数理解析研究所講究録
第 1764 巻 2011 年 80-101

80

INTRODUCTION TO NON-DEGENERATE MIXED FUNCTIONSMUTSUO OKA

1. COMPLEX ANAYTIC HYPERSURFACE SINGULARITIESContents
Chaptcr 1. Complex analytic hypersurafcc singularities
1.1 Milnor fibration
1.2 Weighted homogeneous polynomial
Chapter 2. Mixed function
2.1 Weighted homogeneous polynomlal
2.2 Mixed non-singular point
2.3 Transversality
2.4 Two special cases
2.5 General mixed functions
2.6 Milnor flbration
Chapter 3. Mixed projective curves
3.1 Canonical orientation
3.2 Milnor fibration and Hopf fibration
3.3 Degree of mixed projective hypersurfacc
3.4 Twisted join type polynomial
1.1. Milnor fibration. We first recall the theory of Milnor fibrations for holomorphic functions.
such that $f(0)=0$ .
Let $f(z)$ be a holomorphic function of n-variables
As is well-known, J. Milnor proved that$z_{1},$

Theorem 1. ([15]) There exists a positive number
ping$\epsilon_{0}$$\ldots,$$z_{n}$

such that the argument map-$\varphi:=f/|f|:S_{\epsilon}^{2r\iota-1}\backslash K_{\epsilon}arrow S^{1}$

is a locally trivial fibration for any positive$\epsilon$

with$\epsilon\leq\epsilon_{0}$where$K_{\overline{c}}=f^{-1}(0)\cap S_{e}^{2n-1}$.

We call this the first description of Milnor fibration. Topologically: $Farrow F$ . The characteristic polynomial is defined by $P_{n-1}(t)=$$\det(h$ $-t id)$ , where: $H_{n-1}(F)arrow H_{l-1}(F)$ .
is called monodromy homomorphism.
$S_{\epsilon}\backslash$.

$K_{\epsilon}\cong F\cross I/h$$h_{*}$$h_{*}$

as in the above theorem.
Theorem 2. ([15]) Suppose that is
is an isolated singularity.
Then $K$ is $(n-3)-$ connected. Suppose further that
Then has the homotopy type of a bouquet of spheres of $S^{n-1}$$\epsilon$$s\uparrow,fficientl\uparrow/small$
$O$

$F$

Define$gradf(z):=(\frac{\overline{\partial f}}{\partial z_{1}}, \ldots, \frac{\overline{\partial f}}{\partial z_{n}})$81
M. OKA

Euclidean inncr product and hermitian inner product:$z=(z_{1}, \ldots, z_{n})=x+iy_{)}w=(w_{1}\ldots., w_{n})=u+iv$$(z, w)=z_{1}\overline{w}_{1}+\cdots+z_{\iota}\overline{w}_{\iota}$,$(z, w)_{\mathbb{R}}=(x, u)+(y, v)=\Re(z, w)$$\mathbb{C}^{n}\Leftrightarrow \mathbb{R}^{2n}$,$z\Leftrightarrow(x, y)$$f(z)=g(x, y)+ih(x, y)$$gradg=(\frac{\partial g}{\partial x_{1}}, \ldots, \frac{\partial g}{\partial x_{n}}, \frac{\partial g}{\phi_{1}}, \ldots, \frac{\partial g}{\partial y_{n}})$Tangent spaces:$T_{Z}S_{\epsilon}=\{w\in \mathbb{C}^{r\iota}|\Re(z, w)=0\}$$T_{Z}f^{-I}(t)=\{w\in \mathbb{C}^{n}|(w, gradf(z))=0\}$$= \{w\in \mathbb{C}^{n}|w_{1}\frac{\partial f}{\partial z_{1}}+\cdots+\prime u)_{n}\frac{\partial f}{\partial z_{n}}=0\}$$=\{w\in \mathbb{R}^{2n}| (w, gradg(z))_{\mathbb{R}}=(w, gradh(z))_{R}=0\}$$T_{Z}F=\{w\in \mathbb{C}^{n}|\Re(w, z)=\Re(w, igrad\log(f(z))) =0\},$$F=\varphi^{-1}(1)$Note that$\log f(z)=\log|f(z)|+i\arg(f(z))$ ,$f(z)/|f(z)|=\exp(i\arg(f(z))$ .

Lemma 3. ([15]) $\varphi=f/|f|$ :are linearly independent over R.

$S_{\epsilon}arrow S^{1}$

is a submersion if and only if$\{z$, igrad $\log f(z)\}$1.1.1. Cone theorem and the second fibmtion. Suppose that $V=f^{-1}(0)$ has an
isolated singularity at the origin. Then there exists a positive
such that$S_{r}rhV$ for any $0<r\leq\epsilon$ . Note that$\epsilon>0$$S_{r}rhV\Leftrightarrow T_{Z}S_{r}\supset T_{Z}V${ , grad f(z)}: linearly independent over
Theorem 4. ([15]) Assume
is an isolated singularity of V. Then there exists
a positive number $r_{0}>0$ so that for any $0<r\leq r_{0},$rn V. In particular,.
Cone$\Leftrightarrow\forall z\in S_{r}\cap V,$$\mathbb{C}$

$z$

$O$

$S_{r}$$r,$$(B_{r}, B_{r}\cap V)\cong$

We call such$(S_{r}, K_{r})$$r_{0}$

a stable radius.

Proof. Suppose there does not exists suchfind a real analytic path:such that $p(t)$ is tangent to$p$$[0, \epsilon]arrow V,$$V$$r_{0}$. By Curvc Selection lemma, one can$p(t)\in V\backslash \{O\},$$t\neq 0$. Thus we can write$p(t)=\lambda(t)gradf(p(t)),$ $t>0$.

Then we have a contradiction as follows.
Put $\varphi(z)=(z, z)=\Vert z\Vert^{2}$ and.
$\phi=\sqrt{\varphi}$$\frac{d}{dt}\varphi(p(t))=2\Re(\frac{dp(t)}{dt},p(t))$$=2 \Re(\frac{dp(t)}{dt}, \lambda(t)gradf(p(t)))=2\Re\overline{\lambda}(t)\frac{df(p(t))}{dt}=0$.

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For the proof of the second assertion, we construct a vector field onsuch that, there exists an open neighborhood $U(z)$ such that
For any$T_{Z}f^{-1}(f(z))$ for any $z\in U(z)$ . More precisely, for any $0<\epsilon_{1}<r_{0}$ , there exists. Take $U(z)$ sosuch that $f^{-1}(7T)rhS_{7}$. for any
that$B_{r}\backslash \{O\}$$\chi$$\chi(z)\in$$z\in V\backslash \{O\}$$\epsilon_{1}\leq r\leq r_{0}$$|\eta|\leq\delta(\epsilon_{1}),$$\delta(\epsilon_{1})>0$$U(z)\subset f^{-1}(D_{\delta(\epsilon_{1})})\cap B_{r_{0}}\backslash B_{\text{\’{e}}_{1}}^{o}$$\Re(\chi(z), grad\phi(z))=-1$ .

Then using the integration $\psi(z, t)(\psi(z, 0)=z)$ of , we see that$-1$ and thus starting from $z\in S,.,$. Using this, we get a$\frac{d\psi(\psi(Z,t)}{dt}=$$\chi$$\lim_{tarrow\tau}\cdot|\psi(z, t)|=0$

homeomorphism:$\Psi:(S_{r}, K_{r})\cross[0, r)arrow B_{r}\backslash \{O\},$$\Psi(z.t)=\psi(z, t)$which extends to the homeomorphism Cone $(S_{r}, K_{r})\cong(B_{r}, V)$ .

1.1.2. Second description of Milnor fibration. Take
Theorem 5. Fix$\Delta_{\delta}^{*}\iota s$$r,$$0<r\leq r_{0}$as before. Fix $0<r\leq r_{0}$ .

$r0$. Take $\delta>0sufficicntl\uparrow/small$ so that for. Put
a locally trivial fibration, where$|\eta|\leq\delta,$$E(r, \delta)^{*}=\{z\in B_{r}\backslash V||f(z)|\leq\delta\}$$S_{r}rhf^{-1}(\eta)$

Theorem 6. Two fibrations
lent.

$\square$. Then.

$f$:$anyr \int_{arrow}$$E(r, \delta)^{*}$$\Delta_{\delta}^{*}=\{\eta\in \mathbb{C}|0<|\eta|\leq\delta\}$$\varphi:S_{r}\backslash K_{r}arrow S^{1}$

and

$f$

:

$\partial E(r, \delta)^{*}arrow S_{\delta}^{1}$are equiva-,
and be given posi1.2. Weighted homogeneous polynomials. Let
is called
tive integers with $gcd(a_{1}, \ldots, a,,)=1$ . An analytic function
weighted homogeneous polynomial of type $(a_{1}\ldots..a_{n};c)$ or a weighted homogeneous
if satisfies the functional
polynomial of degree with the weight vector
equality$a_{1},$$\ldots$$c$$a_{n}$$f(z_{1}, \ldots , z_{f\iota})$$(a_{1}, \ldots, a_{n})$$c$$f(t^{a\iota}z_{1}, \ldots, t^{a_{n}}z_{n})=t^{c}f(z_{1}, \ldots, z_{n}))$Definition 7. The

C’$\mathbb{C}$

$a$

$f$

$z\in \mathbb{C}^{\iota},$$t\in \mathbb{C}^{*}$’-action associated with a weighted homogeneous polynomial:$\cross \mathbb{C}^{n}arrow \mathbb{C}^{n}$

Then weighted homogeneous

Note that $V=f^{-1}(0)$ is$\mathbb{C}$,$(t, z)\mapsto t\circ z:=(t^{a_{1}}z_{1}, \ldots, t^{a_{n}}z_{?t}))$$\Leftrightarrow f$’( oz) $=t^{c}f(z)$ .
$t$stable.

Example 8. 1. Let $f(z)$ be a homogeneous polynomial of degree . Then $f(z)$. Thus $f(z)$ is asatisfies the obvious equality:$(1,\cdots,1;c)$,weighted homogeneous polynomial of type2. Let $f(z)=z_{1}^{a_{1}}+\cdots+z_{n}^{a_{n}}$ (Pham-Brrieskorn polynomial). Then it is weighted
and $p_{j}=$homogeneous polynomial of type $(p_{1}, \ldots,p_{n}, c)$ where $c=$ lcm$c$$f(tz_{1}, \cdots , tz_{\iota})=t^{c}f(z_{1}, \cdots, z_{\tau\iota})$$(a_{1}, \ldots, a_{n})$$c/a_{j}$.

Theorem 9. Assume that is a generalized weighted homogeneous polynomial of. Thentype(1) (Euler equality). We have the equality;$f$$(a_{1}, \ldots, a_{n};c)$$cf( z)=\sum_{i=1}^{n}a_{i^{Z}i^{\frac{\partial f}{\partial z_{i}}}}$.

(2) Assume that $c\neq 0$ . The only possible cnttical value of$\mathbb{C}^{n}-f^{-1}(0)arrow$ C’ is a locally trivial
fibmtion.

$f$

is

$0$

and

$f$

:

83
M. OKA(3) Assume that$c>0$ . Then the Milnor fibration of at the origin,
is defincd on any sphere
andt he restriction of the above fibration
over $S^{I}: \int:\int^{-1}(S^{1})arrow S^{1}$ is equivalent to the Milnor fibration of at theorigin; $f/|f|$ :. In particular, the Milnor
for any$f^{-1}(1)$ .
isdiffeomorphic
hypersurface
totheThe hypersurface
fiber
affine$f^{-1}(0)$ is contractible to the origin.
(4) ([Or-Mil]) Assume that the mgin an isolated singular point and$c>$$a_{1},$$\ldots$$f$$a_{n},$$S_{\Xi},$$\epsilon>0$$f$$\epsilon>0$$S_{\epsilon}^{2n-1}-K_{\epsilon}arrow S^{1}$

$0$

$0$

$\iota s$$a_{1},$.

(a) For any $e>0$ , the spheretransversely.

$S_{\epsilon}$$\ldots,$$a_{n},$and the hypersurface $f^{-1}(0)$ intersect$v_{i}\in N,$ $(u_{i}, v_{i})=1$ . $i=1,$ . .
(b) Putting $c/a_{i}=u_{\tau}/v_{i}$ with, , thedivisor of the zeta-function is given as follows.
$n$$u_{i},$$( \zeta(t))=(\frac{1}{v_{1}}\Lambda_{1l_{1}}-1)\cdots(\frac{1}{v_{n}}\Lambda_{\uparrow\iota_{n}}-1)$

In particular,$\mu=\prod_{i=1}^{n}\ell-1)=\prod_{i=1}^{n}(\frac{\tau\iota_{i}}{v_{i}}-1)$.

Proof. (4-a): Euler equality is given by differentiating $t^{c}f(z)=f$ ( oz). Assume
$t$

that

$\lambda z_{i}=\overline{\frac{\partial f}{\partial z_{i}}},$$i=1,$$\ldots,$$n$

Then by Euler equality, we have$cf( z)=\sum_{i=1}^{n}a_{i}z_{i}\frac{\partial f}{\partial z_{i}}=\overline{\lambda}\sum_{i=1}^{n}a_{i}z_{j}\overline{z}_{i}\neq 0$.
$\square$1.2.1. Equivalence of global and local fibmtions. $E=f^{-1}(S^{1})$ and define$\psi:Earrow S_{r}\backslash V$,$\psi(z)=\tau oz,$$\Vert\tau\circ z\Vert=r,$$\tau>0$or$\xi=\psi^{-1}:S_{r}\backslash Varrow E$,$\xi(z)=s(z)\circ z,$ $s(z)=|1/|f(z)|^{1/c}$

Then we havc the commutative diagram:$E$$arrow^{\psi}$$S_{r}\backslash K$$\downarrow f/|f|$$\downarrow f$$S^{1}$$arrow^{id}$$S^{1}$

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INTRODUCTION TO NO-DEGENERATE MIXED FUNCTIONS

2. $MlXED$ FUNCTION
Now we considcr the situation of rcal algebraic variety of codimension 2:$V=\{g(x, y)=h(x,y)=0|z=x+iy\in \mathbb{C}^{\tau\iota}\}$whcre $z_{j}=x_{j}+iy_{j}$ andover the circle. It can be writte as$g,$ $h\in \mathbb{R}[x, y]$$V=\{z\in \mathbb{C}^{n}|f(z,\overline{z})=0\}$

Wc call$f$,. We study when the link is fibered$f( z,\overline{z})=g(\frac{z+\overline{z}}{2}, \frac{z-\overline{z}}{2i})+ih(\frac{z+\overline{z}}{2}, \frac{z-\overline{z}}{2i})$a mixed polynomial.

2.1. Weighted homogeneous polynomials. A mixed polynomial
is called polar weighted homogeneous (respectively mdially weighted homogeneous)(resp.
..
and a non-zero integer
if there exist integers positive integers) such thatand a non-zero integer
positive integers$gcd(p_{1}\ldots., p_{n})=1,$, if$f( z,\overline{z})=\sum_{\nu,\mu}c_{\nu,\mu}z^{\nu}\overline{z}^{\mu}$$p_{1},$$q_{1},$$\ldots,$$,p_{n}$$,$$m_{p}$$m_{r}$$q_{\iota}$$\sum_{j=1}^{n}p_{j}(\nu_{j}-\mu_{j})=rr\iota_{p}$

$($

resp.

$gcd(q_{1},$ $\ldots,$ $q_{n})=1,$$c_{\nu.\mu}\neq 0$$\sum_{j=1}q_{j}(\nu_{j}+\mu_{j})=\gamma r\iota_{7}.)$

Wc sayis a md-polar weighted homogeneous if is radially weighted homo, and is also polar weightcd homogeneous of
gcneous of type, say,. We definc vcctors of rational numbers
type, say $(I)1,$
by $u_{i}=q_{i}/m_{r},$and we call thcm the normalized mdialand(respectively polar) weights.
$\int(z,\overline{z})$$\int$$(q_{1}, \ldots, q_{n};m_{r})$$\ldots$$f$$p_{\iota};rn_{\rho})([23])$$(tl_{1)}\ldots , \prime n_{1})$$v_{i}=p_{i}/m_{\rho}$$(v_{1}, \ldots, v_{n})$Example 10. Let$f=z_{1}^{3}\overline{z}_{1}+z_{2}^{4}\overline{z}_{2}$Using a polar coordinate, we define a polar$(r, \eta)$. Polar $weight:(3,2;6)$ , Radial weight: (5,4;20)

of$\mathbb{R}^{+}\cross S^{1}$$\mathbb{C}||\eta|=1\}$

where-action on$\mathbb{C}$’$7^{\cdot}>0$$(r, \eta)\circ z=(r^{q_{1}}\eta^{\rho_{1}}z_{1}, \ldots, r^{q_{\mathfrak{n}}}\eta^{\rho_{n}}z_{n})$,$\mathbb{C}^{n}$andbywith $S^{1}=\{\eta\in$$\eta\in S^{1}$$re^{i\eta}=(r, \eta)\in \mathbb{R}^{+}\cross S^{1}$$(r, \eta)\circ\overline{z}=\overline{(r,\eta)oz}=(7^{q_{1}}\eta^{-\rho_{1}}\overline{z}_{1}, \ldots, r^{q_{n}}r|^{-\rho_{n}}\overline{Z}_{7l})$.

is rad-polar weightcd homogeneous polynomial. Then
Assume thatfies the functional equality$f(z,\overline{z})$(1)$f((r, \eta)\circ(z,\overline{z}))=r^{m_{r}}\eta^{m_{p}}f(z,\overline{z})$$f$satis-.

This notion was introduced by Ruas-Seade-Verjovsky [27] implicitly and then by
Cisneros-Molina [5].
It is easy to see that such a polynomial defines a global fibration$f:\mathbb{C}^{n}-f^{-1}(0)arrow \mathbb{C}^{*}$For example, put.

$U_{r,\theta}=\{\rho e^{i\eta}|1/r\leq\rho\leq r, -\theta\leq\eta\leq\theta\}$$\Psi:[1/r, r]\cross[-\theta, \theta]\cross f^{-1}(1)arrow f^{-1}(U_{r,\theta})$

Theorem 11. $\varphi=f/|f|$ :,$\Psi(\rho, \theta, z)=(’ C^{\backslash }/m_{p}))\circ z$$S_{r}^{2n-1}\backslash K_{r}arrow S^{1}$$r>0$ and it is equivalent to $f$ : $f^{-1}(S^{1})arrow S^{1}$ .

is a locally trivial fibmtion for any

Proof. First,$\psi_{\theta}$:$\varphi^{-1}(1)arrow\varphi(\exp(i\theta))$,$\psi_{\theta}(z)=(1, \exp(i\theta/m_{p}))$oz

gives the trivialization. Define$\Phi:S_{r}^{2n-1}\backslash K,$.

$arrow f^{-1}(S^{1})$$\Phi(z)=(1/|f(z)|^{1/m_{r}}, 0)\circ z$$\square$85
M. OKA2.2. Mixed non-singular point. Lct
bc a mixcd polynomial and we consider a hypersurface. Put $z_{j}=x_{j}+iy_{j}$ . Then
is a real analytic function of $2n$ variables $(x, y)$ with $x=(x_{1}, \ldots, x_{n})$ and$y=(y_{1}, \ldots, y_{n})$ .
Put $f(z,\overline{z})=g(x, y)+ih(x, y)$ where . arc real analytic
functions. Recall that$f(z,\overline{z})$$V=\{z\in \mathbb{C}^{r\iota};f(z_{\grave{l}}\overline{z})=0\}$$f(z,\overline{z})$

$h$

$g$

.

$\frac{\partial}{\partial z_{j}}=\frac{1}{2}(\frac{\partial}{\partial x_{j}}-i\frac{\partial}{\partial y_{J}})$$\frac{\partial}{0’\overline{z}_{J}}=\frac{1}{2}(\frac{\partial}{\partial x_{J}}+i\frac{\partial}{0y_{J}})$

Thus for a complex valued function , we dcfine$f$$\overline{\partial}z_{j}\partial\perp_{=}$

Wc assume that
two subvariety. Put

$g,$

$h$

;di

,

$+i \frac{\partial h}{\partial z_{J}}$$\lrcorner\partial\underline{\dot{(}})_{L}\partial\overline{z}_{j}\overline{\partial}\overline{z}_{jJ}=+i\frac{\partial}{\partial}\frac{g}{z}-$are non-constant polynomials. Then$d_{N}g( x, y)=(\frac{\text{\^{o}}}{\partial}x_{1}L , .

.

.

, \frac{\text{\^{o}}}{\partial}x_{n}4_{-\#_{y_{1}}^{\partial}}, .

$d_{\mathbb{R}}h( x, y)=(\frac{\partial h}{\partial x_{1}}, \ldots, \frac{\partial h}{\partial x_{n}}, \frac{\partial h}{\partial y_{1}} , .

.

.

$V$

is a real codimension, , \#_{11n}^{\partial_{-}})\in \mathbb{R}^{2n}$, \frac{\partial h}{\partial’y_{n}})\in \mathbb{R}^{2n}$For a complex valued mixed polynomial, we use the notation:$\ldots,\partial)\in \mathbb{C}^{n}$$df(z,\overline{z})=(_{\partial z_{1}\partial z_{n}}^{\lrcorner}\partial;,$.

$\overline{d}f(z,\overline{z})=(\frac{\partial\int}{\partial\overline{z}1}\cdots\cdot\cdot\overline{0}_{\overline{z}_{n}}^{\partial}\perp)\in \mathbb{C}^{n}$

We say that a point $z\in V$ is a mixed-singular point of if and only if:
is surjectivc or equivalently the two vectors $dg_{N}(x, y),$ $dh_{N}(x, y)$are linearly dependent over R.
$V$$df_{z}$$T_{Z}|BC^{n}arrow T_{f(Z)}\mathbb{C}$

Proposition 12. [23] The following two conditions are equivalent.
(1) $z\in V$ is a mixed singular point.
(2)
are linearly dependent over R.
(3) There exists a complex number . $|cx|=1$ such that$d_{CjR},$$clh_{\mathbb{R}}$$\overline{df(z.\overline{z})}=\alpha\overline{d}f(z,\overline{z})$$\alpha$.

2.3. Transversality. We assume again
is a rad-polar weighted homogeneous polynomial as before. First wc obscrve that $f^{-1}(t)$ is mixcd non-singular forany
as :is surjectivc.
$\int(z,\overline{z})$$t\neq 0$$df$$T_{Z}f^{-1}(t)arrow T_{t}\mathbb{C}$

Proposition 13. [23] Let $V=f^{-1}(0)$ . Assume that the mdial weight $q_{j}>0$ for
any . Thenis contractible to the origin O. If further
is an isolated mixed
singularity of $V,$
is smooth.
$V$

$j$

$O$

$V\backslash \{O\}$

Proposition 14. [23] (Tmnsversality) Under the same assumption as in Proposition 13, th sphereintersects tmnsversely with for any$\tau>0$ .
$e$

$V$

$S_{r}=\{z\in \mathbb{C}^{n};\Vert z\Vert=\tau\}$

2.4. Two special cases. There are two cases foc which we know more about the
topology of Milnor fiber.
2.4.1. Casel, Simplicial polynomial. Let$\ldots.c_{s}\neq 0$ . Putpolynomial. Here we assume that$\int(z.\overline{z})=\sum_{j=1}^{9}c_{j}z^{n_{j}}\overline{z}^{m_{j}}$be a mixed$c_{1},$$\hat{f}(w):=\sum_{j=1}^{s}c_{j}w^{n_{j}-m_{j}}$,$w=(w_{1}\ldots., w_{\tau\iota})\in \mathbb{C}^{n}$.

We call
the the associated Laurent polynomial. This polynomial plays
an important role for the determination of the topology of the hypersurface $F=$$f^{-1}(1)$ . Note that$f$

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INTRODUCTION TO NO-DEGENERATE MIXED FUNCTIONS

is a polar weighted homogeneous polynomial of polar
Proposition 15. If$(p_{1}, \ldots,p_{n};m_{p}),\hat{f}(w)$is also a weighted homogeneous Laurent polynoweight type.
in the complex variables
mial of type$f(z,\overline{z})$$(p_{1}, \ldots,p_{?l};m_{\rho})$$’|v_{1)}\ldots,$ $w_{n}$is called simplicial if thc exponent vectors
A mixed polynomial
respectively. In particular, simplicin
independent
are linearly
ity implies that $s\leq n$ . When $s=n$ , we say that is full. Put $n_{j}=(n_{j.1}, \ldots, n_{j,n})$ ,$m_{j}=(m_{j,1}, \ldots,m_{j,n})$ in. Assume that $s\leq n$ . Consider two integral matrix
respectively.
$N=(n_{i,j})$ and $M=(m_{i,j})$ where the k-th row vectors are$\{n_{j}\pm$$f(z,\overline{z})$$m_{j}|j=1,$ $\ldots,$ $s\}$$Z^{n}$$f$$N^{n}$$m_{k}$$n_{k},$

is simplicial,
be a mixed polynomial as above. If
Lemma 16. Let
thenis a polar weighted homogeneous polynomial. In the case $s=n,$
is simplicial if and only if dct $(N\pm M)\neq 0$ .
$f(z,\overline{z})$$\int(z_{j}\overline{z})$$f(z,\overline{z})$$f(z,\overline{z})$2.4.2. Example. Let$B_{a,b}(z,\overline{z})=z_{1}^{a_{1}}\overline{z}_{1}^{b_{1}}+\cdots+z_{n}^{a_{n}}\overline{z}_{n}^{b_{n}},$ $a_{i},$$b_{i}\geq 1,$$f_{a,b}(z,\overline{z})=z_{1}^{a_{1}}\overline{z}_{2}^{b_{1}}+\cdots+z_{n}^{a_{n}}\overline{z}_{1}^{b_{n}},$$b_{i}\geq 1,$$a_{i},$$k(z,\overline{z})=z_{1}^{d}(\overline{z}_{1}+\overline{z}_{2})+\cdots+z_{n}^{d}(\overline{z}_{n}+\overline{z}_{1}),$$\forall i$$\forall i$$d\geq 2$.

The associated Laurent polynomials are$\hat{f_{a.b1}}(w)=u)^{a_{1}}?v_{2}^{-b_{1}}+\cdots+’|l)_{n1}^{(x_{n\prime u)^{-b_{n}}}}$$\hat{k}(w)=w_{1}^{d}(1/w_{1}+1/w2)+\cdots+w_{n}^{d}(1/w_{n}+1/w_{1})$.

Corollary 17. For the polynomial $f_{a,b}$ , the following conditions are equivalent.
(1) $f_{a,b}$ is simplicial.
(2) $f_{a,b}$ is a polar weighted homogeneous polynomial.
.
(3) (SC)$a_{1}\cdots a_{n}\neq b_{1}\cdots b_{\downarrow}$Let
be a polar wcighted homogeneous polynomial of$(q_{1}\ldots..q_{n};m_{r})$and of polar weight typc
radial weight typc
. Lct $F=f^{-1}(1)$ be the fiber.
$f( z,\overline{z})=\sum_{j=1}^{s}c_{j}z^{n_{j}}\overline{z}^{m_{j}}$$(p_{1}\ldots.

, p_{n};m_{p})$2.4.3. Canonical stmtification ofF and the topology of each stmtum. For any subset$I\subset\{1,2, \ldots, n\}$ , we define$\mathbb{C}^{I}=\{z|z_{j}=0, j\not\in I\},$$\mathbb{C}^{*I}=\{z|z_{i}\neq 0 iff i\in I\},$$\mathbb{C}^{*n}=C^{*\{1,\ldots,n\}}$. For simplicity,by the restriction:
and we define mixed polynomials.
Note thatis a non-empty
we write a point ofas . Putis not constantly zero. Now we observe
proper subset of
if and only ifthat the hypersurface $F=f^{-1}(1)$ has the canonical stratification$f^{f}$$\mathbb{C}^{I}$$Z_{I}$$\mathbb{C}^{*I}$$f^{I}=f|_{\mathbb{C}^{I}}$$F^{*I}=\mathbb{C}^{*I}\cap F$$F^{*I}$$f^{I}(z_{I},\overline{z}_{I})$$F=\coprod_{J}F^{*I}$

.

$:=$
. Put
Thus it is essential to determine the topology of each stratum, the open dense stratum and put
whereis theassociated Laurent weighted homogeneous polynomial.
$F^{*1}$$\hat{F}^{*};=f^{-1}(1)\cap \mathbb{C}^{*n}$$F\cap \mathbb{C}^{*n}$$F^{*}$$\hat{f}(w)$is a simplicial polar weighted homogeneous
Theorem 18. [23] Assume that$f(w)$
polynomial and let
be the associated Laurent weighted homogeneous polynowhich gives an
mial. Then there exists a canonical diffeomorphism :$f(z,\overline{z})$$\mathbb{C}^{*n}arrow \mathbb{C}^{*n}$$\varphi$

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M. OKA

isomorphism of the two Milnor fibmtions defined byand$f(z,\overline{z})$$\hat{f}(w)$;$arrow^{f}$$\mathbb{C}^{*n}-f^{-1}(0)$$\mathbb{C}^{*}$$\mathbb{C}^{*}\downarrow id$$\mathbb{C}^{*\iota}-\hat{f}^{-1}(0)\downarrow\varphi$$arrow^{f^{\dot}}$

and it satisfies
monodromy maps.

and$\varphi(F^{*\iota})=\hat{F}^{*n}$is compatible with the respective canonical$\varphi$

Proof. Assume first that $s=n$ for simplicity. Recall that$f( w)=\sum_{j=1}^{n}c_{j}w^{n_{j}-m_{j}}$Let $w=(?1)1\cdots\cdot$ ,of . We construct$\})_{?L})$.

be the complex coordinatcs ofwhich is the ambient space:
so that $\varphi(z)=w$ satisfies$\mathbb{C}^{\prime\iota}$$\hat{F}$$\mathbb{C}^{*\prime\iota}arrow \mathbb{C}$$\varphi$“$\iota$$w(\varphi(z))^{n_{j}-m_{j}}=z^{n_{j}}\overline{z}^{m_{J}}$

thus

,

$\int(\varphi(z))=\int(z)\wedge$.

For the construction of , we use the pol\‘ar coordinates
forC’ andpolar coordinatesfor. Thus $z_{j}=p_{j}$ cxp
and $w_{j}=\xi_{j}\exp(i\eta_{j})$ .
First we take. Put $n_{j}=(n_{j,1}, \ldots.n_{j,n}),$ $m_{j}=(m_{j,1)}\ldots, m_{j,n})$ in. Consider two integral matrix $N=(n_{i,j})$ and $M=(m_{i,j})$ where the k-th
row vector are
respectively. Now taking the logarithm of thc equality, we get an equivalent equality:$(\rho_{j}, \theta_{j})$$\varphi$$(\xi_{j}, 71j)$$z_{j}\in$$(i\theta_{j})$$w_{j}$$\eta j=\theta_{j}$$N^{n}$$n_{k},$$m_{k}$$z^{n_{j}}\overline{z}^{m_{j}}=w^{n_{j}-m_{j}}$$(n_{j1}+7\prime l_{j1})\log\rho_{1}+\cdots+(n_{jn}+7|\iota_{jn})\log p_{7l}$$=(n_{j1}-m_{j1})\log\xi_{1}+\cdots+(n_{jn}-m_{jn})$

for $j=1,$$\ldots,$

$n$

Iog

$\xi_{n}$,. This can be written as(2)$(N+M)(\begin{array}{ll}log \rho_{1}| log \rho_{n}\end{array})=(N-M)(\begin{array}{ll}log \xi_{l}| log \xi_{n}\end{array})$

Put $(N-M)^{-1}(N+M)=(\lambda_{ij})\in$ GL$\varphi:\mathbb{C}^{*n}arrow \mathbb{C}^{*n}$.

$(n, \mathbb{Q})$. Now we define$\varphi$

as follows.

$z=(\rho_{1}\exp(i\theta_{1}), \ldots, \rho_{\iota}\exp(i\theta_{?b}))\mapsto$$w=(\xi_{1}\exp(i\theta_{1}), \ldots, \xi_{n}\exp(i\theta_{n}))$where is given by. It is obvious thatfor $j=1,$
is a real analytic isomorphism of. Let us consider the Milnor fibrations
to
of
and
in the respective ambient tori.
$\xi_{j}=\exp(\sum_{i=1}^{?l}\lambda_{ji}\log\rho_{i})$$\xi_{j}$$\mathbb{C}^{*n}$$f(z,\overline{z})$$\ldots,$$n$$\mathbb{C}^{*n}$$\hat{f}(w)$$\mathbb{C}^{*n}$$f:\mathbb{C}^{*n}\backslash f^{-1}(0)arrow \mathbb{C}^{*}$

Recall that the monodromy maps:$h^{*}$$\hat{h}^{*}$:,$h^{*},\hat{h}^{*}$$F^{*}arrow F^{*}$$\hat{F}^{*}arrow\hat{F}^{*}$,,$\hat{f}:\mathbb{C}^{*n}\backslash f^{-1}(0)arrow \mathbb{C}^{*}$are given asoz$w\mapsto\exp(2\pi i/m_{p})$ ow.
$z\mapsto\exp(2\pi i/rr\iota_{p})$$\varphi$

88
INTRODUCTION TO NO-DEGENERATE MIXED FUNCTIONS. Namely
Recall that the ’-action associated with $f(w)$ is thc polar action of. Thus we have the commutativc dia, cxpowgram:$f(z,\overline{z})$$\mathbb{C}$$\exp i\theta$$(\exp(ip_{1}\theta)tl\prime_{1},$$=$$(ip_{\iota}\theta)\prime w,,)$$\ldots$$\hat{F}_{\alpha}^{*}F_{\alpha}^{*}\downarrow\varphiarrow^{arrow\hat h^{*}h^{*}}$$\hat{F}_{\alpha}^{*}\Gamma_{\alpha}^{*}\downarrow\varphi$

where$F_{a}^{*}=f^{-1}(\alpha)\cap \mathbb{C}^{*n}$2.4.4. Remark. The caseand$\hat{F}_{a}^{*}=f^{-1}(\alpha)\cap \mathbb{C}^{*n}$for$\alpha\in \mathbb{C}^{*}$.

$\square$,. is studied in [27]. In this: $f^{-1}(1)arrow g^{-1}(1)$ is givcn by$f(z.\overline{z})=z_{1}^{a_{1}}\overline{z}_{1}+\cdots+z_{n}^{a_{n}}\overline{z}$case, $g=z_{1}^{a_{1}-1}+\cdots+z_{n}^{a_{n}-1}$ and$\varphi$$w_{j}=z_{j}|z_{j}|^{\frac{2}{a_{j}-1}},$$.)^{\prime=1,\ldots,n}$

We can see that this is a homeomorphism.

2.4.5. Zeta-functions. Now we know that by [19, 20], the inclusion map
is $(n-1)$ -equivalence and $\chi(\hat{F}^{*})=(-1)^{n-1}\det(N-M)$ for $s=n$ and otherwise.
In general, for a diffeomorphism $h:Farrow F$ , the zeta function of is defined by$\hat{F}^{*}arrow \mathbb{C}^{*n}$

$0$

$h$

$\zeta_{h}(t)=\frac{\prod_{j=1}^{\infty}\det(th_{2j-1}-id)}{\prod_{j=0}^{\infty}dct(l_{\text{ノ}}h_{2j}-id)}$where $h_{j}=h_{*}$ : $H_{j}(F)arrow H_{j}(F)$ .
. Thc
has a period
Note also in our casc the monodromy map :$p*$$m_{p}|k$and othcrwise. Thus using the formula ofis
if
fixed point locus of
the zeta function (see, for example [15]),へ$\hat{h}$$\hat{F}^{*}arrow F^{*}$$m_{p}$$(\hat{h})^{k}$$\emptyset$$\zeta_{\hat{h}}.(t)=\exp(\sum_{j=0}^{\infty}(-1)^{n-1}dt^{jm_{\rho}}/(jm_{p}))=(1-t^{m_{p}})^{(-1)^{n}d/m_{p}}$where $d=\det(N-M)$ if $s=n$ and $d=0$ for $s<n$ . Translating this in the: $F^{*}arrow p*$ , we obtain
monodromy$h^{*}$

Corollary 19.
has a homotopy type of CW-complex of dimension $n-1$ andthe inclusion map
is an $(s-1)$ -equivalence. The zeta functionof$(1-t^{m_{p}})^{(-1)^{n}d/m_{p}}$$F^{*}arrow F^{*}$:
with $d=\det(N-M)$ if $s=n$ andis given as$F^{*}$$F^{*}arrow \mathbb{C}^{*n}$$\zeta_{h}\cdot(t)$$h^{*}$$\zeta_{h}\cdot(t)=1$

for $s<n$ .

. Now we are ready to patch together the information of
the strata
for the topology of . First wc introduce the notion of k-convenienceis k-convenientwhich is introduced for holomorphic functions ([20]). We say$I\subset\{1,2,\ldots,n\}$$|I|\geqn-k$followingany.
isobvious by the
if
with
The
for
definition.
2.4.6. Connectivity of$F^{*I}$$F$$F$$f(z,\overline{z})$$f^{I}\not\cong 0$

Proposition 20. [23] Assume thatis a simple polar weighted homogeneouspolynomial with monomials and assume that is k-convenient. Then $k\leq s-1$ .
$f(z,\overline{z})$

$f$

$s$

Now we have the following result about the connectivity of$F$.

is a simple polar weighted homogeneous
Theorem 21. [23] Assume thatpolynomial withis k-convenient. Then $F$ ismonomials and assume that$\min(k, n-2)$ -connected.
$f(z,\overline{z})$

$s$

$f$

89
M. OKA

2.4.7. Join type polynomials. Another special typc of mixed functions are mixed
polynomials of join type. Consider the rad-polar wcighted homogcneous poly$h(w,-w)$ with $z=(z_{1}, \ldots.z_{n})$ and $w=(w_{1)}\ldots, w_{m})_{0}$ Consider
nomials$f(z, w,\overline{z},-w)=g(z,\overline{z})+h(w,-w)$ . Then$g(z,\overline{z}),$Theorem 22. (Cisneros-Molina [5]) The Milnor fiber of is homotopic to the join
of $g^{-1}(1)*h^{-1}(1)$ and the monodromy is also theof the respective monodromy.
’$f$$\gamma oin$andbe the normalized polar weights. Then
Proof. Let
has the nomalized polar weight. The polar weight is given
by multiplying the least common multiple of the denominator. The proof is divided$F_{9}=g^{-1}(g),$ $F_{h}=h^{-1}(1)$ . Let
into three steps. Let$(r_{1}, \ldots, r_{m})$$(p_{1}, \ldots, p_{n})$$f$$(p_{1}, \ldots, p_{n}, r_{1}, \ldots, r_{m})$$F_{f}=f^{-1}(1)\subset \mathbb{C}^{n+m},$$F_{1}=\{(z, w)|g(z,\overline{z})\in \mathbb{R}\}\dot{)}F_{2}=\{(z, w)\in F_{1}|0\leq g(z,\overline{z})\leq 1\}$

Step 1.
odromy.
Step 2.
Step 3.

$F_{1}\subset F$.

is a defoemation retract which is compatiblc with thc mon-is also a deformation retract.
is homotopic to the join $F_{g}*F_{h}$ and the joined monodromy.

$F_{2}\subset F_{1}$$F_{2}$$\square$Corollary 23. Suppose thatand has the homotopy types of bouques of spheres
connected andof dimension $n-1$ and $m-1$ . Then
$F_{1}$$F_{2}$$F_{harrow}isn+m-2$$H_{n+m-1}(F)$$H_{n+m-1}(F)$$\downarrow\cong$$\downarrow\cong$$H_{n-1}(F_{1})\otimes H_{m-1}(F_{2})$$h_{1}\underline{\otimes}h_{2}\rangle$$H_{n-1}(F_{1})\otimes H_{m-1}(F_{2})$2.5. General mixed functions. This section is completely included in [24].
2.5.1. Newton boundary of a mixed function. Suppose that we are given a mixed
analytic function. We always assume that $c_{0,0}=0$ so that
$O\in f^{-1}(0)$ . We call the variety $V=f^{-1}(0)$ the mixed hypersurface. The mdial
Newton polygon(at the origin) of a mixed function
is defined by
the convex hull of$f( z,\overline{z})=\sum_{\nu,\mu}c_{\nu,\mu}z^{\nu}\overline{z}^{\mu}$$\Gamma_{+}(f;z,\overline{z})$$f(z,\overline{z})$ノ $+\mu)+\mathbb{R}^{+n}$$\bigcup_{c_{\nu,\mu}\neq 0}(\iota$.

Hereafter we call
. The Newton
simply the Newton polygon ofboundary
is defined by the union of compact faces of. Observe
that
is nothing but the ordinary Newton boundary if is a complex analytic
function. For a given positive integer vector $P=(p_{1}, \ldots,p_{n})$ , we associate alinear functionondefined by
for $\nu\in\Gamma(f)$ and let$\Delta(P, f)=\triangle(P)$ be the facc where
takes its minimal value. In other words,
gives radial weights for variables
by rdeg $P^{Z}j=r\deg_{P}\overline{z}_{j}=p_{j}$ andrdeg). To distinguish the points on the Newton boundary
and weight vectors, we denote by $N$ the set of integer weight vectors and denote a$N^{++}$
vector $P\in N$ by a column vectors. We denote by
the subset of positiveor strictly positive weight vectors respectivcly. Thus $P={}^{t}(p_{1},$ $\ldots,p_{n})\in N^{++}$(respectively $P\in N^{+}$ ) if and only if $p_{i}>0$ (resp.
) for any $i=1$ , . . , . We
denote the minimal value ofby $d(P;f)$ or simply $d(P)$ . Note that$\Gamma_{+}(f;z,\overline{z})$$f(z,\overline{z})$$\Gamma(f;z,\overline{z})$$\Gamma_{+}(f)$$\Gamma(f)$$f$$p_{P}$$\Gamma(f)$$\ell_{P}(\nu)=\sum_{j=1}^{\tau\iota}p_{j}\nu_{j}$$\ell_{P}$$P$$z_{1},$$\ldots,$$z_{n}$$P^{Z^{\nu}\overline{z}^{\mu}=\sum_{j=1}^{n}p_{j}(t\text{ノ_{}j}}+\mu j$$N^{+},$$p_{i}\geq 0$$p_{P}$$d(P;f)= \min\{r\deg_{P}z^{\nu}z^{\mu}|c.,$.

$\neq 0\}$

$n$

.

90
INTRODUCTION TO NO-DEGENERATE MIXED FUNCTIONS

For a positive weight$P$, we define the face function$f_{P}(z,\overline{z})$$\int_{P}(z,\overline{z})=\sum_{\nu+\mu\in\Delta(P)}c_{\nu,\mu}z^{\nu}\overline{z}^{\mu}$by.

. The Newton
Example 24. Consider a mixed function$:={}^{t}(2,3)$
which have weight vectors
has two facesboundary
and $Q:={}^{t}(1,1)$ respectively. The corresponding invariants are$:=z_{1}^{3}\overline{z}_{1}^{2}+z_{1}^{2}z_{2}^{2}+z_{2}^{3}\overline{z}_{2}$$f$$\Delta_{1},$$\Gamma(f;z,\overline{z})$$P$$\triangle_{2}$$f_{P}(z,\overline{z})=z_{1}^{3}\overline{z}_{1}^{2}+z_{1}^{2}z_{2}^{2},$$d(P;f)=10$$f_{Q}(z,\overline{z})=z_{1}^{2}z_{2}^{2}+z_{2}^{3}\overline{z}_{2},$$d(Q;f)=4$ .

FIGURE 1.

$\Gamma(f)$is a given mixed function
2.5.2. Non-degenemte functions. Suppose that
$P\inN^{++}$, thc facc function
is a radially wcightcd homogeneous. For
with $d=d(P;f)$ .
polynomial of type$f(z.,\overline{z})$$f_{P}(z,\overline{z})$$f(z,\overline{z})$$(p_{1}, \ldots,p_{\iota};d)$isDefinition 25. Let
be a strictly positivc weight vector. We say that
contains no critical point of thenon-degenemte for $P$ , if thc fiber
is a smooth real codimension. In particular,:
mapping
is strongly non-degenerate for
2 manifold or an empty set. We say that:has no critical points. If $\dim\triangle(P)\geq 1$ , we further
if the mapping
is surjective onto .
assume thatis called non-degenemte (respectively strongly nonA mixed function
degenemte) if is non-degenerate (resp. strongly non-degenemte) for any strictly
positive weight vector .
. Then $V=f^{-1}(0)$ is a single
Consider the function$\{O\}$. By the above definition, is a non-degencrate mixed function. To
point
is a true
avoid such an unpleasant situation, we say that a mixed function
non-degenemte function if it satisfies further the non-emptiness condition:is(NE) : For any $P\in N^{++}$ with $\dim\Delta(P, g)\geq 1$ , the fiber
non-empty.
$P$$f(z,\overline{z})$$f_{P}^{-1}(0)\cap \mathbb{C}^{*n}$$f_{P}$$f_{P}^{-1}(0)\cap \mathbb{C}^{*n}$$\mathbb{C}^{*n}arrow \mathbb{C}$$P$$f(z,\overline{z})$$f_{P}$$\mathbb{C}^{*n}arrow \mathbb{C}$$\mathbb{C}$$f_{P}:\mathbb{C}^{*n}arrow \mathbb{C}$$f(z,\overline{z})$

$f$

$P$

$f(z,\overline{z})=z_{1}\overline{z}_{1}+\cdots+z_{n}\overline{z}_{n}$$f$$g(z,\overline{z})$$g_{P}^{-1}(0)\cap \mathbb{C}^{*n}$Remark 26. Assume that $f(z)$ is a holomorphic function. Then $f_{P}(z)$ is aweighted homogeneous polynomial and we have the Euler equality:$d(P;f)f_{P}( z)=\sum_{i=1}^{n}p_{i}z_{i}\frac{\partial f_{P}}{\partial z_{i}}(z)$.

91
M. OKAThus:
has no critical point over . Thus is non-degenerate for
implies is strongly non-degenerate for P. This is also the case if
is apolar weighted homogeneous polynomial.
$f_{P}$$P$$\mathbb{C}^{*n}arrow \mathbb{C}$$f$$\mathbb{C}^{*}$$f$$f_{P}(z.\overline{z})$2.5.3. Isolatedness of the singularities. Let. As we aremainly interested in the topology of a germ of a mixcd hypcrsurface at the origin,
we always assumc that does not have the constant term so that $O\in f^{-1}(0)$ . Put$\int(z.\overline{z})=\sum_{1/,\mu}c_{\nu},z^{\nu}\overline{z}^{\mu}l^{A}$$f$$V=f^{-1}(0)\subset \mathbb{C}^{n}$.

2.5.4. Mixed singular points. We say that $w\in V$ is a mixed singular point if $w$ is a
critical point of the mapping :. We say that is mixed non-singular if
it has no mixed singular points. If is mixcd non-singular, is smooth variety ofreal codimension two. Note that a singular point of (as a point of a real algebraic
variety) is a mixed singular point of but the conversc is not necessarily true. For
example, every point of the spherc
is a mixed singular
$f$

$V$

$\mathbb{C}^{n}arrow \mathbb{C}$

$V$

$V$

$V$

$V$

$S=\{z_{1}\overline{z}_{1}+\cdots+z_{\tau\iota}\overline{z}_{11}=1\}$

point.

2.5.5. Non-vanishing coordinate subspaces. For a subset $J\subset\{1,2, \ldots.n\}$ , we consider the subspacc
and the restriction. Consider thc set$\mathbb{C}^{J}$$f^{J};=f|_{\mathbb{C}^{J}}$$\mathcal{N}\mathcal{V}(f)=\{I\subset\{1\ldots., n\}|f^{I}\not\equiv 0\}$We call$\mathcal{N}\mathcal{V}(f)$.

the set of non-vanishing coordinate subspaces for$V\#=U_{v}^{V\cap \mathbb{C}^{*I}}I\in N(f)$Theorem 27. [24] Assume that
Then there exists a positive number

$f$

. Put

.

is a true non-degenerate mixed function.
such that the following properties are satis-$\int(z,\overline{z})$$r_{0}$

fied.

(1) (Isolatedness of the singularity) The mixed hypersurface
non-singular. In particular, $co\dim_{R}v\#=2$ .
(2) $(bansvC^{1}rsality)$ The spherewith $0<r\leq r_{0}$ intersects$V^{t}\cap B_{r_{O}}$$S_{r}$

We say that$f$is k-convenient if$J\in \mathcal{N}\mathcal{V}(f)$$V\#$is mixed

transversely.

for any $t’\subset\{1\ldots., 7b\}$ with $|J|=$$n-k$ . We say that $f$ is convenient if $f$ is $(n-1)$ -convenient. Notc that $V\#=V\backslash \{O\}$if is convenient. For a given with $0<\ell\leq n$ , we put $W(\ell)=\{z\in \mathbb{C}^{n}||I(z)|\leq\ell\}$
where $I(z)=\{i|z_{i}=0\}$ . Thus
. If is -convenient, $V\cap W(\ell)\subset V\#$ .
$f$$\ell$$W(r\iota-1)=\mathbb{C}^{*r\iota}$$f$$\ell$

Corollary 28. Assume thatis a convenient true non-degenemte mixed
polynomial. Then $V=f^{-1}(0)$ has an isolated mixed singularity at the origin.
$f(z,\overline{z})$Remark 29. The assumption “true” is to make sure that
non-empty.

$V^{*}=f^{-1}(0)\cap \mathbb{C}^{*n}$is2.6. Milnor fibration. In this section, we study the Milnor fibration, assumingthatis a strongly non-degenerate convenient mixed function. We have seenin Theorem 27 that there exists a positive number such that $V=f^{-1}(0)$ is mixed
non-singular except at the origin in the balland the sphereintersects
transversely withfor any $0<r\leq r_{0}$ . The following is a key assertion for which
we need the strong non-degeneracy.
$f(z,\overline{z})$$r_{0}$$S_{r}^{2n-1}$$B_{r_{0}}^{2n}$$V$

Lemma 30. [23] Assume thatis a strongly non-degenerate convenient mixed, there exists positive numfunction. For any fixed positive number with$(a)$ theberssuch that for any
and withfiber$f(z,\overline{z})$$r_{1}$$\delta_{0}\ll r_{1}$$\eta\neq 0,$$|\eta|\leq\delta_{0}$$r_{1}\leq r_{0}$$r$$r_{1}\leq r\leq r_{0},$92
INTRODUCTION TO NO-DEGENERATE MIXED FUNCTIONS$V_{\eta}$

has no mixed singularity inside the ballis tmnsverse and smooth.

$:=f^{-1}(\eta)$$V_{\eta}\cap S_{r}^{2n-1}$$B_{r_{0}}^{2n}$and (b) the intersection

2.6.1. Milnor fibration, the second description. Put$D(\delta_{0})^{*}=\{l\mathfrak{j}\in \mathbb{C}|0<|\eta|\leq\delta_{0}\},$$E(r, \delta_{0})^{*}=f^{-1}(D(\delta_{0})^{*})\cap B_{r}^{2n}$$S_{\delta_{0}}^{1}=\partial D(\delta_{0})^{*}=\{\eta\in \mathbb{C}||?\}|=\delta_{0}\}$.

$\partial F_{\lrcorner}(r, \delta_{0})^{*}=f^{-1}(S_{\delta_{0}}^{1})\cap B_{r}^{2n}$.

By Lemma 30 and thc theorem of Ehresm\‘an ([31]), we obtain thc following description of the Milnor fibration of the second type ([10]).
Theorem 31. (Thc second description of the Milnor fibration) Assume thatis a convenient, strongly non-degenerate mixed function. Take positive numbers
as in Lemma 30. Then :andandsuch thatare locally trivial fibmtions and the topologicaland :isomorphism class does not depend on the choice of and .
$f(z,\overline{z})$$r_{0},$$r\leq r_{0}$$\delta_{0}$$r_{1}$$f$$D(\delta_{0})^{*}$$E(r, \delta_{0})^{*}arrow$$f$$\delta_{0}\ll r_{1}$$\partial E(r, \delta_{0})^{*}arrow S_{\delta_{0}}^{1}$$\delta_{0}$$r$2.6.2. Milnor fibmtion, the first description. We considcr now thc original Milnor
fibration on the sphere, which is defined as follows:$\varphi:S^{2n-1}\backslash K_{r}arrow S^{1}$,$z\mapsto\varphi(z)=f(z,\overline{z})/|f(z,\overline{z})|$where $K_{r}=V\cap S_{r}^{2n-1}$ . The fibrations of this typc for mixed functions and related
topics have been studied by many authors ([27, 28, 6, 29, 26, 3]). But most of theworks treat rather special classes of functions. The mapping can be identified
with $\varphi(z)=-\Re(i\log f(z))$ , taking the argument as a local coordinate of the circle. For
of the tangent space. We use the basis, we use two complex “gradient vectors” defined bya mixed function$\varphi$$\theta$$S^{1}$$T_{Z}\mathbb{C}^{n}\otimes \mathbb{C}$$\{\frac{\partial}{\partial z_{j}}, \frac{\partial}{\partial\overline{z}_{j}}|j=1, \ldots, n\}$$g(z,\overline{z})$$dg=( \frac{\partial g}{\partial z_{1}}, \ldots, \frac{\partial g}{\partial z_{n}})$,$\overline{d}g=(\frac{\partial g}{\partial\overline{z}_{1}}, \ldots, \frac{\partial g}{\partial\overline{z}_{n}})$Take a smooth path $z(t),$ $-1\leq t\leq 1$ with. Then we have$z(O)=w\in \mathbb{C}^{n}\backslash V$.

and put$v=\frac{dZ}{dt}(0)\in$$T_{W}\mathbb{C}^{n}$$-(\Re(i\log f(z(t),\overline{z}(t)))_{t=0}\underline{d}$$(lt$$=- \Re(\sum_{i=1}^{n}i\{\frac{\partial f}{\partial z_{j}}(w^{-}w)\frac{dz_{j}}{dt}(0)+\frac{\partial f}{\partial\overline{z}_{j}}(w^{-}w)\frac{d\overline{z}_{j}}{dt}(0)\}/f(w, w-))$$=\Re(v, i\overline{d\log f}(w^{-}w))+\Re(\overline{v}, i\overline{\overline{d}\log f}(w, w-))$$=\Re(v,$$i$dlog $f(w,\overline{w}))+\Re(v, -i,\overline{d}\log f(w,-w))$$=\Re(v, i(\overline{d\log f}-\overline{d}\log f)(w,-w))$.

Namely we have(3)$- \frac{d}{dt}(\Re(i\log f(z(t),\overline{z}(t)))_{\iota=0}=\Re(v, i(\overline{d\log f}-\overline{d}\log f)(w,\overline{w}))$.

Thus by the same argument as in Milnor [15], we get
Lemma 32. [24] A pointcomplex vectors$z\in S_{r}^{2n-1}\backslash -K_{r}$is a critical point of if and only if theand are linearly dependent over$i(\overline{d\log f}(z,\overline{z})-d\log f(z,\overline{z}))$$\mathbb{R}two$

The key assertion is the following.

$\varphi$$z$93
M. OKA

Lemma 33. [24] Assume thatis a stmngly non-degenemte mixed function. Thenexists a positive number
such that the two complex vectors
dlogandare linearly independent over for
any with $0<r\leq r_{0}$ .
$f(z,\overline{z})$$ther\underline{e}$$i$$\gamma_{0}$$\int(z,\overline{z})-d\log f(z,\overline{z}))$$($$z\in S_{r}\backslash K_{r}$$\mathbb{R}$$r$

Now we are ready to prove thc existence of thc Milnor fibration of the firstdescription.
Theorem 34. [24](Milnor fibration, the first description) Let
be a strongly
non-degenemte convenient mixed function. There exists a positive number
such
that
$f(z,\overline{z})$$r_{0}$$\varphi=f/|f|:S_{r}^{2n-1}\backslash K_{\Gamma}arrow S^{1}$

is a locally trivial fibmtion for any

with $0<r\leq r_{0}$ .

$r$2.6.3. Equivalence of two Milnor fibrations. Take positive numbers
as in Theorem 31. Wc comp\‘are the two fibrations$r,$$\delta_{0}$with$\delta_{0}\ll$$7’$$f:\partial E(r, \delta_{0})arrow S_{\delta_{0}}^{1}$,$\varphi:S_{r}^{2r\iota-1}\backslash K_{r}arrow S^{1}$and we will show that they are isomorphic. However the proof is much more
complicated compared with the case of holomorphic functions. The reason is that
we have to take care of the two vectors
$i$

$($

dlog $f-\overline{d}\log f),$$\overline{d\log f}+\overline{d}\log f$which are not perpendicular. (In the holomorphic casc, thc proof is casy as the two
vectors reduce to the perpendicular vectors) Consider a smooth$v=\frac{dZ(t)}{dt}(0)$curvc $z(t),$ $-1\leq t\leq 1$ , withand
. Put $v=$
. First from (3), we observe that$i\overline{d\log f},$$\overline{d\log f}.$$z(O)=w\in B_{r}^{2\tau\iota}\backslash V$$(v_{1}, .

, .

, v_{n})$$\frac{\log f(z(t),\overline{z}(t))}{dt}|_{t=0}=\sum_{j=1}^{2\iota}(v_{j}\frac{\partial\log f}{\partial z_{j}}(w^{-}w)+\overline{v}_{j}\frac{\partial\log f}{\partial\overline{z}_{j}}(w, w-))$$= \Re(v, (og \int+\overline{d}\log f)(w^{-}w))+i\Re(v, i(\overline{d\log f}-\overline{d}\log f)(w, w))$

Define two vectors on$\mathbb{C}^{n}-V$.

:$v_{1}(z,\overline{z})=\overline{d\log f}(z,\overline{z})+\overline{d}\log f(z,\overline{z})$$v_{2}(z,\overline{z})=i$

$($

dlog

$\int(z,\overline{z})-\overline{d}\log f(z,\overline{z}))$The above equality is translated as(4)$\frac{\log f(z(t),\overline{z}(t))}{dt,}|_{t=0}=\Re(v, v_{1}(w,-w))+i\Re(v, v_{2}(w, w-))$.

The following will play the key role for the equivalence of two fibrations:Lemma 35. [24]labelkey lemma Under the same assumption as in Theorem 34,
there exists a positive number
so that for any with,and$\Vert z\Vert\leq r_{0}$$z$$r_{0}$$f(z,\overline{z})\neq 0$

the three vectors,
are either (i) linearly independent overand the relation can be written as$z$$v_{1}(z,\overline{z})$$\mathbb{R}$,$v_{2}(z,\overline{z})$or (ii) they are linearly dependent over$\mathbb{R}$(5)

and the coefficient$z=av_{1}(z,\overline{z})+bv_{2}(z,\overline{z}),$ $a,$$a$is positive.

$b\in \mathbb{R}$.

94
INTRODUCTION TO NO-DEGENERATE MIXED FUNCTIONS$1|\mathfrak{l}$’1$\mathfrak{l}$

$1$

$1$

1
$1$

FIGURE 2. If$\lambda_{0}\leq 0,$$|\beta|<|\gamma|$

Now we are ready to prove the isomorphism theorem:
Theorem 36. Under the same assumption as in Theorem 34, the two fibmtions$f:\partial E(r, \delta_{0})arrow S_{\delta_{0}}^{1}$,$\varphi:S_{r}^{2n-1}\backslash K_{r}arrow S^{1}$

are topologically isomorphic.
3. MIXED PROJECTIVE CURVES
bc a rad-polar weighted homogeneous mixed polynomial with $z=$andand
. Namely there exist integers
that
such
positive intcgersLet$f(z,\overline{z})$$d_{r},$$d_{p}$$\int(t\circ z, t\circ\overline{z})=t^{d_{r}}f(z,\overline{z})$$\mathbb{R}^{+}\cross S^{1}$

toz,$f(\rho\circ z,\overline{\rho\circ z})=\rho^{d_{\rho}}f(z,\overline{z})$This gives$(p_{1}, \ldots, p_{n})$$(q_{1}, \ldots, q_{n})$$(z_{1}, \ldots, z_{n})\in \mathbb{C}^{n}$,$=(t^{q\iota}z_{1}, \ldots, t^{q_{\hslash}}z_{n})_{\dot{\prime}}t\in \mathbb{R}^{+}$$\rho\circ z=(\rho^{P1}z_{1,\ldots)}\rho^{\rho_{n}}z_{n}),$$|\rho|=1$$\rho\in \mathbb{C},$.

action by$(t, \rho)\circ z=(t^{q_{1}}\rho^{\rho_{1}}z_{1}, \ldots, t^{q_{n}}\rho^{p_{n}}z_{n})$,$t\rho\in \mathbb{R}^{+}\cross S^{1}$.

is strongly polar weighted homogeneous if $p_{j}=q_{j}$ for $j=$
We say that
action whichis in fact thc
action on. Then the associatedis defined by.
$f(z,\overline{z})$$\mathbb{R}^{+}\cross S^{1}$$1_{:}\ldots,$ $n$$\mathbb{C}^{*}$$\mathbb{C}^{n}$$(z, \tau)=((z_{1}, \ldots, n), \tau)\mapsto\tau oz=(z_{1}\tau^{\rho_{1}}, \ldots, z_{n}\tau^{p_{n}}),$$\tau\in \mathbb{C}^{*}$is strongly polar homogeneous if further the weights satisfies theWe sayequalities $q_{j}=p_{j}=1$ for any . A strongly polar weighted homogeneous polynomial$f(z,\overline{z})$$j$$f(z,\overline{z})$(6)satisfies the equality:$f((t, \rho)\circ z, \overline{(t,\rho)\circ z})=t^{d_{r}}\rho^{d_{p}}\int(z.\overline{z})$,$(t, \rho)\in \mathbb{R}^{+}\cross S^{1}$.

is a strongly polar weighted homogeneous polynomial of
Assume thatradial degree and of polar degree respectivcly and let $P=(p_{1}, \ldots,p_{n})$ be theweight vector. Let be the mixed affine hypersurface$f(z,\overline{z})$$d_{p}$$d_{r}$$\overline{V}$$\tilde{V}=f^{-1}(0)=\{z\in \mathbb{C}^{n}|f(z,\overline{z})=0\}$

Let :the fiber. Recall that$\varphi$$S^{2n-1}\backslash Karrow S^{1}$.

be the Milnor fibration with. Thus $F$ is defined by$K=\tilde{V}\cap S^{2n-1}$$\varphi(z)=\int(z,\overline{z})/|f(z,\overline{z})|$$F=\varphi^{-1}(1)=\{z\in S^{2n-1}\backslash K|f(z,\overline{z})>0\}$and let $F$ be

95
M. OKA

We can equivalently consider the global fibration :. Thcn the Milnor
fiber is identified with the hypersurface $f^{-1}(1)$ . The monodromy map : $Farrow F$(in either case) is defined by$\mathbb{C}^{n}-\tilde{V}arrow \mathbb{C}^{*}$

$f$

$h$

$h( z)=(\exp(\frac{2p_{1}\pi i}{d_{p}})z_{1}, \ldots, \exp(\frac{2p_{n}\pi i}{d_{p}})z_{1})$We consider also the wighted projective hypersurface$V=\{(z_{1}:z_{2} :.

.

.

$V$

.

defined by:z_{n})\in \mathbb{C}\mathbb{P}(P)^{\iota-1}|f(z,\overline{z})=0\}$

where CP $(P)^{n-1}$ is the weighted projcctive space defined by thc equivalence inducedby the above C’ action:$w=\tau\circ z$$z\sim w\Leftrightarrow\exists\tau\in \mathbb{C}^{*},$.

It is well-known thatis an orbifold with at most cyclic quotient singularities.
By (6), $z\in f^{-1}(0)$ and $z’\sim z$ , then $z’\in f^{-1}(0)$ . Thus the hypersurface$V=\{[z]\in \mathbb{C}\mathbb{P}^{n-1}(P)|f(z)=0\}$ is well-defined.
Consider the quotient map :or :CIP $(P)^{n-1}$ . For the brevity‘s sake, we denote
the restrictions
and $\pi|K$ : $Karrow V$ by the same . We areinterested in the topology of and the relation with the Milnor fibration.
$\mathbb{C}\mathbb{P}(P)^{n-1}$$\pi$$S^{2n-1}arrow \mathbb{C}\mathbb{P}(P)^{n-1}$$\pi$$\mathbb{C}^{n}\backslash \{O\}arrow$$\pi|F:Farrow \mathbb{C}\mathbb{P}^{n-1}\backslash V$

$\pi$

$V$

3.1. Canonical orientation. It is well known that a complex analytic smoothvariety has a canonical orientation which comes from the complex structure (seefor example p.18, [7] . Let $\tilde{V}=f^{-1}(0)$ be a mixed hypersurface. Take a point. We say that a is a mixed singular point of , if a is a critical point of the$mapp_{\sim}^{ing}f$ :C. Otherwise, a is a mixed regular point. Note that a point
a $\in V$ to be a regular point as a point of a real analytic variety is a necessary
condition but not a sufficient condition for the regularity as a point on a mixed
variety. Recall that a is a mixed singular point if and only if
is surjective. This is equivalent to the existencc of a complex number
withsuch that$)$$a\in\tilde{V}$$\tilde{V}$$\mathbb{C}^{n}arrow$$df_{a}:T_{a}\mathbb{C}^{\tau\iota}arrow T_{f(a)}\mathbb{C}$$\alpha\in \mathbb{C}$$|\mathfrak{a}|=1$i.e.,$\overline{df}(a,\overline{a})=\alpha\overline{d}f(a,\overline{a})$$\overline{\frac{\partial f}{\partial z_{j}}}(a,\overline{a})=(y\frac{\partial f}{\partial\overline{z}_{j}}(a,\overline{a}),$$.i=1,$$\gamma\iota$$\ldots,$([23]). We assert that
Proposition 37. There is a canonical orientation on the smooth part of a mixed
hypersurface.

Proof. Take a regular point a. The normal bundle

of
has a
canonical orientation so that:is an orientation preserving
isomorphism. This gives a canonical orientation on so that the ordered union of
the oriented framesof
if andis the orientation of
only if
is an oriented frame ofwhere
is an oriented
frame of normal vectors.
$\in\tilde{V}$$df_{a}$$\mathcal{N}$$\tilde{V}\subset \mathbb{C}^{n}$$\mathcal{N}_{a}arrow T_{f(a,\overline{a})}\mathbb{C}$$\tilde{V}$$\{v_{1}, \ldots, v_{2n-2}, n_{1}, n_{2}\}$$T_{a}\mathbb{C}^{n}$$\{v_{1}, \ldots , v_{2,\iota-2}\}$$T_{a}\tilde{V}$$\mathbb{C}^{b}$$\{n_{1}, n_{2}\}$$\square$Consider a mixed homogeneous hypersurface
mixed projective hypersurface.
Proposition 38. Let. Thenand only if $\pi(a)\in V$ is a mixed singular point.
$a\in\tilde{V}\backslash \{O\}$See [24] for the proof.

$\tilde{V}$$a\in\tilde{V}$and let$V$

be the corresponding

is a mixed singular point of$\tilde{V}$if96
INTRODUCTION TO NO-DEGENERATE MIXED FUNCTIONS

3.2. Milnor fiberation and Hopf fiberation. Consider the Hopf fibration :and its restriction to the Milnor fiber $F=\{z\in S^{2r\iota-1}|f(z,\overline{z})>$$K=f^{-1}(0)\cap
S^{2n-1}$ be the link. As
is polar weighted, it is easy to. Put
and the covering
is a cyclic covering of ordersee that :$\pi$$S^{2\tau\iota-1}arrow \mathbb{C}\mathbb{P}^{\tau\iota-1}$$0\}$$\int$$\pi$$Farrow \mathbb{C}\mathbb{P}^{n-1}\backslash V$$d_{p}$

transformation is generated by the monodromy map$h;Farrow F$,$z\mapsto\exp(\frac{2\pi i}{d_{p}})\cdot z$.

Thus we have
Proposition 39.

(1)$\chi(F)=d_{p}\chi(\mathbb{C}\mathbb{P}^{n-1}\backslash V)$.

and $\chi(V)=n-\chi(F)/d_{p}$ .
(2)(3) We have the following exact sequence.
$\chi(\mathbb{C}\mathbb{P}^{\iota-1}\backslash V)=n-\chi(V)$.

$1arrow\pi_{1}(F)arrow^{\pi\#}\pi_{1}(\mathbb{C}\mathbb{P}^{n-1}\backslash V)arrow Z/d_{p}Zarrow 1$

The following special cases are uscd later.
be the(1) Suppose $n=2$ . Then $K\subset S^{3}$ is a link. Put
Corollary 40.
number of the components. Then is points and and are related by$7^{\cdot}$$V$$\frac{\chi(F)}{d_{p}}=2-r\cdot$(2) Suppose $n=3$ . Then

$V\subset \mathbb{P}^{2}$

$r$

$r$

$\chi$.

is a curve of genus$1+2g= \frac{\chi(,F)}{d_{\rho}}$

$g$

and

.

be a mixed polar weighted polynomial of two variables andRemark 41. Let. Let be the number of irreducible
let be the number of link components
components of . Then $r\geq s$ . For example, For example, let. Then for $t=0$ , lkn $(f)=1=6$ and for $t=2,$ $s=1$ and$f$$S^{3}$$r$$s$$f(z_{1)}z_{2},\overline{z}_{1},\overline{z}_{2})=$$f$$-2z_{1}^{2}\overline{z}_{1}+z_{2}^{2}\overline{z}_{2}+tz_{1}^{2}\overline{z}_{2}$$1kn(f)=3$ .

Corollary 42. If $d_{p}=1$ , the projection

$\pi$

:

$Farrow \mathbb{C}\mathbb{P}^{n-1}\backslash V$

is a diffeomorphism.

The monodromy map $h:Farrow F$ givcs free $Z/d_{p}Z$ action on
periodic monodromy argumcnt in [15], wc get$F$. Thus using the

Proposition 43. The zeta function of $h:Farrow F$ is given by$\zeta(t)=(1-t^{d_{p}})^{-\chi(F)/d_{p}}$.

In particular, if $d_{\rho}=1,$ $h=id_{F}$ and $\zeta(t)=(1-t)^{-\chi(F)}$ .

3.3. Degree of mixed projective hypersurfaces. Suppose that
$2r,$ $q;n)$ be a strongly polar homogeneous polynomial and let$V=\{z\in \mathbb{C}\mathbb{P}^{n-1}|f(z,\overline{z})=0\}$$f(z,\overline{z})\in\Lambda t(q+$.

of is either empty or $co\dim_{R}\Sigma V\geq 2$ . We
We assume that the singular locus
is canonically oriented so that the union ofhave observed that
$T_{P}V$$\{v_{1,\ldots,2(n-2)\}}?)$, say
and the frame of normal bundle
the oriented frame ofon thewhich is compatible with the local defining complex function. (Recall that is a mixedaffine chart $U_{j}=\{z_{j}\neq 0\}$ is the oriented frame offunction of the variables $u_{i}=z_{i}/z_{j},$ $i\neq j$ defined by$\Sigma V$$V$$V\backslash \Sigma V\subset \mathbb{C}\mathbb{P}^{n-1}$$\{w_{1}, w_{2}\}$$g_{j}$$\mathbb{C}\mathbb{P}^{n-1}$$g_{j}$$g_{j}(u,\overline{u})=f(z,\overline{z})/z_{j}^{q+r}/\overline{z}_{j}^{r}.)$97
M. OKA

Thus it has a fundamental class $[V]\in H_{2n-4}(V;Z)$ by Borel-Haefliger [4], The topological degree of is the integer so that
where , :map andis the homology class of a canonical hyperplane
$V$

$d$

$\iota_{*}[V]=d[\mathbb{C}\mathbb{P}^{7l-2}]$$Varrow \mathbb{C}\mathbb{P}^{r\iota-1}$$l$$[\mathbb{C}\mathbb{P}^{n-2}]$$\mathbb{C}\mathbb{P}^{n-2}isthei.nclusion$

Theorem 44. [22] The topological degree ofthe fundamental class [V] corresponds tosion mappin9 .

$V$

is equal to the polar degree . Namelyby the inclu$q$$q[\mathbb{C}\mathbb{P}^{n-2}]\in H_{2(n-2)}(\mathbb{C}\mathbb{P}^{n-1})$$\iota_{*}$

3.3.1. Residue formula for a monic mixed polynomial. Let
be a mixed polynomial. Put $d= \max\{a+b|c_{a,b}\neq 0\}$ and we calldegree of . We say that is a monic mixed polynomial of degree
unique monomial of radial degrce .

$g(w, \overline{w})=\sum_{a.b}c_{a,b}w^{a}\overline{w}^{b}$

$g$

$g$

$d$

$d$

the radial
if has a$g$$d$Lemma 45. Assume thatis written as$g(w,\overline{w})$is a monic mixed polynomial of degree

$d$

which

$g(w,\overline{w})=c_{0}(\overline{w})w^{q+r}+c_{1}(\overline{w})w^{q+r-1}+\cdots+c_{q+r}$$c_{j}(\overline{w})\in \mathbb{C}[\uparrow\overline{v}],$$j=0,$$rdcg_{\overline{w}^{C’}j}\leq\gamma\cdot,$$c_{0}(\overline{w})=c_{0r}\overline{w}^{r}+\cdots+c_{00},$$\ldots,$$q+7^{\cdot}$$c_{0r}\neq 0$with $d=q+2r$ . Then$\frac{1}{2\pi}l_{|w|=R}$

Gauss $(g)d\theta=q$ ,3.3.2. Mixed projective curves. In this section, wc study basic examples in the projectivc surface. Thus we assume that $n=3$ . We considcr projective curves ofdegree :$\mathbb{C}\mathbb{P}^{2}$$q$$C=\{[z_{1}:z_{2}:z_{3}]\in \mathbb{C}\mathbb{P}^{2}|f(z_{1}, z_{2}, z_{3})=0\}$

where is a strongly polar homogeneous polynomial with pdeg $f=q$ . We haveseen that the topological degree of is by Theorem 44. The genus of is not
an invariant of . Recall that for a differentiable curve of genus , embedded in, with the topological degree , wc have the following Thom’s inequality, which
was conjectured by Thom and proved by for example Kronheimer-Mrowka [13]:
$f$

$C$

$q$

$g$

$C$

$q$

$C$

$g$

$\mathbb{C}\mathbb{P}^{2}$$q$$g \geq\frac{(q-1)(q-2)}{2}$where the right side number is the gcnus of algebraic curves of dcgree , given by the
Pl\"ucker formula. Recall that for a mixed strongly polar homogcneous polynomial,$q$

the genus and the Euler characteristic of the Milnor fiber arc related as follows (Corollary 40):$g= \frac{1}{2}(\frac{\chi(F)}{q}-1)$

where$F=\{(z_{1}, z_{2}, z_{3})\in \mathbb{C}^{3}|f(z_{1}, z_{2}, z_{3_{j}}\overline{z}_{1},\overline{z}_{2},\overline{z}_{3})=1\}$.

Now we will see some examples which shows that $\chi(F)$ is not an invariant of

$q$

.

98
INTRODUCTION TO NO-DEGENERATE MIXED FUNCTIONS

I. Simplicial polynomials. Wc consider the following simplicial polar homogeneous polynomials of polar degree .
$q$$f_{s_{1}}(z,\overline{z})=z_{1}^{q+r}\overline{z}_{1}^{r}+z_{2}^{q+r}\overline{z}_{2}^{r}+z_{3}^{\eta+r}\overline{z}_{3}^{r}$$f_{s_{2}}(z,\overline{z})=z_{1}^{q+r-1}\overline{z}_{1}^{r}z_{2}+z_{2}^{q+r-1}\overline{z}_{2}^{r}z_{3}+z_{3}^{q+r}\overline{z}_{3}^{r}$.

$f_{\hslash_{J}^{a}}(z,\overline{z})=z_{1}^{q+r-1}\overline{z}_{1}^{\tau}.z_{2}+z_{2}^{q+r-1}\overline{z}_{2}.z_{3}+z_{3}^{q+r-1_{\overline{Z}_{3}^{z}Z_{1}}}$$f_{s4}(z,\overline{z})=z_{1}^{q+r+1}\overline{z}_{1}^{r}\overline{z}_{2}+z_{2}^{q+r+1}\overline{z}_{2}^{r}\overline{z}_{3}+z_{3}^{q+r}\overline{z}_{3}^{r}$$f_{s_{5}}(z,\overline{z})=z_{1}^{q+r+1}\overline{z}_{1}^{r}\overline{z}_{2}+z_{2}^{q+r+1}\overline{z}_{2}^{r}\overline{z}_{3}+z_{3}^{q+r+1}\overline{z}_{3}^{r}\overline{z}_{1}$be the corresponding projective curves
and letbe the Milnor fiber of. First, the Euler characteristic of the Milnor fibers and the genera
for $i=1,$
arc given as follows.

Let$C_{s_{i}}$$f_{s_{i}}$$F_{s:}$$5$$\ldots,$$\chi(F_{s}.)=q^{3}-3q^{2}+3q$ ,$\chi(F_{s_{4}})=q(q^{2}+q+1)$$g(C_{s_{i}})= \frac{(q-1)(q-2)}{2},$,$\chi(F_{s_{5}})=q(q^{2}+3q+3)$$i=1,2.3$$g(C_{S4})= \frac{q(q+1)}{2}$,$g(C_{s_{4}})= \frac{(q+2)(q+1)}{2}$arc isomorphic to algebraic plane curves
andIn [21], we have shown thatdefined by the associated homogeneous polynomials of degree :$C_{s_{1}}$$C_{s_{2}}$$q$$q_{s_{1}}(z)=z_{1}^{q-1}z_{2}+z_{2}^{q-1}z_{3}+z_{3}^{q}$$g_{S2}(z)=z_{1}^{q-1}z_{2}+z_{2}^{q-1}z_{3}+z_{3}^{q}$

We also expect that

$C_{s_{3}}$is isotopic to the algebraic curve$z_{1}^{q-1}z_{2}+z_{2}^{q-1}z_{3}+z_{3}^{q-1}z_{1}=0$

as the genus of$C_{s_{3}}$.

,suggests it (see also [21]).

II. We consider the following join type polar homogeneous polynomial.
$h_{j}(z,\overline{z})=g_{j}(w^{-}w)+z_{3}^{q+r_{\overline{Z}_{3}^{r}}}$,$g_{j}(w,-w)=(w_{1}^{q+.7}\overline{w}_{1}^{j}+w_{2}^{q+j}\overline{w}_{2}^{j})(\uparrow v_{1}^{r-j}-\alpha w_{2}^{r-j})(\overline{w}_{1}^{r-j}-\beta\overline{w}_{2}^{r-j})$$0\leq j\leq r$

,

.

of
is connected. The link(
C’ are generic.) The the Milnor fiber
component number of $g=0$ is lkn $(g)=q+2(,.-j)$ . Thus $\chi(F_{g_{j}})=q(q-2+2(’\cdot-j))$by Corollary 40 and $g= \frac{(q-1)(q-2+2(r-j))}{2}$ .
In particular, taking $q=2$ , we obtain $g=r-j$ and thus$\alpha,$$\beta\in$$F_{g_{g}}$Corollary46. For any smooth surface
so that the degree of is 2.

$S$$g_{j}$of genus , there is an embedding$S\subset \mathbb{C}\mathbb{P}^{2}$

$g$

$S$

We observe that the case $q=1$ gives only the trivial casc $g=0$ in this family.

3.4. Twisted join type polynomial. In this section, we introducc a new class of
mixed polar weighted polynomials which we use to construct curves with embedded
be a polar weighted homogeneousdegree 1. For further detail, see [22]. Let
polynomial of n-variables $z=(z_{1}, \ldots, z_{n})$ . Let
be the radial and polar
be the radial and polar weight respectively and let$Q’={}^{t}(q_{1}/d,$and $P’=$
we
call
that
degree respectively. For simplicity,$f(z,\overline{z})$$Q={}^{t}(q_{1},$$d,$$\ldots,$$q_{n}),$$P=\downarrow(p_{1}, \ldots,p_{n})$$q$$\ldots,$$q_{n}/d)$99
M. OKA$\ldots,p_{n}/q)$ the normalized mdial weights andthe normalized polar weights
respectively. Consider the mixed polynomial of (n l)-variables:${}^{t}(p_{1}/q,$$+$$g(z,\overline{z}, w.\overline{w})=f(z,\overline{z})+\overline{z}_{n}w^{a}\overline{w}^{b}$

Consider thc rational numbers$\overline{q}_{n+1},\overline{p}_{n+1}$$\frac{q_{n}}{d}+(a+b)\overline{q}_{\iota+1}=1$,$a>b$ .

,

satisfying$- \frac{p_{n}}{q}+(a-b)\overline{p}_{\iota+1}=1$.

We assume that $q_{?t}<d$ so thatare positive rational numbers. The polynomial is a polar weighted homogeneous polynomial with the normalized radialand polar weights
andrespectively. The radial and polar degrec of are given by lcm $(d. denom(\overline{q}_{n+1}))$and lcm $(q, dcnorn,(\overline{p}_{n+1}))$ wherc. We call,is the dcnominator oftwisted join of. We say that 9 is a polar weighted homogeneousandpolynomial of twisted join type. A twisted join type polynomial behaves differently
than the simple join type, as wc will see below.
We recall thatis callcd to bc l-convenient if the restriction of to eachcoordinate hyperplane $:=f|_{\{z_{i}=0\}}$ is non-trivial for $i=1,$([23])$\overline{q}_{\iota+1},,\overline{p}_{r\iota+1}$$g$$\overline{Q’}=L(q_{1}/d, \ldots , q_{n}/d,\overline{q}_{\tau+1})$$\overline{P’}={}^{t}(p_{1}/q,$$\ldots,$$p_{n}/q,\overline{p}_{n+1})$$g$$dr^{I},\gamma\iota om(x)$$\prime f’\in \mathbb{Q}$

$q$

$a$

$\overline{z}_{n}w^{a}\overline{w}^{b}$$\int(z,\overline{z})$$f$$f(z,\overline{z})$$f_{i}$$\ldots,$

Lemma 47. Assume that $n\geq 2$ and

$f$

$n$

is l-convenient. Then$\phi_{\#}:\pi_{1}((\mathbb{C}^{*})^{n}\backslash F_{f}^{*})\cong Z^{n}\cross Z$

is an isomorphism where$\phi$

is the canonical mapping$\phi:(\mathbb{C}^{*})^{n}\backslash F_{f}^{*}arrow(\mathbb{C}^{*})^{n}\cross(\mathbb{C}\backslash \{1\})$

defined by$\phi(z)=(z, f(z,\overline{z}))$and$F_{f}^{*}$$:=f^{-1}(1)\cap(\mathbb{C}^{*})^{n}$

Put $F_{f_{n}}:=f_{n}^{-1}(1)=F_{f}\cap\{z_{n}=0\}\subset \mathbb{C}^{n-1}$ with.

$f_{\mathcal{T}l}:=f|_{\mathbb{C}^{n}\cap\{z_{n}=0\}}$

Theorem 48. [22] Assume that $n\geq 2$ and is l-convenient andtwisted join polynomial as above. Then
(1) the Milnor fiber of $F_{g}=g^{-1}(1)$ , is simply connected.
(2) The Euler charactentstic of
is given by the formula;$\chi(F_{g})=-(0-b-1)\chi(F_{f})+(c\prime_{l}-b)\chi(F_{f_{n}})$ .
$f$.

$g(z,\overline{z}, w,\overline{w})$

is a

$g,$

$F_{9}$

3.4.1. Construction of a family of mixed curves with polar degree . Now we areready to construct a key family of mixed curves with embedding degree . Recall
the polynomial:
$q$

$q$

$w=(z_{1}, z_{2})$$h_{q.r,j}(w,\overline{w}):=(z_{1}^{q+j}\overline{z}_{1}^{j}+z_{2}^{q+j}\overline{z}_{2}^{j})(z_{1}^{r-j}-\alpha z_{2}^{r-j})(\overline{z}_{1}^{r-j}-\beta\overline{z}_{2}^{r-j}))$.

$h_{q,r},.’(w, w-)$ is l-convenient strongly polar homogeneous polynomial with the radialdegrce $q+r$ and thc polar degree respectively. The constants
are generic,$|\beta|\neq 0,1$ and
For this, it suffices to assume that
. Consider the twisted
join polynomial of 3 variables:$q$$\alpha,$$z_{1},$$z_{2},$ $z_{3}$$s_{q,r,j}(z,\overline{z})=h_{q,r,g}(w^{-}w)+\overline{z}_{2}z_{3}^{q+r}\overline{z}_{3}^{r-1}$,$z=(z_{1}, z_{2}, z_{3})$Letbe the Milnor fiber and let
sponding mixed projective curve:$F_{q,r,j}=s_{q,r,j}^{-1}(1)\subset \mathbb{C}^{3}$$S_{q,r,j}\subset \mathbb{P}^{2}$$S_{q,r,j}=\{[z]\in \mathbb{P}^{2}|s_{q,r,j}(z,\overline{z})=0\}$Note that
and$F_{q,,.,j}$$S_{q,r,j}$$S_{q.?\cdot,j}$.

$\beta$$|\alpha|\neq|\beta|$$|\alpha|,$.
be the corre-.

is a smooth mixed curvc. The following describes the topology of

100
INTRODUCTION TO NO-DEGENERATE MIXED FUNCTIONS

Theorem 49.
$by$ :(1) The Euler chamcteristic of the Milnor fiber$\chi(F_{q,r,j})=q(q^{2}-q+1+2(r-j))$(2) The genus of$S_{q,’\cdot,j}$

is given$F_{q,r,j}$.

is given by:$g(S_{q.r,j})= \frac{q(q-1)}{2}+(r-j)$

Proof. Let$H_{q,r,j}=h_{q,,.,j}^{-1}(1)$. Then by Corollary 40,$\chi(H_{q,r,j})=-q(q-2+2(r-j))$$\chi(H_{q,r,j}\cap\{z_{2}=0\})=q$

and thc assertion follows from Theorcm 48.

口3.4.2. Mixed curves with polar degree 1. We considcr thc case $q=1,$ $j=0$ :$\{\begin{array}{ll}f\iota(w, w-) :=(z_{1}+z_{2})(z_{1}^{r}-\alpha z_{2}^{r})(\overline{z}_{1}^{r}-\beta\overline{z}_{2}^{r})f_{r}(z,\overline{z}) :=h(w,\overline{w})+\overline{z}_{2}z_{3}^{r+1}\overline{z}_{3}^{r-1}S_{r} :=\{[z]\in \mathbb{P}^{2}|f_{r}(z,\overline{z})=0\}.\end{array}$Corollary 50. Let
be the mixed curve as above. Then the embedding degree ofis .
is 1 and the genus of$S_{r}$$S_{r}$$S_{r}$$r$

Proof. Let $F_{r}=f^{-1}(1)$ be the Milnor fiber of . By Theorem 48, we have
. Thus by Corollary 40, the assertion follows immediately.
$f_{r}$$2r+1$

Remark 51. $h,(w, w-)$ can be replaced by
the topology.

$(z_{1}^{r+1}-z_{2}^{r+1})(\overline{z}_{1}-\beta\overline{z}_{2}^{r})$$\chi(F_{r})=$口without changing

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DEPARTMENT oP MATHEMATICSTOKYO UNIVERSITY OF SCIENCE
26 $w_{AKAM}$ lYA-C110, $s_{HINJUKU-KU}$
TOKYO 162-8601
E-mail: oka@rs.kagu.tus.ac.jp