Splittings of singular fibers into Lefschetz fibers

Transcription

1 OP Splittings of singular fibers into Lefschetz fibers Takayuki OKUDA the University of Tokyo Tohoku University January 8, 2017

2 Introduction 1 Splitting of singular fibers 2 Topological monodromy 3 Results 4 Applications (Dehn-twist expression)

3 Introduction 1 Splitting of singular fibers 2 Topological monodromy 3 Results 4 Applications (Dehn-twist expression)

4 Introduction 1 Splitting of singular fibers 2 Topological monodromy 3 Results Applications m 1 (Dehn-twist expression) m } n 1

5 Introduction 1 Splitting of singular fibers 2 Topological monodromy 3 Results 4 Applications (Dehn-twist expression)

6 Interests Splitting of singular fibers

7 Degeneration of Riemann surfaces M : a smooth complex surface : the unit disk in C π : M : a proper surjective holomorphic map s.t. X s := π 1 (s) (s 0) are smooth curves of genus g. X 0 := π 1 (0) is a singular fiber. ( 0 is a unique critical value.) π : M is called a degeneration (or, degenerating family) of Riemann surfaces of genus g. Regard X 0 as the divisor defined by π: X 0 = m i Θ i, where Θ i is an irreducible component with multiplicity m i. 4 / 23

8 Degeneration of Riemann surfaces M : a smooth complex surface : the unit disk in C π : M : a proper surjective holomorphic map s.t. X s := π 1 (s) (s 0) are smooth curves of genus g. X 0 := π 1 (0) is a singular fiber. ( 0 is a unique critical value.) π : M is called a degeneration (or, degenerating family) of Riemann surfaces of genus g. Regard X 0 as the divisor defined by π: X 0 = m i Θ i, where Θ i is an irreducible component with multiplicity m i. 4 / 23

9 Degeneration of Riemann surfaces M : a smooth complex surface : the unit disk in C π : M : a proper surjective holomorphic map s.t. X s := π 1 (s) (s 0) are smooth curves of genus g. X 0 := π 1 (0) is a singular fiber. ( 0 is a unique critical value.) 1 3 π : M is called a degeneration (or, degenerating family) of Riemann surfaces of genus g. 2 Regard X 0 as the divisor defined by π: X 0 = m i Θ i, where Θ i is an irreducible component with multiplicity m i. 4 / 23

10 Degeneration of Riemann surfaces M : a smooth complex surface : the unit disk in C π : M : a proper surjective holomorphic map s.t. X s := π 1 (s) (s 0) are smooth curves of genus g. X 0 := π 1 (0) is a singular fiber. ( 0 is a unique critical value.) π : M is called a degeneration (or, degenerating family) of Riemann surfaces of genus g. Regard X 0 as the divisor defined by π: X 0 = m i Θ i, where Θ i is an irreducible component with multiplicity m i. 4 / 23

11 Degeneration of Riemann surfaces M : a smooth complex surface : the unit disk in C π : M : a proper surjective holomorphic map s.t. X s := π 1 (s) (s 0) are smooth curves of genus g. X 0 := π 1 (0) is a singular fiber. ( 0 is a unique critical value.) π : M is called a degeneration (or, degenerating family) of Riemann surfaces of genus g. Regard X 0 as the divisor defined by π: X 0 = m i Θ i, where Θ i is an irreducible component with multiplicity m i. 4 / 23

12 Degeneration of Riemann surfaces M : a smooth complex surface : the unit disk in C π : M : a proper surjective holomorphic map s.t. X s := π 1 (s) (s 0) are smooth curves of genus g. X 0 := π 1 (0) is a singular fiber. ( 0 is a unique critical value.) π : M is called a degeneration (or, degenerating family) of Riemann surfaces of genus g. Regard X 0 as the divisor defined by π: X 0 = m i Θ i, where Θ i is an irreducible component with multiplicity m i. 4 / 23

13 Splitting of singular fibers 5 / 23 π : M : a degeneration w/ singular fiber X 0 {π t : M t } : a family of deformations of π : M i.e. π 0 : M 0 coincides with π : M. If π t (t 0) has k singular fibers X s1,..., X sk, k 2, = We say that X 0 splits into X s1,..., X sk.

14 Splitting of singular fibers 5 / 23 π : M : a degeneration w/ singular fiber X 0 {π t : M t } : a family of deformations of π : M i.e. π 0 : M 0 coincides with π : M. If π t (t 0) has k singular fibers X s1,..., X sk, k 2, = We say that X 0 splits into X s1,..., X sk.

15 Splittability of singular fibers Fact (1) A Lefschetz fiber and (2) a multiple smooth fiber admit no splittings (i.e. any deformation is equisingular). Such fibers are said to be atomic. Conjecture (from topological aspect) Every singular fiber can split into singular fibers each of which is (1) or (2), in finite steps of deformations. How to construct splitting families Double covering method Moishezon (genus 1 case), Horikawa (genus 2 case), Arakawa-Ashikaga (hyperelliptic case) Barking deformation Takamura (some criterion) 6 / 23

16 Splittability of singular fibers Fact (1) A Lefschetz fiber and (2) a multiple smooth fiber admit no splittings (i.e. any deformation is equisingular). Such fibers are said to be atomic. Conjecture (from topological aspect) Every singular fiber can split into singular fibers each of which is (1) or (2), in finite steps of deformations. How to construct splitting families Double covering method Moishezon (genus 1 case), Horikawa (genus 2 case), Arakawa-Ashikaga (hyperelliptic case) Barking deformation Takamura (some criterion) 6 / 23

17 Splittability of singular fibers Fact (1) A Lefschetz fiber and (2) a multiple smooth fiber admit no splittings (i.e. any deformation is equisingular). Such fibers are said to be atomic. Conjecture (from topological aspect) Every singular fiber can split into singular fibers each of which is (1) or (2), in finite steps of deformations. How to construct splitting families Double covering method Moishezon (genus 1 case), Horikawa (genus 2 case), Arakawa-Ashikaga (hyperelliptic case) Barking deformation Takamura (some criterion) 6 / 23

18 Splittability of singular fibers Fact (1) A Lefschetz fiber and (2) a multiple smooth fiber admit no splittings (i.e. any deformation is equisingular). Such fibers are said to be atomic. Conjecture (from topological aspect) Every singular fiber can split into singular fibers each of which is (1) or (2), in finite steps of deformations. How to construct splitting families Double covering method Moishezon (genus 1 case), Horikawa (genus 2 case), Arakawa-Ashikaga (hyperelliptic case) Barking deformation Takamura (some criterion) 6 / 23

19 From topological aspects Topological monodromy

20 Monodromy of degenerations 8 / 23 π : M : a degeneration of Riemann surfaces of genus g with singular fiber X 0 A loop C in turning around 0 once in positive direction induces a self-homeomorphism φ of a general fiber X s. (monodromy homeomorphism) Identifying X s with Σ g, its isotopy class [φ] MCG(Σ g ) is called the topological monodromy. Theorem (Imayoshi, Shiga-Tanigawa, Earle-Sipe) Every topological monodromy is pseudo-periodic of negative twist.

21 Monodromy of degenerations 8 / 23 π : M : a degeneration of Riemann surfaces of genus g with singular fiber X 0 A loop C in turning around 0 once in positive direction induces a self-homeomorphism φ of a general fiber X s. (monodromy homeomorphism) Identifying X s with Σ g, its isotopy class [φ] MCG(Σ g ) is called the topological monodromy. Theorem (Imayoshi, Shiga-Tanigawa, Earle-Sipe) Every topological monodromy is pseudo-periodic of negative twist.

22 Monodromy of degenerations 8 / 23 π : M : a degeneration of Riemann surfaces of genus g with singular fiber X 0 A loop C in turning around 0 once in positive direction induces a self-homeomorphism φ of a general fiber X s. (monodromy homeomorphism) Identifying X s with Σ g, its isotopy class [φ] MCG(Σ g ) is called the topological monodromy. Theorem (Imayoshi, Shiga-Tanigawa, Earle-Sipe) Every topological monodromy is pseudo-periodic of negative twist.

23 Monodromy of degenerations 8 / 23 π : M : a degeneration of Riemann surfaces of genus g with singular fiber X 0 A loop C in turning around 0 once in positive direction induces a self-homeomorphism φ of a general fiber X s. (monodromy homeomorphism) Identifying X s with Σ g, its isotopy class [φ] MCG(Σ g ) is called the topological monodromy. Theorem (Imayoshi, Shiga-Tanigawa, Earle-Sipe) Every topological monodromy is pseudo-periodic of negative twist.

24 Monodromy of degenerations 8 / 23 π : M : a degeneration of Riemann surfaces of genus g with singular fiber X 0 A loop C in turning around 0 once in positive direction induces a self-homeomorphism φ of a general fiber X s. (monodromy homeomorphism) Identifying X s with Σ g, its isotopy class [φ] MCG(Σ g ) is called the topological monodromy. Theorem (Imayoshi, Shiga-Tanigawa, Earle-Sipe) Every topological monodromy is pseudo-periodic of negative twist.

25 Matsumoto-Montesinos theorem Theorem (Matsumoto-Montesinos, 91/92) top. equiv. classes of conj. classes in MCG g of 1:1 minimal degenerations of pseudo-periodic mapp. classes Riemann surfs. of genus g of negative twist via topological monodromy, for g 2. plefschetz fiber Right-handed Dehn twist 9 / 23

26 Matsumoto-Montesinos theorem Theorem (Matsumoto-Montesinos, 91/92) top. equiv. classes of conj. classes in MCG g of 1:1 minimal degenerations of pseudo-periodic mapp. classes Riemann surfs. of genus g of negative twist via topological monodromy, for g 2. pmultiple smooth fiber Periodic mapping class w/o multiple points 9 / 23

27 Matsumoto-Montesinos theorem Theorem (Matsumoto-Montesinos, 91/92) top. equiv. classes of conj. classes in MCG g of 1:1 minimal degenerations of pseudo-periodic mapp. classes Riemann surfs. of genus g of negative twist via topological monodromy, for g 2. pstellar fiber Periodic mapping class 9 / 23

28 Matsumoto-Montesinos theorem Theorem (Matsumoto-Montesinos, 91/92) top. equiv. classes of conj. classes in MCG g of 1:1 minimal degenerations of pseudo-periodic mapp. classes Riemann surfs. of genus g of negative twist via topological monodromy, for g 2. psingular fiber Pseudo-periodic mapping class of negative twist 9 / 23

29 Degeneration of propeller surfaces 10 / 23 A propeller surface is a Riemann surface Σ g of genus g 2 equipped with Z g -action s.t. Σ g /Z g has genus 1. ω g : a propeller automorphism X g : the singular fiber with monodromy ω g M X g ω g π 0

30 Results Splittability into Lefschetz fibers

31 Case g = 2 12 / 23 Theorem (Y. Matsumoto) π : M : the degeneration of Riemann surfaces of genus 2 with monodromy ω 2 Then its singular fiber X can split into four Lefschetz fibers. Moreover, their vanishing cycles are as depicted below. ω ω 2 X 2

32 Case g = 2 12 / 23 Theorem (Y. Matsumoto) π : M : the degeneration of Riemann surfaces of genus 2 with monodromy ω 2 Then its singular fiber X can split into four Lefschetz fibers. Moreover, their vanishing cycles are as depicted below. ω ω 2 ω 2 = τ 0 τ a τ b τ c

33 Psudo-propeller maps ω (2) 3 X (2) n = 2 γ m = 3 m = 3 1 } n = 2 γ : a separating simple loop on Σ g s.t. Σ g \ γ = Σ m,1 Σn,1 (g = m + n, m 1, n 0) : a psudo-periodic map satisfying ω (n) m X (n) m ω (n) m Σm,1 a periodic map with a fixed pt of rot angle 2π m. ω (n) m Σn,1 id. NOTE: (ω (n) m )m = τ γ : the singular fiber with monodromy ω(n) m 13 / 23

34 ω (2) 3 2π 3 id n = 2 γ Psudo-propeller maps X (2) m = 3 m = 3 1 } n = 2 γ : a separating simple loop on Σ g s.t. Σ g \ γ = Σ m,1 Σn,1 (g = m + n, m 1, n 0) : a psudo-periodic map satisfying ω (n) m X (n) m ω (n) m Σm,1 a periodic map with a fixed pt of rot angle 2π m. ω (n) m Σn,1 id. NOTE: (ω (n) m )m = τ γ : the singular fiber with monodromy ω(n) m 13 / 23

35 Psudo-propeller maps 13 / 23 X (1) X (0) ω (1) 4 ω (0) 5 = ω 5 γ

36 Psudo-propeller maps 13 / 23 X (3) X (4) 1 1 ω (3) 2 γ ω (4) 1 = τ γ

37 Results Theorem (O-Takamura) 1 For any m 2, n 0, the singular fiber X (n) m can split into X(n+1) m 1 and three Lefschetz fibers. 2 For any g 2, we have the following sequence: X (0) g X (1) g 1 X (g 2) 2 X (g 1) 1, where A B means A splits into B and 3 Lefschetz fibers. X (n) m m 2 1 m 1 1 } n X (n+1) m 1 m 1 1 m 2 1 } n / 23

38 Results Theorem (O-Takamura) 1 For any m 2, n 0, the singular fiber X (n) m can split into X(n+1) m 1 and three Lefschetz fibers. 2 For any g 2, we have the following sequence: X (0) g X (1) g 1 X (g 2) 2 X (g 1) 1, where A B means A splits into B and 3 Lefschetz fibers. X (n) m m 2 1 m 1 1 } n X (n+1) m 1 m 1 1 m 2 1 } n / 23

39 Results 15 / 23 X (n) m X(n+1) m 1 g = 2 2 g = 3 g = 4

40 Results 15 / 23 X (n) m X(n+1) m 1 g = 2 2 g = g = 4

41 Results 15 / 23 X (n) m X(n+1) m 1 g = 2 2 g = g = 4

42 Results 15 / 23 X (n) m X(n+1) m 1 g = 2 2 g = g = 4 4 3

43 Results 15 / 23 X (n) m X(n+1) m 1 g = 2 2 g = g =

44 Results 15 / 23 X (n) m X(n+1) m 1 g = 2 2 g = g =

45 Sketch proof of Theorem Θ 0 : the core of X (n) m (the component with multiplicity m) p 1, p 2 : the intersection points of Θ 0 with other components. N 0 : a tubular nbhd of Θ 0 in M (embedded in the normal bundle on Θ 0 in M) 1 Realize the restriction of π to N 0 as a holomorphic function π(z, ζ) = ζ m σ(z), (z, ζ) N 0 where σ is a holomorphic section of N ( m) 0 such 1 that div(σ) = (m 1)p 1 + p 2. 2 Denoting the fiber of N 0 over p i by F pi X, 0 then X (n) m N 0 (= div(π)) = mθ 0 + (m 1)F p1 + F p2, m 1 N 0 m 1 1 m Θ 0 p 1 p 2 Θ+ Θ m 1 16 / 23

46 Sketch proof of Theorem 16 / 23 Θ 0 : the core of X (n) m (the component with multiplicity m) p 1, p 2 : the intersection points of Θ 0 with other components. N 0 : a tubular nbhd of Θ 0 in M (embedded in the normal bundle on Θ 0 in M) 1 Realize the restriction of π to N 0 as a holomorphic function π(z, ζ) = ζ m σ(z), (z, ζ) N 0 where σ is a holomorphic section of N ( m) 0 such that div(σ) = (m 1)p 1 + p 2. Denoting the fiber of N 0 over p i by F pi, then X (n) m N 0 (= div(π)) = mθ 0 + (m 1)F p1 + F p2, N 0 m 1 1 m Θ 0 p 1 p 2 Fp1 Fp2

47 Sketch proof of Theorem 2 Find a meromorphic section ρ of N ( 1) 0 such that div(ρ) = p 1 + p 2 q (q p i ), so that a holomorphic function ( π t on N 0 ) can be defined by π t (z, ζ) = ζ m t σ(z) 1 ζρ(z) (ρ(z)ζ t) ( ) near = ζ m 1 σ(z) ζ = ζ m 1 σ(z) ρ(z) t ρ(z) z = p i near z = q. Then π 1 t (0) = (m 1)Θ 0 + (m 2)F p1 + Ξ, where Ξ is an irreducible component intersecting Θ 0 at q. N 0 N 0 m m Θ 0 p 1 p 2 Fp1 Fp2 Θ 0 1 Ξ p 1 q p 2 Fp1 16 / 23

48 Sketch proof of Theorem 2 Find a meromorphic section ρ of N ( 1) 0 such that div(ρ) = p 1 + p 2 q (q p i ), so that a holomorphic function ( π t on N 0 ) can be defined by π t (z, ζ) = ζ m t σ(z) 1 ζρ(z) (ρ(z)ζ t) ( ) near = ζ m 1 σ(z) ζ = ζ m 1 σ(z) ρ(z) t ρ(z) z = p i near z = q. Then π 1 t (0) = (m 1)Θ 0 + (m 2)F p1 + Ξ, where Ξ is an irreducible component intersecting Θ 0 at q. t = 0 N 0 N 0 m 1 1 m 2 m Θ 0 p 1 p 2 Fp1 Fp2 Θ 0 p 1 q Ξ 1 m 1 t 0 Fp1 16 / 23

49 Sketch proof of Theorem 3 Propagate the deformation π t on N 0 to obtain a deformation π t : M t with X (n+1) m 1 (by deforming tubular nbhds of some irred components). 2 1 Y 1 1 m 1 m 2 1 Ξ m 1 bark m 1 1 t = 0 N 0 N 0 m 1 1 m 2 m Θ 0 p 1 p 2 Fp1 Fp2 Θ 0 p 1 q Ξ 1 m 1 t 0 Fp1 16 / 23

50 Sketch proof of Theorem 16 / 23 4 By taking a good section ρ, we can assume that π t N0 has Lefschetz singularities, and they lie distinct fibers. m 1 2 m 1 1 m 1 1 m 2 1

51 Remark 17 / 23 Remark of Theorem 1 Generalization of Matsumoto s splitting for genus 2. 2 Analogous to adjacency diagrams of singularities: A 6 A 5 A 4 A 3 A 2 A 1 D 6 D 5 D 4 E 6 3 Splitting of a singular fiber into Lefschetz fibers gives a Dehn-twist expression of its topological monodromy.

52 Application Dehn-twist expression of propeller automorphisms

53 Dehn-twist expressions Theorem (Dehn) The mapping class group of a closed Riemann surface Σ MCG(Σ) = Homeo + (Σ)/isotopy is generated by finitely many Dehn twists. Find Dehn-twist expressions of automorphisms φ = τ 1 τ 2 τ k (up to isotopy). Ex. Hyperelliptic involution (J. Birman and H. Hilden) ι = τ 1 τ 2 τ 2g τ 2g+1 τ 2g+1 τ 2g τ 2 τ 1 19 / 23

54 Dehn-twist expressions Theorem (Dehn) The mapping class group of a closed Riemann surface Σ MCG(Σ) = Homeo + (Σ)/isotopy is generated by finitely many Dehn twists. Find Dehn-twist expressions of automorphisms φ = τ 1 τ 2 τ k (up to isotopy). Ex. Hyperelliptic involution (J. Birman and H. Hilden) ι = τ 1 τ 2 τ 2g τ 2g+1 τ 2g+1 τ 2g τ 2 τ 1 19 / 23

55 Dehn-twist expressions 19 / 23 Find Dehn-twist expressions of automorphisms φ = τ 1 τ 2 τ k (up to isotopy).

56 Dehn-twist expressions Find Dehn-twist expressions of automorphisms φ = τ 1 τ 2 τ k (up to isotopy). monodromy monodromy split Geometric expressions 19 / 23

57 Dehn-twist expressions Find Dehn-twist expressions of automorphisms φ = τ 1 τ 2 τ k (up to isotopy). monodromy monodromy split Geometric expressions 19 / 23

58 Dehn-twist expressions Find Dehn-twist expressions of automorphisms φ = τ 1 τ 2 τ k (up to isotopy). monodromy monodromy split Geometric expressions 19 / 23

59 Case g = 2 20 / 23 Theorem (Y. Matsumoto) π : M : the degeneration of Riemann surfaces of genus 2 with monodromy ω 2 Then its singular fiber X can split into four Lefschetz fibers. Moreover, their vanishing cycles are as depicted below. ω ω 2 X 2

60 Case g = 2 20 / 23 Theorem (Y. Matsumoto) π : M : the degeneration of Riemann surfaces of genus 2 with monodromy ω 2 Then its singular fiber X can split into four Lefschetz fibers. Moreover, their vanishing cycles are as depicted below. ω ω 2 ω 2 = τ 0 τ a τ b τ c

61 Case g = 3 21 / 23 Lemma X (0) 3 can split into X (1) 2 and three Lefshetz fibers, and their vanishing cycles are as depicted below. (1) ω2 ω 3 b 1 a 1 c 1 ω 3 = ω (1) 2 τ a1 τ b1 τ c1.

62 Case g = 3 21 / 23 Lemma X (1) 2 can split into X (2) 1 and three Lefshetz fibers, and their vanishing cycles are as depicted below. ω (1) 2 0 b 2 a 2 c 2 ω (1) 2 = τ 0 τ a2 τ b2 τ c2.

63 Case g = 3 21 / 23 Proposition ω 3 = τ 0 τ a2 τ b2 τ c2 τ a1 τ b1 τ c1. 0 a 2 b 2 c 2 a 1 b 1 c 1

64 Dehn-twist expression 22 / 23 Theorem ω g = τ 0 (τ ag 1 τ bg 1 τ cg 1 ) (τ a1 τ b1 τ c1 ). 0 a g 1 b g 1 c g 1 a 1 b 1 c 1

65 Ending Thank you for your attention.