Mixed hypersurface singularities Mutsuo Oka Dept. of Math. Tokyo - - PowerPoint PPT Presentation

Mixed hypersurface singularities

Mutsuo Oka

• Dept. of Math. Tokyo University of Science

FVJ2018 Nha Trang Septenber18-19

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Mixed hypersurface singularities

Lecture I. Mixed function

This lecture is partially based on [19]. We consider the situation of real algebraic variety of codimension 2: V = {(x, y) ∈ R2n|g(x, y) = h(x, y) = 0} where zj = xj + iyj and g, h ∈ R[x, y]. Under the canonical identification Cn ∼ = R2n, z corresponds to x + iy. We study when the link is fibered over the circle. First in the complex Euclidean space, V can be described as V = {z ∈ Cn | f (z, ¯ z) = 0}, f (z, ¯ z) := g(z + ¯ z 2 , z − ¯ z 2i ) + ih(z + ¯ z 2 , z − ¯ z 2i ) We call f a mixed polynomial. f is a complex-valued real analytic function.

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Mixed hypersurface singularities

Weighted homogeneous polynomials

We first study elementary basic cases. A mixed polynomial f (z, ¯ z) = ∑

ν,µ cν,µzν¯

zµ is called polar weighted homogeneous if there exist integers p1, . . . , pn and a non-zero integer mp such that gcd(p1, . . . , pn) = 1, ∑n

j=1 pj(νj − µj) = mp,

if cν,µ ̸= 0. f (z, ¯ z) is called radially weighted homogeneous if there exist positive integers q1, . . . , qn and a non-zero integer mr such that gcd(q1, . . . , qn) = 1,

n

∑

j=1

qj(νj + µj) = mr, if cν,µ ̸= 0. We say f (z, ¯ z) is a mixed weighted homogeneous if f is radially weighted homogeneous of type, say (q1, . . . , qn; mr), and f is also polar weighted homogeneous of type, say (p1, . . . , pn; mp)([10]). We define vectors of rational numbers (u1, . . . , un) and (v1, . . . , vn) by ui = qi/mr, vi = pi/mp and we call them the normalized radial (respectively polar) weights.

Example 1

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Mixed hypersurface singularities

Using a polar coordinate (r, η) of C∗ where r > 0 and η ∈ S1 with S1 = {η ∈ C | |η| = 1}, we define a mixed R+ × S1-action on Cn by (r, η) ◦ z := (rq1ηp1z1, . . . , rqnηpnzn), reiη = (r, η) ∈ R+ × S1. Assume that f (z, ¯ z) is mixed weighted homogeneous polynomial. Then f satisfies the functional equality f ((r, η) ◦ (z, ¯ z)) = rmr ηmpf (z, ¯ z). (1) This notion was introduced by Ruas-Seade-Verjovsky [23] implicitly and then by Cisneros-Molina [8]. It is easy to see that a mixed weighted homogeneous polynomial defines a global fibration f : Cn − f −1(0) → C∗.

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Mixed hypersurface singularities

For example, put Uθ = {ρeiη | θ − π < η < θ + π, ρ ̸= 0} ≡ R+ × (θ − π, θ + π). The triviality over Uθ is given as follows. Ψ : Uθ×f −1(eiθ) → f −1(Uθ), Ψ(ρei(θ+ξ), z) = (ρ1/mr , exp(iξ/mp))◦z

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Mixed hypersurface singularities

Theorem 2

φ = f /|f | : S2n−1

r

\ Kr → S1 is a locally trivial fibration for any r > 0 and it is equivalent to the restriction of the global fibration f : f −1(S1) → S1. Proof First, the triviality of φ is given similarly as follows. For example, over U = (−π, π), ψ : φ−1(1) × U → φ−1(U), ψ(z, θ) = (1, exp(iθ/mp)) ◦ z. Define a fiber preserving diffeomorphism Φ : S2n−1

r

\ Kr → f −1(S1) Φ(z) = (1/|f (z)|1/mr , 0) ◦ z

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Mixed hypersurface singularities

This gives the commutative diagram: S2n−1

r

\ Kr

Φ

− → f −1(S1)   φ   f S1 = S1

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Mixed hypersurface singularities

Mixed singular point for a generic mixed function

Let f (z, ¯ z) be a mixed polynomial and we consider a hypersurface V = {z ∈ Cn; f (z, ¯ z) = 0}. Put zj = xj + iyj. Then f (z, ¯ z) is a real analytic function of 2n variables (x, y) with x = (x1, . . . , xn) and y = (y1, . . . , yn). Put f (z, ¯ z) = g(x, y) + i h(x, y) where g, h are real analytic functions. Recall that

∂ ∂zj = 1 2

(

∂ ∂xj − i ∂ ∂yj

) ,

∂ ∂¯ zj = 1 2

(

∂ ∂xj + i ∂ ∂yj

) We use the notations:

∂f ∂zj = ∂g ∂zj + i ∂h ∂zj , ∂f ∂¯ zj = ∂g ∂¯ zj + i ∂h ∂¯ zj

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Mixed hypersurface singularities

Put dRg(x, y) = ( ∂g

∂x1 , . . . , ∂g ∂xn , ∂g ∂y1 , . . . , ∂g ∂yn ) ∈ R2n

dRh(x, y) = ( ∂h

∂x1 , . . . , ∂h ∂xn , ∂h ∂y1 , . . . , ∂h ∂yn ) ∈ R2n

For a complex valued mixed polynomial, we use the notation: ∂f (z, ¯ z) = ( ∂f

∂z1 , . . . , ∂f ∂zn ) ∈ Cn,

¯ ∂f (z, ¯ z) = ( ∂f

∂¯ z1 , . . . , ∂f ∂¯ zn ) ∈ Cn

We say that a point z ∈ V is a mixed-singular point of V if and

• nly if dfz : TzCn → Tf (z)C is not surjective or equivalently the

two vectors dg(x, y), dh(x, y) are linearly dependent over R. Assume that p ∈ V is a mixed regular point. Then V is a real codimension two subvariety at p.

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Mixed hypersurface singularities

Proposition 0.1

The following two conditions are equivalent.

• 1. z ∈ V is a mixed singular point.
• 2. dg, dh are linearly dependent over R.
• 3. There exists a complex number α, |α| = 1 such that

∂f (z, ¯ z) = α ¯ ∂f (z, ¯ z).

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Mixed hypersurface singularities

First assume that dRg, dRh are linearly dependent at z. Suppose for example that dg(x, y) ̸= 0 and write dh(x, y) = t dg(x, y) for some t ∈ R. This implies that

∂f ∂xj = (1 + ti) ∂g ∂xj , ∂f ∂yj = (1 + ti) ∂g ∂yj ,

thus

∂f ∂zj = (1 + ti)

(

∂g ∂xj − i ∂g ∂yj

) ,

∂f ∂¯ zj = (1 + ti)

(

∂g ∂xj + i ∂g ∂yj

) . Thus ∂f (z, ¯ z) = (1 + ti) (

∂g ∂x1 − i ∂g ∂y1 , . . . , ∂g ∂xn − i ∂g ∂yn

) = 2(1 + ti)∂g(z, ¯ z) ¯ ∂f (z, ¯ z) = (1 + ti) (

∂g ∂x1 + i ∂g ∂y1 , . . . , ∂g ∂xn + i ∂g ∂yn

) = 2(1 + ti)¯ ∂g(z, ¯ z)

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Mixed hypersurface singularities

Here ∂g = ( ∂g

∂z1 , . . . , ∂g ∂zn ) and ¯

∂g = ( ∂g

∂¯ z1 , . . . , ∂g ∂¯ zn ). As g is a real

valued polynomial, using the equality ∂g(x, y) = ¯ ∂g(x, y) we get ∂f (z, ¯ z) = 1 − ti 1 + ti ¯ ∂f (z, ¯ z). Thus it is enough to take α = 1−ti

1+ti .

Conversely assume that ∂f (z, ¯ z) = α¯ ∂f (z, ¯ z) for some α = a + bi with a2 + b2 = 1. Using the notations dxg = ( ∂g ∂x1 , . . . , ∂g ∂xn ), dyg = ( ∂g ∂y1 , . . . , ∂g ∂yn ), etc, we get (1 − a)dxg + b dyg = −b dxh − (1 + a)dyh −b dxg + (1 − a)dyg = (a + 1)dxh − b dyh.

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Mixed hypersurface singularities

Solving these equations assuming a ̸= 1, we get dR g = (dxg, dyg) = −2b (1 − a)2 + b2 dR h which proves the assertion. If a = 1, the above equations implies that dh = 0 and the linear dependence is obvious.

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Mixed hypersurface singularities

Transversality

We assume again f (z, ¯ z) is a mixed weighted homogeneous polynomial as before. First we observe that

Proposition 0.2

f −1(t) is mixed non-singular for any t ̸= 0 and z ∈ f −1(t) as dfz : Tzf −1(t) → TtC is surjective. Let V = f −1(0). Assume that the radial weight qj > 0 for any j. Then V is contractible to the

• rigin 0. If further 0 is an isolated mixed singularity of V , V \{0}

is smooth. We omit the proof of the first assertion. A canonical deformation retract βt : V → V is given as βt(z) = t ◦ z, 0 ≤ t ≤ 1. (More precisely, the action is defined for t > 0 but it is easy to see β0(z) = limt→0 βt(z) = 0.) Then β1 = idV and β0 is the contraction to 0. Assume that z ∈ V \{0} is a singular point. Consider the decomposition into real analytic functions f (z) = g(x, y) + ih(x, y).

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Mixed hypersurface singularities

Using the radial R+-action, we see that g(r ◦ (x, y)) = rmr g(x, y), h(r ◦ (x, y)) = rmr h(x, y). (2) This implies that g(x, y), h(x, y) are weighted homogeneous polynomials of (x, y) and the Euler equality can be restated as mr g(x, y) = ∑n

j=1 pj

( xj

∂g ∂xj (x, y) + yj ∂g ∂yj (x, y)

) mr h(x, y) = ∑n

j=1 pj

( xj ∂h

∂xj (x, y) + yj ∂h ∂yj (x, y)

) . Differentiating the equalities (2) in r, we get ∂g ∂xj (r◦(x, y)) = rmr−qj ∂g ∂xj (x, y), ∂h ∂xj (r◦(x, y)) = rmr−qj ∂h ∂xj (x, y). This implies that these differentials are also weighted homogeneous polynomials of degree mr − qj.

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Mixed hypersurface singularities

Thus the jacobian matrix ( ∂(g, h) ∂(xi, yi)(r ◦ (x, y)) ) is the same with the jacobian matrix at z = (x, y) up to scalar multiplications in the column vectors by rmr−q1, . . . , rmr−qn, rmr−q1, . . . , rmr−qn respectively. Assume that (x, y) ∈ V \ {0} is a mixed singular point. Then any points of the

• rbit r ◦ (x, y), r > 0 are singular points of V . This is a

contradiction to the assumption that 0 is an isolated singular point

• f V , as limr→0 r ◦ (x, y) = 0.
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Mixed hypersurface singularities

Proposition 0.3 (Transversality)

Under the same assumption as in Proposition 0.2, the sphere Sτ = {z ∈ Cn; ∥z∥ = τ} intersects transversely with V for any τ > 0. Proof: Let ϕ(x, y) = ∥z∥2 = ∑n

j=1(x2 j + y2 j ). Then Sτ intersects

transversely with V if and only if the gradient vectors dRg, dRh, dRϕ are linearly independent over R. Note that dRϕ(x, y) = 2(x1, y1, . . . , xn, yn). Suppose that the sphere S∥z∥ is tangent to V at z = (x, y) ∈ V . Then we have for example, a linear relation dg(x, y) = α dh(x, y) + β dϕ(x, y) with some α, β ∈ R. Note that the tangent vector vr to the R+-oribit is tangent to V and it is written vr = (q1x1, . . . , qnxn, q1y1, . . . , qnyn) as a real vector. Then we have

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Mixed hypersurface singularities

0 =

dg(r◦(x,y) dr

|r=1 = ∑n

j=1 qj

( xj

∂g ∂xj (x, y) + yj ∂g ∂yj (x, y)

) = (vr(x, y), dg(x, y))R = (vr(x, y)R, α dh(x, y)) + (vr(x, y), β dϕ(x, y))R = 2β ∑n

j=1 qj(x2 j + y2 j )

as (vr(x, y), dh(x, y))R = 0 by the same reason. This is the case

• nly if β = 0 which is impossible as V \{0} is non-singular by

Proposition 0.2.

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Mixed hypersurface singularities

Newton boundary (General case)

Suppose that a general mixed polynomial f (z, ¯ z) = ∑

ν,µ cν,µzν¯

zµ is given. Newton bounary is defined as the boundary of the convex hull ∪

cν,µ̸=0(ν + µ + RN +). f is strongly non-degenerate if for any

face ∆, f∆ : C∗n → C is surjective and no critical point where f∆ = ∑

ν+µ∈∆ cν,µzν¯

zµ.

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Mixed hypersurface singularities

Theorem 3 ([13])

Assume that f (z, ¯ z) is strongly non-degenerate convenient mixed

• function. Then there exists a positive number r0 such that for any

0 < r ≤ r0, there exists δ(r) and

• 1. 0 is the only mixed critical point in Br0 and
• 2. Sτ ⋔ f −1(η) for any η, |η| ≤ δ(r).

In particular, there are two Milnor fibrations, and they are equivalent: Spherical Milnor fibration f /|f | : Sr0 \ K → S1 and Tubular Milnor fibration f : ∂E(r0, δ) → D∗

δ

where K = f −1(0) ∩ Sr0, ∂E(r0, δ) := {z | z ∈ Br0, 0 ̸= |f (z)| ≤ δ}.

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Mixed hypersurface singularities

Lecture II. Simplicial polynomial

Let f (z, ¯ z) = ∑s

j=1 cj znj¯

zmj be a mixed polynomial. Here we assume that c1, . . . , cs ̸= 0. Put ˆ f (w) :=

s

∑

j=1

cj wnj−mj, w = (w1, . . . , wn) ∈ Cn. We call ˆ f 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

Proposition 0.4

If f (z, ¯ z) is a polar weighted homogeneous polynomial of polar weight type (p1, . . . , pn; mp), ˆ f (w) is also a weighted homogeneous Laurent polynomial of type (p1, . . . , pn; mp) in the complex variables w1, . . . , wn.

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Mixed hypersurface singularities

A mixed polynomial f (z, ¯ z) is called simplicial if the exponent vectors {nj ± mj | j = 1, . . . , s} are linearly independent in Zn (or equivqlently in Rn) respectively. In particular, simplicity implies that s ≤ n. When s = n, we say that f is full. Put nj = (nj,1, . . . , nj,n), mj = (mj,1, . . . , mj,n) in Nn. Assume that s ≤ n. Consider two integral matrix N = (ni,j) and M = (mi,j) where the k-th row vectors are nk, mk respectively.

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Mixed hypersurface singularities

Lemma 4

Let f (z, ¯ z) be a mixed polynomial as above. If f (z, ¯ z) is simplicial, then f (z, ¯ z) is a polar weighted homogeneous polynomial. In the case s = n, f (z, ¯ z) is simplicial if and only if det(N ± M) ̸= 0. Proof: Let P′ = t(p′

1, . . . , p′ n), Q′ = t(q1,′ . . . , qn)′ be possible

normalized polar and radial weights. Then they must satisfy the equalities: p′

1(nj1 − mj1) + · · · + p′ n(njn − mjn) = 1, j = 1, . . . , s

q′

1(nj1 + mj1) + · · · + q′ n(njn + mjn) = 1, j = 1, . . . , s.

By the assumption, these equations have solutions.

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Mixed hypersurface singularities

Example

Let Ba,b(z, ¯ z) = za1

1 ¯

zb1

1 + · · · + zan n ¯

zbn

n , ai, bi ≥ 1, ∀i

fa,b(z, ¯ z) = za1

1 ¯

zb1

2 + · · · + zan n ¯

zbn

1 , ai, bi ≥ 1, ∀i

The associated Laurent polynomials are

• Ba,b(w) = wa1−b1

1

+ · · · + wan−bn

n

• fa,b(w) = wa1

1 w−b1 2

+ · · · + wan

n w−bn 1

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Corollary 5

For the polynomial fa,b, the following conditions are equivalent.

• 1. fa,b is simplicial.
• 2. fa,b is a mixed weighted homogeneous polynomial.
• 3. (SC)

a1 · · · an ̸= b1 · · · bn. Let f (z, ¯ z) = ∑s

j=1 cj znj¯

zmj be a mixed weighted homogeneous polynomial of radial weight type (q1, . . . , qn; mr) and of polar weight type (p1, . . . , pn; mp). Let F = f −1(1) be the fiber.

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Canonical stratification of F and the topology of each stratum

For any subset I ⊂ {1, 2, . . . , n}, we define CI = {z | zj = 0, j / ∈ I}, C∗I = {z | zi ̸= 0 iff i ∈ I}, and we define mixed polynomials f I by the restriction: f I = f |CI . In particular, we denote C∗n for C∗{1,...,n}. For simplicity, we write a point of CI as zI. Put F ∗I = C∗I ∩ F. Note that F ∗I is a non-empty proper subset of C∗I if and only if f I(zI, ¯ zI) is not constantly zero. Now we observe that the hypersurface F = f −1(1) has the canonical stratification F = ⨿I F ∗I. Thus it is essential to determine the topology of each stratum F ∗I.

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Mixed hypersurface singularities

Put F ∗ := F ∩ C∗n, the open dense stratum and put ˆ F ∗ := ˆ f −1(1) ∩ C∗n where ˆ f (w) is the associated Laurent weighted homogeneous polynomial.

Theorem 6

Assume that f (z, ¯ z) is a simplicial mixed weighted homogeneous polynomial and let ˆ f (w) be the associated Laurent weighted homogeneous polynomial. Then there exists a canonical diffeomorphism φ : C∗n → C∗n which gives an isomorphism of the two toric Milnor fibrations defined by f (z, ¯ z) and ˆ f (w): C∗n − f −1(0)

f

− → C∗   φ   id C∗n − ˆ f −1(0)

ˆ f

− → C∗ and it satisfies φ(F ∗n) = ˆ F ∗n and φ is compatible with the respective canonical monodromy maps.

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Mixed hypersurface singularities

Proof: Assume first that s = n for simplicity. Recall that ˆ f (w) =

n

∑

j=1

cjwnj−mj. Let w = (w1, . . . , wn) be the complex coordinates of Cn which is the defining space of ˆ f . We construct φ : C∗n → C∗n so that φ(z) = w satisfies w(φ(z))nj−mj = znj¯ zmj, thus ˆ f (φ(z)) = f (z). For the construction of φ, we use the polar coordinates (ρj, θj) for zj ∈ C∗ and polar coordinates (ξj, ηj) for wj. Thus zj = ρj exp(iθj) and wj = ξj exp(iηj). First we take ηj = θj.

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Mixed hypersurface singularities

Put nj = (nj,1, . . . , nj,n), mj = (mj,1, . . . , mj,n) in Nn. Consider two integral matrix N = (ni,j) and M = (mi,j) where the k-th row vector are nk, mk respectively. Now taking the logarithm of the equality znj¯ zmj = wnj−mj, we get an equivalent equality: (nj1 + mj1) log ρ1 + · · · + (njn + mjn) log ρn = (nj1 − mj1) log ξ1 + · · · + (njn − mjn) log ξn for j = 1, . . . , n. This can be written as (N + M)    log ρ1 . . . log ρn    = (N − M)    log ξ1 . . . log ξn    (3)

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Mixed hypersurface singularities

Put (N − M)−1(N + M) = (λij) ∈ GL(n, Q). Now we define φ as follows. φ : C∗n → C∗n, z = (ρ1 exp(iθ1), . . . , ρn exp(iθn)) → w = (ξ1 exp(iθ1), . . . , ξn exp(iθn)) where ξj is given by ξj = exp(∑n

i=1 λji log ρi) for j = 1, . . . , n. It is

• bvious that φ is a real analytic isomorphism of C∗n to C∗n. Let us

consider the Milnor fibrations of f (z, ¯ z) and ˆ f (w) in the respective ambient tori C∗n. f : C∗n\f −1(0) → C∗, ˆ f : C∗n\ˆ f −1(0) → C∗ Recall that the monodromy maps h∗, ˆ h∗ are given as h∗ : F ∗ → F ∗, z → exp(2πi/mp) ◦ z ˆ h∗ : ˆ F ∗ → ˆ F ∗, w → exp(2πi/mp) ◦ w.

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Mixed hypersurface singularities

Recall that the C∗-action associated with ˆ f (w) is the polar action

• f f (z, ¯

z). Namely exp iθ ◦ w = (exp(ip1θ)w1, . . . , exp(ipnθ)wn). Thus we have the commutative diagram: F ∗

α h∗

− → F ∗

α

  φ   φ ˆ F ∗

α ˆ h∗

− → ˆ F ∗

α

where F ∗

α = f −1(α) ∩ C∗n and ˆ

F ∗

α = ˆ

f −1(α) ∩ C∗n for α ∈ C∗. For the case s < n, just fix λji so that log ξj = ∑n

i=1 λji log ρi) for

j = 1, . . . , n.

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Mixed hypersurface singularities

Remark

The case mixed Brieskorn f (z, ¯ z) = za1

1 ¯

z1 + · · · + zan

n ¯

zn is studied in [23]. In this case, g = za1−1

1

+ · · · + zan−1

n

and φ : f −1(1) → g−1(1) is given by wj = zj|zj|

2 aj −1 , j = 1, . . . , n

We can see that this is a homeomorphism, extended to Cn → Cn.

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Mixed hypersurface singularities

Zeta-functions

Now we know that by [14], the inclusion map ˆ F ∗ ֒ → C∗n is (n − 1)-equivalence and χ( ˆ F ∗) = (−1)n−1 det(N − M) for s = n and 0 otherwise. In general, for a diffeomorphism h : F → F, the zeta function of h is defined by ζh(t) = ∏∞

j=1 det(th2j−1 − id)

∏∞

j=0 det(th2j − id)

where hj = h∗ : Hj(F) → Hj(F). Note that in our case the monodromy map ˆ h : ˆ F ∗ → ˆ F ∗ has a period mp. The fixed point locus of (ˆ h)k is F ∗ if mp | k and ∅

• therwise.
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Mixed hypersurface singularities

Thus using the formula of the zeta function (see, for example [6]), ζˆ

h∗(t) = exp( ∞

∑

j=0

(−1)n−1d tjmp/(jmp)) = (1 − tmp)(−1)nd/mp where d = det (N − M) if s = n and d = 0 for s < n. Translating this in the monodromy h∗ : F ∗ → F ∗, we obtain

Corollary 7

F ∗ has a homotopy type of CW-complex of dimension n − 1 and the inclusion map F ∗ ֒ → C∗n is an (s − 1)-equivalence and if s = n, χ(F ∗) = (−1)nd or 0 and the zeta function ζh∗(t) of h∗ : F ∗ → F ∗ is given as (1 − tmp)(−1)nd/mp with d = det (N − M) if s = n and ζh∗(t) = 1 for s < n.

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Mixed hypersurface singularities

Join type polynomials

Another special type of mixed functions are mixed polynomials of join type. Consider the mixed weighted homogeneous polynomials g(z, ¯ z), h(w, ¯ w) with z = (z1, . . . , zn) and w = (w1, . . . , wm). Consider f (z, w, ¯ z, ¯ w) = g(z, ¯ z) + h(w, ¯ w). Then

Theorem 8 (Cisneros-Molina [8], see also [9])

The Milnor fiber of f is homotopic to the join of the respective Milnor fibers g−1(1) ∗ h−1(1) and the monodromy is also the join

• f the respective monodromy.
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Mixed hypersurface singularities

Proof

: The proof is completely parallel to that of [9]. Let (p1, . . . , pn) and (r1, . . . , rm) be the normalized polar weights. Then f has the nomalized polar weight (p1, . . . , pn, r1, . . . , rm). The polar weight is given by multiplying the least common multiple of the denominator. Let mr,g, mp,g (respectively mr,h, mp,h) be the radial and polar degree of g (resp.

• f h). We denote the associated R+ × S1-action as

(reiθ, (z, w)) → (reiθ ◦ (z, w) = (reiθ ◦ z, reiθ ◦ w). For the brevity’s sake, we define complex valued functions for η = r exp(iθ), ξg(η) := r1/mr,g eiθ/mp,g , ξh(η) := r1/mr,heiθ/mp,h defined on C \ {η ≤ 0} where the argument is chosen on the interval (−π, π). By defintion, we have g(ξg(η) ◦ z) = ηg(z), h(ξh(η) ◦ w) = ηh(w).

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The proof of the assertion is divided into three steps. Let Ff = f −1(1) ⊂ Cn+m, Fg = g−1(g), Fh = h−1(1). Consider the identification (z, w) ∼ (z′, w) z, z′ ∈ g−1(0), w ∈ Fh (z, w) ∼ (z, w′) z ∈ Fg, w, w′ ∈ h−1(0).

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Let ˜ F be the quotient space of F by the above equivalence

• relation. The corresponding equivalence class is denoted as

[∗, w], [z, ∗] repectively. For the topology, we consider two subsets ˜ F ′ = {[z, w] | g(z) ̸= 0} and ˜ F ′′ = {[z, w] | h(w) ̸= 0} and canonical projections π1 : ˜ F ′ → Cn \ g−1(0) and π2 : ˜ F ′′ → Cm \ h−1(0). We put ˜ F the weakest topology in which π1, π2 are continuous. Let F1 = {[z, w] | g(z) ∈ R} = {(z, w) | g(z), h(w) ∈ R}, F2 = {[z, w] ∈ F1 | 0 ≤ g(z), h(w) ≤ 1}.

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Proposition 0.5

Step 1. The canonical mapping π : F → ˜ F is an homotopy equivalence. Proof: Put Nh,2ε := {(z, w) ∈ F | |h(w, ¯ w)| ≤ 2ε} Ng,2ε := {(z, w) ∈ F | |g(z)| ≤ 2ε}. Consider a homotopy H : F × [0, 1] → F defined as follows.

• 1. If (z, w) ∈ F \ Nh,2ε, Ht(z, w) = (z(t), w(t)) where

w(t) = ξh(1 − tρ(|h(w)|)) ◦ w z(t) = ξg(1 − h(w(t)) ◦ z.

• 2. If (z, w) ∈ F \ Ng,2ε, Ht(z, w) = (z(t), w(t)) where

z(t) = ξg(1 − tρ(|g(z)|)) ◦ z w(t) = ξh(1 − h(z(t)) ◦ w.

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Mixed hypersurface singularities

Observe that Ht = id on F \ Ng,2ε ∪ Nh,2ε and H0 = id and H1 collapses z × h−1(0) to (z, 0) and g ∈ (0) × w to (0, w). We define a canonical mapping ˜ H1 : ˜ F → F associated with H1. It is easy to see that H also associate ˜ H : ˜ F × [0, 1] → ˜ F so that ˜ H([z, w], t) = [z(t), w(t)]. Then it is easy to see that ˜ H1 ◦ π ≃ idF by H and π ◦ ˜ H1 ≃ id˜

F by ˜

• H. For more detail, see Lemma 1, [9].
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ε 2ε 1

Figure: Function ρ(t)

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Mixed hypersurface singularities

Step 2. F1 ⊂ ˜ F is a deformation retract which is compatible with the monodromy. We define deformation homotopy [z(t), w(t)], 0 ≤ t ≤ 1 as follows.

• 1. For [z, w] with g(z) ̸= 0, 1, put

z(t) := ξg (ℜg(z) + (1 − t)iℑg(z) g(z) )

• z,

w(t) := ξh (ℜh(w) + (1 − t)iℑh(w) h(w) )

• w.

Note that g(z(t)) = ℜg(z) + (1 − t)iℑg(z) h(w(t)) = ℜh(w) + (1 − t)iℑh(w).

• 2. For [z, w] with g(z) = 0 or 1, z(t) ≡ z, w(t) ≡ w.
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Step 3. F2 ⊂ F1 is also a deformation retract. A deformation retract can be defined similarly. Step 4. F2 is homeomorphic to the join Fg ∗ Fh and the monodromy map is the joined respective monodromy maps. We define ψ : F2 → Fg ∗ Fh by ψ([z, w]) =      [ξg(1/g(z)) ◦ z, ξh(1/h(w)) ◦ w, g(z)], 0 < g(z) < 1 [∗, w, 0], g(z) = 0 [z, ∗, 1], g(z) = 1.

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For the more detail, see [8, 9].

Corollary 9

Suppose that F1 and F2 has the homotopy types of bouques of µg-spheres of dimension n − 1 (resp. µh-spheres of dimension m − 1). Then F is n + m − 2 connected and F is homotopic to a bouquet of µgµh-spheres of dimension n + m − 1 Hn+m−1(F)

h∗

− → Hn+m−1(F)   ∼ =   ∼ = Hn−1(F1) ⊗ Hm−1(F2)

h1∗⊗h2∗

− → Hn−1(F1) ⊗ Hm−1(F2)

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Lecture III. Moduli space of strongly polar weighted homogeneous polynomials and number of zeros without sign

Introduction: Consider a mixed polynomial of one variable f (z, ¯ z) = ∑

ν,µ aν,µzν ¯

zµ. We denote the set of roots of f by V (f ). Assume that z = α is an isolated zero of f = 0. Put f (z, ¯ z) = g(x, y) + ih(x, y) with z = x + iy. A root α is called simple if the Jacobian J(g, h) is not vanishing at z = α. We call α an orientation preserving or positive root (respectively orientation reversing, or negative), if the Jacobian J(g, h) is positive (resp. negative) at z = α. There are two basic questions.

• 1. Determine the number of roots with sign.
• 2. Determine the number of roots without sign.
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Number of roots with sign

Let C be a mixed projective curve of polar degree d defined by a strongly mixed homogeneous polynomial F(z, ¯ z), z = (z1, z2, z3) of radial degree dr = d + 2s and let L = {z3 = 0} be a line in P2. Recall that F(z, ¯ z) is strongly mixed weighted homogeneous if it is polar and radial weighted homogeneous with respect to the same weight vector ([11]). In particular, if this weoght is (1, . . . , 1), we say f is mixed strongly homogeneous. We assume that L intersects C transversely.

Proposition 0.6 (Theorem 4.1, [11])

With the hypothesis above, the fundamental class [C] is mapped to d[P1] and thus the intersection number [C] · [L] is given by d. This is also given by the number of the roots of F(z1, z2, 0) = 0 in P1 counted with sign.

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The following is useful to compute the number of zeros with sign

• f such polynomials. Let f (z, ¯

z) be a given mixed polynomial of

• ne variable, we consider the filtration by the degree:

f (z, ¯ z) = fd(z, ¯ z) + fd−1(z, ¯ z) + · · · + f0(z, ¯ z). Here fℓ(z, ¯ z) := ∑

ν+µ=ℓ cν,µzν ¯

zµ. that we have a unique factorization of fd as follows. fd(z, ¯ z) = czp¯ zq ∏s

j=1(z + γj ¯

z)νj, p + q + ∑s

j=1 νj = d, c ∈ C∗.

where γ1, . . . , γs are mutually distinct non-zero complex numbers. We say that f (z, ¯ z) is admissible at infinity if |γj| ̸= 1 for j = 1, . . . , s . For non-zero complex number ξ, we put ε(ξ) = { 1 |ξ| < 1 −1 |ξ| > 1 and we consider the following integer:

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β(f ) := p − q +

s

∑

j=1

ε(γj)νj,

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The following equality holds.

Theorem 10 ([12])

Assume that f (z, ¯ z) is an admissible mixed polynomial at infinity. Then the total number of roots with sign is equal to β(f ). Remark: Here if α is a non-simple root, we count the number with

• multiplicity. The multiplicity is defined by the local rotation

number at α of the normalized Gauss mapping Sε(α) → S1, z → f (z)/|f (z)|. We are more interested in the total number of V (f ) which we denote by ρ(f ), which is the cardinality of ♯V (f ) for particular classes of mixed polynomials ignoring the sign. The notion of the multiplicity is not well defined for a root without

• sign. Thus we assume that roots are all simple. The problem is

that ρ(f ) is not described by the highest degree part fd, which was the case for the number of roots with sign β(f ). We will give an example of mixed polynomial below ρ(f ) = n2.

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Example 11

Let us consider the Chebycheff polynomial Tn(x). It has two critical values 1 and -1 and the roots of Tn(x) = 0 are in the interval (−1, 1). Consider a polynomial F(x, y) = (y − Tn(x) + i(x − aTn(by)), a, b ≫ 1. By the assumption a, b ≫ 1, F = 0 has n2 roots in (−1, 1) × (−1, 1). Consider F as a mixed polynomial by substituting x, y by x = (z + ¯ z)/2, y = −i(z − ¯ z)/2. This example gives an extreme case for which the possible complex roots (by Bezout theorem) of ℜF = ℑF = 0 are all real roots.

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Mixed hypersurface singularities

Figure: Roots of F(x, y) = 0, n = 5, a = 3/2, b = 2

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The above example shows implicitly that the behavior of the number of roots without sign behaves very violently if we do not assume any assumption on f . Consider a mixed polynomial of one variable f (z, ¯ z) = ∑

ν,µ aν,µzν ¯

zµ. Put degz f := max{ν | aν,µ ̸= 0} deg¯

z f := max{µ | aν,µ ̸= 0}

deg f := max{µ + ν | aν,µ ̸= 0} We call degz f , deg¯

z f , deg f the holomorphic degree , the

anti-holomorphic degree and the mixed degree of f respectively. We consider the following subclasses of mixed polynomials: L(n + m; n, m) := {¯ zmq(z) − p(z) | degz q(z) = n, degz p(z) ≤ n}, Lhs(n + m; n, m) := {r(¯ z)q(z) − p(z) | deg¯

z r(¯

z) = m, degz q(z) = n, degz p(z) ≤ n}, M(n + m; n, m) := {f (z, ¯ z) | deg f = n + m, degz f = n, deg¯

z f = m

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Mixed hypersurface singularities

where p(z), q(z) ∈ C[z], r(¯ z) ∈ C[¯ z]. We have canonical inclusions: L(n + m; n, m) ⊂ Lhs(n + m; n, m) ⊂ M(n + m; n, m). The classes L(n + m; n, m), Lhs(n + m; n, m) come from harmonic functions ¯ zm − p(z) q(z), r(¯ z) − p(z) q(z) as their numerators. Especially L(n + 1; n, 1) corresponds to the lens equation. We call ¯ zm − p(z)

q(z) = 0 a generalized lens equation

and and r(¯ z) − p(z)

q(z) = 0 a harmonically splitting lens type equation

• respectively. The corresponding numerators are called a

generalized lens polynomial and a harmonically splitting lens type polynomial respectively. The polynomials which attracted us in this paper are these classes. We thank to A. Galligo for sending us their paper where we learned this problem ([3]).

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Mixed hypersurface singularities

Lens equation

The following equation is known as the lens equation. L(z, ¯ z) = ¯ z −

n

∑

i=1

σi z − αi = 0, σi, αi ∈ C∗. (4) We identify the left side rational function with the mixed polynomial given by its numerator L(z, ¯ z)

n

∏

i=1

(z − αi) ∈ (n + 1; n, 1). throughout this paper. The real and imaginary part of this polynomial are polynomials of x, y of degree n + 1. Unlike the previous example, ρ(f ) is much more smaller than (n + 1)2. This type of equation is studied by astrophysicists.

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Mixed hypersurface singularities

For more explanation from astrophysical viewpoint, see for example Petters-Werner [20]. The lens equation can be written as L(z, ¯ z) := ¯ z − φ(z), φ(z) = p(z) q(z) ̸= 0, (5) deg p(z) ≤ n, deg q(z) = n. It is known that

Proposition 0.7

The number of zeros ρ(L) of L is bounded by 5n − 5 by Khavinson-Neumann [4]. Bleher-Homma-Ji-Roeder have determined the exact range of ρ(L):

Theorem 12 (Theorem 1.2,[2])

Suppose that the lens equation has only simple solutions. Then the set of possible numbers of solutions is equal to {n − 1 + 2k | 0 ≤ k ≤ 2n − 2} = {n − 1, n + 1, · · · , 5n − 7, 5n − 5}.

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The estimation in Proposition 0.7 are optimal. Rhie gave an explicit example of f which satisfies ρ(f ) = 5n − 5 ( See Rhie [21]). Thus the inequality ρ(f ) ≤ 5(n − 1) is optimal. The minimum of ρ is n − 1 and it can be obtained for example by ¯ zzn − 1. In the proof of Proposition 0.7, the following principle in complex dynamics plays a key role.

Lemma 13

Let r be an rational function on P1. If z0 is an attracting or rationally neutral fixed point, then z0 attracts some critical point

• f r.

Elkadi and Galligo studied this problem from computational point

• f view to construct such a mixed polynomial explicitly and

proposed the similar problem for generalized lens polynomials L(n + m; n, m) ([3]).

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Moduli space of strongly weighted homogeneous polynomials

Consider a strongly mixed weighted homogeneous polynomial F(z, ¯ z) of two variables z = (z1, z2) with polar weight P = t(p, q), gcd(p, q) = 1 and let dp, dr be the polar and radial degrees

• respectively. See [11] for definition. This is equivalent to the

equality (7) holds. Let C∗ × C2 → C2, (ρ, (z1, z2)) → ρ ◦

P (z1, z2) := (z1ρp, z2ρq)

(6) be the associated C∗-action.

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Recall that F satisfies the Euler equality: F(r exp(θi) ◦

P (z, ¯

z)) = rdr exp(dpθi)F(z, ¯ z). (7) A strongly mixed homogeneous polynomial is the case where the weight is the canonical weight 1 := t(1, 1). Consider the global Milnor fibration F : C2 \ F −1(0) → C∗ and let M = {z ∈ C2|F(z, ¯ z) = 1} be the Milnor fiber. We assume further that F is convenient. Namely F|z1=0, F|z2=0 are not identically zero. By the convenience assumption and the strong mixed weighted homogenuity, we can find some integers n, r such that dp = npq, dr = (n + 2r)pq and we can write F(z1, ¯ z1, z2, ¯ z2) as a linear combination of monomials zν1

1 zν2 2 ¯

zµ1

1 ¯

zµ2

2

where the summation satisfies the equality (ν1 + µ1)p + (ν2 + µ2)q = dr (ν1 − µ1)p + (ν2 − µ2)q = dp.

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In particular, we see that the coefficients of z(n+r)q

1

¯ zrq

1 and

z(n+r)p

2

¯ zrp

2 are non-zero and any other monomials satisfies

ν1, µ1 ≡ 0 mod q, ν2, µ2 ≡ 0 mod p. The monodromy mapping h : M → M is defined by h : M → M, z → exp(2πi/npq)◦

Pz = exp(2πi/nq)z1, exp(2πi/np)z2).

Thus there exists a strongly mixed homogeneous polynomial G(w, ¯ w), w = (w1, w2) of polar degree n and radial degree (n + 2r) such that F(z, ¯ z) = G(zq

1 , ¯

zq

1 , zp 2 , ¯

zp

2 ).

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Mixed hypersurface singularities

The curve F = 0 is invariant under the C∗-action given by (6). Let P1(P) be the weighted projective line which is the quotient space

• f C2 \ {0} by the action (6). It has two singular points A = [1, 0]

and B = [0, 1] (if p, q ≥ 2) and the complement U := P1(P) \ {A, B} is isomorphic to C∗ with coordinate z := zq

1 /zp 2 . Note that z is well defined on z2 ̸= 0. The zero locus

V (F) of F in P1(P) does not contain A, B and it is defined on U by the mixed polynomial f (z, ¯ z) = 0 where f is defined by the equality: f (z, ¯ z) := F(z, ¯ z)/(z(n+r)p

2

¯ zrp

2 )

= c zn+r ¯ zr + ∑

i,j

ai,jzi ¯ zj where the summation is taken for i ≤ n + r, j ≤ r and i + j < n + 2r and c ̸= 0 is the coefficient of z(n+r)q

1

¯ zrq

1 in F. Note

also that g(z) = f (z) where g(w) := G(w1, ¯ w1, w2, ¯ w2)/(wn+r

2

¯ wr

2),

w = w1/w2. Thus in these affine coordinates z, w, we have

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Mixed hypersurface singularities

This implies that f (z) = g(z), the number of points of V (f (z)) and V (g(w)) are equal in their respective projective spaces and f = g ∈ M(n + 2r; n + r, r). The associated C∗-action to G(w, ¯ w) is the canonical linear action and we simply denote it as ρ ◦ w instead of ρ ◦

1 w. Let M(G) be

the Milnor fiber of G and let P1 be the usual projective line. The monodromy mapping hG : M(G) → M(G) of G is given by hG(w) = exp(2πi/n) ◦ w.

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Mixed hypersurface singularities

Then we have a canonical diagram C2

φq,p

− → C2 ↑ ↑ M

φq,p

− → M(G)   π   π′ P1(P) \ V (f )

¯ φq,p

− → P1 \ V (g) π is a Z/dpZ-cyclic covering branched over {A, B},( dp = npq) while π′ is a Z/nZ-cyclic covering without any branch locus. φq,p is defined φq,p(z1, z2) = (zq

1 , zp 2 ) which satisfies

φq,p(ρ ◦

P z) = ρpq ◦ (zq 1 , zp 2 ), ρ ∈ S1 and thus φq,p ◦ h = hG ◦ φq,p

as we have φq,p(h(z)) = φ((exp(2πi/nq)z1, exp(2πi/np)z2) = (exp(2πi/n)zq

1 , exp(2πi/n)zp 2 ) = hG(φq,p(z)).

The mapping ¯ φq,p is canonically induced by φq,p and we observe that ¯ φq,p gives a bijection of ¯ φq,p : P1(P) \ {A, B} → P1 \ { ¯ A, ¯ B} and it induces an bijection between V (f ) and V (g). Here

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Mixed hypersurface singularities

Recall that by [13, 1], we have

Proposition 0.8

• 1. χ(M(G)) = n(2 − ρ(g)).
• 2. χ(M) = −npqρ(f ) + n(p + q).
• 3. The links KF := F −1(0) ∩ S3 and KG := G −1(0) ∩ S3 have

the same number of components and it is given by ρ(f ). proof: The assertion follows from a simple calculation of Euler

• characteristics. (1) is an immediate result that M(G) π′

− →P1 \ V (G) is an n-fold cyclic covering. (2) follows from the following. π : M ∩ C∗2 → P1(P) \ ({A, B} ∪ V (F)) is an npq-cyclic covering while M ∩ {z1 = 0} and M ∩ {z2 = 0} are np and nq points respectively. Thus χ(M) = χ(M ∩ C∗2) + χ(M ∩ {z1 = 0}) + χ(M ∩ {z2 = 0}) = npq(−ρ(f )) + np + nq

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The link components of KF and KG are S1 invariant and the assertion (3) follows from this observation. The correspondence F(z, ¯ z) → f (z) is reversible. Namely we have

Proposition 0.9

For a given f (z, ¯ z) ∈ M(n + m; n, m) and any weight vector P = t(p, q), we can define a strongly mixed weighted homogeneous polynomial of two variables z = (z1, z2) with weight P by F(z, ¯ z) := f (zq

1 /zp 2 , ¯

zq

1 /¯

zp

2 )zpn 2 ¯

zpm

2 .

The polar degree and the radial degree of F are (n − m)pq and (n + m)pq respectively. The coefficient of znq

1 ¯

zqm

1

in F is the same as that of zn¯

• zm. If f has a non-zero constant term, F is

convenient polynomial. The correspondence F(z, ¯ z) → f (z, ¯ z), f (z, ¯ z) → F(z, ¯ z) are inverse of the other.

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proof

: In fact, the monomial zi ¯ zj, i + j ≤ n + m, i ≤ n, j ≤ m changes into zqi

1 ¯

zjq

1 zp(n−i) 2

¯ zp(m−j)

2

. In particular, zn¯ zm → zqn

1 ¯

zqm

1 ,

1 → zpn

2 ¯

zpm

2 .

It is well-known that the Milnor fibration of a weighted homogeneous polynomial h(z) ∈ C[z1, . . . , zn] with an isolated singularity at the origin is described by the weight and the degree by Orlik-Milnor [7]. This assertion is not true for a mixed weighted homogeneous polynomials.

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Let ˜ M(n + m; n, m; P), ˜ Lhs(n + m; n, m; P), ˜ L(n + m; n, m; P) be the space of strongly mixed weighted homogeneous convenient polynomials of two variables with weight P = (p, q), gcd(p, q) = 1 and with isolated singularity at the origin which corresponds to M(n + m; n, m), Lhs(n + m; n, m), L(n + m; n, m) respectively through Proposition 0.8 and Proposition 0.9. For P = (1, 1), we simply write as ˜ M(n + m; n, m), ˜ Lhs(n + m; n, m), ˜ L(n + m; n, m)

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Proposition 0.10

The moduli spaces ˜ M(n + m; n, m; P), ˜ Lhs(n + m; n, m; P), ˜ L(n + m; n, m; P) are isomorphic to the moduli spaces M(n + m; n, m), Lhs(n + m; n, m), L(n + m; n, m) respectively. As the above moduli spaces do not depend on the weight P (up to isomorphism), we only consider hereafter strongly mixed homogeneous polynomials. Assume that two polynomials F1, F2 are in a same connected component. Then their Milnor fibrations are equivalent. Thus

Corollary 14

Assume that F1, F2 ∈ ˜ M(n + m; n, m) have different number of link components ρ(f1), ρ(f2). Then they belongs to different connected components of ˜ M(n + m; n, m). In particular, the number of the connected components of ˜ M(n + m; n, m) is not smaller than the cardinality of the set {ρ(f ) | f ∈ M(n + m; n, m)}.

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Remark

: For a fixed number ρ of link components, we do not know if the subspace of the moduli space with link number ρ is connected or not.

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Example 15

Consider a strongly mixed homogeneous polynomial F of polar degree 1 and radial degree 3. Namely f ∈ M(3; 2, 1). Its possible link components are 1,3,5. 1 and 3 are given in Example 59, [13]. An example of 5 components are given by Bleher-Homma-Ji-Roeder ([2]). For example, we can take f (z) = ¯ z(z2 − 1/2) − z + 1/30 F(z, ¯ z) = ¯ z1(z2

1 − z2 2/2) − (z1z2 − z2 2/30)¯

z2

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Open problems

For the connectivity of the Milnor fiber of a general mixed polynomial with isolated singularity, nothing is known.

Conjecture 0.11

Assume that f (z, ¯ z) is a convenient strongly non-degenerate mixed

• function. Then
• 1. Milnor fiber M has a homotopy type of n − 1 dimensional

CW-complex and

• 2. M is n − 2 connected.
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A weaker conjecture is

Conjecture 0.12

Under the same assumption and n ≥ 3, Milnor fiber is 1-connected. We can prove the following very modest assertion.

Proposition 0.13 ([18])

Under the same assumption, the Milnor fiber is connected.

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• V. Blanloeil, M. Oka.

Topology of strongly polar weighted homogeneous links. SUT J.Math. 51 ,no.1(2015), 119–128.

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harmonic rational functions and its application to gravitational lensing, Int. Math. Res. Not. IMRN, 8, 2245–2264.

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• Proc. Amer. Math. Soc., 131(2):409–414, 2003.
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• J. Milnor.

Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton, N.J., 1968.

• J. Milnor and P. Orlik.

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On the homotopy types of hypersurfaces defined by weighted homogeneous polynomials. Topology, 12:19–32, 1973.

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Topology of polar weighted homogeneous hypersurfaces. Kodai Math. J., 31(2):163–182, 2008.

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On mixed Brieskorn variety. Comtemporary Math., 538,389-399,20, 2011

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