A^{*}$. In particular, it is known that

$\Omega^{*}\otimes_{\mathbb{L}^{*}}\mathbb{Z}\simeq{\rm
CH}^{*},\quad\Omega^{*}\otimes_{\mathbb{L}^{*}}\mathbb{Z}[\beta,\beta^{-1}]\simeq
K^{0}[\beta,\beta^{-1}],$

where $\mathbb{L}$ denotes the Lazard ring, which is isomorphic to the
absolute ring, $\Omega^{*}(k)$ for ${\rm Spec}(k)$ [LM07].

A remarkable approach to re-interpreting $\Omega_{*}$ geometrically was later
found by Levine-Pandharipande [LP09]. Briefly, let us review the geometric
construction.

Let $Y\in{\rm\bf Sch}_{k}$. Two morphisms $f:X\to Y$ and
$f^{\prime}:X^{\prime}\to Y$ are said to be isomorphic if there exists an
isomorphism $\varphi:X\to X^{\prime}$ so that $f=f^{\prime}\circ\varphi$. Let
$\mathcal{M}(Y)$ denote the set of isomorphism classes over $Y$ of proper
morphisms $f:X\to Y$ with $X\in{\rm\bf Sm}_{k}$ which is graded by the
dimension of $X$. The set $\mathcal{M}(Y)$ is a monoid where the addition of
isomorphism classes of $f:X_{1}\to Y$ and $g:X_{2}\to Y$ is defined by the
disjoint union

$f+g:X_{1}\sqcup X_{2}\to Y.$

Let $\mathcal{M}_{*}^{+}(Y)$ be the graded group completion of
$\mathcal{M}_{*}(Y)$.

Now, consider a family of morphisms to $Y$; let

$F:M\to Y\times\mathbb{P}^{1}$

be a proper morphism with $M\in{\rm\bf Sm}_{k}$ of pure dimension. Put
$pr_{1}$ and $pr_{2}$ the projections to the first and second factors of
$Y\times\mathbb{P}^{1}$, respectively. Assume that

$\pi:=pr_{2}\circ F:M\to\mathbb{P}^{1}$

is a double-point degeneration over $0\in\mathbb{P}^{1}$, that means that

$\pi^{-1}(0)=A\cup B$

where $A$ and $B$ are smooth Cartier divisors intersecting transversely in
$M$. By restricting the map $pr_{1}\circ F$, we obtain maps $A\to Y$ and $B\to
Y$. Let

$D=A\cap B$

and $N_{A/D}$ the normal bundles of $D$ in $A$. Set

$\pi_{P}:P:=\mathbb{P}(N_{A/D}\oplus\mathcal{O}_{D})\to D.$

Here it does not matter to distinguish $A$ and $B$, because it holds that

$N_{A/D}\otimes N_{B/D}\simeq\mathcal{O}_{D}$

and thus the resulting $\mathbb{P}^{1}$-bundles are isomorphic. Composing
$\pi_{P}$ with $pr_{1}\circ F|_{D}$, we have a map $P\to Y$. Let
$p\in\mathbb{P}^{1}$ be an arbitrary regular value of $\pi$ and $X_{1}$ the
regular fiber at $p$, and it yields a map $pr_{1}\circ F:X_{1}\to Y$. Then an
associated double-point relation over $Y$ is defined by

$[A\to Y]+[B\to Y]-[P\to Y]-[X_{1}\to Y]\in\mathcal{M}_{*}^{+}(Y).$

Let $\mathcal{R}_{*}(Y)$ be the subgroup generated by all double-point
relations over $Y$. As a special case, if $\pi$ is smooth over
$0,\infty\in\mathbb{P}^{1}$, we may view $\pi$ as a double-point degeneration
with $\pi^{-1}(0)=A\cup\emptyset$ and $D=\emptyset$, then naïve cobordism is
also a double-point relation:

$[\pi^{-1}(0)\to Y]-[\pi^{-1}(\infty)\to Y]\in\mathcal{R}_{*}(Y).$

The main theorem of [LP09] gives a geometric presentation of the algebraic
cobordism theory of [LM07]:

######  Theorem 5.2.

([LP09, Thm.0.1]) There is a canonical isomorphism:

$\Omega_{*}(Y)\simeq\mathcal{M}_{*}^{+}(Y)/\mathcal{R}_{*}(Y).$

###  5.3. Formal group law

Let $A^{*}$ be an oriented cohomology theory on ${\rm\bf Sm}_{k}$. For a line
bundle $L$ on $X$ in ${\rm\bf Sm}_{k}$, we have defined
$c_{1}(L):=s^{*}s_{*}(1)\in A^{*}(X)$. In case of $A^{*}={\rm CH}^{*}$, the
first Chern class is additive, i.e., $c_{1}(L\otimes
L^{\prime})=c_{1}(L)+c_{1}(L^{\prime})$, but it is not so for other cases.
Namely, there is a unique power series

$F_{A}(u,v)=u+v+\sum_{i,j\geq 1}a_{ij}u^{i}v^{j}\in A^{*}(k)[[u,v]]$

with $a_{ij}\in A^{1-i-j}(k)$ such that

$c_{1}(L\otimes M)=F_{A}(c_{1}(L),c_{1}(M))$

for any line bundles $L$ and $M$ on $X$. The series should satisfy
$F_{A}(u,v)=F_{A}(v,u)$ and $F_{A}(u,F(v,w))=F_{A}(F(u,v),w)$ (thus $a_{ij}$
satisfy that $a_{ij}=a_{ji}$ and some additional relations caused by the
latter property), and then $F$ is called a commutative formal group law (of
rank one).

In particular, for $A^{*}={\rm CH}^{*}$ (additive) and
$K^{0}[\beta,\beta^{-1}]$ (multiplicative), respectively,

$F_{ad}(u,v)=u+v,\qquad F_{ml}(u,v)=u+v-\beta uv.$

The universal theory $A^{*}=\Omega^{*}$ corresponds to the universal formal
group law $F_{\mathbb{L}}$. A computation shows that
$a_{11}=-[\mathbb{P}^{1}]$,
$a_{12}=a_{21}=[\mathbb{P}^{1}]^{2}-[\mathbb{P}^{2}]$ and so on [LM07, §2.5].
The double-point relation measures the difference of $F_{\mathbb{L}}$ from the
additive one $F_{ad}=u+v$; for divisors $A$, $B$ and $X_{1}$ in the above
setting, $X_{1}$ is linearly equivalent to $A+B$, and thus the double-point
relation means

$[P\to D]=A+B-F_{\mathbb{L}}(A,B)=-\sum a_{ij}A^{i}B^{j}.$

###  5.4. Cohomology operations

We refer to a recent work of A. Vishik [Vi19] on cohomology operations in
algebraic geometry; results which we will later make use of are algebro-
geometric counterpart to well-known facts/theorems in complex cobordism
theory.

Let $A^{*}$ and $B^{*}$ be oriented cohomology theories on ${\rm\bf Sm}_{k}$.
Fix $\kappa,\ell\in\mathbb{Z}$. A cohomology operation $G:A^{\kappa}\to
B^{\ell}$ means a natural transformation when $A^{\kappa}$ and $B^{\ell}$ are
regarded as functors from ${\rm\bf Sm}_{k}^{op}$ to pointed sets. That is, for
$Y\in{\rm\bf Sm}_{k}$,

$G_{Y}:A^{\kappa}(Y)\to B^{\ell}(Y)$

is a map between two sets so that $G_{Y}(0)=0$ and for any morphism
$f:Y^{\prime}\to Y$ in ${\rm\bf Sm}_{k}$, the following diagram commutes:

$\textstyle{A^{\kappa}(Y^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\scriptstyle{G_{Y^{\prime}}}$$\textstyle{A^{\kappa}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G_{Y}}$$\textstyle{B^{\ell}(Y^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{B^{\ell}(Y)}$

An operation $G$ is additive if the maps $G_{Y}$ are homomorphisms of abelian
groups. A main result in [Vi19] is the following theorem:

######  Theorem 5.3.

([Vi19, Thm.1.1]) Fix $\kappa,\ell\in\mathbb{Z}$. Let $B^{*}$ be an oriented
cohomology theory on ${\rm\bf Sm}_{k}$. Any family of homomorphisms

$G:\Omega^{\kappa}((\mathbb{P}^{\infty})^{\times l})\to
B^{\ell}((\mathbb{P}^{\infty})^{\times l})$

for $l\in\mathbb{Z}_{\geq 0}$ commuting with the action of the permutation
group $\mathfrak{S}_{l}$ and the pullbacks for specific morphisms listed below
extends to a unique additive operation $G:\Omega^{\kappa}\to B^{\ell}$ on
${\rm\bf Sm}_{k}$:

  * -

partial diagonals
$(\mathbb{P}^{\infty}\stackrel{{\scriptstyle\Delta}}{{\to}}\mathbb{P}^{\infty}\times\mathbb{P}^{\infty})\times\cdots$;

  * -

partial Segre embeddings
$(\mathbb{P}^{\infty}\times\mathbb{P}^{\infty}\stackrel{{\scriptstyle{\rm
s}}}{{\to}}\mathbb{P}^{\infty})\times\cdots$;

  * -

$({\rm
Spec}(k)\hookrightarrow\mathbb{P}^{\infty})\times(\mathbb{P}^{\infty})^{\times
r}$ for any $r$;

  * -

partial projection $(\mathbb{P}^{\infty})^{\times
l}\to(\mathbb{P}^{\infty})^{\times k}$ for any $k<l$.

Based on this theorem, a complete description of universal operations
$\Omega^{\kappa}\to\Omega^{\ell}$ is obtained as follows [Vi19, Thm.3.18
(Thm.6.1)]:

######  Theorem 5.4.

([Vi19]) Let $\varphi:\Omega^{\kappa}\to\Omega^{\ell}$ an additive operation
on ${\rm\bf Sm}_{k}$. Then
$\varphi:\Omega^{\kappa}\to\Omega^{\ell}\otimes_{\mathbb{Z}}\mathbb{Q}$ is
uniquely written as a $\mathbb{L}\otimes_{\mathbb{Z}}\mathbb{Q}$-linear
combination of Landweber-Novikov operations with multi-indices
$I=(i_{1},\cdots,i_{k})$

$s_{I}:\Omega^{\kappa}(Y)\to\Omega^{\ell}(Y)\otimes_{\mathbb{Z}}\mathbb{Q},\quad[f:X\to
Y]\mapsto f_{*}(c_{I}(f)).$

######  Remark 5.5.

(Pullback in $\Omega^{*}$) A difficult step in the construction of the
algebraic cobordism $\Omega^{*}(-)$ is to extend the pullback operations for
arbitrary morphisms in ${\rm\bf Sm}_{k}$ other than smooth morphisms. That was
done in Levine-Morel [LM07]. On the other hand, in geometric construction of
Levine-Pandharipande [LP09], the pullback was explicitly described not for all
morphisms but for certainly nice ones by moving by translation. The class of
such nice morphisms include $X\to\prod\mathbb{P}^{N_{i}}$ in ${\rm\bf Sm}_{k}$
[LP09, §8.2-8.3].

###  5.5. Multi-singularity loci cobordism classes

Let $\underline{\eta}=(\eta_{1},\cdots,\eta_{r})$ be a multi-singularity type.
To define $m_{\underline{\eta}}(f)\in{\rm CH}^{*}(X)$ and
$n_{\underline{\eta}}(f)\in{\rm CH}^{*}(Y)$ (Definition 4.9), we have used the
fundamental class of the geometric subset in the Chow group of
$X^{[[n]]}\in{\rm\bf Sch}_{k}$:

$[\Xi(X;\underline{\eta})]\in{\rm CH}_{*}(X^{[[n]]}).$

Instead, fixing a scheme structure, we may take the class of the structure
sheaf

$[\mathcal{O}_{\Xi(X;\underline{\eta})}]\cdot\beta^{d}\in
G_{0}(X^{[[n]]})[\beta,\beta^{-1}]$

with $d=\dim\Xi(X;\underline{\eta})$) in the Grothendieck group of coherent
sheaves on $X^{[[n]]}$ with parameter $\beta$. It is then straightforward to
define $K$-theoretic multi-singularity loci classes

$m^{K}_{\underline{\eta}}(f)\in
K^{0}(X)[\beta,\beta^{-1}],\;\;\;n^{K}_{\underline{\eta}}(f)\in
K^{0}(Y)[\beta,\beta^{-1}]$

in entirely the same way as in ${\rm CH}^{*}$ (e.g., see [KS04, §2.2] for a
summary of Intersection Theory in $K$-theory).

More generally, we are concerned with algebraic cobordism multi-singularity
loci classes.

######  Definition 5.6.

($\Omega$-assignment) Suppose that to every smooth scheme $X$, one assigns a
distinguished bordism class

$\mu_{\Xi}(X;\underline{\eta})\in\Omega_{*}(X^{[[n]]}).$

We call it an $\Omega$-assignment $\mu$ for the type $\underline{\eta}$, if
the following properties are satisfied:

  1. (1)

the image of $\mu_{\Xi}(X;\underline{\eta})$ in ${\rm CH}_{*}(X^{[[n]]})$ via
the natural transformation is the fundamental class
$[\Xi(X;\underline{\eta})]$;

  2. (2)

For any closed embedding $\iota:X^{\prime}\hookrightarrow X$ in ${\rm\bf
Sm}_{k}$, with $X^{\prime}$ and $X$ connected, it holds that

$\mu_{\Xi}(X^{\prime};\underline{\eta})=(\iota^{[[n]]})^{!}(\mu_{\Xi}(X;\underline{\eta}))\in\Omega_{*}(X^{\prime[[n]]})$

where $\iota^{[[n]]}:X^{\prime[[n]]}\to X^{[[n]]}$ is the regular embedding
obtained in Proposition 3.2 and $(\iota^{[[n]]})^{!}$ is the refined Gysin
map;

  3. (3)

The assignment to the disjoint union of $X_{1}$ and $X_{2}$ is of the form

$\mu_{\Xi}(X_{1}\sqcup X_{2};\underline{\eta})=\sum_{I\sqcup
J}\;\mu_{\Xi}(X_{1};\underline{\eta}_{I})\times\mu_{\Xi}(X_{2};\underline{\eta}_{J})$

in certain components of the algebraic cobordism group of the disjoint union

$\bigoplus_{I\sqcup J}\;\Omega_{*}(X_{1}^{[[I]]}\times
X_{2}^{[[J]]})\subset\Omega_{*}((X_{1}\sqcup X_{2})^{[[n]]})$

where the summand runs over all partition $I\sqcup J=\\{1,\cdots,r\\}$ ($I$ or
$J$ can be empty), $[[I]]:=\sum_{i\in I}n(\eta_{i})$, and $\times$ is the
cross product in $\Omega_{*}$.

To see the existence of such an $\Omega$-assignment, we first need to give
$\mu_{\Xi}(X;\underline{\eta}_{J})$ (for any $J$) for every connected smooth
scheme $X$ so that (1) and (2) are satisfied. That is achieved by Hironaka’s
resolution of singularities (see Remark 5.8 below). Then, we take the equality
in (3) as the definition of $\mu_{\Xi}(X_{1}\sqcup X_{2};\underline{\eta})$
for the disjoint union of connected $X_{1}$ and $X_{2}$, and iteratively use
this to define $\mu_{\Xi}(X;\underline{\eta})$ for $X$ having multiple
connected components.

######  Definition 5.7.

(Multi-singularity loci cobordism classes) Suppose that we are given an
$\Omega$-assignment $\mu$ for type $\underline{\eta}$. For an arbitrary proper
morphism $f:X\to Y$ of codimension $\kappa$ in ${\rm\bf Sm}_{k}$, we define
the multi-singularity loci cobordism classes associated to $\mu$ by

$\displaystyle m^{\Omega}_{\underline{\eta};\mu}(f)$ $\displaystyle:=$
$\displaystyle{\rm
pr}_{1*}((f^{[[n]]})^{!}(\mu_{\Xi}(X;\underline{\eta})\times[id_{Y}]))\,\in\Omega^{\ell-\kappa}(X),$
$\displaystyle n^{\Omega}_{\underline{\eta};\mu}(f)$ $\displaystyle:=$
$\displaystyle\bar{f}_{*}((f^{[[n]]})^{!}(\mu_{\Xi}(X;\underline{\eta})\times[id_{Y}]))\,\in\Omega^{\ell}(Y)$

where $(f^{[[n]]})^{!}$ is the refined Gysin map of the Hilbert extension map
in $\Omega^{*}$ [LM07, §6.6].

For short, we will often write $m^{\Omega}_{\underline{\eta}}(f)$ and
$n^{\Omega}_{\underline{\eta}}(f)$ by omitting the letter $\mu$ in the
subscript if it is easily understood. Obviously, changing the coefficients as
to be in ${\rm CH}^{*}$ via the natural transformation, we recover
$m_{\underline{\eta}}(f)$ and $n_{\underline{\eta}}(f)$ defined in Definition
4.9 (also for $K$-theory version as well).

Emphasized is that multi-singularity loci cobordism classes is not uniquely
determined by the type $\underline{\eta}$, for there are many different
choices of $\Omega$-assignments.

######  Remark 5.8.

(Resolution of singularities) Such an $\Omega$-assignment can be constructed
by equivariant resolution of singularities as briefly explained below. Let $X$
be a connected smooth scheme. Note that $\Xi^{\circ}(X;\underline{\eta})$ is a
locally closed smooth subscheme of $X^{[[n]]}$ given as an orbit of the action
of local isomorphisms of $X$ at ordered $r$ points. Thus singularities of the
Zariski closure $\Xi(X;\underline{\eta})$ are located along some adjacent
strata in $\Xi(X;\underline{\eta})-\Xi^{\circ}(X;\underline{\eta})$, and they
are determined by the dimension of $X$ and local data of more complicated
multi-singularity types than $\underline{\eta}$. Namely, any desingularization
of $\Xi(X;\underline{\eta})$ are essentially indebted on those local data
only. For an open set $U\subset\mathbb{A}^{m}$, choose an embedded resolution
of the locus $\Xi(U;\underline{\eta})\subset U^{[[n]]}$ so that it is
equivariant with respect to the groupoid action of local isomorphisms at $r$
points. By glueing such finitely many patches for $X=\bigcup U_{\alpha}$, we
obtain an assignment $\mu_{\Xi}(X;\underline{\eta})\in\Omega_{*}(X^{[[n]]})$.
When the dimension $m$ goes up, additional desingularizations are created
along new adjacent strata. In particular, we may first construct a
desingularizaton of $\Xi(\mathbb{P}^{N};\underline{\eta})$ and go to higher
$N$ step by step, that produces a projective limit system
$\\{\mu_{\Xi}(\mathbb{P}^{N};\underline{\eta})\\}_{N\in\mathbb{N}}$. From this
system, we obtain $\mu_{\Xi}(X;\underline{\eta})$ via the Gysin homomorphism
of the inclusion of any smooth $X\subset\mathbb{P}^{N}$, that yields an
$\Omega$-assignment which we seek for (at least in case that $X$ is quasi-
projective).

The $\Omega$-assignment itself is just a bit hard technical matter in theory.
Truly interesting is the case that we are given a nice resolution of the
geometric subset of a specific singularity type. For instance, there are at
least two interesting results on Thom polynomials valued in $\Omega^{*}$ as
far as the author knows.

######  Example 5.9.

(Degeneracy loci cobordism class) Hudson [Hu12] has established the algebraic
cobordism Thom-Porteous formula by using the Bott-Samelson resolution of the
closures of Schubert cells (also see [HPM20]). Those satisfy the above
properties, i.e., they define an $\Omega$-assignment. Note that since the
theory $\Omega^{*}$ established in [LM07, LP09], there have appeared many
successful works by several authors on Schubert Calculus in $\Omega^{*}$ and
also in other oriented cohomology theory, e.g., [HV11, HPM20].

######  Example 5.10.

(Double-point loci cobordism class) Based on [Mc76, Ro73], Audin [Au80]
presented the double-point formula in complex cobordism $MU$, which should
also be valid in $\Omega^{*}$. In this case, we consider
$\underline{\eta}=A_{0}^{2}$ ($n=2$) and the geometric subset
$\Xi(A_{0}^{2})=X^{[[2]]}={\rm Bl}(X\times X)$ which is smooth, thus we have
an $\Omega$-assignment for the type $A_{0}^{2}$. The relative Hilbert scheme
provides a desingularization of the double-point locus for a generic map
(Remark 3.10).

##  6\. Proof of Theorems

In this section we prove Theorems 4.17 and 4.18. For the simplicity, we first
discuss $m_{\underline{\eta}}(f)$ and $n_{\underline{\eta}}(f)$ defined in
${\rm CH}^{*}$. At the last step, we will switch to algebraic cobordism
$\Omega^{*}$ in order to properly employ Vishik’s theorem (Theorem 5.4). The
$\Omega^{*}$-version will be stated in Theorem 6.19. A summary of Intersection
Theory in §2.3 is available in $\Omega_{*}$ as well (see [LM07, Chap.6]), and
therefore, most of our arguments work in the $\Omega^{*}$-setting without any
change, unless specifically mentioned.

###  6.1. Double-point degeneration

To begin with, we need to precisely know the behavior of multi-singularity
loci classes under double-point degeneration.

Let $\underline{\eta}=(\eta_{1},\cdots,\eta_{r})$ be a multi-singularity type
of maps with codimension $\kappa=l-m$ and $\ell:=\ell(\underline{\eta})\leq
l$. Let

$F:M\to\tilde{Y}:=Y\times\mathbb{P}^{1}$

be a proper morphism in ${\rm\bf Sm}_{k}$ with $\dim M=m+1$ and $\dim Y=l$
such that $\pi=pr_{2}\circ F:M\to\mathbb{P}^{1}$ is a double-point
degeneration over $0\in\mathbb{P}^{1}$ and $1\in\mathbb{P}^{1}$ is a regular
value. Now set

$X_{0}:=\pi^{-1}(0)=A\cup B,\quad X_{1}=\pi^{-1}(1)$

and $D:=A\cap B$, $P:=\mathbb{P}(N_{A/D}\oplus\mathcal{O}_{D})$ as same as
before. We denote the restriction maps by

$g_{A}:A\to Y,\quad g_{B}:B\to Y,\quad h_{P}:P\to Y,\quad h_{X_{1}}:X_{1}\to
Y,$

and also, by taking disjoint unions,

$g:A\sqcup B\to Y,\quad h:P\sqcup X_{1}\to Y.$

Those maps are of codimension $\kappa$, thus target multi-singularity loci
classes of type $\underline{\eta}$ are defined. The following proposition is
essential in our argument. We will also treat the same claim in the
$\Omega^{*}$-setting.

######  Proposition 6.1.

Under the above setting of an arbitrary double-point degeneration, we have

$n_{\underline{\eta}}(g)=n_{\underline{\eta}}(h)\in{\rm CH}^{\ell}(Y).$

Before starting the proof, we check the claim in simplest examples, that gives
a clue as to what the proof should clarify.

######  Example 6.2.

(Double-point loci class)  Put $Y=\mathbb{P}^{2}$ and let $M$ be the closure
of the family of affine plane curves $xy-t=0$ in $Y\times\mathbb{P}^{1}$.
Denote the embedding by

$F:M\to\tilde{Y}=Y\times\mathbb{P}^{1}.$

Set $X_{0}=M\cap(Y\times\\{0\\})=A\cup B$, where $A$ and $B$ are transverse
lines in $Y$, and $X_{1}=M\cap(Y\times\\{1\\})$ a smooth conic. Put $D=A\cap
B$, the double-point, and $P=\mathbb{P}^{1}=\mathbb{P}(T_{D}Y)$. Now, we apply
the double-point formula in §2.1.2 to maps of codimension $\kappa=1$,
$g:A\sqcup B\to Y$ and $h:P\sqcup X_{1}\to Y$. Obviously,

$n_{A_{0}^{2}}(g)=g_{*}m_{A_{0}^{2}}(g)=2\in{\rm CH}^{2}(Y).$

On the other hand, there is no double-point of $h_{X_{1}}$ and $h_{P}:P\to Y$
is highly degenerate. Even so, the double-point formula makes sense for
arbitrary maps [Fu84, §9.3]. Since ${h_{P}}_{*}(1)=0$ and $h_{P}^{*}TY$ is
trivial, we see

$\displaystyle m_{A_{0}^{2}}(h_{P})$ $\displaystyle=$ $\displaystyle
h_{P}^{*}{h_{P}}_{*}(1)-c_{1}(h_{P}^{*}TY-TP)$ $\displaystyle=$
$\displaystyle-c_{1}(-TP)=c_{1}(T\mathbb{P}^{1}).$

Thus $n_{A_{0}^{2}}(h)=h_{*}m_{A_{0}^{2}}(h)=2$.

######  Example 6.3.

(Degeneracy loci class)  Let $M$, $Y$, $X_{0}$, $X_{1}$ and the map $F$ be the
same as in Example 6.2. Composing (the first factor of) $F$ with a generic
linear projection to a fixed line $Y^{\prime}$ in $Y=\mathbb{P}^{2}$, we
obtain a proper morphism

$F^{\prime}:M\to\tilde{Y}^{\prime}=Y^{\prime}\times\mathbb{P}^{1}.$

We consider $g:A\sqcup B\to Y^{\prime}$ and $h:P\sqcup X_{1}\to Y^{\prime}$,
which are maps of codimension $\kappa=0$. Apply the simplest Thom-Porteous
formula $m_{A_{1}}=c_{1}$ to those maps. Note that it is available for
arbitrary maps [Fu84, §14.4]. Since $g_{A}$ and $g_{B}$ are non-singular, we
see that

$n_{A_{1}}(g)=g_{*}m_{A_{1}}(g)=0\in{\rm CH}^{1}(Z).$

On the other hand, for $h_{P}^{*}TY^{\prime}$ is trivial, we see

$\displaystyle m_{A_{1}}(h_{P})$ $\displaystyle=$ $\displaystyle
c_{1}(h_{P}^{*}TY^{\prime}-TP)$ $\displaystyle=$ $\displaystyle
c_{1}(-TP)=-c_{1}(T\mathbb{P}^{1}).$

Then $n_{A_{1}}(h_{P})={h_{P}}_{*}m_{A_{1}}(h_{P})=-2$, which is caused by an
excess intersection of $h_{P}$. There are two simple branch points for the
projection of the conic $X_{1}$ to the line $Y^{\prime}$; in other words, the
singularity locus for $F^{\prime}:M\to\tilde{Y}^{\prime}$ meets $X_{1}$ at two
points. Therefore, we see $n_{A_{1}}(h_{X_{1}})=2$. Hence

$n_{A_{1}}(h)=n_{A_{1}}(h_{X_{1}})+n_{A_{1}}(h_{P})=2-2=0.$

###  6.2. Refined intersection

We prove Proposition 6.1 through several steps. We use the same setting and
notations as in §6.1.

To begin with, we fix some terminologies: consider a commutative diagram with
two vector bundles $E\to B$ and $F\to B^{\prime}$:

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\varphi}}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{B^{\prime}}$

We call $\bar{\varphi}:E\to F$ a vector bundle morphism over $\varphi$ if it
is linearly isomorphic fiberwise (i.e., $E$ is identified with the induced
bundle $\varphi^{*}F$ on $B$), while we call $\bar{\varphi}:E\to F$ a vector
bundle map over $\varphi$ if it is linear fiberwise and the rank of the linear
maps on fibers can vary, i.e., it is a section of ${\rm
Hom}_{k}(E,\varphi^{*}F)\to B$.

####  6.2.1. Quadratic map

First, we recall an elementary fact which is equivalent to that the degree
$\int c_{1}(T\mathbb{P}^{1})=2$:

######  Lemma 6.4.

There is a vector bundle morphism
$\bar{\psi}:T\mathbb{P}^{1}\to\mathcal{O}_{\mathbb{P}^{1}}(1)$ over the
quadratic map $\psi:\mathbb{P}^{1}\to\mathbb{P}^{1}$,
$\psi([z_{0};z_{1}])=[z_{0}^{2};z_{1}^{2}]$.

Set $L:=N_{A/D}$; its dual $L^{*}$ is isomorphic to $N_{B/D}$ and $L\oplus
L^{*}$ is the normal bundle $N_{M/D}$ of $D$ in $M$. As seen before, a double-
point degeneration yields the $\mathbb{P}^{1}$-bundle

$\pi_{P}:P:=\mathbb{P}(L\oplus\mathcal{O}_{D})\to D.$

Now we consider another $\mathbb{P}^{1}$-bundle

$\pi_{Q}:Q:=\mathbb{P}(L\oplus L^{*})\to D,$

which is the exceptional fiber of the blow-up of $M$ along $D$. We introduce
two line bundles,

$T\pi_{P}:=\ker d\pi_{P}\to P,\quad\nu^{*}:=\mathcal{O}_{Q}(1)\to Q,$

the relative (fiber) tangent bundle of $\pi_{P}$ and the dual to the
tautological line bundle $\nu\,(:=O_{Q}(-1))$ of $Q$, respectively.

######  Lemma 6.5.

There is a vector bundle morphism $\bar{\psi}:T\pi_{P}\to\nu^{*}$ over a map
$\psi:P\to Q$ which is a family of fiberwise quadratic maps commuting with
projections to $D$.

$\textstyle{T\pi_{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\psi}}$$\textstyle{\nu^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{\pi_{P}}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{Q}}$$\textstyle{D}$

Proof : We build $\psi$ and $\bar{\psi}$ as follows. Take a local
trivialization of $L$ and let $(z_{0},z_{1})$ and $(u_{0},u_{1})$ be linear
coordinates of fibers of $L\oplus\mathcal{O}_{D}$ and $L\oplus L^{*}$ subject
to the splitting, respectively. Transitions are $(z_{0},z_{1})\mapsto(\alpha
z_{0},z_{1})$ and $(u_{0},u_{1})\mapsto(\alpha u_{0},\alpha^{-1}u_{1})$ with
$\alpha\in GL(1)$. We set locally
$[u_{0};u_{1}]=\psi([z_{0};z_{1}]):=[z_{0}^{2};z_{1}^{2}]$ in homogeneous
coordinates of fibers. This is compatible with the transition

$\psi([\alpha z_{0};z_{1}])=[\alpha^{2}z_{0}^{2};z_{1}^{2}]=[\alpha
z_{0}^{2};\alpha^{-1}z_{1}^{2}],$

thus $\psi:P\to Q$ is well-defined. Applying Lemma 6.4 fiberwise, we get the
vector bundle morphism $\bar{\psi}$. More precisely, putting $z=z_{0}/z_{1}$,
$u=u_{0}/u_{1}$ ($z_{1},u_{1}\not=0$), take local sections $\partial_{z}$ of
$T\pi_{P}$ and $s_{0}:=u_{1}$ of $\nu^{*}$, then set
$\bar{\psi}(\partial_{z}):=s_{0}$. Put $z^{\prime}=\alpha z$ and
$u_{1}^{\prime}=\alpha^{-1}u_{1}$, then we have
$\partial_{z^{\prime}}=\alpha^{-1}\partial_{z}$ and
$\bar{\psi}(\partial_{z^{\prime}})=\alpha^{-1}s_{0}$. That is also for
$w=z_{1}/z_{0}$, $v=u_{1}/u_{0}$, $s_{1}:=-u_{0}$ and
$\bar{\psi}(\partial_{w}):=s_{1}$. Hence, $\bar{\psi}$ is compatible with the
transition. Note that $\partial_{w}=-z^{2}\partial_{z}$ and $s_{1}=-us_{0}$
($u=\psi(z)=z^{2}$). Thus $\bar{\psi}$ is well-defined. $\Box$

There are two global sections corresponding to $z=0,\infty$ in the above
construction; we denote them by

$D_{0},\;D_{\infty}\subset P\to D.$

The fiberwise quadratic map ${\psi}:P\to Q$ sends them to sections

$D^{\prime}_{0},\;D^{\prime}_{\infty}\subset Q\to D,$

indeed, ${\psi}$ is a double covering branched along $D^{\prime}_{0}$ and
$D^{\prime}_{\infty}$.

####  6.2.2. Desingularization

Consider the projectivized cotangent bundle

$\tilde{\pi}:\mathbb{P}(T^{*}M)\to M$

which is also the Grassmannian bundle of $m$-planes in $TM$ (note that $\dim
M=m+1$). We denote a point of $\mathbb{P}(T^{*}M)$ by the pair
$q=(x,\lambda_{q})$ of a point $x=\tilde{\pi}(q)\in M$ and a hyperplane
$\lambda_{q}\subset T_{x}M$. At every point $x\in M-D$, we take the tangent
space at $x$ of the hypersurface $X_{\pi(x)}$

$T_{x}X_{\pi(x)}=\ker d\pi_{x}\subset T_{x}M.$

Collect those hyperplanes and also limiting hyperplanes: we denote the Zariski
closure by

$\tilde{M}:={\rm
Closure}\\{\;(x,T_{x}X_{\pi(x)})\in\mathbb{P}(T^{*}M)\;|\;x\in M-D\;\\}$

with the natural projection

$\tilde{\pi}:\tilde{M}\to M.$

######  Lemma 6.6.

$\tilde{\pi}:\tilde{M}-\tilde{\pi}^{-1}(D)\to M-D$ is isomorphic and
$\tilde{M}$ is non-singular. In particular, $\tilde{\pi}^{-1}(D)$ is
identified with $Q$ defined previously, and its normal bundle in $\tilde{M}$
is isomorphic to $\nu=O_{Q}(-1)$.

Proof : Take an affine chart $(U,(x,y,u_{2},\cdots,u_{m}))$ of $M$ centered at
a point of $D$ such that $X_{0}\cap U$ is defined by $xy=0$, i.e., locally,
$(u_{2},\cdots,u_{m})$ are coordinates of $D\cap U$ and $x,y$ are coordinates
normal to $D$ in $A$ and $B$, respectively. The tangent space $TX_{t}$ at
$q=(x_{0},y_{0},u)\in X_{t}\cap U$ with $t\not=0$ is given by
$x_{0}dy+y_{0}dx=0$ in $T_{q}M$, i.e., it is the direct sum of $T_{q}D$ and
the line spanned by $x_{0}\partial_{x}-y_{0}\partial_{y}$. Then we can
identify $\tilde{\pi}^{-1}(U)$ with the blow-up ${\rm Bl}_{D\cap U}U$ of $U$
along $D\cap U$ by $(q,[x_{0};-y_{0}])\mapsto(q,[x_{0};y_{0}])$, and the claim
follows. $\Box$

Let $\tilde{X}_{0}$ denote the schematic preimage
$\tilde{\pi}^{-1}(X_{0})\subset\tilde{M}$; it consists of irreducible
components $\tilde{A}$ and $\tilde{B}$ (the strict transforms of $A$ and $B$,
respectively) and $Q$:

$[\tilde{X}_{0}]=[\tilde{A}]+[\tilde{B}]+2[Q].$

Those components have transverse intersection:

$\tilde{A}\cap Q=D_{0}^{\prime},\quad\tilde{B}\cap
Q=D_{\infty}^{\prime},\quad\tilde{A}\cap\tilde{B}=\emptyset.$

Put $\tilde{Y}:=Y\times T$ with $T:=\mathbb{P}^{1}$. We set

$\tilde{F}:=F\circ\tilde{\pi}:\tilde{M}\to\tilde{Y}$

and also $F_{1}:=pr_{1}\circ\tilde{F}:\tilde{M}\to Y$ and
$F_{2}:=pr_{2}\circ\tilde{F}:\tilde{M}\to T$.

For any $t\not=0$, $X_{t}$ is isomorphic to $\tilde{X}_{t}$. Denote the
restriction by

$f_{t}:\tilde{X}_{t}\to Y\,(=Y\times\\{t\\}).$

Let $i_{g}:\tilde{A}\sqcup\tilde{B}\hookrightarrow\tilde{M}$,
$\iota:\tilde{X}_{t}\hookrightarrow\tilde{M}$ and $j:Q\to\tilde{M}$ be the
inclusion and $i_{h}=(j\circ\psi)\sqcup\iota:P\sqcup\tilde{X}_{t}\to\tilde{M}$
($t\not=0$) where $\psi:P\to Q$ is the quadratic map in Lemma 6.5. Put

$g=F_{1}\circ i_{g}:\tilde{A}\sqcup\tilde{B}\to Y,\quad h:=F_{1}\circ
i_{h}:P\sqcup\tilde{X}_{t}\to Y.$

Here $h$ depends on $t\,(\not=0)$, but this simplification may not cause any
confusion.

The multi-singularity loci classes of type $\underline{\eta}$ for maps $g$ and
$h$ ($t\not=0$) are defined (Definition 4.9):

$m_{\underline{\eta}}(g)\in{\rm CH}^{\ell-\kappa}(\tilde{A})\oplus{\rm
CH}^{\ell-\kappa}(\tilde{B}),$ $m_{\underline{\eta}}(h)\in{\rm
CH}^{\ell-\kappa}(P)\oplus{\rm CH}^{\ell-\kappa}(\tilde{X}_{t}).$

We compare these classes in ${\rm CH}_{l-\ell}(\tilde{M})={\rm
CH}^{\ell-\kappa+1}(\tilde{M})$ through the pushforward of $i_{g}$ and
$i_{h}$. Below, as a notational convention, we may drop the letter $\iota_{*}$
for the pushforward of inclusions $\iota$ to the ambient space, if it is
easily understood from the context.

The latter class $m_{\underline{\eta}}(h)$ is the sum of parts supported on
$P$ and $\tilde{X}_{t}$. Since $h_{P}:P\to Y$ is degenerate along the fibers
of $P\to D$, the multi-singularity loci class supported on $P$ contains an
excess intersection. We denote the class by

$m_{\underline{\eta}}(h)^{P}\in{\rm CH}_{l-\ell}(P).$

On the other hand, for any regular value $t$, we denote by

$m_{\underline{\eta}}(h)^{X_{t}}\in{\rm CH}_{l-\ell}(\tilde{X}_{t})$

the multi-singularity loci class supported on $\tilde{X}_{t}$. The image in
${\rm CH}_{l-\ell}(\tilde{M})$ does not depend on $t\not=0$. Obviously, we
have

######  Lemma 6.7.

$\displaystyle(i_{h})_{*}m_{\underline{\eta}}(h)=\iota_{*}m_{\underline{\eta}}(h)^{X_{t}}+j_{*}\psi_{*}m_{\underline{\eta}}(h)^{P}\in{\rm
CH}_{l-\ell}(\tilde{M})$.

####  6.2.3. Dynamic intersection

Put $T^{*}=T-\\{0\\}(=\mathbb{P}^{1}-\\{0\\})$ and

$\tilde{M}^{*}:=\tilde{M}-\tilde{X}_{0}=\tilde{M}\times_{T}T^{*},\quad\tilde{Y}^{*}:=Y\times
T^{*}.$

Also we set

$\mathcal{M}_{P}^{\circ}:=(P\times_{k}T^{*})\sqcup\tilde{M}^{*}\;\mbox{
(disjoint)}$

as a smooth $T^{*}$-scheme, and define a proper morphism over $T^{*}$

$\tilde{F}^{\circ}:=(h_{P}\times
id_{T^{*}})\sqcup\tilde{F}:\mathcal{M}_{P}^{\circ}\to\tilde{Y}^{*}$

to be the family

$h_{t}:P\sqcup\tilde{X}_{t}\to Y\qquad(t\in T^{*}).$

Then we have the multi-singularty loci class for the map $\tilde{F}^{\circ}$
of type $\underline{\eta}$:

$m_{\underline{\eta}}(\tilde{F}^{\circ})=\alpha+\beta\in{\rm
CH}_{l-\ell+1}(\mathcal{M}_{P}^{\circ})$

where we denote

$\alpha:=m_{\underline{\eta}}(\tilde{F}^{\circ})^{\tilde{M}^{*}}\in{\rm
CH}_{l-\ell+1}(\tilde{M}^{*})$

and

$\beta:=m_{\underline{\eta}}(\tilde{F}^{\circ})^{P\times_{k}T^{*}}\in{\rm
CH}_{l-\ell+1}(P\times_{k}T^{*}).$

We take the intersection product

$\alpha_{t}:=\tilde{X}_{t}\cdot\alpha=\iota^{*}(\alpha)\in{\rm
CH}_{l-\ell}(\tilde{X}_{t})$

($t\not=0$) where $\iota:\tilde{X}_{t}\hookrightarrow\tilde{M}^{*}$ is the
inclusion. Also we set $\beta_{t}:=(P\times_{k}\\{t\\})\cdot\beta\in{\rm
CH}_{l-\ell}(P)$.

######  Lemma 6.8.

It holds that

$\alpha_{t}=m_{\underline{\eta}}(h)^{X_{t}}\in{\rm
CH}_{l-\ell}(\tilde{X}_{t}),\quad\beta_{t}=m_{\underline{\eta}}(h)^{P}\in{\rm
CH}_{l-\ell}(P).$

Proof : The inclusion $Y=Y\times\\{t\\}\hookrightarrow\tilde{Y}^{*}$
($t\not=0$) is transverse to the map
$\tilde{F}^{\circ}:\mathcal{M}_{P}^{\circ}\to\tilde{Y}^{*}$, and the fiber
square gives $h:P\sqcup\tilde{X}_{t}\to Y$. The lemma follows from Proposition
4.13. $\Box$

Recall the basic construction of dynamic intersection in §2.3.1. Write
$\alpha=\sum n_{i}[\mathcal{Y}_{i}]$ where $\mathcal{Y}_{i}$ are irreducible
subvarieties of $\tilde{M}^{*}$ of dimension $l-\ell+1$, and take the Zariski
closure $\overline{\mathcal{Y}_{i}}\subset\tilde{M}$. Let

$\tilde{\alpha}:=\sum n_{i}[\overline{\mathcal{Y}_{i}}]\in{\rm
CH}_{l-\ell+1}(\tilde{M}).$

Then the limit intersection class is defined, and it actually satisfies

$\lim_{t\to
0}\alpha_{t}=\tilde{X}_{0}\cdot\tilde{\alpha}=(\tilde{A}+\tilde{B}+2Q)\cdot\tilde{\alpha}\in{\rm
CH}_{l-\ell}(\tilde{X}_{0}).$

Note that $\tilde{X}_{0}\cdot\tilde{\alpha}$ and
$\alpha_{t}=\tilde{X}_{t}\cdot\tilde{\alpha}$ ($t\not=0$) coincide in
$\tilde{M}$, for $[\tilde{X}_{0}]=[\tilde{X}_{t}]\in{\rm
CH}_{l-\ell}(\tilde{M})$.

Also for $\beta=\sum m_{k}[\mathcal{Y}^{\prime}_{k}]$ where
$\mathcal{Y}^{\prime}_{k}$ are irreducible subvarieties of $P$, we put

$\tilde{\beta}:=\sum m_{k}[\overline{\mathcal{Y}^{\prime}_{k}}]\in{\rm
CH}_{l-\ell+1}(P\times_{k}T).$

The limit class is defined and $\lim_{t\to 0}\beta_{t}=\beta_{t}\in{\rm
CH}_{l-\ell}(P)$ ($t\not=0$).

Comparing the limit of $\alpha_{t}$ and that of $\beta_{t}$, we obtain the
following main lemma, which will be shown later in §6.2.6. The lemma
introduces a certain refined intersection class $\delta_{D}$, which is the
contribution to multi-singularity loci classes of $h$ and $g$ from bi-
singularities lying on the same fiber of $D_{0}^{\prime}\sqcup
D_{\infty}^{\prime}\to D$ (cf. Example 6.2).

######  Lemma 6.9.

There is a class $\delta_{D}\in{\rm CH}_{l-\ell}(D^{\prime}_{0}\sqcup
D^{\prime}_{\infty})$ satisfying that

(1) $\displaystyle
j_{*}\psi_{*}m_{\underline{\eta}}(h)^{P}=j_{*}\delta_{D}-2\,Q\cdot\tilde{\alpha}$
(2)
$\displaystyle(i_{g})_{*}m_{\underline{\eta}}(g)=j_{*}\delta_{D}+(\tilde{A}+\tilde{B})\cdot\tilde{\alpha}$

in ${\rm CH}_{l-\ell}(\tilde{M})$ where $j:Q\to\tilde{M}$ is the inclusion.

Proof of Proposition 6.1: It follows from Lemmas 6.8 and 6.9 that

$(i_{g})_{*}m_{\underline{\eta}}(g)-j_{*}\psi_{*}m_{\underline{\eta}}(h)^{P}=(\tilde{A}+\tilde{B}+2Q)\cdot\tilde{\alpha}=\tilde{X}_{0}\cdot\tilde{\alpha}=\tilde{X}_{t}\cdot\tilde{\alpha}=\iota_{*}m_{\underline{\eta}}(h)^{X_{t}}$

in ${\rm CH}_{l-\ell}(\tilde{M})$. Hence, by Lemmas 6.7, we see

$(i_{g})_{*}m_{\underline{\eta}}(g)=(i_{h})_{*}m_{\underline{\eta}}(h).$

Since $g=F_{1}\circ i_{g}$ and $h=F_{1}\circ i_{h}$, we have

$\displaystyle n_{\underline{\eta}}(g)$ $\displaystyle=$ $\displaystyle
g_{*}m_{\underline{\eta}}(g)$ $\displaystyle=$
$\displaystyle(F_{1})_{*}(i_{g})_{*}m_{\underline{\eta}}(g)$ $\displaystyle=$
$\displaystyle(F_{1})_{*}(i_{h})_{*}m_{\underline{\eta}}(h)$ $\displaystyle=$
$\displaystyle h_{*}m_{\underline{\eta}}(h)$ $\displaystyle=$ $\displaystyle
n_{\underline{\eta}}(h).$

This completes the proof of Proposition 6.1. $\Box$

######  Remark 6.10.

We have defined $\tilde{\alpha}\in{\rm CH}_{l-\ell+1}(\tilde{M})$ by dynamic
intersection. Note that $\tilde{\alpha}$ is entirely different from the multi-
singularity loci class $m_{\underline{\eta}}(\tilde{F})\in{\rm
CH}_{l-\ell+1}(\tilde{M})$ for the map
$\tilde{F}:\tilde{M}\to\tilde{Y}=Y\times T$. Indeed, the divisor $Q$ always
becomes an irreducible component of the singular locus of $\tilde{F}$ and thus
it contributes to some part of $m_{\underline{\eta}}(\tilde{F})$, while
$\tilde{\alpha}$ does not have such a component.

####  6.2.4. Jet section

Let $q\in\tilde{M}$ and $k$ sufficiently large. Regarding $\lambda_{q}$ as the
germ of a hyperplane in a local chart of $M$ centered at $\tilde{\pi}(q)$, we
denote by $\Psi_{q}$ the $k$-jet of the composed map-germ

$\textstyle{(\lambda_{q},0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{incl\;\;\;\;}$$\textstyle{(M,\tilde{\pi}(q))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{(Y,f_{1}(q)).}$

In other words, it is an element of the $k$-jet space

$\Psi_{q}\in J^{k}(\lambda_{q},T_{F_{1}(q)}Y)=\bigoplus_{i=1}^{k}{\rm
Hom}({\rm Sym}^{i}(\lambda_{q}),T_{F_{1}(q)}Y).$

Let $H\to\tilde{M}$ be the tautological vector bundle of rank $m$, i.e., the
bundle whose fiber at $q\in\tilde{M}$ is $\lambda_{q}$, then we have a jet
section

$\Psi:\tilde{M}\to J^{k}(H,F_{1}^{*}TY).$

Note that the restriction of $\Psi$ to $\tilde{X}_{t}$ is nothing but the jet
section of $f_{t}:X_{t}\to Y$ at $\tilde{\pi}(q)\in X_{t}$ ($t\not=0$), while
the restriction to $Q$ is entirely different from the jet section of
$F_{1}\circ\pi_{Q}:Q\to Y$. Indeed, it follows from Lemma 6.6 that

$H|_{Q}\simeq\nu\oplus\pi_{Q}^{*}TD.$

We compare the section $\Psi|_{Q}$ over $Q$ with the $k$-jet extension of
$h_{P}=F_{1}\circ\psi:P\to Y$, which is a section of $J^{k}(TP,h_{P}^{*}TY)$
over $P$. Since the tangent bundle $TP$ splits as

$TP=T\pi_{P}\oplus\pi_{P}^{*}TD,$

we associate a bundle morphism

$\textstyle{\bar{\psi}\oplus
id_{TD}:TP\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\nu^{*}\oplus\pi_{Q}^{*}TD}$

over $\psi:P\to Q$ using $\bar{\psi}$ in Lemma 6.5. This induces an
isomorphism between the tangent space $T_{p}P$ and the vector space
$\lambda_{q}$ at every $p\in P$ and $q=\psi(p)$, if we identify $\nu_{q}$ and
$\nu_{q}^{*}$ (fibers at $q$). Combining it with the jet section $\Psi_{p}$ we
have the $k$-jet of a map-germ $(P,p)\to(Y,h_{P}(p))$ which we may regard as a
deformation of the jet of $h_{P}$ at $p$. However, we can not do this
globally, for $\nu\not\simeq\nu^{*}$ in general. So we now suppose to take a
section $s_{\nu}$ of ${\rm Hom}(\nu^{*},\nu)$ over $Q$ – indeed, $s_{\nu}$ may
be the zero section. We then think of the bundle map

$(s_{\nu}\circ\bar{\psi})\oplus id_{TD}:TP\to\nu\oplus\pi_{Q}^{*}TD=H$

over $\psi:P\to Q$. Combining this map with the section $\Psi:\tilde{M}\to
J^{k}(H,F_{1}^{*}TY)$, we obtian a section

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J^{k}(TP,h_{P}^{*}TY).}$

In other words, the obtained jet section assigns to each $p\in P$ the $k$-jet
of the germ of composed map

$\textstyle{(P,p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\lambda_{q},0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi\;\;\;}$$\textstyle{(Y,h_{P}(p)).}$

It actually coincides with the $k$-jet of $h_{P}$ at $p$ if $s_{\nu}=0$.

For a mono-singularity type $\eta_{1}$, the singularity loci class
$m_{\eta_{1}}(\Psi)\in{\rm CH}^{*}(\tilde{M})$ is defined. Then the
intersection product $Q\cdot m_{\eta_{1}}(\Psi)$ is equal to the loci class
$m_{\eta_{1}}(\Psi|_{Q})\in{\rm CH}^{*}(Q)$. Also we obtain the loci class
$m_{\eta_{1}}(h_{P})\in{\rm CH}^{*}(P)$ of $h_{P}:P\to Y$, which actually
contains excess intersection. The following lemma formulates a general
property suggested by Example 6.3.

######  Lemma 6.11.

$(j\circ\psi)_{*}\beta_{t}=-2\,Q\cdot\tilde{\alpha}\in{\rm
CH}^{*}(\tilde{M})$.

Proof : Let $L_{Q}$ be the line bundle over $\tilde{M}$ associated to the
divisor $Q$, then $c_{1}(L_{Q})=[Q]$ and $\nu=j^{*}L_{Q}$. The map
$\bar{\psi}$ between the total spaces of line bundles $T\pi_{P}$ and
$\nu^{*}(\subset L_{Q}^{*})$ sends a section to a section (with degree two),
and its Gysin-image in ${\rm CH}^{*}(\nu^{*})={\rm CH}^{*}(\tilde{M})$ equals
$2c_{1}(L_{Q}^{*})=-2[Q]$. The class $m_{\eta_{1}}(h_{P})$ is obtained by the
pullback of $m_{\eta_{1}}(\Psi|_{Q})$ via ${\psi}$ together interchanging
$\nu$ and $\nu^{*}$. Hence, by the projection formula,
$(j\circ\psi)_{*}m_{\eta_{1}}(h_{P})=j_{*}\psi_{*}(-\psi^{*}m_{\eta_{1}}(\Psi))=-j_{*}(\psi_{*}(1)\cdot
m_{\eta_{1}}(\Psi))=-2\,Q\cdot m_{\eta_{1}}(\Psi)$. $\Box$

######  Remark 6.12.

Lemma 6.11 can also be viewed in another way. The map $h_{P}:P\to Y$ is the
composition of $\psi:P\to\tilde{M}$, $\tilde{F}:\tilde{M}\to Y\times T$ and
$Y\times T\to Y$. As mentioned in Remark 6.10, the mono-singularity loci class
$m_{\eta_{1}}(\tilde{F})$ contains some part supported on $Q$.

####  6.2.5. Refined intersection

Recall that $m_{\underline{\eta}}(h)$ is the image via ${\rm pr}_{1*}$ of the
intersection product of the Hilbert extension map

$h^{[[n]]}:(P\sqcup\tilde{X}_{t})^{[[n]]}\to((P\sqcup\tilde{X}_{t})\times
Y)^{[[n]]}$

with the geometric subset $V:=\Xi(P\sqcup\tilde{X}_{t};\underline{\eta})\times
Y$ (Definition 4.9). The class $m_{\underline{\eta}}(h)^{P}$ is the part
supported on $P$, that is, it comes from the intersection product on connected
components of $(P\sqcup\tilde{X}_{t})^{[[n]]}$ whose first factor (via ${\rm
pr}_{1}$) is mapped to $P$, i.e., the union of

$None$ $P^{[[J]]}\times\tilde{X}_{t}^{[[K]]}\quad\mbox{($J\sqcup
K=\\{1,\cdots,r\\}$ with $1\in J$)}.$

A linear vector field of fiber tangents
$\mbox{\boldmath$v$}\in\Gamma(T\pi_{P})$ over $P$ is defined by

$\mbox{\boldmath$v$}(z)=z\partial_{z}\;\;\;(\mbox{or}\;\;\;\mbox{\boldmath$v$}(w)=-w\partial_{w})$

using the fiber coordinate $z$ or $w$ appearing in local trivialization of
$\pi_{P}:P\to D$ (indeed, $v$ is well-defined on the entire space $P$). We
also denote by the same letter $v$ as a section of $TP$ obtained via the
natural inclusion $T\pi_{P}\subset TP$. Notice that the zero locus of $v$
coincides with $D_{0}\sqcup D_{\infty}\subset P$. Furthermore, $v$ is extended
to a section on $P\sqcup\tilde{X}_{t}$ so that it assigns the zero on
$\tilde{X}_{t}$. Then we introduce a section on each component

$s_{\mbox{\tiny$\mbox{\boldmath$v$}$}}:P^{[[J]]}\times\tilde{X}_{t}^{[[K]]}\to
N_{h_{P}}$

by assigning to every $(x^{|J|},z_{I})\in P^{[[J]]}$ the derivation (see §3.4)

$s_{\mbox{\tiny$\mbox{\boldmath$v$}$}}(x^{|J|},z_{I}):=Dh_{P}^{[[|J|]]}\left(\bigoplus_{p\in{\rm
Supp}(I)}\mbox{\boldmath$v$}(p)\right)\in(N_{h_{P}})_{(x,z_{I})}$

and the zero to other $(x^{|K|},z_{I})\in\tilde{X}_{t}^{[[K]]}$ and by taking
the direct sum.

The normal cone $C_{V}$ associated to $V$ lives in the normal bundle of the
Hilbert extension map $h^{[[n]]}$ which is isomorphic to $(h^{*}TY)^{[[n]]}$.
We denote the normal bundle over the union of components $(c)$ by $N_{h_{P}}$.
Then by the definition,

$m_{\underline{\eta}}(h)^{P}={\rm
pr}_{1*}s_{\mbox{\tiny$\mbox{\boldmath$v$}$}}^{*}[C_{V}]\quad\in{\rm
CH}_{l-\ell}(P).$

Note that $D_{0}\sqcup D_{\infty}$ is always contained in the preimage of
$s_{\mbox{\tiny$\mbox{\boldmath$v$}$}}$ of the normal cone $C_{V}$. We
distinguish irreducible components of
$s_{\mbox{\tiny$\mbox{\boldmath$v$}$}}^{-1}(C_{V})$ as follows.

Let $Z\subset(P\sqcup\tilde{X}_{t})^{[[n]]}$ be the closed subset consisting
of $(x,z_{I})$ such that the support of $I$ contains two distinct points
$p,p^{\prime}\in D_{0}\sqcup D_{\infty}$ with
$\pi_{P}(p)=\pi_{P}(p^{\prime})\in D$ and the first factor ${\rm pr}_{1}(x)\in
P$ is $p$. The components of
$s_{\mbox{\tiny$\mbox{\boldmath$v$}$}}^{-1}(C_{V})$ which are contained in $Z$
define the refined intersection class

$(s_{\mbox{\tiny$\mbox{\boldmath$v$}$}}^{*}[C_{V}])^{Z}\in{\rm
CH}_{l-\ell}(Z).$

Since ${\rm pr}_{1}(Z)$ is contained in $D_{0}\sqcup D_{\infty}$, we put

$\delta_{D}:={\rm
pr}_{1*}((s_{\mbox{\tiny$\mbox{\boldmath$v$}$}}^{*}[C_{V}])^{Z})\in{\rm
CH}_{l-\ell}(D_{0}\sqcup D_{\infty}).$

By the natural identification via $\psi$, we may also think of $\delta_{D}$ as
a class in ${\rm CH}_{l-\ell}(D^{\prime}_{0}\sqcup D^{\prime}_{\infty})$.

####  6.2.6. Proof of Lemma 6.9

First we show the equality

(1) $\displaystyle
j_{*}\psi_{*}m_{\underline{\eta}}(h)^{P}=j_{*}\delta_{D}-2\,Q\cdot\tilde{\alpha}.$

By the definition, the class $\delta_{D}$ contributes to
$m_{\underline{\eta}}(h)^{P}\in{\rm CH}_{l-\ell}(P)$, and the sum of
components of $s_{\mbox{\tiny$\mbox{\boldmath$v$}$}}^{-1}(C_{V})$ which are
not contained in $Z$ corresponds to, via ${\rm pr}_{1*}$, the residual part of
$m_{\underline{\eta}}(h)^{P}$ off $\delta_{D}$. Push the residual class to $Q$
via $\psi$. Through the interpretation of $h_{P}$ by the jet section $\Psi$
and a similar argument as in the proof of Lemma 6.11, we can see that the
obtained class in $Q$ is just the intersection class of $\tilde{\alpha}$ and
$\psi_{*}[P]=2[Q]$ with the opposite sign: $-2\,Q\cdot\tilde{\alpha}\in{\rm
CH}_{l-\ell}(Q)$. This implies (1).

Next, we show

(2)
$\displaystyle(i_{g})_{*}m_{\underline{\eta}}(g)=j_{*}\delta_{D}+(\tilde{A}+\tilde{B})\cdot\tilde{\alpha}.$

We remark that

  * (a)

at every point $p\in D_{0}$ (resp. $p^{\prime}\in D_{\infty}$), the tangent
space $T_{p}P$ is naturally identified with $\lambda_{q}=T_{q}\tilde{A}$
(resp. $\lambda_{q^{\prime}}=T_{q^{\prime}}\tilde{B}$) where $q=\psi(p)$ and
$q^{\prime}=\psi(p^{\prime})$;

  * (b)

thus, punctual Hilbert schemes are isomorphic, ${\rm
Hilb}^{n}(\mathcal{O}_{P,p})\simeq{\rm Hilb}^{n}(\mathcal{O}_{\tilde{A},q})$
and ${\rm Hilb}^{n}(\mathcal{O}_{P,p^{\prime}})\simeq{\rm
Hilb}^{n}(\mathcal{O}_{\tilde{B},q^{\prime}})$;

  * (c)

also the jets of $h_{P}:P\to Y$ at $p$ and $p^{\prime}$ are identified with
the jets of $g:\tilde{A}\sqcup\tilde{B}\to Y$ at $q$ and $q^{\prime}$,
respectively.

Let $W$ be a component of $s_{\mbox{\tiny$\mbox{\boldmath$v$}$}}^{-1}(C_{V})$
contained in $Z$. For $(x,z_{I})\in W$, assume that $I$ is supported at
$p={\rm pr}_{1}(x)\in D_{0}$, $p^{\prime}\in D_{\infty}$ (or $p\in
D_{\infty}$, $p^{\prime}\in D_{0}$) with $\pi_{P}(p)=\pi_{P}(p^{\prime})$, and
other $p_{3},\cdots,p_{r}$ in $P\sqcup\tilde{X}_{t}$. From the third remark
(c), the singularity type of $h_{P}$ at $p$ (resp. $p^{\prime}$) and that of
$g$ at $q$ (resp. $q^{\prime}$) are the same, say $\eta_{1}$ (resp.
$\eta_{2}$). Other points $p_{j}$ correspond to other mono-singularity types
$\eta_{j}$’s in $\underline{\eta}$ ($3\leq j\leq r$). If
$p_{j}\in\tilde{X}_{t}$, then it has the limit in $Q$ or
$\tilde{A}\sqcup\tilde{B}$ as $t\to 0$.

  * (d)

If some $p_{j}$ lies on $P$ and $\psi(p_{j})\in Q$ lies on some component
$\overline{\mathcal{Y}}_{k}$ of the cycle representing the class
$\tilde{\alpha}$, then there is some $\hat{p}_{j}\in\tilde{X}_{t}$ in the
locus satisfying that there are $\hat{p},\hat{p}^{\prime}\in D_{0}\sqcup
D_{\infty}$ with $\pi_{P}(\hat{p})=\pi_{P}(\hat{p}^{\prime})$ and other points
such that the multi-germ of $h$ at those points is of type $\underline{\eta}$.
This condition characterizes another component of
$s_{\mbox{\tiny$\mbox{\boldmath$v$}$}}^{-1}(C_{V})$ contained in $Z$, say
$W^{\prime}$, which is different from $W$. From a similar argument as the
above proof of the equality (1) using Lemma 6.11, we can see that ${\rm
pr}_{1*}[W]=-{\rm pr}_{1*}[W^{\prime}]$, and hence, the contributions from
those components $W$ and $W^{\prime}$ to the refined class $\delta_{d}$ cancel
each other out.

  * (e)

If every $p_{j}$ lies on $\tilde{X}_{t}$ and has the limit in
$\tilde{A}\sqcup\tilde{B}$, the limit of $p$ (the first entry), regarded as in
$D_{0}^{\prime}\sqcup D_{\infty}^{\prime}$, belongs to the
$\underline{\eta}$-multi-singularity locus of $g:\tilde{A}\sqcup\tilde{B}\to
Y$. Namely, it is a point in the $\eta_{1}$-locus of $g$ accompanied by other
points in $\tilde{A}\sqcup\tilde{B}$ of prescribed types with the same image
in $Y$. This condition characterizes the component $W$, that is, if $W$
consists of points $(x,z_{I})$ with this condition, ${\rm pr}_{1*}[W]$
contributes to the part of the loci class $m_{\underline{\eta}}(g)$ supported
on $D_{0}\sqcup D_{\infty}$.

  * (f)

Besides from (d) and (e), if all $p_{j}$ are paired so that each pair is just
the fiber of $D_{0}\sqcup D_{\infty}\to D$ at a point of $D$, then ${\rm
pr}_{1*}[W]$ also contributes to the part of $m_{\underline{\eta}}(g)$
supported on $D_{0}\sqcup D_{\infty}$.

By the argument in (d), we may consider only components $W$ in cases (e) and
(f). Consequently, the refined class $\delta_{D}$ becomes part of
$m_{\underline{\eta}}(g)$. The remaining part
$m_{\underline{\eta}}(g)-\delta_{D}$ should come from the
$\underline{\eta}$-singularities of $g$ off $D_{0}\sqcup D_{\infty}^{\prime}$,
that is, the intersection $(\tilde{A}+\tilde{B})\cdot\tilde{\alpha}$. Thus we
obtain (2). This completes the proof. $\Box$

####  6.2.7. Algebraic cobordism setting

We prove a stronger version of Proposition 6.1. Let us choose and fix an
$\Omega$-assignment $\mu$ associated to $\underline{\eta}$. We will show that

$n_{\underline{\eta};\mu}^{\Omega}(g)=n_{\underline{\eta};\mu}^{\Omega}(h)\;\;\in\Omega^{*}(Y).$

The proof essentially goes along the same line as seen above but with some
technical remarks. A main point is to lift up the equality
$[\tilde{X}_{t}]=[\tilde{A}]+[\tilde{B}]+2[Q]$ in ${\rm CH}^{*}(\tilde{M})$ to
the level of algebraic cobordism. Let

$i_{0}:D^{\prime}_{0}\hookrightarrow Q,\quad
i_{\infty}:D^{\prime}_{\infty}\hookrightarrow Q,\quad
j:Q\hookrightarrow\tilde{M}$

be the inclusions, and $D^{\prime}_{0},D^{\prime}_{\infty}$ are naturally
identified with $D$ (e.g., $P\to D^{\prime}_{0}$ is identified with $P\to D$
through $D_{0}^{\prime}\simeq D$). We then set

$\Theta:=[P\to Q]-(i_{0})_{*}[P\to D^{\prime}_{0}]-(i_{\infty})_{*}[P\to
D^{\prime}_{\infty}]\in\Omega^{*}(Q).$

Note that the cobordism class $\Theta$ becomes to be $2Q$ after applying
$\Omega^{*}\to{\rm CH}^{*}$.

######  Proposition 6.13.

In $\Omega^{*}(\tilde{M})$, it holds that

$[\tilde{X}_{t}\to\tilde{M}]=[\tilde{A}\to\tilde{M}]+[\tilde{B}\to\tilde{M}]+j_{*}\Theta.$

Proof : We refer to §3.1 and §3.2 in [LM07]. We set

$E:=\tilde{A}+\tilde{B}+2\tilde{Q},$

which is a strict normal crossing divisor in $\tilde{M}$ [LM07, Def.3.1.4].
Note that $\tilde{A}\cap\tilde{B}=\emptyset$. Put
$L_{A}:=O_{\tilde{M}}(\tilde{A})$, $L_{B}:=O_{\tilde{M}}(\tilde{B})$,
$L_{Q}:=O_{\tilde{M}}(Q)$, line bundles of those divisors.

First, by [LM07, Prop.3.1.9], we have

$[\tilde{X}_{t}\to\tilde{M}]=|O_{\tilde{M}}(E)|=c_{1}(L_{A}\otimes
L_{B}\otimes L_{Q}^{\otimes 2})\;\;\in\Omega^{*}(\tilde{M}).$

Since the singular locus $|E|_{\rm sing}$ is just $Q$, it follows from [LM07,
Lem.3.2.2] that there exists some $\theta\in\Omega^{*}(Q)$ such that

$None$
$[\tilde{X}_{t}\to\tilde{M}]=[\tilde{A}\to\tilde{M}]+[\tilde{B}\to\tilde{M}]+j_{*}\theta\;\;\in\Omega^{*}(\tilde{M}).$

We show $\theta=\Theta$ below. Expanding $\theta$ by the formal group law, we
may write777 Actually, $\theta^{\prime}$ is the sum of terms
$a_{ijk}c_{1}(j^{*}L_{A})^{i}c_{1}(j^{*}L_{B})^{j}c_{1}(j^{*}L_{Q})^{k}$ with
$a_{ijk}\in\mathbb{L}$, $i+j\geq 1$, $k\geq 1$ and $a_{i00}=a_{0j0}=0$ in the
expansion by the formal group law $F_{\mathbb{L}}$.

$\theta=2[id_{Q}]+\theta^{\prime}\;\;\;\in\Omega^{*}(Q)$

for some $\theta^{\prime}\in\Omega^{*}(Q)\cdot\gamma$, where
$\gamma:=c_{1}(O_{Q}(1))=c_{1}(j^{*}L_{Q})$, and $id_{Q}$ is the identity map
$Q\to Q$. Note that

$\tilde{\pi}_{*}[\tilde{X}_{t}\to\tilde{M}]=[{X}_{t}\to M]=[A\to M]+[B\to
M]-[P\to M]\;\;\in\Omega^{*}(M)$

by the double-point relation over $M$. Comparing this equality with $(1)$, we
get

$\tilde{\pi}_{*}j_{*}\theta=-[P\to M].$

Here, we may take the normal bundle of $D$ in $M$, instead of $M$, and then

$None$ $(\pi_{Q})_{*}\theta=-[P\to D]$

where $\pi_{Q}=\tilde{\pi}|_{Q}:Q\to D$ is the projection.

Since $\pi_{Q}:Q\to D$ is a projective bundle, $\Omega^{*}(Q)$ is a free
$\Omega^{*}(D)$-module of rank $2$ [HV11, Vi07]:

$None$
$\Omega^{*}(Q)\simeq\Omega^{*}(D)\cdot[id_{Q}]\oplus\Omega^{*}(D)\cdot\gamma$

Now $\psi:P\to Q$ is a $2$-to-$1$ surjective map branched along
$D_{0}^{\prime}\sqcup D_{\infty}^{\prime}$. It then follows from (3) that
there is an element $\xi\in\Omega^{*}(D)\cdot\gamma$ such that

$[P\to Q]=2[id_{Q}]+\xi\;\;\;\in\Omega^{*}(Q)$

(this also follows from the degree formula in [LM07]). Hence, $\theta=[P\to
Q]+\theta^{\prime}-\xi$. For $(\pi_{Q})_{*}[P\to Q]=[P\to D]$ and $(2)$, we
see

$[P\to Q]+\theta=2[P\to Q]+\theta^{\prime}-\xi\in\ker(\pi_{Q})_{*}.$

Here $\theta^{\prime}-\xi\in\Omega^{*}(D)\cdot\gamma$ and
$(\pi_{Q})_{*}(\theta^{\prime}-\xi)=-2[P\to D]$. Thus we may write it by

$\theta^{\prime}-\xi=-2(i_{0})_{*}[P\to D_{0}^{\prime}]=-(i_{0})_{*}[P\to
D_{0}^{\prime}]-(i_{\infty})_{*}[P\to D_{\infty}^{\prime}].$

That implies that $\theta=\Theta$. $\Box$

Note that every ingredient defined in ${\rm CH}^{*}$ in the previous
subsections can be formulated in algebraic cobordism setting – not only multi-
singularity loci classes $m^{\Omega}_{\underline{\eta};\mu}$ but also the
classes $\tilde{\alpha}$, $\tilde{\beta}$ and $\delta_{D}$ are defined in
$\Omega^{\ell-\kappa}(\tilde{M})$, $\Omega^{\ell-\kappa}(P)$ and
$\Omega^{*}(D_{0}^{\prime}\sqcup D^{\prime}_{\infty})$, respectively. In
entirely the same way as in §6.2.5, we get a cobordism version of Lemma 6.9:

######  Lemma 6.14.

In $\Omega^{\ell-\kappa}(\tilde{M})$, it holds that

(1) $\displaystyle
j_{*}\psi_{*}m^{\Omega}_{\underline{\eta};\mu}(h)^{P}=j_{*}\delta_{D}-\Theta\cdot\tilde{\alpha}$
(2)
$\displaystyle(i_{g})_{*}m^{\Omega}_{\underline{\eta};\mu}(g)=j_{*}\delta_{D}+([\tilde{A}\to\tilde{M}]+[\tilde{B}\to\tilde{M}])\cdot\tilde{\alpha}$

Consequently, we see that

$(i_{g})_{*}(m^{\Omega}_{\underline{\eta};\mu}(g))=(i_{h})_{*}(m^{\Omega}_{\underline{\eta};\mu}(h))\in\Omega^{\ell-\kappa+1}(\tilde{M}).$

Pushing the both sides to $Y$ via $(F_{1})_{*}$ yields

$n^{\Omega}_{\underline{\eta};\mu}(g)=n^{\Omega}_{\underline{\eta};\mu}(h)\in\Omega^{*}(Y).$

This completes the proof of Proposition 6.1 valued in $\Omega^{*}$. $\Box$

###  6.3. Target multi-singularity loci classes

We prove Theorem 4.17 along the line mentioned at the step (v) in
Introduction.

####  6.3.1. Residual class operation

######  Definition 6.15.

Let $f:X\to Y$ be a morphism of codimension $\kappa$ in ${\rm\bf Sm}_{k}$.

  1. (1)

For a type $\eta$ of mono-germs, we put

$\Delta n_{\eta}(f):=f_{*}(m_{\eta}(f))=n_{\eta}(f)\;\;\in{\rm
CH}^{\ell(\eta)}(Y).$

  2. (2)

For a multi-singularity type $\underline{\eta}=(\eta_{1},\cdots,\eta_{r})$
with $\ell:=\ell(\underline{\eta})$ $(r\geq 2)$, we recursively define

$\Delta n_{\underline{\eta}}(f):=n_{\underline{\eta}}(f)-\sum\,\Delta
n_{J_{1}}(f)\cdots\Delta n_{J_{s}}(f)\;\;\in{\rm CH}^{\ell}(Y)$

where the summand runs over all partitions into a disjoint union of at least
two non-empty unordered subsets, $\\{1,\cdots,r\\}=J_{1}\sqcup\cdots\sqcup
J_{s}$ $(s\geq 2)$.

Let $f:X_{1}\to Y$ and $g:X_{2}\to Y$ be morphisms in ${\rm\bf Sm}_{k}$ with
$\dim X_{1}=\dim X_{2}$, and $f+g:X_{1}\sqcup X_{2}\to Y$ be the disjoint sum
of two morphisms.

For a mono-singularity type $\alpha$ ($r=1$), the geometric subset
$\Xi(X_{1}\sqcup X_{2};\alpha)$ is the disjoint union of
$\Xi(X_{1};\alpha)\subset X_{1}^{[[n]]}$ and $\Xi(X_{2};\alpha)\subset
X_{2}^{[[n]]}$ (Lemma 4.3), and thus it follows from the definition of
degeneracy class of maps (Defnition 4.9) that

$n_{\alpha}(f+g)=n_{\alpha}(f)+n_{\alpha}(g).$

As a double-check, this equality follows from the fact that the Thom
polynomial for mono-singularity type is a polynomial of $c_{i}(f)$’s.

In case of $r=2$, for a bi-germ $\underline{\eta}=(\alpha,\beta)$, we can show
that

$n_{\alpha\beta}(f+g)=n_{\alpha\beta}(f)+n_{\alpha\beta}(g)+n_{\alpha}(f)n_{\beta}(g)+n_{\alpha}(g)n_{\beta}(f)$

and therefore

$\displaystyle\Delta n_{\alpha\beta}(f+g)$ $\displaystyle=$ $\displaystyle
n_{\alpha\beta}(f+g)-n_{\alpha}(f+g)\cdot n_{\beta}(f+g)$ $\displaystyle=$
$\displaystyle
n_{\alpha\beta}(f+g)-(n_{\alpha}(f)+n_{\alpha}(g))(n_{\beta}(f)+n_{\beta}(g))$
$\displaystyle=$ $\displaystyle
n_{\alpha\beta}(f)-n_{\alpha}(f)n_{\beta}(f)+n_{\alpha\beta}(g)-n_{\alpha}(g)n_{\beta}(g)$
$\displaystyle=$ $\displaystyle\Delta n_{\alpha\beta}(f)+\Delta
n_{\alpha\beta}(g).$

More generally, the following proposition holds:

######  Proposition 6.16.

The class $\Delta n_{\underline{\eta}}(f)$ depends only on the cobordism class
of $f$, and it yields a well-defined group homomorphism

$\Delta n_{\underline{\eta}}:\Omega^{\kappa}(Y)\to{\rm
CH}^{\ell}(Y),\quad[f]\mapsto\Delta n_{\underline{\eta}}(f).$

In particular, $n_{\underline{\eta}}(f)$ also depends only on the cobordism
class of $f$.

Proof : Note that $\Delta n_{\underline{\eta}}:\mathcal{M}^{\kappa}(Y)\to{\rm
CH}^{\ell}(Y)$ is well-defined as a map between sets, because
$n_{\underline{\eta}}(f)$ is invariant under the action of isomorphisms of the
source spaces (Lemma 4.12 (2)). First, we show

Claim: $\Delta n_{\underline{\eta}}$ is additive on the monoid
$\mathcal{M}^{\kappa}(Y)$.

We use the induction on $r$. In mono-singularity case ($r=1$), the claim is
clear as seen above. Let $r\geq 2$ and suppose that the claim holds for any
$r^{\prime}$-tuple multi-singularities with $r^{\prime}<r$.

Now let $\underline{\eta}=(\eta_{1},\cdots,\eta_{r})$, a multi-singularity
type. Let $f:X_{1}\to Y$ and $g:X_{2}\to Y$ be maps in ${\rm\bf Sm}_{k}$ with
the same codimension $\kappa$ and the same target space.

We recall Lemma 4.3: in $(X_{1}\sqcup X_{2})^{[[n]]}$

$\Xi(X_{1}\sqcup X_{2};\underline{\eta})=\bigsqcup_{I\sqcup
J}\Xi(X_{1};\underline{\eta}_{I})\times\Xi(X_{2};\underline{\eta}_{J}).$

Consider the intersection product of the Hilbert extension map

$(f+g)^{[[n]]}:(X_{1}\sqcup X_{2})^{[[n]]}\to((X_{1}\sqcup X_{2})\times
Y)^{[[n]]}$

with the subscheme $\Xi(X_{1}\sqcup X_{2};\underline{\eta})\times Y$. Pushing
it out to $Y$, we have

$n_{\underline{\eta}}(f+g)=n_{\underline{\eta}}(f)+n_{\underline{\eta}}(g)+\sum_{I\sqcup
J}n_{I}(f)\cdot n_{J}(g),$

where $I\sqcup J=\\{1,\cdots,r\\}$ and both $I$ and $J$ are non-empty. It is
for $n_{I}(f)\cdot n_{J}(g)\in{\rm CH}^{\ell}(Y)$ is the pull back of
$n_{I}(f)\times n_{J}(g)\in{\rm CH}^{*}(Y\times Y)$ via the diagonal embedding
$Y\hookrightarrow Y\times Y$. By Definition 6.15, it holds that

$n_{I}(f)=\sum\Delta n_{I_{1}}(f)\cdots\Delta n_{I_{t}}(f)$

where the summand runs over all partitions of $I$ into non-empty unordered
subsets ($t\geq 1$), and $n_{J}(g)$ is also. Therefore, we see that

$\sum_{I\sqcup J}n_{I}(f)\cdot n_{J}(g)=\sum_{(J_{i};h_{i})}\Delta
n_{J_{1}}(h_{1})\cdots\Delta n_{J_{s}}(h_{s})$

where $J_{1}\sqcup\cdots\sqcup J_{s}\;(s\geq 2)$ are partitions of
$\\{1,\cdots,r\\}$ into non-empty unordered subsets and for each $i$, $h_{i}$
denotes either $f$ or $g$ but all $h_{1},\cdots,h_{s}$ are not the same
simultaneously. Now, since each $J_{i}$ satisfies $|J_{i}|<r$, the induction
hypothesis implies that for each $1\leq i\leq s$,

$\Delta n_{J_{i}}(f+g)=\Delta n_{J_{i}}(f)+\Delta n_{J_{i}}(g).$

Thus we see

$\displaystyle\Delta n_{\underline{\eta}}(f+g)$
$\displaystyle=n_{\underline{\eta}}(f+g)-\sum\,\Delta
n_{J_{1}}(f+g)\cdots\Delta n_{J_{s}}(f+g)$
$\displaystyle=n_{\underline{\eta}}(f)+n_{\underline{\eta}}(g)+\sum_{I\sqcup
J}n_{I}(f)\cdot n_{J}(g)-\sum\prod(\Delta n_{J_{i}}(f)+\Delta n_{J_{i}}(g))$
$\displaystyle=\left(n_{\underline{\eta}}(f)-\sum\,\prod\Delta
n_{J_{i}}(f)\right)+\left(n_{\underline{\eta}}(g)-\sum\,\prod\Delta
n_{J_{i}}(g)\right)$ $\displaystyle=\Delta n_{\underline{\eta}}(f)+\Delta
n_{\underline{\eta}}(g).$

Hence, the claim is proven.

It immediately yields a group homomorphism

$\Delta n_{\underline{\eta}}:\mathcal{M}^{\kappa+}(Y)\to{\rm CH}^{\ell}(Y).$

Recall that $\mathcal{R}(Y)$ is generated by the double-point relation:

$[g:A\sqcup B\to Y]-[h:X_{1}\sqcup P\to Y]\in\mathcal{M}^{\kappa+}(Y).$

Proposition 6.1 says that $n_{\underline{\eta}}(g)=n_{\underline{\eta}}(h)$.
From Definition 6.15, by the induction on $r$, we see that

$\Delta n_{\underline{\eta}}(g-h)=\Delta n_{\underline{\eta}}(g)-\Delta
n_{\underline{\eta}}(h)=0,$

i.e., the homomorphism $\Delta n_{\underline{\eta}}$ vanishes on
$\mathcal{R}(Y)$. Thus we have a well-defined group homomorphism

$\Delta n_{\underline{\eta}}:\Omega^{\kappa}(Y)\to{\rm
CH}^{\ell}(Y),\quad[f]\mapsto\Delta n_{\underline{\eta}}(f).$

This completes the proof. $\Box$

######  Proposition 6.17.

$\Delta n_{\underline{\eta}}:\Omega^{\kappa}(Y)\to{\rm CH}^{\ell}(Y)$ is an
additive cohomology operation.

Proof : Thanks to Vishik’s result (Theorem 5.3), it suffices to show that for
$Y=\prod\mathbb{P}^{n_{j}}$, product of projective spaces, $\Delta
n_{\underline{\eta}}$ commutes with the pullback

$\varphi^{*}:\Omega^{*}(Y)\to\Omega^{*}(Y^{\prime})$

for any morphisms $\varphi:Y^{\prime}\to Y$ listed in Theorem 5.3. Let $Y$ and
$\varphi$ be so. Take an arbitrary cobordism class $[f:X\to
Y]\in\Omega^{\kappa}(Y)$. By moving by translation, $f$ can be linearly
deformed to be transverse to $\varphi$ so that the cobordism class does not
change (Remark 5.5). So we may assume $f$ itself admits the transversality,
then the fibre square

$f^{\prime}:X\times_{Y}Y^{\prime}\to Y^{\prime}$

exists in ${\rm\bf Sm}_{k}$ and satisfies $\varphi^{*}([f])=[f^{\prime}]$
(Remark 5.5). Lemma 4.13 says that $n_{\underline{\eta}}$ satisfies the
naturality for base-change. From Definition 6.15 combined with the induction
on $r$, we see that $\Delta n_{\underline{\eta}}$ also satisfies the same
base-change property: $\Delta
n_{\underline{\eta}}([f^{\prime}])=\varphi^{*}(\Delta
n_{\underline{\eta}}([f]))$. Hence we have

$\Delta n_{\underline{\eta}}\circ\varphi^{*}=\varphi^{*}\circ\Delta
n_{\underline{\eta}}$

for any $[f]\in\Omega^{\kappa}(Y)$. This completes the proof. $\Box$

We immediately have

######  Corollary 6.18.

For each multi-singularity type $\underline{\eta}$,

$n_{\underline{\eta}}:\Omega^{\kappa}(Y)\to{\rm CH}^{\ell}(Y)$

is a well-defined cohomology operation (not additive if $r\geq 2$).

####  6.3.2. Theorem in cobordism setting

Now we switch from ${\rm CH}^{*}$ to $\Omega^{*}$. Let
$\underline{\eta}=(\eta_{1},\cdots,\eta_{r})$ be a multi-singularity type of
map-germ with codimension $\kappa$. Suppose that we are given an
$\Omega$-assignment $\mu$ for the type $\underline{\eta}$ (Definition 5.6).
For an arbitrary proper morphism $f:X\to Y$,
$n_{\underline{\eta};\mu}^{\Omega}(f)\in\Omega^{\ell}(Y)$ is defined
(Definition 5.7). We show the following result:

######  Theorem 6.19.

(Target cobordism Thom polynomials) For every $J\subset\\{1,\cdots,r\\}$,
there is a residual polynomial $R_{J}$ of $c_{1},c_{2},\cdots$ with
coefficients in $\mathbb{L}\otimes_{Z}\mathbb{Q}$ such that for an arbitrary
proper morphism $f:X\to Y$ with codimension $\kappa$, the target multi-
singularity loci cobordism class is expressed by

$n_{\underline{\eta};\mu}^{\Omega}(f)=\sum f_{*}(R_{J_{1}})\cdots
f_{*}(R_{J_{s}})$

in $\Omega^{\ell}(Y)_{\mathbb{Q}}$, where the summand runs over all
partitions.

Proof : It immediately follows that

  * -

Proposition 6.1 holds also in $\Omega^{*}$-setting (proven in §6.2.7);

  * -

the class $\Delta n_{\underline{\eta};\mu}^{\Omega}(f)$ is also defined
(Definition 6.15);

  * -

Propositions 6.16 and 6.17 in $\Omega$-setting are shown in entirely the same
way, and thus

$\Delta
n_{\underline{\eta};\mu}^{\Omega}:\Omega^{\kappa}(Y)\to\Omega^{\ell}(Y)$

is a well-defined additive operation.

Now, we are ready to employ Theorem 5.4 (Vishik [Vi19, Thm.1.2]). We conclude
that the operation $\Delta n_{\underline{\eta};\mu}^{\Omega}$ is uniquely
written by a linear combination of Landweber-Novikov operations:

$\Delta n_{\underline{\eta};\mu}^{\Omega}=\sum
a_{I}s_{I}\quad(a_{I}\in\mathbb{L}\otimes_{\mathbb{Z}}\mathbb{Q}).$

Put $R_{\underline{\eta}}:=\sum a_{I}c^{I}$, then

$f_{*}(R_{\underline{\eta}}(f))=\Delta
n_{\underline{\eta};\mu}^{\Omega}([f]).$

Hence, by the induction on $r$ and the definition of $\Delta
n_{\underline{\eta};\mu}^{\Omega}([f])$, we get the desired expression of the
class $n_{\underline{\eta};\mu}^{\Omega}(f)$. Namely, we have shown the
existence of the (target) cobordism Thom polynomial for an $\Omega$-assignment
$\mu$ associated to $\underline{\eta}$. $\Box$

####  6.3.3. Proof of Theorem 4.17

For any cohomology theory $A^{*}$ obtained from $\Omega^{*}$, we define the
$A^{*}$-multi-singularity loci classes associated to an $\Omega$-assignment
$\mu$. In particular, in case of $A^{*}={\rm CH}^{*}$ (and also
$K^{0}[\beta,\beta^{-1}]$), the classes depend only on $\underline{\eta}$, not
the choice of $\mu$:

$\textstyle{\Omega^{\kappa}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n_{\underline{\eta};\mu}^{\Omega}}$$\scriptstyle{n_{\underline{\eta}}}$$\textstyle{\Omega^{\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\otimes_{\mathbb{L}}\mathbb{Z}}$$\textstyle{{\rm
CH}^{\ell}}$

In Theorem 6.19, it is shown that $n_{\underline{\eta};\mu}^{\Omega}$ is
expressed by a universal multi-singularity Thom polynomial. After reducing to
${\rm CH}^{*}$ (or $K^{0}[\beta,\beta^{-1}]$), such universal polynomials for
any choices of $\mu$ coincide with each other. This completes the proof of
Theorem 4.17. $\Box$

###  6.4. Source multi-singularity loci classes

We prove Theorem 4.18 by using basic materials of the restriction method due
to R. Rimányi [Ri01, Ri02, Ka03, Ka06, RS98, Oh06, Oh12]. The proof uses
Thoerem 4.17 but also relies on a different nature which comes from local
symmetry of singularities and equivariant cohomology. That suggests there
should be a comprehensive new theory, see §7.1.8 and §7.1.9. Note that in the
statement of Theorem 4.18, the case of $\kappa<0$ is excluded (the main reason
is mentioned in §7.1.5).

Now, fix $\kappa=l-m\geq 0$. We work with ${\rm CH}^{*}_{\mathbb{Q}}$
throughout this section.

######  Remark 6.20.

(Quasi-homogenious normal forms) We say that a multi-singularity type
$\underline{\eta}=(\eta_{1},\cdots,\eta_{r})$ is quasi-homogeneous if every
entry $\eta_{i}$ admits quasi-homogeneous polynomial normal forms in suitable
local coordinates (under $\mathcal{K}$-equivalence). For instance, considering
the classification list [Mat71, §5] and a standard construction of stable
unfoldings of finitely determined map-germs [Mat69], it turns out that every
locally stable map-germ in Mather’s nice dimensions is of quasi-homogeneous
type, see [MN20]. Below we only consider such singularity types (cf.
admissible maps in Theorem §4.18).

####  6.4.1. Symmetry of singularity type

We denote the group of formal right-left equivalence by

$\hat{\mathcal{A}}_{S}:=Aut(\hat{\mathcal{O}}_{\mathbb{A}^{m},S})\times
Aut(\hat{\mathcal{O}}_{\mathbb{A}^{l},0})$

($S\subset\mathbb{A}^{m}$ with $|S|=s$), which acts on the space of map-germs
$f:(\mathbb{A}^{m},S)\to(\mathbb{A}^{l},0)$ by
$(\sigma,\varphi).f:=\varphi\circ f\circ\sigma^{-1}$. If $s=1$ (the case of
mono-germs), we simply write it by $\hat{\mathcal{A}}$. Note that any quasi-
homogeneous germ has the isotropy subgroup in $\hat{\mathcal{A}}_{S}$ which
contains the non-trivial maximal torus, because there is at least a non-
trivial $GL(1)$-action on the source and the target space preserving the map-
germ [Ri01, Ka03].

Mono-singularities: Let $\xi$ be a quasi-homogeneous mono-singularity type of
codimension $\kappa$, and set $m^{\prime}:=\ell(\xi)-\kappa$ and
$l^{\prime}:=\ell(\xi)$. Consider a mono-germ of locally stable map of that
type; abusing the notation, we denote it by

$\xi:(\mathbb{A}^{m^{\prime}},0)\to(\mathbb{A}^{l^{\prime}},0).$

Abusing the notation, we use the same letter $\xi$ to mean the singularity
type as well. The symmetry group $G(\xi)$ of the singularity type $\xi$ is
defined to be the isotopy subgroup of $\hat{\mathcal{A}}$ for $\xi$, i.e., it
consists of elements $(\sigma,\varphi)\in\hat{\mathcal{A}}$ with
$(\sigma,\varphi).\xi=\xi$; in other words, it is defined by the group of
automorphims of the quotient formal algebra of $\xi$. Since any locally stable
map-germ is $(l^{\prime}+1)$-determined [Mat69], we only have to work at the
level of $(l^{\prime}+1)$-jets. Thus we may assume that the symmetry group is
realized as a subgroup of a certain linear algebraic group. There are
linearized representations $\lambda_{0}:G(\xi)\to GL(m^{\prime})$ and
$\lambda_{1}:G(\xi)\to GL(l^{\prime})$ into general linear groups acting on
the source and the target spaces $\mathbb{A}^{m^{\prime}}$ and
$\mathbb{A}^{l^{\prime}}$, respectively. By the algebraic Borel construction
[To99], there exist the classifying space $BG(\xi)$ of the group $G(\xi)$ as
an ind-variety, and it admits the universal principal $G(\xi)$-bundle
$EG(\xi)\to BG(\xi)$. Put

$E_{0}(\xi):=EG(\xi)\times_{\rho_{0}}\mathbb{A}^{m^{\prime}},\quad
E_{1}(\xi):=EG(\xi)\times_{\rho_{1}}\mathbb{A}^{l^{\prime}},$

which are vector bundles of rank $m^{\prime}$ and $l^{\prime}$ over $BG(\xi)$
associated to the representations $\lambda_{0}$ and $\lambda_{1}$,
respectively. Since the germ $\xi$ is invariant under the action of $G(\xi)$,
we can construct a morphism $f_{\xi}$ between total spaces of those bundles so
that the restriction on each fiber is $\mathcal{A}$-equivalent to $\xi$:

$\textstyle{E_{0}(\xi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\xi}}$$\scriptstyle{\pi_{0}}$$\textstyle{E_{1}(\xi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\textstyle{BG(\xi)}$

######  Remark 6.21.

By Remark 6.20, $G(\xi)$ has the non-trivial maximal torus, thus especially,
the top Chern classes of $E_{0}(\xi)$ and $E_{1}(\xi)$ are non-zero divisors.
Since $BGL(1)=\mathbb{P}^{\infty}$, we may think of $BG(\xi)$ as a projective
smooth variety of dimension high enough.

Multi-singularities: Let $\underline{\xi}$ be a quasi-homogenous multi-
singularity type of codimension $\kappa$, and set
$m^{\prime}:=\ell(\underline{\xi})-\kappa$ and
$l^{\prime}:=\ell(\underline{\xi})$, i.e.,

$\underline{\xi}=(\xi_{1},\xi_{2},\cdots,\xi_{s}),\quad\xi_{i}:(\mathbb{A}^{m^{\prime}},0)\to(\mathbb{A}^{l^{\prime}},0).$

Here, especially the first entry $\xi_{1}$ is distinguished from others;
denote the rest $(s-1)$-tuple by
$\underline{\xi}/\xi_{1}:=(\xi_{2},\cdots,\xi_{s})$. Let
$Aut(\underline{\xi})\subset\mathfrak{S}_{s}$ denote the subgroup of
permutations of $\\{1,\cdots,s\\}$ preserving the collection
$\underline{\xi}$, and we regard $Aut(\underline{\xi}/\xi_{1})\subset
Aut(\underline{\xi})$ as a subgroup fixing $1$ (see Proposition 4.15)888For
instance, if $\underline{\xi}=(A_{0},A_{1},A_{0},A_{0},A_{2})$,
$Aut(\underline{\xi})=\\{\,\sigma\in\mathfrak{S}_{5}\,|\,\sigma(2)=2,\sigma(5)=5\,\\}\simeq\mathfrak{S}_{3}$
and
$Aut(\underline{\xi}/\xi_{1})=\\{\,\sigma\,|\,\sigma(1)=1,\sigma(2)=2,\sigma(5)=5\,\\}\simeq\mathfrak{S}_{2}$.
.

The semi-direct product $\hat{\mathcal{A}}_{S}\ltimes\mathfrak{S}_{s}$ acts on
the space of all ordered $s$-tuples of mono-germs. Let
$G(\underline{\xi})\subset\hat{\mathcal{A}}_{S}\ltimes Aut(\underline{\xi})$
denote the isotropy subgroup at $\underline{\xi}$. Also we define the smaller
isotropy subgroup $G(\underline{\xi}/\xi_{1})$ by the intersection of
$G(\underline{\xi})$ with $\hat{\mathcal{A}}_{S}\ltimes
Aut(\underline{\xi}/\xi_{1})$. Let $\lambda_{0}$ be the linearized
representation of $G(\underline{\xi}/\xi_{1})$ on the source domain
$\mathbb{A}^{m^{\prime}}$ of the first (fixed) mono-germ $\xi_{1}$, and
$\lambda_{1}$ be the representation of $G(\underline{\xi})$ on the common
target domain $\mathbb{A}^{l^{\prime}}$ of the multi-germ $\underline{\xi}$.
By the Borel construction again, we associate the universal vector bundles

$X_{\underline{\xi}}:=EG(\underline{\xi}/\xi_{1})\times_{\lambda_{0}}\mathbb{A}^{m^{\prime}},\quad
Y_{\underline{\xi}}:=EG(\underline{\xi})\times_{\lambda_{1}}\mathbb{A}^{l^{\prime}}$

with the base spaces being the classifying spaces
$BG(\underline{\xi}/\xi_{1})$ and $BG(\underline{\xi})$, respectively. The
inclusion map between the two groups induces a morphism

$\rho_{\underline{\xi}}:BG(\underline{\xi}/\xi_{1})\to BG(\underline{\xi}),$

which is a $d$-to-$1$ covering with

$d:=\deg_{1}(\underline{\xi})=\\#{\rm Aut}(\underline{\xi})/\\#{\rm
Aut}(\underline{\xi}/\xi_{1}).$

In fact, the algebraic Borel construction uses finite dimensional
approximation of the classifying spaces [To99], so we may regard
$\rho_{\underline{\xi}}$ as a covering of schemes of sufficiently high
dimension. Furthermore, over the map $\rho_{\underline{\xi}}$, we obtain a
universal singular map $f_{\underline{\xi}}$ between the total spaces of the
above vector bundles

$\textstyle{X_{\underline{\xi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\underline{\xi}}}$$\scriptstyle{\pi_{0}}$$\textstyle{Y_{\underline{\xi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\textstyle{BG(\underline{\xi}/\xi_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{\underline{\xi}}}$$\textstyle{BG(\underline{\xi})}$

by the gluing construction such that the restriction of $f_{\underline{\xi}}$
to every fiber is $\mathcal{A}$-equivalent to the mono-germ $\xi_{1}$. Note
that $f_{\underline{\xi}}$ itself is proper, for $\kappa\geq 0$. Thus the
pushforward $(f_{\underline{\xi}})_{*}$ is defined.

######  Lemma 6.22.

$(f_{\underline{\xi}})_{*}:{\rm
CH}^{*}(X_{\underline{\xi}})_{\mathbb{Q}}\to{\rm
CH}^{*+\kappa}(Y_{\underline{\xi}})_{\mathbb{Q}}$ is injective.

Proof : The groups $G(\underline{\xi}/\xi_{1})$ and $G(\underline{\xi})$ share
the common non-trivial maximal torus, and thus ${\rm
CH}^{*}(X_{\underline{\xi}})_{\mathbb{Q}}\simeq{\rm
CH}^{*}(Y_{\underline{\xi}})_{\mathbb{Q}}$ via the pullback
$f_{\underline{\xi}}^{*}$, moreover, those are isomorphic to ${\rm
CH}^{*}(BG(\underline{\xi}))_{\mathbb{Q}}$. Note that
$f_{\underline{\xi}*}(f_{\underline{\xi}}^{*}\omega)=f_{\underline{\xi}*}(1)\cdot\omega$
and the top Chern classes of $X_{\underline{\xi}}$ and $Y_{\underline{\xi}}$
are non-zero divisors (Remark 6.21). Further, the zero section of
$X_{\underline{\xi}}$ is sent via $f_{\underline{\xi}}$ to the zero section of
$Y_{\underline{\xi}}$, i.e., the restriction map between the zero sections is
identical to $\rho_{\underline{\xi}}$. Hence, we have an obvious relation
between the top Chern classes

$f_{\underline{\xi}*}(1)\cdot
c_{m^{\prime}}(X_{\underline{\xi}})=d\,c_{l^{\prime}}(Y_{\underline{\xi}})$

in ${\rm CH}^{l^{\prime}}(BG(\underline{\xi}))_{\mathbb{Q}}$ (cf. [Ka06,
Lem.4.1]), and especially, $f_{\underline{\xi}*}(1)$ is a non-zero divisor.
The injectivity of $f_{\underline{\xi}*}$ follows from this relation. $\Box$

The constructed universal maps $f_{\xi}$ and $f_{\underline{\xi}}$ are
building blocks of the restriction method (Remark 6.25).

####  6.4.2. Stratifications

From now on, we consider an admissible map in ${\rm\bf Sm}_{k}$

$f:X\to Y\quad(m=\dim X,\;\;l=m+\kappa=\dim Y,\;\;\kappa\geq 0).$

That is, $f$ is a proper locally stable map which has only quasi-homogeneous
singularity types and satisfies ‘stratawise normal Euler condition’. The
latter condition will be used in a Mayer-Vietoris argument arising in the
final step of §6.4.4.

Since $f$ is locally stable, we obtain a stratification of each of $X$ and $Y$
into several multi-singularity loci in a coherent way subject to $f$. Every
stratum is reduced, locally closed, but not assumed to be connected. Also
every stratum satisfies the frontier condition – its closure is the union of
some strata of lower dimension (called boundary strata). These statifications
are indexed by multi-singularity types in the following sense999In case of
$\kappa>0$, $M_{A_{0}}$ for mono-type $A_{0}$ of immersion-germ is the unique
open stratum in $X$, and every positive codimensional stratum (of type
$A_{0}^{2}$, $A_{1}$ and so on) is adjacent to it; especially,
$M_{A_{0}}^{\circ}\to N_{A_{0}}^{\circ}$ is isomorphic. In case of $\kappa=0$,
we also include the type $A_{0}$, although our original definition of
$\underline{\xi}$ excludes this type, and we always assume that our
equidimensional proper map $f:X\to Y$ is dominant; especially,
$M_{A_{0}}^{\circ}\to N_{A_{0}}^{\circ}$ is a $d$-to-$1$ covering, where $d$
is the degree of $f$. . The source manifold is stratified as

$X=\bigsqcup M_{\underline{\xi}}^{\circ}$

where the union is taken over all mono or semi-ordered multi-singularity types
of codimension $\kappa$ (that means that the first entry is distinguished and
the others are non-ordered). The target manifold is stratified as

$Y=(Y-f(X))\;\sqcup\;\bigsqcup N_{\underline{\xi}}^{\circ}$

where the union is taken over all non-ordered mono/multi-singularity types.
Here, we abuse the notation so that for each semi-ordered/non-ordered type, an
ordered multi-singularity type $\underline{\xi}$ is selected as a
representative. The number of multi-singularity types $\underline{\xi}$ with
$\ell(\underline{\xi})\leq l$ are finite, so the strata are finite.

Note that $\dim M_{\underline{\xi}}^{\circ}=\dim
N_{\underline{\xi}}^{\circ}=l-\ell(\underline{\xi})$ and the restriction of
$f$ to every stratum

$f:M_{\underline{\xi}}^{\circ}\to N_{\underline{\xi}}^{\circ}$

is either an isomorphism or a finite-to-one covering (the degree is
$\deg_{1}(\underline{\xi})$). If an entry $\xi_{j}$ with $j\not=1$ is of
different type from $\xi_{1}$, the preimage
$f^{-1}(N_{\underline{\xi}}^{\circ})\subset X$ contains another stratum
$M_{\underline{\sigma\xi}}^{\circ}$ with some $\sigma\in Aut(\underline{\xi})$
with $\sigma(1)=j$.

We denote the embeddings of those loci by

$i_{\underline{\xi}}:M_{\underline{\xi}}^{\circ}\hookrightarrow X,\quad
j_{\underline{\xi}}:N_{\underline{\xi}}^{\circ}\hookrightarrow
Y\quad(j_{\underline{\xi}}=f\circ i_{\underline{\xi}})$

and their normal bundles, respectively, by

$p_{0}:{\rm N}_{0}(\underline{\xi})\to M_{\underline{\xi}}^{\circ},\quad
p_{1}:{\rm N}_{1}(\underline{\xi})\to N_{\underline{\xi}}^{\circ}$

with the zero-sections $s_{0}:M_{\underline{\xi}}^{\circ}\to{\rm
N}_{0}(\underline{\xi})$ and $s_{1}:N_{\underline{\xi}}^{\circ}\to{\rm
N}_{1}(\underline{\xi})$. Below we often denote the normal bundles by ${\rm
N}_{0}$ and ${\rm N}_{1}$ for short if it does not make any confusion. Since
the structure groups of those vector bundles are reduced to the groups
$G(\underline{\xi}/\xi_{1})$ and $G(\underline{\xi})$, respectively, there are
classifying maps101010We simplify our notation of classifying maps as noted in
the footnote of §2.1.1. such that those bundles are induced from
$X_{\underline{\xi}}$ and $Y_{\underline{\xi}}$, respectively; let $\rho_{0}$
and $\rho_{1}$ denote the bundle morphisms. In a canonical way, we obtain (the
germ at the zero-section of) a locally stable map111111Intuitively,
$\varphi_{f,\underline{\xi}}$ is thought of as an algebro-geometric
alternative to the restriction map of $f:X\to Y$ to a tubular neighborhood of
$M_{\underline{\xi}}^{\circ}$ in $X-\partial M_{\underline{\xi}}$.

$\varphi_{f,\underline{\xi}}:{\rm N}_{0}\to{\rm N}_{1}$

such that the germ of $\varphi_{f,\underline{\xi}}$ at $x_{1}\in
M_{\underline{\xi}}^{\circ}$ (zero-section) is isomorphic to the germ
$f:(X,x_{1})\to(Y,f(x_{1}))$. In fact, $\varphi_{f,\underline{\xi}}$ is
induced from the universal singular map $f_{\underline{\xi}}$, i.e., we have
the following commutative diagram (we set
$B_{0}:=BG(\underline{\xi/\xi_{1}})$, $B_{1}:=BG(\underline{\xi})$):

$\textstyle{{\rm
N}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{f,\underline{\xi}}}$$\scriptstyle{p_{0}}$$\scriptstyle{\rho_{0}}$$\textstyle{X_{\underline{\xi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\underline{\xi}}}$$\textstyle{{\rm
N}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{1}\qquad}$$\textstyle{Y_{\underline{\xi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{M_{\underline{\xi}}^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\underline{\xi}}}$$\scriptstyle{f}$$\textstyle{B_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{\underline{\xi}}}$$\textstyle{Y}$$\textstyle{N_{\underline{\xi}}^{\circ}\;\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\underline{\xi}}}$$\textstyle{B_{1}}$

At every $p\in M_{\underline{\xi}}^{\circ}$, the tangent space at $s_{0}(p)$
of the total space ${\rm N}_{0}$ splits into the tangent
$T_{p}M_{\underline{\xi}}^{\circ}$ and the normal ${\rm N}_{0}|_{p}$, thus,
especially, we have

$s_{0}^{*}T{\rm N}_{0}=i_{\underline{\xi}}^{*}TX,\quad
s_{0}^{*}\varphi_{f,\underline{\xi}}^{*}T{\rm
N}_{1}=j_{\underline{\xi}}^{*}f^{*}TY.$

Set $\partial
M_{\underline{\xi}}:=M_{\underline{\xi}}-M_{\underline{\xi}}^{\circ}$ (the
union of boundary strata). We let $i_{\underline{\xi}}^{*}$ denote the
composition ${\rm CH}^{q}(X)_{\mathbb{Q}}\to{\rm CH}^{q}(X-\partial
M_{\underline{\xi}})_{\mathbb{Q}}\to{\rm
CH}^{q}(M_{\underline{\xi}}^{\circ})_{\mathbb{Q}}$ of the flat pullback (of
the open embedding) and the Gysin map (of the closed embedding).

######  Lemma 6.23.

It holds that

$s_{0}^{*}\rho_{0}^{*}c(f_{\underline{\xi}})=i_{\underline{\xi}}^{*}c(f)\in{\rm
CH}^{*}(M_{\underline{\xi}}^{\circ})_{\mathbb{Q}}.$

Also $s_{I}(f_{\underline{\xi}})$ and $s_{I}(f)$ coincide through the pullback
of $\rho_{1}\circ s_{1}$ and $j_{\underline{\xi}}$.

Proof : Since ${\rm N}_{0}$ and ${\rm N}_{1}$ are induced from the universal
bundles $X_{\underline{\xi}}$ and $Y_{\underline{\xi}}$, respectively, we see

$\rho_{0}^{*}c(\rho_{\underline{\xi}}^{*}Y_{\underline{\xi}}-X_{\underline{\xi}})=c(f^{*}{\rm
N}_{1}-{\rm N}_{0})\in{\rm CH}^{*}(M_{\underline{\xi}}^{\circ})_{\mathbb{Q}}$

(here we abuse the notation $\rho_{0}$ to mean the classifying map
$M_{\underline{\xi}}^{\circ}\to B_{0}$ as well). Over the total space ${\rm
N}_{0}$, the tangent bundle $T{\rm N}_{0}$ (resp.
$\varphi_{f,\underline{\xi}}^{*}T{\rm N}_{1}$) splits into the pullback via
$p_{0}^{*}$ of two factors, the tangent bundle $TM_{\underline{\xi}}^{\circ}$
of the base space and the normal bundle ${\rm N}_{0}$ itself (resp. $f^{*}{\rm
N}_{1}$). Then we may write

$c(\varphi_{f,\underline{\xi}}):=c(\varphi_{f,\underline{\xi}}^{*}T{\rm
N}_{1}-T{\rm N}_{0})=p_{0}^{*}c(f^{*}{\rm N}_{1}-{\rm N}_{0})$

in ${\rm CH}^{*}({\rm N}_{0})_{\mathbb{Q}}$ by canceling the factor
$TM_{\underline{\xi}}^{\circ}$. That is the same for the tangent bundle
$TX_{\underline{\xi}}$ and $f_{\underline{\xi}}^{*}TY_{\underline{\xi}}$ over
$X_{\underline{\xi}}$, and we get

$c(f_{\underline{\xi}}):=c(f_{\underline{\xi}}^{*}TY_{\underline{\xi}}-TX_{\underline{\xi}})=\pi_{0}^{*}c(\rho_{\underline{\xi}}^{*}Y_{\underline{\xi}}-X_{\underline{\xi}})$

in ${\rm CH}^{*}(X_{\underline{\xi}})_{\mathbb{Q}}$ (here we may regard the
base space $B_{0}$ as a finite dimensional approximation in the algebraic
Borel construction [To99]). In particular, we have

$c(\varphi_{f,\underline{\xi}})=\rho_{0}^{*}c(f_{\underline{\xi}})\in{\rm
CH}^{*}({\rm N}_{0})_{\mathbb{Q}}.$

The rest is a straightforward calculation:

$\displaystyle s_{0}^{*}\rho_{0}^{*}c(f_{\underline{\xi}})$ $\displaystyle=$
$\displaystyle s_{0}^{*}c(\varphi_{f,\underline{\xi}})$ $\displaystyle=$
$\displaystyle s_{0}^{*}c(\varphi_{f,\underline{\xi}}^{*}T{\rm N}_{1}-T{\rm
N}_{0})$ $\displaystyle=$ $\displaystyle
c(s_{0}^{*}\varphi_{f,\underline{\xi}}^{*}T{\rm N}_{1}-s_{0}^{*}T{\rm N}_{0})$
$\displaystyle=$ $\displaystyle c(j_{\underline{\xi}}^{*}TY-
i_{\underline{\xi}}^{*}TX)$ $\displaystyle=$ $\displaystyle
i_{\underline{\xi}}^{*}c(f^{*}TY-TX)$ $\displaystyle=$ $\displaystyle
i_{\underline{\xi}}^{*}c(f).$

Furthermore, LN classes are also, for
$\rho_{1}^{*}\circ(f_{\underline{\xi}})_{*}=(\varphi_{f,\underline{\xi}})_{*}\circ\rho_{0}^{*}$.
$\Box$

####  6.4.3. Restriction equations

Let $f:X\to Y$ be as above in §6.4.2 and take the stratifications of $X$ and
$Y$ according to multi-singularity types.

Pick a type $\underline{\eta}=(\eta_{1},\cdots,\eta_{r})$ such that
$M_{\underline{\eta}}^{\circ}\not=\emptyset$ for $f$, and put
$\ell:=\ell(\underline{\eta})$. The closure $M_{\underline{\eta}}$ has the
right dimension $m-\ell+\kappa$ and represents the source loci class
$m_{\underline{\eta}}(f)\in{\rm CH}^{\ell-\kappa}(X)_{\mathbb{Q}}$ up to the
multiplicity (Proposition 4.15). For the simplicity, we may assume that
$M_{\underline{\eta}}$ is one of the highest codimensional strata, so it is a
smooth closed subscheme $M_{\underline{\eta}}=M_{\underline{\eta}}^{\circ}$.
Then we have the so-called restriction equations [Ri01, FR02]

$None$
$i_{\underline{\xi}}^{*}m_{\underline{\eta}}(f)=\left\\{\begin{array}[]{cr}\\#{\rm
Aut}(\underline{\eta}/\eta_{1})\cdot c_{\ell-\kappa}({\rm
N}_{0}(\underline{\xi}))&(\underline{\xi}=\underline{\eta})\\\
0&(\underline{\xi}\not=\underline{\eta})\end{array}\right.$

in ${\rm CH}^{\ell-\kappa}(M_{\underline{\xi}}^{\circ})_{\mathbb{Q}}\simeq{\rm
CH}^{\ell-\kappa}({\rm N}_{0}(\underline{\xi}))_{\mathbb{Q}}$ (this is almost
obvious, for $m_{\underline{\eta}}(f)$ is represented by
$[M_{\underline{\eta}}]\in{\rm CH}_{m-\ell+\kappa}(X)_{\mathbb{Q}}$). Recall
that $\varphi_{f,\underline{\xi}}:{\rm N}_{0}(\underline{\xi})\to{\rm
N}_{1}(\underline{\xi})$ is deduced from the universal singular map
$f_{\underline{\eta}}:X_{\underline{\eta}}\to Y_{\underline{\eta}}$.

######  Lemma 6.24.

For any $\underline{\xi}$, it holds that

$i_{\underline{\xi}}^{*}m_{\underline{\eta}}(f)=s_{0}^{*}\rho_{0}^{*}m_{\underline{\eta}}(f_{\underline{\xi}})\in{\rm
CH}^{\ell-\kappa}(M_{\underline{\xi}}^{\circ})_{\mathbb{Q}}.$

Proof : If $\underline{\xi}=\underline{\eta}$, the zero section is the stratum
of type $\underline{\eta}$ for the map $f_{\underline{\eta}}$, thus
$m_{\underline{\eta}}(\varphi_{f,\underline{\eta}})={s_{0}}_{*}(1)\in{\rm
CH}^{\ell-\kappa}({\rm N}_{0}(\underline{\eta}))_{\mathbb{Q}}$, and thus

$i_{\underline{\eta}}^{*}m_{\underline{\eta}}(f)=c_{top}({\rm
N}_{0}(\underline{\eta}))=s_{0}^{*}{s_{0}}_{*}(1)=s_{0}^{*}m_{\underline{\eta}}(\varphi_{f,\underline{\eta}})=s_{0}^{*}\rho_{0}^{*}m_{\underline{\eta}}(f_{\underline{\eta}}).$

If $\underline{\xi}\not=\underline{\eta}$, then the both sides are zero,
because $f_{\underline{\xi}}$ does not have $\underline{\eta}$-singular
points. $\Box$

####  6.4.4. Proof of Theorem 4.18

Let $f:X\to Y$ be an admissible map of codimension $\kappa(\geq 0)$ in
${\rm\bf Sm}_{k}$. Take the stratification as above, and put

$\hat{m}_{\underline{\eta}}(f):=\sum R_{J_{1}}(f)f^{*}n_{J_{2}}(f)\cdots
f^{*}n_{J_{s}}(f)$

in ${\rm CH}^{\ell-\kappa}(X)_{\mathbb{Q}}$, where the summand runs over all
possible partitions $\\{1,\cdots,r\\}=J_{1}\sqcup\cdots\sqcup J_{s}$ with
$1\in J_{1}$. By Lemma 4.12 (1) and a remark after Theorem 4.17, we see

$f_{*}m_{\underline{\eta}}(f)=n_{\underline{\eta}}(f)=f_{*}\hat{m}_{\underline{\eta}}(f)\in{\rm
CH}^{\ell}(Y)_{\mathbb{Q}}.$

In general $f_{*}$ may not be injective, so this formula is insufficient to
imply the theorem.

First, we claim that for any types $\underline{\xi}$ it holds that

$None$
$i_{\underline{\xi}}^{*}{m}_{\underline{\eta}}(f)=i_{\underline{\xi}}^{*}\hat{m}_{\underline{\eta}}(f)\;\in{\rm
CH}^{\ell-\kappa}(M_{\underline{\xi}}^{\circ})_{\mathbb{Q}}.$

In fact, applying the above relation to the universal singular map
$f_{\underline{\xi}}$, we have

$f_{\underline{\xi}*}m_{\underline{\eta}}(f_{\underline{\xi}})=n_{\underline{\eta}}(f_{\underline{\xi}})=f_{\underline{\xi}*}\hat{m}_{\underline{\eta}}(f_{\underline{\xi}})\in{\rm
CH}^{\ell}(Y_{\underline{\xi}})_{\mathbb{Q}}.$

Since $f_{\underline{\xi}*}$ is injective by Lemma 6.22, we see

$m_{\underline{\eta}}(f_{\underline{\xi}})=\hat{m}_{\underline{\eta}}(f_{\underline{\xi}})\in{\rm
CH}^{\ell-\kappa}(X_{\underline{\xi}})_{\mathbb{Q}}.$

Pulling back the both sides via $\rho_{0}\circ s_{0}$,

$i_{\underline{\xi}}^{*}{m}_{\underline{\eta}}(f)=s_{0}^{*}\rho_{0}^{*}m_{\underline{\eta}}(f_{\underline{\xi}})=s_{0}^{*}\rho_{0}^{*}\hat{m}_{\underline{\eta}}(f_{\underline{\xi}})=i_{\underline{\xi}}^{*}\hat{m}_{\underline{\eta}}(f)$

where the first equality follows from Lemma 6.24 and the third one from Lemma
6.23. So the claim $(*)$ is proven.

Recall the excision property [Fu84, §1.8]: for a closed subscheme $W$ of $X$,
we have the exact sequence

$\textstyle{{\rm
CH}_{q}(W)_{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$$\textstyle{{\rm
CH}_{q}(X)_{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{{\rm
CH}_{q}(X-W)_{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $i_{*}$ is the pushforward of the inclusion and $j^{*}$ is the flat
pullback. Initially we take $W$ to be the complement to the open stratum
$M_{A_{0}}^{\circ}$, and then replace $X$ and $W$, respectively, by
$M_{\underline{\xi}}$ and $\partial
M_{\underline{\xi}}:=M_{\underline{\xi}}-M_{\underline{\xi}}^{\circ}$ (the
union of boundary strata) according to the adjacency of strata.

Inductively using the excision, it turns out that the class
$\hat{m}_{\underline{\eta}}(f)\in{\rm CH}^{\ell-\kappa}(X)_{\mathbb{Q}}={\rm
CH}_{m-\ell+\kappa}(X)_{\mathbb{Q}}$ is supported on ${\rm
CH}_{m-\ell+\kappa}(M_{\underline{\eta}})_{\mathbb{Q}}$, i.e., for any other
type $\underline{\xi}$ than $\underline{\eta}$, the class vanishes in ${\rm
CH}_{m-\ell+\kappa}(M_{\underline{\xi}}^{\circ})_{\mathbb{Q}}$. Indeed, if
$\hat{m}_{\underline{\eta}}(f)$ is localized to a class $c\in{\rm
CH}_{m-\ell+\kappa}(M_{\underline{\xi}})_{\mathbb{Q}}$, then it is pulled back
to $j^{*}c\in{\rm
CH}_{m-\ell+\kappa}(M_{\underline{\xi}}^{\circ})_{\mathbb{Q}}$ and

$s_{0}^{*}s_{0*}j^{*}c=i_{\underline{\xi}}^{*}\hat{m}_{\underline{\eta}}(f)=0$

by $(r)$ and $(*)$, where
$s_{0}:M_{\underline{\xi}}^{\circ}\hookrightarrow{\rm N}_{0}(\underline{\xi})$
is the zero section. Then $j^{*}c=0$ by our assumption on $f$ that
$c_{top}({\rm N}_{0}(\underline{\xi}))$ is a non-zero divisor (indeed,
$s_{0}^{*}s_{0*}$ is just the multiplication by the top Chern class). Hence,
by the excision, $c\in{\rm CH}_{m-\ell+\kappa}(\partial
M_{\underline{\xi}})_{\mathbb{Q}}$. Since $\partial M_{\underline{\xi}}$
consists of several strata of codimension higher than that of
$\underline{\xi}$, the induction proceeds.

Finally, $(r)$ and $(*)$ in the case of $\underline{\xi}=\underline{\eta}$
shows that $i_{\underline{\eta}}^{*}\hat{m}_{\underline{\eta}}(f)$ equals
$\\#{\rm Aut}(\underline{\eta}/\eta_{1})\cdot[M_{\underline{\eta}}^{\circ}]$
in ${\rm CH}_{m-\ell+\kappa}(M_{\underline{\eta}})_{\mathbb{Q}}$. Thus we
conclude that $m_{\underline{\eta}}(f)$ and $\hat{m}_{\underline{\eta}}(f)$
coincide. $\Box$

######  Remark 6.25.

(Restriction method) In case of $\kappa\geq 0$, we have seen that
$m_{\underline{\eta}}(f)$ is expressed by a unique polynomial of variables
$c_{i}(f)$ and $s_{I}(f)$, that is the source multi-singularity Thom
polynomial of type $\underline{\eta}$. This fact makes it possible to
determine the precise form of the Thom polynomial practically. Namely, the
above formula $(r)$ taken for all $\underline{\xi}$ gives an overdetermined
system of linear equations of unknown coefficients of the polynomial for
$\underline{\eta}$, and all that remains is to solve it. That is the strategy
of the restriction method, and there have already been done many explicit
computations of Thom polynomials, see e.g., [Ri01, FR02, Ka03, Ka06, Oh16].
This method brought a striking breakthrough in this field, because people had
previously thought that to get the Thom polynomial for a mono/multi-
singularity type is more or less equivalent to explicitly find a nice
desingularization of the singularity locus of that type (in general, the
latter problem is hard). Indeed, the problem is now understood separately in
two aspects, the hidden combinatorics in computation and the existence of
universal expression. The computation strategy well performs with equivariant
geometry, while our proof of the existence of multi-singularity Thom
polynomials heavily relies on the existence of desingularizations of geometric
subsets ($\Omega$-assignments) that is guaranteed by Hironaka’s resolution of
singularities.

##  7\. Perspectives

###  7.1. Further problems

Finally, we collect some of further research directions and explain each of
them briefly.

####  7.1.1. Thom series and geometric subsets

For mono-singularity types, Thom polynomials have been considered so that
$\kappa=\dim Y-\dim X$ is involved as a parameter – those are often called
Thom series [FR12, BS12, Ka17]. Also for multi-singularity types, it is very
natural to expect that there is a universal series containing $\kappa$ as a
parameter, say multi-singularity Thom series (cf. §2.1.2). Indeed, by our
definition, all multi-singularity loci classes of type $\underline{\eta}$ for
any $\kappa$ are determined at once by the invariant stratum
$\Xi(X;\underline{\eta})\subset X^{[[n]]}$ which depends on the type of local
algebra. Here $Y$ and $\kappa$ are sort of auxiliary data. In this sense, the
expected theory of multi-singularity Thom series is tied with the geometry of
the invariant stratification of Hilbert schemes.

####  7.1.2. Universal Segre clsses

Let $\underline{\eta}$ be a multi-singularity type of map-germs. Suppose that
$f:X\to Y$ (in ${\rm\bf Sm}_{k}$) is a proper locally stable map. Set
$V:=N_{\underline{\eta}}(f)$ for short and $\iota:V\hookrightarrow Y$ the
inclusion. If $V$ is smooth, we may consider the Chern classes $c(TV)$ of the
tangent bundle and its pushforward into $Y$. In general, however, the locus
$V$ is singular along adjacent strata, and then, an alternative to $c(TV)$ is
the Chern-Schwartz-MacPherson class (CSM class) for singular varieties
[Mac74], denoted by $c^{\rm SM}(V)\in{\rm CH}_{*}(V)$. More suitably, we make
use of the Segre-Schwartz-MacPherson class (SSM class) for
$\iota:V\hookrightarrow Y$;

$s^{{\rm SM}}(V,Y):=\iota^{*}c(TY)^{-1}\frown c^{\rm SM}(V)\;\in{\rm
CH}_{*}(V)$

(in case that $V$ is smooth, then $s^{{\rm SM}}(V,Y)=c(TV-\iota^{*}TY)$).
Consider the SSM class in ${\rm CH}^{*}(Y)$ via the pushforward $\iota_{*}$
(for $Y$ is smooth), the leading term is nothing but the class $[V]$, thus it
is expressed by the Thom polynomial of type $\underline{\eta}$ in $Y$. Namely,
taking account of the multiplicity (Proposition 4.15), we have

$\\#{\rm Aut}(\underline{\eta})\cdot\iota_{*}s^{{\rm
SM}}(V,Y)=n_{\underline{\eta}}(f)+h.o.t.\in{\rm CH}^{*}(Y).$

In particular, the Euler characteristic of the locus is given by

$\chi(V)=\frac{1}{\\#{\rm Aut}(\underline{\eta})}\int_{Y}c(TY)\cdot
s^{SM}(V,Y).$

A general theory of universal SSM classes for singularity loci has been
proposed in the author’s previous work [Oh16, Oh06, Oh08]. It was well
established for mono-singularity case in [Oh06]. Afterwards, the theory has
been featured in variants of Schubert Calculus, e.g., equivalriant conormal
Schubert Calculus, by several authors in relation with the theory of stable
envelopes due to Okounkov and others, see e.g., [AMS17, FRW21].

Also for multi-singularity case, it is natural to expect that each higher
order term of the SSM class (in $Y$) is universally expressed in terms of
Landweber-Novikov classes, which we call target higher Thom polynomial,
provisionally. That would possibly be verified in a similar way as in the
present paper, that will be discussed somewhere else. In fact, assuming such
universal expressions, some higher Thom polynomials have been computed by the
restriction method [Oh16, PP21, NO25] and applied successfully to the study on
vanishing topology of quasi-homogeneous map-germs [Oh16, PP21]. Also this is
related to Chern class of the sheaf of logarithmic vector fields for a free
divisor (discriminant) [Li18, Oh16].

####  7.1.3. Generating series

In [Ka06] Kazarian presented a generating series for target multi-singularity
Thom polynomials, that comes from the combinatorial structure of residual
polynomials:

$1+\sum_{\underline{\eta}}n_{\underline{\eta}}\cdot\frac{t^{\underline{\eta}}}{|{\rm
Aut}(\underline{\eta})|}=\exp\left(\sum_{\underline{\eta}}f_{*}(R_{\underline{\eta}})\cdot\frac{t^{\underline{\eta}}}{|{\rm
Aut}(\underline{\eta})|}\right).$

Also, in a similar way, the author made a conjectural generating series for
target higher Thom polynomials (universal SSM classes) in [Oh16, §6.4]. It
would be very natural to expect that those generating series are directly
related to an invariant stratification of $X^{[n]}$ mentioned in §7.1.1. For
example, in [Oh08, Ca10], a generating series (zeta function) of the CSM
classes of $X^{[[n]]}$ is presented in relation with the symmetric product
$S^{n}X$. Its relative version might give some clue for future work.

####  7.1.4. Bivariant theory

As mentioned before, we expect that Theorem 4.18 on the existence of source
multi-singularity Thom polynomials should be true for any codimension $\kappa$
and for arbitrary maps. An interesting approach is related to Grothendieck’s
bivariant theory. From our approach using cohomology operations, an optimistic
hope is that there is a suitable bivariant algebraic cobordism theory which
admits a bivariant analogue to Vishik’s theorem so that there exists a certain
cohomology operation

$m_{\underline{\eta},\mu}^{\Omega}:\Omega^{0}(X\stackrel{{\scriptstyle
f}}{{\to}}Y)\to\Omega^{\ell-\kappa}(X\stackrel{{\scriptstyle f}}{{\to}}Y)$

expressed in terms of Chern classes and Landweber-Novikov classes which
satisfies that the class $[id:X\to X]$ is sent to the source loci class
$m_{\underline{\eta},\mu}^{\Omega}(f)\in\Omega^{\ell-\kappa}(X)$. That leads
to the desired solution for the existence of the source multi-singularity Thom
polynomial in the same principle as in the proof of Theorem 4.17. This picture
may be discussed better in a wider perspecitve of classifying stacks or
derived algebraic geometry (§7.1.8, §7.1.9).

####  7.1.5. Critical cobordism

We have seen in Remark 6.25 that in case of non-negative codimension
$\kappa\geq 0$, the restriction method is available for computing certain
muti-singularity Thom polynomials. However, in case of $\kappa<0$, Theorem
4.17 is not enough for fully supporting the use of this method – indeed, in
this case, the universal singular map $f_{\underline{\xi}}$ is always
improper, so Theorem 4.17 can not be applied to it.

This trouble was noticed by Kazarian in [Ka06] and he introduced localized
Thom polynomials (still its existence is a conjecture). Indeed, our true
object is the restriction map on the critical scheme of $f$

$f_{C}:C(f)\to Y$

and it should be enough to assume the properness of $f_{C}$, not that of $f$.
Note that $C(f)$ is not always smooth, thus it does not directly yield a
cobordism class in $Y$. Then it is natural to pick up a specified cobordism
class $[\tilde{f}_{C}:\tilde{C}(f)\to Y]$ by using a canonical
desingularization of the first Thom-Boardman singularity, i.e., we take the
simplest degeneracy loci class in $\Omega^{*}(X)$ for the bundle map $df:TX\to
f^{*}TY$ in the sense of [Hu12] and push it out to $\Omega^{*}(Y)$. We need to
characterize this cobordism class and connect it with
$m_{\underline{\eta}}(f)$ and $n_{\underline{\eta}}(f)$, and seek for a new
variant of algebraic cobordism theory, say it provisionally, critical
cobordism, for this restricted class of proper maps
$\tilde{f}_{C}:\tilde{C}(f)\to Y$ without starting with $f:X\to Y$.

####  7.1.6. Legendre singularities

A particular case of the expected critical cobordism is an algebraic version
of Legendre cobordism theory (cobordism theory for Legendre maps due to
Arnol’d and Vassiliev [Va88]). Let $Y$ and $L$ be in ${\rm\bf Sm}_{k}$ such
that $\dim Y=\dim L+1$. The projectived cotangent bundle
$\pi:\mathbb{P}(T^{*}Y)\to Y$ possesses a canonically defined contact
structure (i.e., maximally non-integrable hyperplane field). An immersion
$\bar{\varphi}:L\to\mathbb{P}(T^{*}Y)$ is said to be Legendrian if it is
integrable (i.e., the image $d\varphi(T_{x}N)$ at every point $x\in N$ is
included in the contact hyperplane), and the composed map
$\varphi:=\pi\circ\bar{\varphi}:L\to Y$ is called a Legendre map [AGV85].
Legendre maps form a special class of maps. For instance, if $f:X\to Y$
($\kappa<0$) is appropriately generic and $\dim{\rm coker}\,df_{x}\leq 1$,
then the critical locus $L:=C(f)$ is smooth and admits a Legendre immersion
$L\to\mathbb{P}(T^{*}Y)$ in a canonical way so that its Legendre map, i.e.,
the composition with $\pi$, coincides with the restriction $f_{C}:L\to Y$.
Legendre singularities has been well explored in Arnol’d’s school in the
context of the classification of hypersurface singularities (note that it
differs from the classification of ICIS). Legendrian Thom polynomials for
mono-singularities, that live in the Chow ring of Lagrange Grasmannian, have
been well studied, see [Va88, Ka03, MPW11]. So we have enough material to
build new theories of algebraic Legendre cobordism and muti-singularity
Legendrian Thom polynomials.

####  7.1.7. Multi-singularities in $\mathcal{A}$-classification

There are many different versions of classification theories of map-germs.
Legendre singularities mentioned above is one of them. Also
$\mathcal{A}$-classification is quite important in singularity theory of maps.
For locally stable germs, $\mathcal{A}$-classification and
$\mathcal{K}$-classification are equivalent [Mat69], but for unstable germs,
$\mathcal{A}$-classification gets to be much finer. Such unstable
singularities appear in families of mappings, e.g., morphisms between total
spaces of manifold bundles over the common base, and one seeks for universal
expression of the cohomology class represented by the bifurcation locus of
prescribed type defined as a subscheme of the base. Thom polynomials in
$\mathcal{A}$-classification of mono-germs has been studied in [SO18], while
there is a large room for developing multi-singularity Thom polynomials in
$\mathcal{A}$-classification. For instance, bifurcation loci of a map from a
universal family of curves [Mu83] to another manifold are quite common objects
in algebraic geometry (epsecially the theory of Gromov-Witten invariants).

####  7.1.8. Classifying stacks for multi-singularities

The set of all (semi-ordered) multi-singularity types of map-germs of
codimension $\kappa$ admits a poset-structure via the adjacency relation:
$\underline{\xi}<\underline{\tau}$ if $\underline{\xi}$ is adjacent to
$\underline{\tau}$, i.e., $M_{\underline{\xi}}^{\circ}\subset{\rm
Closure}(M_{\underline{\tau}}^{\circ})-M_{\underline{\tau}}^{\circ}$ for a
stable unfolding of type $\underline{\xi}$. Let $\kappa>0$ be fixed. With
respect to this partial order, suppose that we are given a poset $\mathcal{T}$
consiting of several multi-singularity types. A $\mathcal{T}$-map $f:X\to Y$
is defined to be a proper map in ${\rm\bf Sm}_{k}$ having only singularities
of type in $\mathcal{T}$. We may expect that there is a ‘universal
$\mathcal{T}$-map’

$F_{\mathcal{T}}:\mathcal{X}_{\mathcal{T}}\to\mathcal{Y}_{\mathcal{T}}$

between certain Artin stacks with global stratifications of quotient stacks

$\mathcal{X}_{\mathcal{T}}:=\bigsqcup_{\underline{\xi}\in\mathcal{T}}\,X_{\underline{\xi}},\;\;\mathcal{Y}_{\mathcal{T}}:=\bigsqcup_{\underline{\xi}\in\mathcal{T}}\,Y_{\underline{\xi}}$

such that for any $\mathcal{T}$-map $f:X\to Y$ there exists a morphism
(classifying map) $\rho:Y\to\mathcal{Y}_{\mathcal{T}}$ whose fiber product
with $F_{\mathcal{T}}$ recovers $f$ in a certain sense (perhaps, up to
cobordism):

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\rho}}$$\scriptstyle{f}$$\textstyle{\mathcal{X}_{\mathcal{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{\mathcal{T}}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\mathcal{Y}_{\mathcal{T}}}$

In differential topology, it has been made up as the Thom-Pontrjagin
construction for singular maps due to Rimányi-Szücs [RS98]. In algebraic
geometry, the counterpart has not yet been found so far – building blocks,
i.e., quotient stacks $X_{\underline{\xi}}$, $Y_{\underline{\xi}}$ and the
universal singular map $f_{\underline{\xi}}$, have been obtained, but it is
totally unclear how to glue them (perhaps, for this glueing, the Hilbert
schemes should take a key role). As for Intersection Theory on Artin stacks,
see e.g., [Kr99].

In this picture, one may assign to the cobordism class of $f:X\to Y$ the
pullback $\rho^{*}[BG_{\underline{\eta}}]\in{\rm CH}^{\ell}(Y)$, that should
define our cohomology operation $n_{\underline{\eta}}$. It is also interesting
to relate the universal map $F_{\mathcal{T}}$ with an expected bivariant
theory (cf. §7.1.4).

####  7.1.9. Motivic homotopy theory

There is a ‘bigger’ theory of algebraic cobordism due to Morel and Voevodsky,
denoted by ${\rm MGL}^{p,q}(-)$ with bi-degree, as an algebraic counterpart to
stable homotopy theory in topology. If the base field is of characteristic
$0$, there is a natural transformation

$\Omega^{\kappa}(Y)\to{\rm MGL}^{2\kappa,\kappa}(Y)$

(for $Y\in{\rm\bf Sm}_{k}$) and it is known to be isomorphic (M. Levine).
Beyond this background, it would be very meaningful to pursue a direct analogy
of Thom’s original idea [Th71] on the Thom-Pontrjagin construction in
algebraic geometry side, as Kazarian suggested in [Ka03]. In fact, as the
classifying space of algebraic cobordism, the infinite loop space of motivic
Thom spectrum $\Omega_{\rm T}^{\infty}\Sigma^{\kappa}_{\rm T}{\rm MGL}_{k}$ is
recently argued in the context of derived algebraic geometry, e.g., see §1.4
of [EHK20]. The cohomology of the loop space must contain all the information
of universal polynomials associated to multi-singularity types. The space
should also be related to $\mathcal{Y}_{\mathcal{T}}$ mentioned in §7.1.8; it
approximates the loop space as an Artin stack presentation.

####  7.1.10. Direct approach via tautological integrals

There is still room for thinking about a constructive approach which directly
generalizes the proof of double-point formula (cf. §2.1.2). Consider
inductively the class

$R_{\underline{\eta}}(f):=m_{\underline{\eta}}(f)-\sum
R_{J_{1}}(f)f^{*}n_{J_{2}}(f)\cdots f^{*}n_{J_{s}}(f)$

in ${\rm CH}^{\ell-\kappa}(X)$, where the sum runs over the partition with
$J_{1},J_{2}\not=\emptyset$. This class should be understood as the
‘unnecessary contribution from the smallest diagonal’, which is given by the
excess intersection of $f^{[[n]]}$ along fibers over the diagonal of the
Hilbert-Chow morphism, and the problem is to show that $R_{\underline{\eta}}$
is indeed uniquely written as a polynomial of Chern classes $c_{i}=c_{i}(f)$.
That should be directly related to tautological integrals on Hilbert schemes
[Re17, Be17, BS12, Be23] and also non-associative Hilbert schemes [Ka17].

####  7.1.11. Real singularities

Enumerative theory of multi-singularities for real algebraic, real analytic or
$C^{\infty}$ maps is somewhat mysterious. First we apply the classical Borel-
Heafliger theorem (cf. Matszangosz [Mts19]). That is, taking fixed point sets
of the involution for a complex algebraic variety $X(\mathbb{C})$, we may
obtain mod.2 enumerative formulas for multi-singularities of real maps
$f:X(\mathbb{R})\to Y(\mathbb{R})$ after replacing Chern classes
$c_{i}=c_{i}(f)$ by the Stiefel-Whitney classes $w_{i}=w_{i}(f)$ in
$H_{*}(X(\mathbb{R});\mathbb{Z}_{2})$. Also it is quite meaningful to seek for
formulas with integer coefficients using Pontryagin classes, but known results
are less, comparing with the case of complex singularities. Moreover, as a
very new direction, it sounds interesting to seek for ‘multiple-point
formulas’ in the Chow-Witt cohomology [Le19].

####  7.1.12. Complex links and quantum invariants

For the link of a complex plane curve singularity, the so-called Hilbert
scheme invariants comes up in relation with quantum polynomial invariants of
the link in the $3$-sphere $S^{3}$. There has not been found any higher
dimensional analogue so far. That may be related to the Euler product-type
expression of the generating series of weighted Euler characteristics of
strata of Hilbert schemes, or of the CSM classes of those strata [Oh08, Ca10]
(see §7.1.3). A hope is that for a quasi-homogeneous isolated complete
intersection singularity (or hypersurface singularity), we may find Hilbert
scheme invariants for the complex link via torus-localization of higher Thom
polynomials (universal SSM classes for multi-singularities) mentioned in
§7.1.2.

###  7.2. Concluding remark

We have developed a general enumerative theory of singularities of maps by
solving affirmatively Kazarian’s conjecture [Ka03, Ka06] in the context of
algebraic geometry over an algebraically closed filed of characteristic $0$.
We have shown the existence of unique universal polynomials which express the
source and the target multi-singularity loci classes for prescribed type
(Theorems 4.17 and 4.18). Our result also holds suitably for any oriented
cohomology theory $A^{*}$ (Theorem 6.19). Main technical ingredients are
Hilbert extension maps, desingularization of geometric subsets
($\Omega$-assignments), cohomology operations on algebraic cobordism
$\Omega^{*}$, and equivariant geometry with symmetries of singularities. We
remark that in Theorem 4.18 about the source multi-singularity loci classes,
we only consider a certainly nice case with $\kappa\geq 0$, thus the full
scope of the conjecture, including the setup over a ground field of positive
characteristic, still remains open.

Thom polynomial theory connects local and global natures of singularities of
maps. Local classification theory of singularities of maps is essentially the
matter of commutative algebra and has been thoroughly studied for several
decades, while the global homotopical nature in algebraic geometry has been
developed much later – it was quite recent that key materials have become
available, such as algebraic Borel construction, algebraic version of Thom
spectra, algebraic coboridisms and algebraic cohomology operations, and so on.
Now, on the basis of these recent achievements in algebraic geometry, we have
viewed a new horizon in enumerative geometry of singularities.

### Acknowledgement

The author would like to thank organizers and participants of the conference
“Characteristic classes for singular spaces” (Kiel, Sept. 2023), where he had
an opportunity to give a series of lectures on the topic of this paper. This
work was partly supported by JSPS KAKENHI Grant Numbers JP17H02838 and
JP23H01075.

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