Introduction Construction Further results Non-K¨ ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Aoyama Gakuin University 2016.3.8 ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction Construction Further results 1 Introduction Problem and Motivation Main Theorem 2 Construction The Matsumoto-Fukaya fibration Holomorphic models 3 Further results ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction Problem and Motivation Construction Main Theorem Further results Our problem A complex mfd ( M, J ) is said to be K¨ ahler if there exists a symplectic form ω compatible with J , i.e., (1) ω ( u, Ju ) > 0 for any u ̸ = 0 ∈ TM , (2) ω ( u, v ) = ω ( Ju, Jv ) for any u, v ∈ TM . Problem ahler complex structure on R 2 n ? Is there any non-K¨ ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction Problem and Motivation Construction Main Theorem Further results Our problem A complex mfd ( M, J ) is said to be K¨ ahler if there exists a symplectic form ω compatible with J , i.e., (1) ω ( u, Ju ) > 0 for any u ̸ = 0 ∈ TM , (2) ω ( u, v ) = ω ( Ju, Jv ) for any u, v ∈ TM . Problem ahler complex structure on R 2 n ? Is there any non-K¨ If n = 1 , the answer is “No”. ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction Problem and Motivation Construction Main Theorem Further results Our problem A complex mfd ( M, J ) is said to be K¨ ahler if there exists a symplectic form ω compatible with J , i.e., (1) ω ( u, Ju ) > 0 for any u ̸ = 0 ∈ TM , (2) ω ( u, v ) = ω ( Ju, Jv ) for any u, v ∈ TM . Problem ahler complex structure on R 2 n ? Is there any non-K¨ If n = 1 , the answer is “No”. If n ≥ 3 ,“Yes” (Calabi-Eckmann). ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction Problem and Motivation Construction Main Theorem Further results Our problem A complex mfd ( M, J ) is said to be K¨ ahler if there exists a symplectic form ω compatible with J , i.e., (1) ω ( u, Ju ) > 0 for any u ̸ = 0 ∈ TM , (2) ω ( u, v ) = ω ( Ju, Jv ) for any u, v ∈ TM . Problem ahler complex structure on R 2 n ? Is there any non-K¨ If n = 1 , the answer is “No”. If n ≥ 3 ,“Yes” (Calabi-Eckmann). Then, what about if n = 2 ? ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction Problem and Motivation Construction Main Theorem Further results Calabi-Eckmann’s construction H 1 : S 2 p +1 → C P p , H 2 : S 2 q +1 → C P q : the Hopf fibrations. H 1 × H 2 : S 2 p +1 × S 2 q +1 → C P p × C P q is a T 2 -bundle. The Calabi-Eckmann manifold M p,q ( τ ) is a complex mfd diffeo to S 2 p +1 × S 2 q +1 s.t. H 1 × H 2 is a holomorphic torus bundle ( τ is the modulus of a fiber torus). E p,q ( τ ) : the top dim cell of the natural cell decomposition. If p > 0 and q > 0 , then it contains holomorphic tori. So, it is diffeo to R 2 p +2 q +2 and non-K¨ ahler. ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction Problem and Motivation Construction Main Theorem Further results Calabi-Eckmann’s construction H 1 : S 2 p +1 → C P p , H 2 : S 2 q +1 → C P q : the Hopf fibrations. H 1 × H 2 : S 2 p +1 × S 2 q +1 → C P p × C P q is a T 2 -bundle. The Calabi-Eckmann manifold M p,q ( τ ) is a complex mfd diffeo to S 2 p +1 × S 2 q +1 s.t. H 1 × H 2 is a holomorphic torus bundle ( τ is the modulus of a fiber torus). E p,q ( τ ) : the top dim cell of the natural cell decomposition. If p > 0 and q > 0 , then it contains holomorphic tori. So, it is diffeo to R 2 p +2 q +2 and non-K¨ ahler. This argument doesn’t work if p = 0 or q = 0 . ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction Problem and Motivation Construction Main Theorem Further results Non-K¨ ahlerness and holomorphic curves Lemma (1) If a complex manifold ( R 2 n , J ) contains a compact holomorphic curve C , then it is non-K¨ ahler. Proof. Suppose it is K¨ ahler. Then, there is a symp form ω compatible ∫ with J . Then, C ω > 0 . On the other hand, ω is exact. By ∫ ∫ Stokes’ theorem, C ω = C dα = 0 . This is a contradiction. ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction Problem and Motivation Construction Main Theorem Further results Main Theorem 0 < ρ 1 < 1 , 1 < ρ 2 < ρ − 1 { } ⊂ R 2 . Let P = 1 Theorem For any ( ρ 1 , ρ 2 ) ∈ P , there are a complex manifold E ( ρ 1 , ρ 2 ) diffeomorphic to R 4 and a surjective holomorphic map f : E ( ρ 1 , ρ 2 ) → C P 1 such that the only singular fiber f − 1 (0) is an immersed holomorphic sphere with one node, and the other fiber is either a holomorphic torus or annulus. ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction The Matsumoto-Fukaya fibration Construction Holomorphic models Further results The Matsumoto-Fukaya fibration f MF : S 4 → C P 1 is a genus- 1 achiral Lefschetz fibration with only two singularities of opposite signs. ( ) F 1 : the fiber with the positive singularity ( z 1 , z 2 ) �→ z 1 z 2 ( ) F 2 : the fiber with the negative singularity ( z 1 , z 2 ) �→ z 1 ¯ z 2 ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction The Matsumoto-Fukaya fibration Construction Holomorphic models Further results The Matsumoto-Fukaya fibration f MF : S 4 → C P 1 is a genus- 1 achiral Lefschetz fibration with only two singularities of opposite signs. ( ) F 1 : the fiber with the positive singularity ( z 1 , z 2 ) �→ z 1 z 2 ( ) F 2 : the fiber with the negative singularity ( z 1 , z 2 ) �→ z 1 ¯ z 2 S 4 = N 1 ∪ N 2 , where N j is a tubular nbd of F j , ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction The Matsumoto-Fukaya fibration Construction Holomorphic models Further results The Matsumoto-Fukaya fibration f MF : S 4 → C P 1 is a genus- 1 achiral Lefschetz fibration with only two singularities of opposite signs. ( ) F 1 : the fiber with the positive singularity ( z 1 , z 2 ) �→ z 1 z 2 ( ) F 2 : the fiber with the negative singularity ( z 1 , z 2 ) �→ z 1 ¯ z 2 S 4 = N 1 ∪ N 2 , where N j is a tubular nbd of F j , = R 4 ( X is a nbd of − sing), N 1 ∪ ( N 2 \ X ) ∼ ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction The Matsumoto-Fukaya fibration Construction Holomorphic models Further results The Matsumoto-Fukaya fibration f MF : S 4 → C P 1 is a genus- 1 achiral Lefschetz fibration with only two singularities of opposite signs. ( ) F 1 : the fiber with the positive singularity ( z 1 , z 2 ) �→ z 1 z 2 ( ) F 2 : the fiber with the negative singularity ( z 1 , z 2 ) �→ z 1 ¯ z 2 S 4 = N 1 ∪ N 2 , where N j is a tubular nbd of F j , = R 4 ( X is a nbd of − sing), N 1 ∪ ( N 2 \ X ) ∼ In the smooth sense, f is a restriction of f MF . ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction The Matsumoto-Fukaya fibration Construction Holomorphic models Further results The Matsumoto-Fukaya fibration 2 Originally, it is constructed by taking the composition of the Hopf fibration H : S 3 → C P 1 and its suspension Σ H : S 4 → S 3 . f MF = H ◦ Σ H . ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction The Matsumoto-Fukaya fibration Construction Holomorphic models Further results The Matsumoto-Fukaya fibration 2 Originally, it is constructed by taking the composition of the Hopf fibration H : S 3 → C P 1 and its suspension Σ H : S 4 → S 3 . f MF = H ◦ Σ H . How to glue ∂N 2 to ∂N 1 is as the following pictures (in the next page). ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction The Matsumoto-Fukaya fibration Construction Holomorphic models Further results Gluing N 1 and N 2 ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction The Matsumoto-Fukaya fibration Construction Holomorphic models Further results Kirby diagrams Figure: The Matsumoto-Fukaya fibration on S 4 . ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
Introduction The Matsumoto-Fukaya fibration Construction Holomorphic models Further results Kirby diagrams 2 Figure: The map f on S 4 \ X ∼ = R 4 . ahler complex structures on R 4 Naohiko Kasuya j.w.w. Antonio J. Di Scala and Daniele Zuddas Non-K¨
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