Splittings of singular fibers and vanishing cycles

Transcription

1 OP Splittings of singular fibers and vanishing cycles Takayuki OKUDA the University of Tokyo Kanazawa University Satellite Plaza January 18, 2017

2 Preliminary Degenerations and their splitting deformations

3 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. 3 / 19

4 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. 3 / 19

5 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. 3 / 19

6 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. 3 / 19

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. 3 / 19

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. 3 / 19

9 Splitting of singular fibers 4 / 19 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.

10 Splitting of singular fibers 4 / 19 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.

11 Splittability of singular fibers 5 / 19 How to construct splittings Double covering method Moishezon (genus 1 case), Horikawa (genus 2 case), Arakawa-Ashikaga (hyperelliptic case) Barking deformation Takamura (some criterion) Fact (Atoms of singular fibers) (1) A Lefschetz fiber and (2) a multiple smooth fiber admit no splittings (i.e. any deformation is equisingular). Conjecture (from a topological viewpoint) Every singular fiber can split into singular fibers each of which is (1) or (2), in finite steps of deformations.

12 Splittability of singular fibers 5 / 19 How to construct splittings Double covering method Moishezon (genus 1 case), Horikawa (genus 2 case), Arakawa-Ashikaga (hyperelliptic case) Barking deformation Takamura (some criterion) Fact (Atoms of singular fibers) (1) A Lefschetz fiber and (2) a multiple smooth fiber admit no splittings (i.e. any deformation is equisingular). Conjecture (from a topological viewpoint) Every singular fiber can split into singular fibers each of which is (1) or (2), in finite steps of deformations.

13 Splittability of singular fibers 5 / 19 How to construct splittings Double covering method Moishezon (genus 1 case), Horikawa (genus 2 case), Arakawa-Ashikaga (hyperelliptic case) Barking deformation Takamura (some criterion) Fact (Atoms of singular fibers) (1) A Lefschetz fiber and (2) a multiple smooth fiber admit no splittings (i.e. any deformation is equisingular). Conjecture (from a topological viewpoint) Every singular fiber can split into singular fibers each of which is (1) or (2), in finite steps of deformations.

14 Splittability of singular fibers 5 / 19 How to construct splittings Double covering method Moishezon (genus 1 case), Horikawa (genus 2 case), Arakawa-Ashikaga (hyperelliptic case) Barking deformation Takamura (some criterion) Fact (Atoms of singular fibers) (1) A Lefschetz fiber and (2) a multiple smooth fiber admit no splittings (i.e. any deformation is equisingular). Conjecture (from a topological viewpoint) Every singular fiber can split into singular fibers each of which is (1) or (2), in finite steps of deformations.

15 Splittability of singular fibers 5 / 19 How to construct splittings Double covering method Moishezon (genus 1 case), Horikawa (genus 2 case), Arakawa-Ashikaga (hyperelliptic case) Barking deformation Takamura (some criterion) Fact (Atoms of singular fibers) (1) A Lefschetz fiber and (2) a multiple smooth fiber admit no splittings (i.e. any deformation is equisingular). Conjecture (from a topological viewpoint) Every singular fiber can split into singular fibers each of which is (1) or (2), in finite steps of deformations.

16 Topological classification π : M is topologically equivalent to another degeneration π : M df ori. preserving homeomorphisms H : M M, h: s.t. h π = π H. Theorem (Terasoma) Top. equivalent degenerations are deformation equivalent. M π H M π h The top. classes of degenerations are completely determined by their topological monodromies. Theorem (Imayoshi, Shiga-Tanigawa, Earle-Sipe) Every topological monodromy is pseudo-periodic of negative twist. 6 / 19

17 Topological classification π : M is topologically equivalent to another degeneration π : M df ori. preserving homeomorphisms H : M M, h: s.t. h π = π H. Theorem (Terasoma) Top. equivalent degenerations are deformation equivalent. M π H M π h The top. classes of degenerations are completely determined by their topological monodromies. Theorem (Imayoshi, Shiga-Tanigawa, Earle-Sipe) Every topological monodromy is pseudo-periodic of negative twist. 6 / 19

18 Topological classification π : M is topologically equivalent to another degeneration π : M df ori. preserving homeomorphisms H : M M, h: s.t. h π = π H. Theorem (Terasoma) Top. equivalent degenerations are deformation equivalent. M π H M π h The top. classes of degenerations are completely determined by their topological monodromies. Theorem (Imayoshi, Shiga-Tanigawa, Earle-Sipe) Every topological monodromy is pseudo-periodic of negative twist. 6 / 19

19 Topological classification π : M is topologically equivalent to another degeneration π : M df ori. preserving homeomorphisms H : M M, h: s.t. h π = π H. Theorem (Terasoma) Top. equivalent degenerations are deformation equivalent. M π H M π h The top. classes of degenerations are completely determined by their topological monodromies. Theorem (Imayoshi, Shiga-Tanigawa, Earle-Sipe) Every topological monodromy is pseudo-periodic of negative twist. 6 / 19

20 Topological classification 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 7 / 19

21 Topological classification 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 7 / 19

22 Topological classification 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 7 / 19

23 Topological classification 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 7 / 19

24 Topics Splittability into Lefschetz fibers

25 Degeneration of propeller surfaces 9 / 19 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 ω 5 π 0

26 Case g = 2 10 / 19 Theorem (Y. Matsumoto) π : M : the degeneration of Riemann surfaces of genus 2 with monodromy ω 2 Then its singular fiber X 2 can split into four Lefschetz fibers. Moreover, their vanishing cycles are as depicted below. ω ω 2 X 2

27 Case g = 2 10 / 19 Theorem (Y. Matsumoto) π : M : the degeneration of Riemann surfaces of genus 2 with monodromy ω 2 Then its singular fiber X 2 can split into four Lefschetz fibers. Moreover, their vanishing cycles are as depicted below. ω ω 2

28 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 11 / 19

29 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 11 / 19

30 Psudo-propeller maps 11 / 19 X (1) X (0) ω (1) 4 ω (0) 5 = ω 5 γ

31 Psudo-propeller maps 11 / 19 X (3) X (4) 1 1 ω (3) 2 γ ω (4) 1 = τ γ

32 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) 1 m 1 m 2 m 1 1 } n / 19

33 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) 1 1 m 1 m 2 m 1 1 } n / 19

34 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) 1 1 m 1 m 2 m 1 1 } n / 19

35 Results 13 / 19 X (n) m X(n+1) m 1 g = 2 2 g = 3 g = 4

36 Results 13 / 19 X (n) m X(n+1) m 1 g = 2 2 g = g = 4

37 Results 13 / 19 X (n) m X(n+1) m 1 g = 2 2 g = g = 4

38 Results 13 / 19 X (n) m X(n+1) m 1 g = 2 2 g = g = 4 4 3

39 Results 13 / 19 X (n) m X(n+1) m 1 g = 2 2 g = g =

40 Results 13 / 19 X (n) m X(n+1) m 1 g = 2 2 g = g =

41 Remark 14 / 19 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 A splitting of a singular fiber into Lefschetz fibers gives a Dehn-twist expression of its topological monodromy.

42 Case g = 2 (bis) 15 / 19 Theorem (Y. Matsumoto) 1 The singular fiber X 2 can split into four Lefschetz fibers, and their vanishing cycles are as depicted below. 2 ω 2 = τ 0 τ a τ b τ c. ω ω 2 0 b a c

43 Case g = 2 (bis) 15 / 19 Theorem (Y. Matsumoto) 1 The singular fiber X 2 can split into four Lefschetz fibers, and their vanishing cycles are as depicted below. 2 ω 2 = τ 0 τ a τ b τ c. ω ω 2 0 b a c

44 Case g = 3 16 / 19 Proposition 1 X (0) 3 can split into X (1) 2 and three Lefshetz fibers, and their vanishing cycles are as depicted below. 2 ω 3 = ω (1) 2 τ a1 τ b1 τ c1. (1) ω2 ω 3 b 1 a 1 c 1

45 Case g = 3 16 / 19 Proposition 1 X (0) 3 can split into X (1) 2 and three Lefshetz fibers, and their vanishing cycles are as depicted below. 2 ω 3 = ω (1) 2 τ a1 τ b1 τ c1. (1) ω2 ω 3 b 1 a 1 c 1

46 Case g = 3 16 / 19 Proposition 1 X (1) 2 can split into four Lefshetz fibers (including X (2) 1 ), and their vanishing cycles are as depicted below. 2 ω (1) 2 = τ 0 τ a1 τ b1 τ c1. ω (1) 2 0 b 2 a 2 c 2

47 Case g = 3 16 / 19 Proposition 1 X (1) 2 can split into four Lefshetz fibers (including X (2) 1 ), and their vanishing cycles are as depicted below. 2 ω (1) 2 = τ 0 τ a1 τ b1 τ c1. ω (1) 2 0 b 2 a 2 c 2

48 Case g = 3 16 / 19 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

49 Dehn-twist expression 17 / 19 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

50 Results 18 / 19 f MCG(Σ g ) : a periodic mapping class of order m b(f) := # {branch points of Σ g Σ g /f} p(f) := # {propeller points of f} h(f) fixed points of f with rotation angle ±1/m r(f) := q j /l j Z + : the total valency sum Theorem (O) X : the singular fiber equipped with periodic monodromy f Suppose f satisfies at least one of the following: b(f) p(f) r(f). b(f) p(f) 2, r(f) = 1 and genus(σ g /f) = 0. Then X can splits into Lefschetz fibers.

51 Results 18 / 19 f MCG(Σ g ) : a periodic mapping class of order m b(f) := # {branch points of Σ g Σ g /f} p(f) := # {propeller points of f} h(f) fixed points of f with rotation angle ±1/m r(f) := q j /l j Z + : the total valency sum Theorem (O) X : the singular fiber equipped with periodic monodromy f Suppose f satisfies at least one of the following: b(f) p(f) r(f). b(f) p(f) 2, r(f) = 1 and genus(σ g /f) = 0. Then X can splits into Lefschetz fibers.

52 Results 18 / 19 Remark of Theorem 1 If m = 2, 3, then X can split into Lefschetz fibers. 2 genus # of periodic m.c # of periodic m.c. satisfying ( ) # of powers of periodic m.c. satisfying ( )

53 Ending Thank you for your attention.