Arc spaces and monodromy - Lama

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Arc spaces and monodromy
Michel Raibaut
November 23, 2011
Abstract
In this talk through the proof of a nice theorem, we present technics of computation of motivic measures.
Let X be a smooth complex algebraic variety and f : X → C be a regular non constant function.
1 Classical objects
1.1 Monodromy
Let n ≥ 2,
C z
f : ↓ ↓
C z n
• 0 is a singular point.
• 0 is a critical value.
↓ f
•
•
↓ f
↓ f
Proposition 1.1.
f : C ∗ → C ∗
z ↦→ z n is a covering of degree n.
Proposition 1.2. We have the following phenomena
• Vanishing cycles : n−1 cycles disappear .
• Monodromy : π 1 (C ∗ ) f −1 (1).
1.2 Milnor fiber of a function at a point
Theorem 1.3 (Milnor (1968)[12], Lé (1976)[11]). Let X be a smooth algebraic complex variety
and
f : X → C a regular non constant function and x 0 ∈ f −1 (0).
There exists 1 >> ε >> η > 0 such that
x 0
f : B(x 0 ,ε)∩f −1 (D(0,η)\{0}) → D(0,η)\{0}
• •
is a locally trivial topological fibration. 0
↓ f
1

•
•
Definition 1.4. Following this theorem there is two definitions :
1. The Milnor fiber of f at x 0 denoted by F x0 is the fiber of this fibration.
2. The monodromy of f at x 0 is the action π 1 (D(0,η)\{0}) F x0 .
The monodromy induces a quasi-unipotent operator
T x0 : H ∗ (F x0 ,Q) → H ∗ (F x0 ,Q).
The eigenvalues of this automorphism are roots of unity (monodromy theorem).
1.3 Lefschetz numbers of the monodromy
Definition 1.5. For any natural number n, we consider the Lefschetz numbers
Λ(T n x 0
) := ∑ q≥0(−1) q Trace(T n x 0
,H q (F x0 ,Q))
of the n-th iterate of the monodromy T x0 .
Proposition 1.6 (A’Campo (73) [1], A’Campo-Deligne (75)) [2]). Let f : X → C be a regular non constant
morphism on a smooth algebraic variety. We have the following equalities : Λ(T m x 0
) = 0 for 0 < m < µ with µ
the multiplicity of f at x 0 .
2 Lefschetz numbers and jet-spaces
Denef-Loeser in [6] consider for all number n ≥ 1
X n,1 (f) := {ϕ ∈ L n (X) | ϕ(0) = x 0 andf(ϕ(t)) = 1.t n modt n+1 }.
Theorem 2.1 (Denef-Loeser (2002) [6], Hrushovski-Loeser (preprint 11/2011) [9]).
For every integer n ≥ 1 : Λ(T n x 0
) = χ c (X n,1 (f)).
Remarks 2.2. Note that :
1. The theorem of A’Campo and Deligne is a corollary of this theorem.
0
x 0
↓ f
2. The proof of Denef-Loeser, we are going to present is a comparison of the computation of each part of
this equality on a log-resolution.
3. Hrushovski-Loeser gave a proof without the use of a log-resolution !
4. Let X n = {ϕ ∈ L n (X) | f(ϕ) = acf(ϕ)t n +...andϕ(0) = x 0 }. ThetorusC ∗ actson X n byλ.ϕ(t) = ϕ(λt).
The function angular component acf is a topological locally trivial fibration and its fiber is X n,1 .
Proof. Strategy
1. Motivic integration
2. Main idea : stratification of arcs !
where
˜ X n,1 := {ϕ ∈ L(X) | f(ϕ(t)) = 1.t n +...}
µ( ˜ X n,1 ) = [X n,1(f)]
L nd
Y L(Y)\L(E) ⊃ Y n,1 = ⊔ I ⊔ (ki) Y (ki)
h ↓ ↓ h ∞ ↓ ↓
X L(X)\L(f −1 (0)) ⊃ ˜X n,1 = ⊔ I ⊔ (ki) X (ki)
2
1.t n

• h is a log-resolution of (X,f −1 (0)) adapted to x 0 .
• in the following we will forget L(E) and L(f −1 (0)) which have measure 0.
• h ∞ is a bijection by properness criterium
3. Computation of µ(Y k ) : Find a Zariski locally trivial fibration !
4. By the change variables formula we compute µ( ˜X k ).
5. By summation we will obtain µ( ˜ X n,1 ) and then [X n,1 (f)].
6. A comparison with the A’Campo formula [2] gives Eu c (X n,1 ) = Λ(T n x 0
).
We can start the proof :
1. Motivic measure (cf talks of G.Fichou):
a) We call Grothendieck group denoted by K 0 (Var C ) the group
• generated by the isomorphisms classes [X], for all X algebraic variety,
• with the relations [X] = [F]+[X \[F] for all F closed subset of X.
This group has a unique structure of ring defined by [X]×[Y] := [X ×Y] for all varieties.
We denote by L the class [A 1 ] and by M the localization K 0 (Var C )[L −1 ].
b) The main relation : Fiber product.
If π : E → B is a Zariski locally trivial fibration of fiber F then [E] = [B]×[F].
c) Let X be a smooth complex algebraic variety of pure dimension d. A subset A ⊂ L(X) is called
cylindrical or constructible if and only if there exists an integer n, and a constructible subset of
L n (X) such that A = πn −1(C).
Proposition 2.3. The family of cylindrical objects of L(X) is a Bool Algebra.
By [5][Lemma 4.1] (cf talks of A.Reguera), if A ⊂ L(X) is cylindrical at level n, then the truncation map
π m+1 (A) → π m (A) is a Zariski locally trivial fibration and
[π m+1 (A)]
[L md ]
= [π n(A)]
[L nd ]
=: µ(A).
This function µ is a measure and µ(A) is called motivic measure of A.
In our context, ˜Xn,1 is a cylindrical subset and µ( ˜X n,1 ) = [π( ˜X n,1)]
L nd .
2. We consider (Y,E,h) a log-resolution of (X,f −1 (0)) adapted to x 0 namely
(a) h : Y → X is proper morphism with Y smooth,
(b) E = h −1 (f −1 (0)) = ∪ i∈A E i is a normal crossing divisor with irreducible components E i smooth,
(c) h : Y \E → X \f −1 (0) is an isomorphism,
(d) h −1 (x 0 ) = ∪ i∈A(x0)E i .
Notations 2.4. We stratify the normal crossing divisor E = ⊔≠I⊂A E 0 I where E0 I = ∩ i∈IE i \ ∪ j∈J E j .
We denote by N i and ν i , the multiplicity of f ◦h and Jach against E i so : divJach = ∑ i∈A (ν i −1)E i
and div(f ◦h) = ∑ i∈A N i(f)E i .
Let I be a subset of A(x 0 ) non empty, let (k i ) ∈ N ∗I simply denoted by k. We define
⎧
⎫
⎨ ord t ψ ∗ E i = k i , i ∈ I, ⎬
Y k :=
⎩ ψ ∈ L(Y) ord t ψ ∗ E i = k i , i /∈ I,
f ◦h(ψ) = 1.t n ⎭
+....
In particular, the order of each arc in Y k against f ◦h and Jac(h) is constant :
∀ψ ∈ Y k , ord t f ◦h = ∑ i∈I
k i N i (f)&ord t Jac(h) = ∑ i∈I(ν i −1)k i
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3. Computation of µ(Y k )
(a) Y k is a cylindrical subset of L(Y), so µ(Y k ) = [π n (Y k )]/L nd .
(b) We haveto compute [π n (Y k )] and for this we will find a Zariskilocally trivial fibration. We only have
local data, so : Let U be an affine subset of Y such that f ◦h |E 0
I ∩U is u× ∏ ı∈I zNi i (f), where for all
i in I, z i is the equation of E i ∩U and u is a unit. The divisor E is a normal crossing divisor and Y
is smooth so there exists (z i ) i/∈I other coordinates such that (z i ) i∈{1,...,d} is a system of coordinates
on U and induces an étale map z : U → A d .
(c) This étale map induces an isomorphism
≃
L n (U) → U × A d L n (A d )
.
ϕ ↦→ (ϕ(0),z(ϕ))
In this way, on the right side we have linearized the problem :
⎛
y ∈ U ∩EI 0 , (a jt kj +∗t kj+1 +...+∗t n ) j∈I
(b j +∗t...+∗t n ) j/∈I
π n (Y k,ϕ(0)∈U∩E 0
I
) ≃
⎜ such that a j ≠ 0,j ∈ I
⎝ z(y) = (0,b)
u(y) ∏ i∈I aNi i = 1
⎞
⎟
⎠
↓ (a,y)
{(a,y) ∈ C ∗I ×E 0 I ∩U | u(y)∏ a Ni
i = 1}
≃
C ∗|I|−1 ×({(a,y) ∈ C ∗ ×E 0 I ∩U | u(y)agcd(Ni)i∈I = 1} =: Ẽ◦ I ∩U)
where the last isomorphism is simply given by a change of coordinates in the torus. We can check
easily that (a,y) is a Zariski trivial fibration of fiber A nd−∑ i∈I ki .
We can glue this construction and we obtain :
[π n (Y k )] = [Ẽ0 I ](L−1)|I|−1 L nd−∑ i∈I ki .
4. Let X˜
k be equal h(Y k ), by the change variable formula of Kontsevich [10] and Denef-Loeser [5]
∫ ∫
µ(Y k ) = 1dµ X = L −ordt Jach d µY = µ(Y k )L −∑ i∈I (νi−1)ki .
X˜
k Y k
Themainpointistoprovethath : π n (Y k ) → π n (X k )isaZariskilocallytrivialfibrationoffiberA ordt Jach .
We obtain
[π n ( X ˜ k )] = [Ẽ0 I ](L−1)|I|−1 L nd−∑ i∈I kiνi .
5. By additivity
[X n,1 ] =
∑
∅̸=I⊂A(x 0)
[Ẽ0 I](L−1) |I|−1 ∑
∑
(k i ) ∈ N ∗
i k iN i (f) = n
L −∑ i∈I kiνi .
6. If we apply the Euler characteristic with compact support by noting that Eu c (L−1) = 0 then we obtain
∑
Eu c (X n,1 ) = N i (f)Eu c (Ei 0 ) = Λ(Tn x 0
),
i ∈ A(x 0 )
N i (f) | n
because in that case Ẽ0 i is a covering of degree N i(f) of Ei 0 . A’Campo in [2] proved the last equality.
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3 Motivic Milnor fiber at a point
Denef-Loeser [4] introduced the motivic zeta function of f at x 0
Z f,x0 (T) = ∑ n
µ( ˜ X n,1 )T n ∈ M[[T]].
By the previous part and summation of geometrical series, this series is rational !
We denote by
Z f,x0 (T) =
∑
S f,x0 := − lim
T→∞ Z f,x 0
(T) =
∅̸=I⊂A(x 0)[Ẽ0 I](L−1) |I|−1∏ i∈I
∑
∅̸=I⊂A(x 0)
This motive is called the Motivic Milnor fiber of f at x 0 [4].
Remarks 3.1. Some precisions :
L −νi T Ni(f)
1−L −νi T Ni(f).
(−1) |I|−1 [Ẽ0 I ](L−1)|I|−1 ∈ M.
1. Iff hascombinatorial data wecanusethem in orderto classifyarcsinstead ofthe Hironaka’sresolution
of singularities. Non-degenerate polynomial map is treated by Guibert [8] and the quasi-ordinary case is
treated by González-Pérez-González-Villa [7].
2. There is also a motivic Milnor fiber in R verifying an A’Campo formula (Comte-Fichou [3]).
3. More generally we can work in the relative equivariant setting with varieties of this type
Z ˆµ → f −1 (0) where the fibers of that structural morphism are invariant under the action.
As an example µ n acts on X n,1 , by λ.ϕ(t) = ϕ(λt) and ϕ ↦→ ϕ(0) induces a variety Xn
µn → f −1 (0). We
denote by Mˆµ f −1 (0)
the correspondant Grothendieck ring. In the same way Denef-Loeser define a global
motivic zeta function which is also rational
Z f (T) := ∑ n≥1
[X n (f)]L −nd T n = ∑
∅̸=I≠A[Ẽ0 I ](L−1)|I|−1∏ i∈I
L −νi T Ni(f)
1−L −νi T Ni(f).
The limit S f in Mˆµ f −1 (0) is the analogous of the nearby cycle sheaf of Deligne ψ f(Q X ). For this reason
it’s called motivic nearby cycles. The construction is compatible to the restriction : S f,x0 is the fiber
of S f in x 0 like the cohomology groups of the Milnor fiber F x0 are isomorphic to the cohomology group
of ψ f (Q X ) x0 .
4. The log-canonical threshold lct(X,(f)) = minν i /N i which is the smallest pole of Z f .
5. Monodromy conjecture : If L r is a pole of Z f then there exist x ∈ f −1 (0) such that e 2iπr is an
eigenvalue of the monodromy of a Milnor fiber of f at x.
References
[1] Norbert A’Campo. Le nombre de lefschetz d’une monodromie. Indag.Math., 35:113–118, 1973.
[2] Norbert A’Campo. La fonction zêta d’une monodromie. Comment. Math. Helv., 50:233–248, 1975.
[3] G. Comte and G. Fichou. Grothendieck ring of semialgebraic formulas and motivic real milnor fibres.
arXiv:1111.3181, 2011.
[4] Jan Denef and François Loeser. Motivic Igusa zeta functions. J. Algebraic Geom., 7(3):505–537, 1998.
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[5] Jan Denef and François Loeser. Germs of arcs on singular algebraic varieties and motivic integration.
Invent. Math., 135(1):201–232, 1999.
[6] Jan Denef and François Loeser. Lefschetz numbers of iterates of the monodromy and truncated arcs.
Topology, 41(5):1031–1040, 2002.
[7] PedroDGonzález Pérezand Manuel González Villa. Motivic milnor fiber ofaquasi-ordinaryhypersurface.
1105.2480.
[8] Gil Guibert. Espaces d’arcs et invariants d’Alexander. Comment. Math. Helv., 77(4):783–820, 2002.
[9] E. Hrushovski and F. Loeser. Monodromy and the lefschetz fixed point formula. arXiv:1111.1954, 2011.
[10] Kontsevich. Lecture at orsay. Décembre 7, 1995.
[11] Dung Trang Le. Some remarks on relative monodromy. In Real and complex singularities (Proc. Ninth
Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), pages397–403.Sijthoff and Noordhoff, Alphen
aan den Rijn, 1977.
[12] John Milnor. Singular points of complex hypersurfaces. Ann. of math. Studies, Princeton Univ.Press,
Princeton, 61, 1968.
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