witt cycle modules_ , Doc. Math. 26 (2021), 617–659.
  * [FH23] J. Fasel and O. Haution, _The stable Adams operations on Hermitian K-theory_ , To appear in Geom. Topol., 2023.
  * [FM81] W. Fulton and R. MacPherson, _Categorical framework for the study of singular spaces_ , Mem. Amer. Math. Soc. 31 (1981), no. 243, vi+165.
  * [Ful98] William Fulton, _Intersection theory_ , second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR MR1644323 (99d:14003)
  * [Gra66] J. W. Gray, _Fibred and cofibred categories_ , Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer-Verlag New York, Inc., New York, 1966, pp. 21–83.
  * [Har66] R. Hartshorne, _Residues and duality_ , Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64, With an appendix by P. Deligne.
  * [Huy16] D. Huybrechts, _Lectures on K3 surfaces_ , Cambridge Studies in Advanced Mathematics, vol. 158, Cambridge University Press, 2016.
  * [Kha21] A. A. Khan, _Voevodsky’s criterion for constructible categories of coefficients_ , https://www.preschema.com/papers/six.pdf, 2021.
  * [Kne77] M. Knebusch, _Symmetric bilinear forms over algebraic varieties_ , Conference on Quadratic Forms—1976 (Proc. Conf., Queen’s Univ., Kingston, Ont., 1976), Queen’s Papers in Pure and Appl. Math., No. 46, Queen’s Univ., Kingston, ON, 1977, pp. 103–283.
  * [Lev09] M. Levine, _Comparison of cobordism theories_ , J. Algebra 322 (2009), no. 9, 3291–3317.
  * [LM07] M. Levine and F. Morel, _Algebraic cobordism_ , Springer Monographs in Mathematics, Springer, Berlin, 2007.
  * [LZ17] Y. Liu and W. Zheng, _Enhanced six operations and base change theorem for higher artin stacks_ , arXiv: 1211.5948, 2017.
  * [Mor04] F. Morel, _An introduction to $\mathbb{A}^{1}$-homotopy theory_, Contemporary developments in algebraic $\mathrm{K}$-theory, ICTP Lect. Notes, XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, pp. 357–441.
  * [MV99] F. Morel and V. Voevodsky, _$\mathbb{A}^{1}$ -homotopy theory of schemes_, Publ. Math., Inst. Hautes Étud. Sci. 90 (1999), 45–143.
  * [PPR09] I. Panin, K. Pimenov, and O. Röndigs, _On Voevodsky’s algebraic $K$-theory spectrum_, Algebraic topology. The Abel symposium 2007. Proceedings of the fourth Abel symposium, Oslo, Norway, August 5–10, 2007, Berlin: Springer, 2009, pp. 279–330.
  * [PW18a] I. Panin and C. Walter, _On the algebraic cobordism spectra $\mathbf{MSL}$ and $\mathbf{MSp}$_, arXiv: 1011.0651, 2018.
  * [PW18b] by same author, _On the algebraic cobordism spectra $\mathbf{MSL}$ and $\mathbf{MSp}$_, arXiv: 1011.0651, 2018.
  * [PW22] by same author, _Quaternionic Grassmannians and Borel classes in algebraic geometry_ , St. Petersbg. Math. J. 33 (2022), no. 1, 136–193.
  * [Qui69] D. Quillen, _On the formal group laws of unoriented and complex cobordism theory_ , Bull. Amer. Math. Soc. 75 (1969), 1293–1298.
  * [Rav86] D. C. Ravenel, _Complex cobordism and stable homotopy groups of spheres_ , Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986.
  * [Rio10] J. Riou, _Algebraic $K$-theory, ${\bf A}^{1}$-homotopy and Riemann-Roch theorems_, J. Topol. 3 (2010), no. 2, 229–264.
  * [Sch10] M. Schlichting, _Hermitian $K$-theory of exact categories_, J. K-Theory 5 (2010), no. 1, 105–165. MR MR2114168 (2005m:19004)
  * [Vis05] A. Vistoli, _Grothendieck topologies, fibered categories and descent theory_ , Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 1–104.
  * [Voe] V. Voevodsky, _The Milnor Conjecture_ , Preprint, December 20, 1996, K-theory Preprint Archives, http://www.math.uiuc.edu/K-theory/0170/.