数理解析研究所講究録1033 巻 1998 年 7-20

7Chern number formula for ramified coverings
Takeshi Izawa
Department of Mathematics, Faculty of Science
Hokkaido University, Sapporo 060, Japan
July 1st, 1997
Abstract
For a ramified covering $f:\mathrm{Y}arrow X$ between compact complex manifolds, weestablish a formula relating the Chern numbers of and $X$ . We obtain the formulaby localizing characteristic classes via the \v{C}ech-de Rham cohomology theory. As
corollaries,we deduce generalizations of the Riemann-Hurwitz formula and a formula of Hirzebruch for the signature, as well as formulas for other invariants suchas the Todd genus.
$\mathrm{Y}$1

Introduction

Let $f:Yarrow X$ be a ramified co.vering between -dimensional compact complex manifolds
be the ramification divisor of , andwith covering multiplicity . Let
the branch locus of . We assume that the ramification divisor and theirreducible component of the branch locus are all non-singular. Our main result is
$n$

$f$

$R_{f}=\Sigma_{i}r_{i}R_{i}$

$\mu$

$f$

$B_{f}=\Sigma_{i}b_{i}B_{i}$$c_{1}^{N_{1}}\cdots C^{N}(nYn)-\mu\cdot cc^{N}(1N1\ldots xn^{n})$$= \sum_{i}(H_{\tau}^{(N_{1}\cdots N)}n(R.1(CLR.))\wedge[R_{i}]-\frac{b_{i}(r_{i}+1)}{r_{i}}H_{T}n((N_{1}\cdots N)B\dot{.}c_{1}(LB_{*}.))-[B_{i}])$$= \sum_{i}\sum_{\alpha=0}^{n-1}\frac{b_{i}(1-(r_{i}+1)^{\alpha}+1)}{r_{i}(r_{i}+1)^{\alpha}}P_{\alpha}(_{C_{1}}(B_{i})\cdots Cn-1(Bi))\cdot c_{1}(LB)^{\alpha}-[B_{i}]$In the above,$\Sigma_{i=1}^{n}\dot{i}N_{i}=n$$H_{\xi}^{(N_{1}}\ldots(N_{\mathfrak{n}})l)=l-1$.

.

and we set formally,$((_{i=1} \prod^{n}(ci(\xi)+Ci-1(\xi)\cdot l)N*\cdot)-C_{1}^{N_{1}}\cdots c_{n^{n}}(N\xi)\mathrm{I}=\sum_{\alpha\simeq 0}^{n-1}P\alpha(_{C_{11}}\cdots cn-)l\alpha$is the coefficient of of $H(l)$ as a polynomial in .
where
We prove the formula for Chern numbers by applying the framework of the localizationof characteristic classes based on the \v{C}ech de-Rham cohomology theory. ([L1], [L2],$P_{\alpha}$$l^{\alpha}$

$l$

8

[LS].) Our methods of proof are very elementary and computational. Classically, allsorts of topological invariants can be calculated as the integral value of differential forms
through the de Rham theorem, which gives the representation of cohomology classes
and describe the explicit corespondence in the Poincar\’e duality. The \v{C}ech-de Rham
cohomology theorey plays the same role for relative cohomology groups as the Alexander
duality. So applying this analogy, we can localize Chern classes at the ramification set,
which gives us more specific geometric information about what is caused by degenerancyof holomorphic maps.

2

Preliminaries

2.1\v{C}ech-de Rham cohomology theory

First we will give a brief sketch of the \v{C}ech-de Rham cohomology theory. (see [BT], [L1],[L2], [S].)1 Definition.
Let $X$ be an -dimensional-manifold andan open covering of $X$ ,whose index set is a countable ordered set such thatis totallyordered if. Let us consider the de Rham Comple..X of sheaves of germsof smooth forms on $X$$C^{\infty}$$n$$\mathcal{U}=\{U_{\alpha}\}_{\alpha\in I}$$(\alpha_{0}, \ldots, \alpha_{p})\in I^{p+1}$$I$$U_{\alpha_{0}}\cap\ldots\cap U_{\alpha_{p}}\neq\phi$$0arrow A^{0_{arrow A^{1_{arrow Aarrow}}}}dd2dA^{3}arrow\ldots$

Now letbe the group of \v{C}ech cochains of degree with values in. The\v{C}echcommutativity of the two operators, thecoboundary operator and the exterior
derivative , gives rise to a double complex’ $\{C^{pq}=C^{p}(\mathcal{U}, A^{q});\delta, d\}$ . The associatedis defined bysingle complex$C^{p}(\mathcal{U}, A^{q})$$A^{q}$$p$$\delta$$d$$(A^{\cdot}(\mathcal{U}), D)$$A^{r}( \mathcal{U})=\bigoplus_{p+q=r}C^{p}(\mathcal{U}, A^{q})$$D=\delta+(-1)pd$ .

We call the cohomology groupsof this associate single complex, the \v{C}ech-de
Rham cohomology groups of $X$ . This cohomology is canonically isomorphic to the classical de Rham cohomology.
$H^{r}(A^{\cdot}(\mathcal{U}))$$([\mathrm{B}\mathrm{T}])$

2 Product structure.
We also define a product structureas$A^{r}(\mathcal{U})\cross A^{s}(\mathcal{U})arrow A^{r+s}(\mathcal{U})$$( \sigma\cdot\tau)_{\alpha_{0}\cdots\alpha_{p}}=\nu=\sum(p0-1)^{(-}r\nu)(p-\nu)\sigma\alpha 0\cdots\alpha_{\nu}$A$\tau_{\alpha_{\nu}\cdots\alpha_{p}}$.

Then it induces the cup product structure for the cohomology of the \v{C}ech de-Rhamcomplex, which is, via the above isomo.rphism, compatible with the usuall product in de.
. .
Rham cohomology.

93 Integration.
Next we define the integration on the \v{C}ech-de Rham cohomology group which is. Supposecompatible with the usual integration on the de Rham cohomology group
now that the manifold $X$ is oriented. Before making our definition, we introduce thefollowing concept.
of -dimensional manifolds
Definition. Let and $X$ be as above. A familywith piecewise smooth boundary in $X$ is called a system of honey-comb cells adapted
to if:.
(1).
(2) Int $(R_{\alpha})\cap Int(R_{\beta})=\phi$ if$(n-p)$-dimensional manifold with piecewise smooth boundis an(3).
ary for any
has no boundary.
is maximal,(4) If$([\mathrm{L}1])$$n$$\{R_{\alpha}\}_{\alpha\in I}$$\mathcal{U}$$R_{\alpha}$$\mathcal{U}$$R_{\alpha}\subset U_{\alpha},$$X= \bigcup_{\alpha}R_{\alpha}$$\alpha\neq\beta$$R_{\alpha_{0}\cdots\alpha_{p}}= \bigcap_{\nu=0^{R_{\alpha}}\nu}^{p}$$(\alpha_{0}\cdots\alpha_{p})\in I^{p+1}$$(\alpha_{0}\cdots\alpha_{p})$$R_{\alpha_{0}\cdots\alpha_{p}}$an orientation by the following rules.
We also givehas the same orientation as $X$ .
(1) Each
for a permutation .
(2)
.
(3)
$R_{\alpha_{0}\cdots\alpha_{p}}$$R_{\alpha}$$R_{\alpha \mathrm{o}(0)\cdots\alpha_{p}()}p=sgn(\rho)\cdot R_{\alpha_{0}}\ldots\alpha_{p}$$\rho$$\partial R_{\alpha_{0}\cdots\alpha_{p}}=\Sigma_{\alpha}R_{\alpha\alpha_{p}}0\cdots\alpha$Now suppose that $X$ is compact, andto . We define the integration on$\{R_{\alpha}\}_{\alpha\in I}$$A^{n}(\mathcal{U})$$\mathcal{U}$a system of honey-comb cells adaptedas:$\int_{\mathrm{x}^{:A^{n}((\mathcal{U}}}A^{\cdot}))arrow \mathrm{C}$,,$\int_{X}\sigma=\sum_{P^{=}0}^{n}(\alpha_{0}\cdots\alpha\sum_{\mathrm{p}\in \mathrm{p}+I1}\int_{R_{\alpha_{0p}}}\cdots\alpha\sigma_{\alpha_{0}}\ldots\alpha_{p})$$\sigma\in A^{n}(\mathcal{U})$.

Then we see, from the fact that this integration is independent of the choice of the systemof honey-comb cells for $D$ -cocycles and it vanishes for $D$ -coboundaries, that it induces
the integration on the cohomology group$\int_{x^{:H^{n}((\mathcal{U}}}A^{\cdot}))arrow \mathrm{C}$,which is compatible with the usual integration on the de Rham cohomology.
4 Alexander-Lefschetz duality.
Finally, we describe the Alexander duality in terms of the \v{C}ech-de Rham cohomology([L1] [L2] [S]). We suppose that $X$ is the same as above, and let $S\subset X$ be a compact
subse-t of $X$ which admits a regular neighborhood, $U_{0}=X-S$ , and a reguler neighand consider the \v{C}ech-de Rham cohomology
borhood of . Now we set$X$
associated with the covering . We setof
so that we have the exact sequence$U_{1}$$S$$\mathcal{U}=\{U_{0}, U_{1}\}$$\mathcal{U}$$A^{r}(\mathcal{U}, U_{0})=\mathrm{k}\mathrm{e}\mathrm{r}(A^{r}(\mathcal{U}).arrow A^{r}(U_{0}.))=$$\{(\sigma_{0,1,01}\sigma\sigma)|\sigma_{0}=0\}$$0arrow A^{r}(\mathcal{U}, U_{0})arrow A^{r}(\mathcal{U})arrow A^{r}(U_{0})arrow 0$.

10Then we conclude $H^{r}(A(\mathcal{U}, U_{0}))\cong Hr(x,x_{-}s;\mathrm{c})$ from the de Rham theorem andthe five lemma.
Letbe a system of honey-comb celles adapted to . Then we still have theintegration$\{R_{0}, R_{1}\}$$\mathcal{U}$$\int_{\mathrm{x}^{:A^{n}}}(u, U\mathrm{o})arrow \mathrm{C}$

given

,

$\dot{\mathrm{b}}\mathrm{y}$$\int_{\mathrm{x}}\sigma=\int_{R_{1}}\sigma 1+\int R01\sigma_{01}$for$\sigma=(.0, \sigma_{1}, \sigma 01)\in A^{n}(\mathcal{U}, U\mathrm{o})$

mol.o

,

. This again induces the integration on the relative coho-$\mathrm{g}\mathrm{y}$$\cdot$$\int_{\mathrm{x}^{:H^{n}((U_{0}}}A^{\cdot}\mathcal{U},))arrow \mathrm{C}$.

.

The cup product induces the pairing
lowed by the integration, gives a bilinier pairing, which fol-$A^{r}(\mathcal{U}, U\mathrm{o})\cross A^{n-r}(U_{1}.)arrow A^{n}(\mathcal{U}, U\mathrm{o})$$A^{r}(\mathcal{U}, U_{0})\cross A^{n}-r(U_{1})arrow \mathrm{C}$,which induces the Alexander duality$H^{r}(X,x-s;^{\mathrm{c})\dot{H}}\cong k(A^{\cdot}(\mathcal{U}, U_{0}))\cong H^{n}-r(U1;\mathrm{C})^{*}\cong H_{n-k}(S;\mathrm{c})$.

5 Chern-Weil theory for \v{C}ech-de Rham cohomology.
We recall some fundamental results of the Chern-Weil theory, the differential geometric treatment of characteristic classes (see [GH]).
Let $X$ be an -dimensional-manifold and $\pi:Earrow X$ a-complex vector$X$$c_{i}(E)$bundle of rank over . Then the -th Chern classinis representedby$C^{\infty}$

$n$

$r$

$C^{\infty}$$H_{DR}^{2i}(X:\mathrm{c})$$\dot{i}$$c_{i}( \nabla)=(\frac{\sqrt{-1}}{2\pi})^{i}P^{i}(\ominus)$,

where we denote by
the -th elementary symmetric polynomial, and the curvaturematrix of a connection on in terms of some frame for . Then there is the following
well-known result for invariant polynomials determined by connection forms.
$P^{i}$$\dot{i}$$\nabla$$\ominus$

$E$

$E$

$([\mathrm{B}])$

Suppose that $\pi:Earrow X$ is a-complex vector bundle of rank over $X$, and, connections on E. Then there exists $P^{i}(\nabla_{0}\cdots\nabla_{p})\in A^{2(n-i})-p(X)$ such that
$C^{\infty}$$\nabla_{0},$$\cdots,$$r$$\nabla_{p}$$dP^{i}( \nabla_{0}\cdot\cdot\phi\nabla p)=\sum^{p}(-1)j-p-1Pj=1i(\nabla 0\cdots\check{\nabla}j\ldots\nabla)p$.

The immediate construction of the above secondary term is given as follows. Let us
consider the trivial extensionof the vector bundle over,$(1-t_{1} -. .. -t_{p})\nabla_{0}+$andthe canonical projection. We takeas a connection on
and we set$E\cross \mathrm{R}^{p}arrow X\cross \mathrm{R}^{p}$$E$$\tilde{\pi}:X\cross \mathrm{R}^{p}arrow X$$t_{1}\nabla_{1}+\cdots+t_{p}\nabla_{p}$$\tilde{\nabla}=$$E\cross \mathrm{R}^{p}$$P^{i}(\mathrm{v}_{0p}\ldots\nabla)=\tilde{\pi}*(P^{i}(\tilde{\nabla}))$,$X\cross \mathrm{R}^{p}$11

then it has the desired property. Here ” ” means the integration along the fibers. By
applying the above result for invariant polynomials determind by connection forms, we
can express the -th Chern classinas follows. ([L1], [L2], [LS].) Let
be a connection onover ,$\tilde{\pi}_{*}$$c_{i}(E)$$\dot{i}$$\nabla_{\alpha}$$E|_{U_{\alpha}}$$H^{2i}(A^{\cdot}(\mathcal{U}))$$U_{\alpha}$$H_{DR}^{2i}(X;\mathrm{R})\cong H^{2i}(A^{\cdot}(\mathcal{U}))$$c_{i}(E)rightarrow[((_{C_{i}}(\nabla_{\alpha})_{\alpha}, ((Ci(\nabla_{\overline{\alpha}})_{\overline{\alpha}}\in I\mathrm{p}))_{p})]$.

In particular, for the case where the covering is given by, the \v{C}ech-de
Rham cocycle
represents the i-th Chern class of $E$ .
$\mathcal{U}=\{U_{0}, U_{1}\}$$(c_{i}(\nabla 0), c_{i}(\nabla_{1}),$$c_{i}(\nabla_{0}, \nabla_{1}))$

6 Correspondence between fundamental classes and cohomology classes ofdivisors.
Let $X$ be an -dimensional compact complex manifold, and $D$ a divisor on $X$ , withlocal defining functions
over some open coveringof $X$ . Then,defines naturally a complex line bundle
which has the system of transition functions. We know that, in the Poincar\’e duality, the Chern classrepresents
theof the fundamental class of the divisor $D$ ,$n$$\{f_{\alpha}\}$$D=\{f_{\alpha}, U_{\alpha}\}$$\{U_{\alpha}\}$$L_{D}$$\{g_{\alpha\beta}=f\alpha/f_{\beta}\}$$c_{1}(L_{D})$$\mathrm{d}\mathrm{u},\mathrm{a}\mathrm{l}$$H_{DR}^{2}$$(X; \mathrm{C})\cong H_{2n-2}(X;\mathrm{c})$$c_{1}(L_{D})rightarrow[D]$$( \int_{\mathrm{x}^{C_{1}(L}}D)\wedge\varphi=\int_{D}\varphi$

,

,

,

$\forall_{\varphi\in Z}2n-2(X))$.

Here, we find a more specific correspondence between the fundamental homology classand the Chern class of $D$ in the Alexander duality, by localizing the Chern class in termsof the \v{C}ech de Rham cohomology theory. For simplification, here we assume that thedivisor $D$ is non-singular. (Indeed the following discussion can be applied to the generalcase. (Originally due to [S].)Let $X$ be an -dimensional complex manifold, $D$ a compact non-singular divisor on$X$ , and $L_{D}arrow X$ theassociated line bundle of $D$ . If $D$ is given by local defining
functions, then those functions clearly give a section, whoseofzero locus coincides with $D$ itself. We set $U_{0}=X-D,$ $\pi:U_{1}arrow D$ a sufficiently small
tubular neighborhood,a closed disk bundle over $D$ which is contained in , andthe complement of the interior of .
We consider the coveringwith a system of honey-comb cells
adapted to . Then as is discussed in the previous sections, the class$n$$\{f_{\alpha}\}$$f_{D}=(f_{\alpha}, U_{\alpha})$$R_{1}$$L_{D}$$U_{1}$$R_{0}$$R_{1}$$\mathcal{U}=\{U_{0}, U_{1}\}$$\{R_{0}, R_{1}\}$$\mathcal{U}$$c_{1}(L_{D})=(_{C}1(\nabla 0), C_{1}(\nabla_{1}),$ $C_{1}(\nabla 0, \nabla_{1}))$in the \v{C}ech-de Rham cohomology can be localized at $D$ , by taking antion
as the connectiononso that.
Now let us consider the pairng$\nabla_{f_{D}}$$\nabla_{0}$$U_{0}$$c_{1}(,\nabla_{f_{D}})=0$$A^{2}(\mathcal{U}, U\mathrm{o})\cross A2n-2(U_{1})arrow \mathrm{C}$,$f_{D}$-trivial connec-

12and compute$\int_{X}c_{1}(LD)\cdot\tau_{1^{--}}\int_{R_{1}}c_{1}(\nabla_{1})\wedge\tau 1+\int_{R_{01}}C_{1}(\nabla_{0}, \nabla 1)\wedge \mathcal{T}1$are expressed as cocycles
for $\tau_{1}\in A^{2n-2}(U_{1})$ . We note that the elements ofvanishes.
whose component on$\pi:U_{1}arrowD$and $D$ have the same homotopy type.
is a deformation retract,
Since
So we have $H^{2n-2}(U_{1})\cong_{H^{2}()}n-2D$ , which implies$A^{2}(\mathcal{U}, U_{0})$$U_{0}$$U_{1}$$\tau_{1}=\pi^{*}\theta+d\rho$

for some $\theta\in A^{2n-2}(D),$we compute$\int_{R_{1}}C_{1}(\nabla_{1})$A$\mathrm{a}\mathrm{n}\mathrm{d}\rho\sim\in A^{2n-3}(U_{1})$,. Using the Stokes’ theorem and$\partial R_{1}=-R_{01}$,,$\tau_{1}=\int_{R_{1}}c_{1}(\nabla_{1})\wedge\pi^{*}\theta+\int_{R_{1}}c_{1}(\nabla_{1})\wedge d\rho=\int_{R_{1}}c_{1}(\nabla_{1})\wedge\pi^{*}\theta-\int_{R_{01}}c_{1}(\nabla_{1})\wedge\rho$$\int_{R_{01}}c_{1}(\nabla 0, \nabla_{1})$A$\tau_{1}$

A

A

$=$

$\int_{R_{01}}.C1(\nabla_{0}, \nabla_{1})$$=$$\int_{R_{01}}C_{1}(\nabla 0, \nabla_{1})\wedge\pi^{*}\theta+\int_{R_{01}}c_{1}(\nabla_{1})\wedge\rho+\int_{\partial R_{01}}C_{1}(\nabla 0, \nabla 1)\wedge\rho$$\pi^{*}\theta+\int_{R_{01}}C_{1}(\nabla 0, \nabla 1)$$d\rho$.

Hence we haveA$\int_{R_{1}}C_{1}(\nabla_{1})$Let$\nabla_{N_{D}}$$\tau_{1}+\int_{R_{01}}C_{1}(\nabla 0, \nabla_{1})$A$\tau_{1}=\int_{R_{1}}C_{1}(\nabla_{1})$Aofbe a connection on the normal bundle
as the connection, we can take$N_{D}$$L_{D}|_{U_{1}}\cong\pi^{*}N_{D}$$\int_{R_{1}}C_{1}(\nabla_{1})$A$\pi^{*}\theta=\int_{R_{1}}c_{1}(\pi^{*}\nabla_{N})D$$D$$\nabla_{1}$$\pi^{*}\nabla_{N_{D}}$A$\pi^{*}\theta=\int_{R_{1}}\pi^{*}$(

A$\pi^{*}\theta+\int_{R_{01}}c1(\nabla 0, \nabla 1)$$\pi^{*}\theta$.

, and also
. Sinceonso that we have$c_{1}(\nabla_{N}D)$$L_{D}|_{D}\cong N_{D}$$L_{D}|_{U_{1}}$A )$\theta$$= \int_{D}c_{1}(\nabla_{N_{D}})$A $\theta=0$ ,because the last term is the integration of a $2n$ -form on a $(2n-2)$ -dimensional submanifold.
. Since the question
Next, we compute the boundary integralAa neighborhood of , we set
is purely local, for any fixed point $p\in D$ , and$U_{p}=\pi^{-1}(V_{p})$ , and take a local coordinate system
around sufficiently small so$D=\{z_{1}=0\}$on
has a non-vanishing,that we may assume that
andby
. If we give a trivialization ofgives a section on for
section. Then, then on $U_{p}-D$$f_{D}=Z_{1}=z1^{\cdot}\pi^{*}SN$ ,$\int_{R_{01}}c_{1}(\nabla 0, \nabla 1)$$\pi^{*}\theta$$V_{p}\subset D$$(U_{p}, z)$$U_{p},$$s_{N}$$U$$\pi^{*}s_{N}$$U_{p}\subset U_{1}$

$p$

$p$

$N_{D}|_{V_{\mathrm{p}}}$$L_{D}$$L_{D}$$\pi^{*}s_{N}$ofand therefore the connection form
$df_{D}/f_{D}=dz_{1}/z_{1}$ of the Cauchy kernel on
let$\theta_{f_{D}}$$\tilde{\theta}=(1-t)\theta f_{D}+t\theta 1$

has the form
with respect to the frame,. To compute the secondary term$\nabla_{f_{D}}$$U$$\pi^{*}s_{N}$$c_{1}(\nabla 0, \nabla_{1})$13.

$c_{1}(\nabla 0, \nabla 1)=\tilde{\pi}*(d\tilde{\theta}-\tilde{\theta}\wedge\tilde{\theta})=\theta f-\theta 1$

Now by the Cauchy integral formula, we have$\int_{R_{01}\cap U_{\mathrm{p}}}c1(\nabla 0, \nabla 1)$A$\pi^{*}\theta=\int_{D\cap U_{p}}\theta=\int_{D\cap U}\tau 1$,

which implies$\int_{R_{01}}c_{1}(\nabla 0, \nabla 1)$A$\pi^{*}\theta=\int_{D}\theta=\int_{D}\tau_{1}$.

To organize the results of the above calculation, we obtain the correspondence$H^{2}(X,x_{-}D;\mathrm{c})\cong H2(A^{\cdot}(\mathcal{U}, U_{0}))\cong H_{2n-2}(D;\mathrm{c})$$(\mathrm{o}, C_{1}(\nabla_{1}),$.

$C1(\nabla 0, \nabla 1))rightarrow[D]$We remark that the above correspondence is more precise than that of the Poincar\’eduality. (1): We do not need the compactness of the ambient space X. (2): The dual of
the Chern class is found in $H.(D)$ , which indicates explicitly the location of singularities.

3

Proof of the main theorem

In this section, we give the proof of the main theorem. Let $X$ and $Y$ be n-dimensional
compact complex manifolds, and $f:Yarrow X$ a ramified covering with covering multiplicity . If gives a simple-sheeted covering, then we see that $c_{*}(Y)-$$\mu c_{*}(X)=0$ , which suggests us that the gap is brought about the ramification. So we
expect that the difference of the Chern classes can be localized at the ramification set.
We recall some basic facts about ramified coverings.
The ramification divisorof is defined as the analytic hypersurface defined by$\{\det(df)=0\}$ . Let
be the irreducible decomposition of . Then we have
$\mu$

$f$

$(\cdot \mathrm{u}\mathrm{n}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{d})\mu$$f$$R_{f}$$R_{f}= \sum r_{i}R_{i}$$R_{f}$$r_{i}+1=[\mathcal{O}_{Y},y:f*\mathcal{O}x_{f(},y)]$,the degree of integral extention
over
for a generic pointwords, indicates the number of deceasing of sheets at .
The branch locus
of is defined by the direct imageofbe the irreducible decomposition of . Then we have$\mathcal{O}_{Y,y}$$f^{*}\mathcal{O}_{\mathrm{x}},f(y)$

$y$

on

$R_{i}$. In other$R_{i}$$r_{i}$$f$$B_{f}$$f^{*}R_{f}$$B_{f}= \sum b_{i}B_{i}$$R_{f}$

under

$f$

. Let

$B_{f}$$b_{i}=\mu-\# f-1(X)$for a generic point on .
Now we assume that the ramification divisor of and theible components ofthe branch locus of are all non-singular. Here we remark that the branch locus possibly
has some self-intersection between other components. It followes from the as samptionthat the ramificcation divisor of is, that:is non-degenerate
so that it gives the un-ramified covering over , with covering multiplicity.
$x$$B_{i}$$\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\dot{\mathrm{c}}$

$f$

$f$

$f$

$\mathrm{n}\mathrm{o}\mathrm{n}-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\dot{\mathrm{a}}\mathrm{r}$$B_{i}$$f|_{R}$.

$R_{i}arrow B_{i}$$r_{i}/b_{i}$14First let us consider the case where the ramification divisor
has only one component, hence the branch locus
also does. We set $R_{f}=r\cdot R$ , and $B_{f}=b\cdot B$ .
Let $\omega:V_{1}arrow B$ be a tubular neighborhood of $B$ , and we take a covering
with, $U_{0}=\mathrm{Y}-R$ , and $\pi:U_{1}arrow R$ , a tubular neighborhood ofofsuch that$U_{1}\subset f^{-1}(V_{1})$ . We consider the \v{C}ech-de Rham cohomology of $Y$ associate with thecovering of , and set, in, that$R_{f}$$B_{f}$$\mathcal{U}=$$\{U_{0}, U_{1}\}$$R$$\mathrm{Y}$$H_{DR}^{2i}(\mathrm{Y})\cong H^{2i}(A^{\cdot}(\mathcal{U}))$$\mathcal{U}$$c_{i}(T\mathrm{Y})rightarrow(c_{i}(\nabla 0), C_{i}(\nabla_{1}),$ $c_{i}(\nabla 0, \nabla 1))$$c_{i}(f^{*\tau x})rightarrow(c_{i}(\tilde{\nabla}0), C_{i}(\tilde{\nabla}_{1}),$,$ci(\tilde{\nabla}0,\tilde{\nabla}1))$.

Since $df:TYarrow TX$ gives a bundle homomorphism outside the ramification, and
since
and
are tubular neighborhoods of and respectively, we have$U_{1}$$R$$V_{1}$$B$$T\mathrm{Y}|_{YR}-\cong f^{*}\tau X|Y-R$,$TY|_{U_{1}}\cong\pi^{*}N_{R}\oplus\pi^{*}TR\cong L_{R}|_{U_{1}}\oplus\pi^{*}TR$,$f^{*\tau x}|_{U_{1}}\cong f^{*}(\omega^{*}N_{B}\oplus\omega^{*}TB)\cong f^{*}(L_{B}\oplus\omega^{*}\tau B)$.

In particular on $U_{1}-R,$ $L_{R}\cong f^{*}LB$ are isomorphic as trivial bundles. Thus we can takeconnections on each neighborhood as follows:$\nabla_{0}=\tilde{\nabla}_{0}$such that$\tilde{\nabla}_{0}|_{V_{1^{-B}}}$$=$$\nabla_{f^{n}f}\oplus Bf**\omega\nabla\tau B$$\nabla_{0}|_{U_{1}-R}$$=$$\nabla_{f^{*}f_{B^{\oplus\nabla\tau R}}}\pi^{*}$$=$$\nabla_{f_{R}}\oplus\pi^{*}\nabla\tau R$

,

,

and
$\tilde{\nabla}_{1}=f^{*}\nabla_{L}\oplus Bf*\omega\nabla*\tau B$$\nabla_{1}=\nabla_{L_{R^{\oplus}}}\pi^{*}\nabla TR$,.

In the above, for a non-singular divisor $D$ we denote by, the section ofand$D$ , the$D$-trivial connection, and a connection of the tangent bundle of respectively.
Next we do local computation for secondary terms. Notation and choice of local
neighborhood and frames are the same as in 3.1.
$f_{D},$$\nabla_{f_{D}}$$f_{D}$$\tilde{A}=$

$=$

$=$

$(1-t)+t$$\nabla_{TD}$

15

Thus
$P^{i}(d\tilde{A}-\tilde{A}\wedge\tilde{A})$$=P^{i}(d\pi^{*}\theta_{TR^{-}}\pi^{*}\theta\tau 0R$A$\pi^{*}\theta_{TR}d\tilde{\theta}-\tilde{\theta}0$$\tilde{\theta})$A$=$$(d\tilde{\theta}-\tilde{\theta}\wedge\tilde{\theta})$A$P^{i-1}$($d\pi*\theta_{T}R^{-}\pi*\theta_{\tau R}$A$\pi^{*}\theta_{TR}$)$+P^{i}(d\pi^{*}\theta\tau R-\pi^{*}\theta_{\tau}R^{\wedge\theta_{\tau R}}\pi*)$

Since only$\tilde{\theta}$.

involves the fiber coordinate , it follows from the projection formula that$t$$c_{i}(\nabla 0, \nabla 1)$$=$$\tilde{\pi}_{*}P^{i}${($d\tilde{A}-\tilde{A}$

A )A

$=$

$=$

$c_{1}(\nabla_{f_{R}}, \nabla L_{R})$To express the secondary term of$c_{1}^{N_{1}}\cdot$..

$\tilde{A}$$\tilde{\theta}$$\tilde{\pi}_{*}$$(d\tilde{\theta}-\tilde{\theta}$

A
A)$P^{i-1}(\pi^{*}(d\theta R^{-}\theta_{R}\wedge\theta_{R}))$$\pi^{*}C_{i-1}(R)$}.

$c_{n}^{N_{n}}(\mathrm{Y})\in H^{2n}(A^{\cdot}(\mathcal{U}))$, we set$H_{\xi}^{(N_{1}\cdots N_{N})}(l)=l^{-1}(n \prod_{i=1}(ci(\xi)+ci-1.(\xi)\cdot l)N.\cdot-c_{1}^{N..N}c(1.n\xi n))=\sum_{\alpha=0}^{n-1}P_{\alpha}(C_{1n}\ldots C-1)l^{\alpha}.\cdot$

Then the \v{C}ech-de Rham class of$c_{1n}^{N_{1}N_{n}}\ldots c(Y)$is represented by,$( \pi^{*}(c_{1}^{N_{1}\ldots N_{n}}C)n(R),\prod_{i=1}^{n}(\pi^{*}ci(R)+\pi^{*}Ci-1(R)C_{1}(L_{R}))^{N_{*}},\cdot c_{1}(\nabla_{f_{R}},\nabla_{L_{D}})\wedge H(TN_{1}R\ldots N_{n})(c_{1}(L_{R})))$

This can be proved by induction for the number of indeterminate as follows. Here weremark that the degree of the class is not necessarily equal to , the dimension of theambient spaces. It follows from the inductive hypothesis that$c_{i}$$n$$c_{1}^{N_{1}}\cdots C_{k}^{N}k(Y)=$$( \pi^{*}(C_{1}^{N_{1}\ldots N_{k}}c_{k})(R),\prod_{=}(\pi^{*}Ci(R)+\pi^{*}ci-1(i1kR)c_{1}(LR))N:,$$c1(\nabla_{f}, \nabla L_{R})R$A$H_{TR}^{()}k(CN_{1}\cdots N1(LR))\mathrm{I}$$c_{k1}^{N_{k}}++1(\mathrm{Y})=$$(C_{k+1}^{N_{k+1}}(R),$$(\pi^{*}c_{k+}1(R)+\pi^{*}Ck(R)C1(L_{R}))^{N_{k}}+1,$

Thus, the secondary term of$c_{1}^{N_{1}}\cdots c_{k}^{N}+1k+1(\nabla 0, \nabla 1)$$c_{1}^{N_{1}}\cdots c_{k}N_{k+1}(\mathrm{Y})$is$c_{1}(\nabla_{fR}, \nabla_{L_{R}})\wedge HT(N_{k1}R+)(C1(L_{R})))$.

16$=c_{1^{1}}^{N}\cdots c_{k}N_{k}(\nabla_{0})\wedge c_{k}(Nk+1\nabla 0, \nabla+11)+C_{1}\cdot\cdot C^{N_{k}}(N_{1}.\nabla_{0}, \nabla k1)$$=c_{1}(\nabla_{f_{r}}, \nabla L_{R})$$=c_{1}(\nabla_{f_{R}}, \nabla_{L_{D}})$A$c_{k1}^{N_{k+1}}(+\nabla_{1})$A $c_{1}(L_{R})^{-1}(_{i=1}^{k+1} \prod(\pi^{*}Ci(R)+\pi C_{i1}-(*R)C_{1}(LR))^{N}:-cN_{1}\ldots 11k+C^{N_{k+}}1(R)\mathrm{I}$A$H_{T}^{(N_{1}\cdots N_{k+}}\mathrm{x}1$)$(c1(L_{R}))$,

which completes the induction.
, from our assumption that the ramifiIn particular for the case when
, thus $f^{*}c_{1}(L_{B})=(r+1)\cdot c_{1}(L_{R})$ .
cation divisor has degree we have
$TR\congf^{*\tau}B$ that $c_{i}(R)=f^{*}c_{i}(B)$ .
Since $f|_{R}:Rarrow B$ is non-degenerate, it follows from
Therefore we have
$n= \sum_{i=1}^{n}\dot{i}\cdot N_{i}$$f^{*}L_{B}=(L_{R})^{\otimes r+1}$$r$$H_{T}^{(N_{1}\cdots N)}n(RC_{1}(L_{R}))-[R]$By calculating the$\check{\mathrm{C}}\mathrm{e}\mathrm{c}\mathrm{h}- \mathrm{d}\dot{\mathrm{e}}$$=$$H_{f^{*}T}^{(N_{1}\cdots N_{n}})(B(r+1)^{-1}\cdot c1(f*(L_{B})))\wedge[(b/r)\cdot B]$$=$$\sum_{\alpha=0}^{n-1}\frac{b}{r(r+1)^{\alpha}}P_{\alpha}(c1\ldots C)n-1^{\cdot}C_{1}(LB)^{\alpha}-[B]$Rham class of$c_{1}^{N_{1}}\cdots c_{n}Nn(f^{*\tau X})$.

all the same, we obtain$c_{1}^{N_{1}}\cdots C_{n}^{N}(nTY)-C_{1^{1}n}^{N\ldots N}Cn(f*TX)=$( $(***),$$0,$$c_{1}(\nabla_{f_{R}}, \nabla L_{R})$A $(H_{TR}^{(}n(N_{1}\cdots N)c_{1}(LR))-(r+1)H(N_{1,\tau}\cdots N_{n})(_{C_{1}}f*B(f^{*}LB)))\mathrm{I}$ .

since it vanishes by evaluating on $R$ because of overdegree,(We omit the component on
which gives integration of $2n$ -forms on hypersurface, as observed in 3.1. )Now, as discused in 3.1, it follows from the correspondence of the Alexander duality that$U_{1}$$c_{1}^{N_{1}}\cdot’\cdot C_{n}^{N}(nTY)-[Y]-\mu\cdot c_{1^{1}n}^{N}\ldots C(N_{n}TX)\sim[X]$$= \int_{R}c_{1}(\nabla_{f}\nabla_{L})R’ R$

A $(H_{\tau R}^{(N_{1}}\ldots N_{n})(C_{1}(LR))-(r+1)H^{(}*B(fTN_{1}\cdots N_{n})f*c_{1}(LB)))$$=H_{\tau^{N_{1}\cdots N_{k+}}}^{()}R1(_{C_{1}(}L_{R}))-[R]-(r+1)H^{()}k+1(_{C_{1}}\tau^{N_{1}\cdots N}B(L_{B}))\sim[(b/r)\cdot B]$$= \sum_{\alpha=0}^{n-1}\frac{b(1-(r+1)^{\alpha+}1)}{r(r+1)^{\alpha}}P\alpha(c1(B)\cdots cn-1(B))\cdot c1(LB)^{\alpha}-[B]$.

We assumed that the ramification divisor of is non-singular, so we can assume thatthe tubular neighborhoods of irreducible components of the divisor do not intersect eachother. Hence taking independent sum we conclude:$f$

Theorem [Chern number formula for ramified coverings]
between compact
Let $f:Yarrow X$ be a ramified covering with covering multiplicity$man\dot{i}foldS$
divisorthe$n,$dimension
complexof , and $B_{f}=$ramificationofthe branch locus of . We assume that the ramification divisor and the irreducible$\mu$$R_{f}= \sum_{i}r_{i}R_{i}$$\sum_{i}b_{i}B_{i}$

$f$

$f$

17

components$N_{i}$$B_{i}$. Then:of the branch locus$B_{f}$

are all non-singular, and suppose that$n=\Sigma_{i+1}^{n}i$.

$c_{1}^{N_{1}}\cdots C_{n^{n}}N(\tau Y)-[Y]-\mu\cdot c_{1}^{N_{1}}\cdots c_{n}^{N}n(\tau X)\sim[x]$$= \sum_{i}(H_{\tau}(N_{1}.\cdots N_{n})(R.C1(LR\dot{.}))arrow[R_{i}]-(r_{i}+1)\cdot H^{(N}1N_{n}(c_{1}(LB\cdot)*)\tau B\dot{.}\cdots)-[B_{i}])$,$= \sum_{i}\sum_{\alpha=0}^{1}n-\frac{b_{i}(1-(ri+1)^{\alpha+}1)}{r_{i}(r_{i}+1)^{\alpha}}P_{\alpha}(C1(B_{i})\cdots Cn-1(Bi))\cdot c1(L_{B})^{\alpha}:-[B_{i}]$where we set$H_{\xi}^{(N_{1}\cdots N_{n})}(l)=l^{-1}(_{i=1} \prod^{n}(_{C_{i}(}\xi)+C_{i1}-(.\xi)\cdot l)N\dot{.}-C^{N_{1}}\cdots C_{n}^{N}(1\mathfrak{n}\xi))=\sum_{\alpha=0}^{n-1}P_{\alpha}(_{C_{1}}\cdots c_{n}-1)l^{\alpha}$

3.1.

Applications

In this section, we give some applications of our formula.
The result for the top Chern class implies the generalized Riemann-Hurwitz formula$\chi(Y)-\mu\cdot\chi(x)=-\sum_{i}bi$. $x(Bi)$ ,which is a special case of the formula proved by Y.Yomdin, [Y].

In case that$(\mathrm{n}=2)$:

The result for the second Chern class inplies$c_{2}(TY)-[Y]- \mu\cdot c_{2}(Tx)-[X]=-\sum_{i}bi$. $x(Bi)$ .

We remark that the more general formula is proved for algebraic cases. (see [I].)We can also deduce the formula for the square of the first Chern classes as follows:$c_{1}(TY)^{2_{\wedge}}[Y]-.

\mu\cdot c1(TX)2\wedge[X]=-\sum_{i}(2b_{i}\cdot\chi(Bi)+\frac{b_{i}(r_{i}+2)}{r_{i}+1}Bi$.

$B_{i})$.

Now from the fact that the signature of the surface is expressed by $L_{1}=(1/3)p_{1}=$, (The calculation for and -genus is found in [H1]), we obtain:$\frac{1}{3}(-2c_{2}+c_{1}^{2})$$\mathrm{T}$$\mathrm{L}$Theorem [The formula for signature for ramified coverings]
Let $f:Yarrow X$ be a ramified covering between compact complex analytic surfaces withcovering multiplicity
the ramification divisor of , andthe$\mu,$$R_{f}= \sum_{i}r_{i}R_{i}$$f$$B_{f}= \sum_{i}b_{i}B_{i}$18branch locus of . We assume that ramification divisor and irreducible components
are all non-singular. Thenthe branch locus$f$$B_{i}$

of$B_{f}$

Sign $(Y)-\mu\cdot S\dot{i}gn(X)$$=$$\frac{1}{3}(p_{1}(Y)-\mu\cdot p_{1}(x))$$=$$\frac{1}{3}\{(c_{1}(Y)2-\mu\cdot c1(X)^{2})-2(c_{2}(Y)-\mu\cdot c_{2}(x))\}$$=$$- \sum_{i}\frac{b_{i}(r_{i}+2)}{3(r_{i}+1)}B_{i}\cdot B_{i}$.

Originally, the formula for signature for cyclic coverings isfollows.

$\mathrm{f}_{\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{u}}1\mathrm{a}\mathrm{t}\mathrm{e}\dot{\mathrm{d}}$for 4-manifold as

Theorem [Hirzebruch [H1]] Let $X$ be a compact oriented differentiable manifold ofdimension 4 without boundary on which the cyclic groupsof order acts by orientation preserving diffeomorphisms. Suppose that is differential submanifold of $X$ , not. Thenoperates freely onnecessarily connected, and has codimension 2. And$G_{n}$$n$$Y$$X-\mathrm{Y}$$G_{n}$

Sign $(X)-n \cdot s\dot{i}gn(x/cn)=-\frac{n^{2}-1}{3n}Y’\cdot Y’$ .
where$\mathrm{Y}^{r}$is the branch locus in $X/G_{n}$ .

We can also deduce the formula for the Todd genus, which isTheorem Under the$sam\dot{e}$$T(\mathrm{Y})-\mu\cdot T(x)$$\frac{1}{12}(c_{2}+c_{1}^{2})$:assumption of the above theorem,$=$$\frac{1}{12}\{(c_{2}(Y)-\mu\cdot C2(X))+(c_{1}(Y)^{2}-\mu\cdot c1(x)^{2})\}$$=$$- \sum_{i}(\frac{b_{i}}{2}T_{1}(Bi)+\frac{b_{i}((r_{i}+1)2-1)}{12r_{i}(r_{i}+1)}.Bi$.

$B_{i})$.

In general, however, the calculation for the -genus or the -genus is more complicated, as examples we introduce formulas for the cases $n=3,4,5$ , and 6. (Also see [H1].):$\mathrm{T}$$\mathrm{L}$$(\mathrm{n}=3)$$T_{3}= \frac{1}{24}C_{1}c_{2}$,$H(l)= \frac{1}{24}((C_{2}+c^{2})1+c_{1}l)$.

$T( \mathrm{Y})-\mu\cdot T(x)---\sum_{i}(b_{i^{\frac{T_{2}(B_{i})}{2}+}}\frac{b_{i}(1-(r_{i}+1)^{2})}{r_{i}(r_{i}+1)}\int_{B}i\frac{T_{1}(B_{i})}{12}\cdot C1(N_{B}):\mathrm{I}\cdot$$(\mathrm{n}=4)$:$T_{4}= \frac{1}{720}\cdot(-C4+c_{3}c1+3c_{2}^{2}+4C2C_{1}-c124)$,

19$H(l)= \frac{1}{720}(15C2c1+5(C2+c_{1})2l-l^{3})$$T(Y)-\mu\cdot T(x)$$=$.

$- \sum_{i}\frac{T_{3}(B_{i})}{2}$$+ \sum_{i}\frac{b_{i}(1-(r_{i}+1)^{2})}{r_{i}(r_{i}+1)}\int_{B}:\frac{T_{2}(B_{i})}{12}\cdot C_{1}(NB_{i})$$+ \sum_{i}\frac{b_{i}(1-(r_{i}+1)^{3})}{r_{i}(r_{i}+1)^{2}}\int_{B_{*}}\frac{c_{1}^{3}(N_{B_{i}})}{720}$.

We can also define the signature for $n=4$ , as.

$L_{2}= \frac{1}{45}(7p2-p_{1})2\frac{1}{45}=(14c4^{-}14C_{3^{C}1}+3C^{2}+24c_{2}C^{2}-1c_{1})4$$H(l)= \frac{1}{45}((-10c_{2}+5c_{1})2l-l^{3})$

Sign

.

$(Y)- \mu\cdot S\dot{i}gn(x)=-\sum_{i}\frac{b_{i}(1-(r_{i}+1)^{2})}{r_{i}(r_{i}+1)}\int_{B}:\frac{L_{1}(B_{i})}{3}\cdot C_{1}(LB:)$$- \sum_{i}\frac{b_{i}(1-(r_{i}+1)^{4})}{r_{i}(r_{i}+1)^{3}}\int_{B}\dot{.}\frac{c_{1}(L_{B}\dot{.})^{3}}{45}$$(\mathrm{n}=5)$:$T_{5}= \frac{1}{1440}(-c_{4}C_{1}+c3C221^{+3C)}C_{2^{C}1}1-c23$$H(l)= \frac{1}{1440}\{(-c_{4}+C_{3}c_{1}+3C_{2^{+c}}^{2}.

$T(Y)-\mu\cdot T(x)$$=$,4C_{2}C^{24}-1)1+5C_{21}cl-c_{1}l3\}$.

$- \sum_{i}\frac{T_{4}(B_{i})}{2}$$+ \sum_{i}\frac{b_{i}(1-(r_{i}+1)^{2})}{r_{i}(r_{i}+1)}\int_{B_{i}}\frac{T_{3}(B_{i})}{12}\cdot c1(NB_{*}.)$.

$- \sum_{i}\frac{b_{i}(1-r_{i}+1^{4})}{r_{i}(r_{i}+1)^{3}}\int_{B}i\frac{T_{1}(B_{i})}{720}\cdot c_{1}^{3}(NB_{i})$$(\mathrm{n}=6)$:$L_{3}= \frac{1}{3^{3}\cdot 5\cdot 7}(62p_{3^{-}}13p2p_{1}+2p_{1}^{3})$$= \frac{1}{3^{3}\cdot 5\cdot 7}(-124c_{6}+124c_{5^{C}}1-72c4^{C_{2}}-26c4c12$+62 $C_{3^{-52c}21}^{2}3^{Cc}+26c_{3}c_{1}^{3}+10c_{2}^{3}+11C_{2}^{2}c_{1}-212C2C_{1}^{4}+2c_{1}^{6}$ ).
$H(l)= \frac{1}{3^{3}\cdot 5\cdot 7}\{(98c4-98c_{3}C_{1}+21c_{22}+28Cc-2271C_{1}4)\cdot l+(14C2-7c_{1})2.

l^{35}+2l\}$$S_{\dot{i}}gn(Y)- \mu\cdot Sign(x)=-\sum_{i}\frac{b_{i}(1-(r_{i}+1)^{2})}{r_{i}(r_{i}+1)}\int_{B_{i}}\frac{L_{2}(B_{i})}{3}\cdot C1(L_{B}.)$.

20$- \sum_{i}\frac{b_{i}(1-(r_{i}+1)^{4})}{r_{i}(r_{i}+1)^{3}}\int_{B}$.

$\frac{L_{1}(B_{i})}{45}arrow c\mathrm{i}(L_{B})^{3}$:.
$- \sum_{i}\frac{b_{i}(1-(r_{i}+1)^{6})}{r_{i}(r_{i}+1)^{5}}\int_{B}\dot{.}\frac{2c_{1}(L_{B}.)^{5}}{945}$

Acknowlegement: I would like to express my gratitude to my advisor Professor T. Suwa
for giving me an opportunity to study in this direction and teaching me how to do. Brasselet, M.
mathematics. I would also like to express my gratitude to Professors
Kwiecins. , D. Lehmann and J. Seade for stimulating discussions and encouragement.
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[BT]

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[H1]

F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer-Verlag, New
York, Heidelberg, Berlin, 1966.

[H2]

F. Hirzebruch, The signature of ramified coverings, Global Analysis (Papers inhonor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 253-265[I]

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968-996.

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des
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T. Suwa, Lectures at Hokkaido University, 1994-1995.

[Y]in a critical set of a complete inY. Yomdin, The structure of strata
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Pure Math., Amer. Math. Soc., Providence, R.I., 1983, 663-665.

$th\acute{e}or\dot{i}e$$Var\dot{i}\acute{e}t\acute{e}sstrat_{\dot{i}fi\acute{e}}es\mathrm{C}^{\infty}$$R\acute{s}\dot{i}dueS$$SouS^{- var\dot{i}}\acute{e}t\acute{e}s$invariantes d’un feuilletage singulier,$\mu=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$S\dot{i}n-$