1 1. R 2n non-kähler complex structure n = 2 n = 1 complex curve Kähler No n 3 Calabi Eckmann Yes complex structure 2 Hopf h p : S 2p+1 CP p, h

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1 Non-Kähler complex structures on R 4 ( ) 1. Antonio J. Di Scala (Politecnico di Torino), Daniele Zuddas (KIAS) [1] R 4 non-kähler complex surfaces Kähler 1. (M, J) complex manifold M complex structure J symplectic form ω (M, J) Kähler ω J 2 (1) 0 u T M ω(u, Ju) > 0 (tamedness) (2) u, v T M ω(u, v) = ω(ju, Jv) (J-invariance) complex manifold Kähler complex structure ω Kähler non-kähler complex manifold Kähler Hermite Kähler Kähler [3] [9] [11] Siu[16] 2. Compact complex surface Kähler first Betti number b 1 compact complex surface b 1 Kähler non-kähler compact Kähler manifold Betti number b 2j+1 Hodge theory non-compact connected open orientable 4-manifold Kähler complex structure b 1 Stein manifolds non-compact complex manifold non-kähler 3. compact holomorphic curve complex manifold non-kähler Kähler manifold (M, J) compact complex curve C J symplectic form ω ω > 0 C nkasuyaoasi.aoyama.ac.jp

complex structure ω Kähler non-kähler complex manifold Kähler Hermite Kähler Kähler [3] [9] [11] Siu[16] 2.

2 1 1. R 2n non-kähler complex structure n = 2 n = 1 complex curve Kähler No n 3 Calabi Eckmann Yes complex structure 2 Hopf h p : S 2p+1 CP p, h q : S 2q+1 CP q h p,q : S 2p+1 S 2q+1 CP p CP q T 2 fiber bundle modulus τ elliptic curve S(τ) CP p CP q {U i U j } (0 i p, 0 j q) U i U j S(τ) h p,q holomorphic T 2 fiber bundle S 2p+1 S 2q+1 complex structure Calabi-Eckmann manifold M p,q (τ) M p,q (τ) open subset R 2n (n 3) non-kähler complex structure S 2p+1 S 2q+1 M p,q (τ) open subset E p,q (τ) p > 0, q > 0 E p,q (τ) h p,q elliptic fiber R 2(p+q+1) 3 non-kähler (n 3 n = p+q +1, p > 0, q > 0 ) R 4 non-kähler complex structure M 0,1 (τ) Hopf surface E 0,1 (τ) C 2 open subset n = 2 S 4 S 2 genus-one achiral Lefschetz fibration Matsumoto-Fukaya fibration 4 positive singularity negative singularity 1 negative singularity 4-ball R 4 fibration holomorphic complex structure holomorphic fibration regular fiber elliptic curve 3 R 4 non-kähler complex structure 4. 1 < ρ 2 < ρ 1 1 (ρ 1, ρ 2 ) complex manifold E(ρ 1, ρ 2 ) surjective holomorphic map f : E(ρ 1, ρ 2 ) CP 1 (1) E(ρ 1, ρ 2 ) R 4.

p,q (τ) open subset R 2n (n 3) non-kähler complex structure S 2p+1 S 2q+1 M p,q (τ) open subset E p,q (τ) p > 0, q > 0 E p,q (τ) h p,q elliptic fiber R 2(p+q+1) 3 non-kähler (n 3 n = p+q +1, p > 0, q

3 (2) f 1 (0) f singular fiber node 1 immersed holomorphic sphere. (3) f regular fiber 2 embedded holomorphic torus embedded holomorphic annulus complex manifold E(ρ 1, ρ 2 ) Matsumoto- Fukaya fibration negative Lefschetz singularity R 4 2 E(ρ 1, ρ 2 ) 2 complex structure biholomorphic biholomorphism Matsumoto-Fukaya fibration complex manifold R 4 complex manifold E(ρ 1, ρ 2 ) (ρ 1, ρ 2 ) (ρ 1, ρ 2) E(ρ 1, ρ 2 ) E(ρ 1, ρ 2) biholomorphic E(ρ 1, ρ 2 ) compact holomorphic curve R 4 non-kähler complex structure connected open orientable 4-manifold non-kähler complex structure Kähler complex structure CP 2 complex structure E(ρ 1, ρ 2 ) 1 blow up E(ρ 1, ρ 2 ) 3 complex manifold strictly pseudoconcave boundary overtwisted contact 3-sphere E(ρ 1, ρ 2 ) overtwisted contact 3-sphere concave holomorphic filling overtwisted contact 3-manifold concave holomorphic filling E(ρ 1, ρ 2 ) compact complex surface overtwisted contact manifold convex holomorphic filling 4 E(ρ 1, ρ 2 )

structure biholomorphic biholomorphism Matsumoto-Fukaya fibration complex manifold R 4 complex manifold E(ρ 1, ρ 2 ) (ρ 1, ρ 2 ) (ρ 1, ρ 2) E(ρ 1, ρ 2 ) E(ρ 1, ρ 2) biholomorphic E(ρ 1, ρ 2 ) compact

4 2. The Matsumoto-Fukaya fibration 1980 S 4 S 2 genus-one achiral Lefschetz fibration [10] Hopf fibration H : S 3 CP 1 suspension ΣH : S 4 S 3 f MF := H ΣH f MF regular fiber 2-torus suspension 2 pinched point Lefschetz singularity torus fibration f MF : S 4 S 2 Matsumoto- Fukaya fibration f MF 2 singular fiber singularity F 1, singularity F 2 S 4 F 1 tubular neighborhood N 1 F 2 tubular neighborhood N 2 S 2 S 2 = D 1 D 2 f MF (F j ) D j, D 1 = D 2 2 disk N 1 := f 1 MF (D 1), N 2 := f 1 MF (D 2) f MF N j f j (j = 1, 2) f 1 : N 1 D 1 singularity 1 genus-one Lefschetz fibration f 2 : N 2 D 2 singularity 1 genus-one achiral Lefschetz fibration Monodromy vanishing cycle right-handed Dehn twist, left-handed Dehn twist N 1 N 2 orientation reversing diffeomorphic N 1 N 2 diffeomorphism Kirby diagram Matsumoto-Fukaya fibration Kirby diagram 1 [13], Figure : The Matsumoto-Fukaya fibration on S 4. diagram 0-handle 2 1-handle once punctured torus 4 2-handle framing 0 2-handle once punctured torus torus torus fibration regular fiber 1-handle framing 1, framing handle Lefschetz singularity vanishing cycle 2-handle 1-handle framing 1 2-handle 2-handle N 1 N 2 diagram

neighborhood N 2 S 2 S 2 = D 1 D 2 f MF (F j ) D j, D 1 = D 2 2 disk N 1 := f 1 MF (D 1), N 2 := f 1 MF (D 2) f MF N j f j (j = 1, 2) f 1 : N 1 D 1 singularity 1 genus-one Lefschetz fibration f 2 : N

5 singularity 2 vanishing cycle T 2 fiber 1 meridian N 1 N 2 T 2 fiber longitude 1 multiplicity-1 logarithmic transformation 2 2: The gluing of N 1 and N 2. N 1 N 2 = S 4 N 2 negative singularity X = B 4 X X D 2 negative singularity 1 annulus fibration (monodromy left-handed Dehn twist) negative Lefschetz singularity standard model B 4 N 1 (N 2 \X) R 4 N 2 \X singularity D 2 trivial annulus fibration A D 2 (A annulus) 5. A D 2 N 1. t D 2 = D 1 = S 1, A {t} f 1 1 (t) = T 2 thickened meridian, t S 1 1 T 2 longitude 1. R 4. R 4 N 1 A D 2 E(ρ 1, ρ 2 ) f : E(ρ 1, ρ 2 ) CP 1 f MF N 1 (N 2 \X) complex manifold

$Lefschetz singularity standard model B 4 N 1 (N 2 \X) R 4 N 2 \X singularity D 2 trivial annulus fibration A D 2 (A annulus) 5. A D 2 N 1.$

6 N 1 (N 2 \X) = R 4 Kirby diagram 3 1 X vanishing cycle framing 1 2-handle, 3-handle, 4-handle 3: The map f on S 4 \X = R E(ρ 1, ρ 2 ) complex manifold E(ρ 1, ρ 2 ) 5 2 N 1 A D 2 complex structure (r) := {z C z < r}, (r 1, r 2 ) := {z C r 1 < z < r 2 }. ρ 0, ρ 1, ρ 2 0 < ρ 0 < ρ 1 < 1 < ρ 2 < ρ 1 1 complex structure A D 2 holomorphic annulus holomorphic disk (1, ρ 2 ) (ρ 1 0 ) N 1 genus-one Lefschetz fibration f 1 : N 1 D 1 elliptic fibration complex structure elliptic surface I 1 singular fiber (0, ρ 1 ) elliptic fibration, n Z π : C (0, ρ 1 )/Z (0, ρ 1 ) n (z, w) = (zw n, w) (ρ 1 ) singular elliptic fibration g 1 : W (ρ 1 ) I 1 singular fiber ([4]) W (1, ρ 2 ) (ρ 1 0 ), E(ρ 1, ρ 2 ) (1, ρ 2 ) (ρ 1 1, ρ0 1 ) W (1, ρ 2 ) (ρ 0, ρ 1 ) biholomorphic φ : (ρ 0, ρ 1 ) C ( 1 φ(w) = exp 4πi (log w)2 1 ) 2 log w

complex structure elliptic surface I 1 singular fiber (0, ρ 1 ) elliptic fibration, n Z π : C (0, ρ 1 )/Z (0, ρ 1 ) n (z, w) = (zw n, w) (ρ 1 ) singular elliptic fibration g 1 : W (ρ 1 ) I 1 singular

7 φ(re i(θ+2π) ) = re iθ φ(re i(θ) ) = wφ(w) (1) φ 1 Z C π holomorphic section φ Y := {(zφ(w), w) C (ρ 0, ρ 1 ) z (1, ρ 2 )} Y Z V := Y/Z φ holomorphic section (1, ρ 2 ) (ρ 0, ρ 1 ) biholomorphic V W biholomorphism j j : V = (1, ρ 2 ) (ρ 0, ρ 1 ) (1, ρ 2 ) (ρ 1 1, ρ 1 0 ); (z, w) (z, w 1 ).. E(ρ 1, ρ 2 ) := W j ( (1, ρ2 ) (ρ 1 0 ) ) R 4 V (1, ρ 2 ) fiber φ φ (1) w 0 1 (1, ρ 2 ) elliptic curve longitude 1 (1) φ w 2π w (1, ρ 2 ) elliptic curve C /Z C meridian, longitude W (1, ρ 2 ) (ρ 1 0 ) 5 E(ρ 1, ρ 2 ) R 4 f (ρ 1 ), (ρ 1 0 ) W g 1, (1, ρ 2 ) (ρ 1 0 ) 2nd factor (ρ 1 ) (ρ 1 0 ) biholomorphism (ρ 0, ρ 1 ) (ρ 1 1, ρ 1 0 ); w w 1 CP 1 f : E(ρ 1, ρ 2 ) CP 1 E(ρ 1, ρ 2 ) f 4. E(ρ 1, ρ 2 ) E(ρ 1, ρ 2 ) f [1], [2] compact holomorphic curve 6. E(ρ 1, ρ 2 ) compact holomorphic curve f compact fiber

. E(ρ 1, ρ 2 ) := W j ( (1, ρ2 ) (ρ 1 0 ) ) R 4 V (1, ρ 2 ) fiber φ φ (1) w 0 1 (1, ρ 2 ) elliptic curve longitude 1 (1) φ w 2π w (1, ρ 2 ) elliptic curve C /Z C meridian, longitude W (1, ρ 2 ) (ρ 1

8 Proof. i: C E(ρ 1, ρ 2 ) compact holomorphic curve C compact Riemann surface, i holomorphic immersion f i: C CP 1 constant map f i compact Riemann surface holomorphic map branched covering map constant map C E(ρ 1, ρ 2 ) CP 1 contractible space E(ρ 1, ρ 2 ) = R 4 null-homotopic branched covering map f i constant map compact holomorphic curve f 1 (w) (w (ρ 1 )) w 0 f 1 (w) modulus 1 log w elliptic curve 2πi R 4 non-kähler complex structure 7. (ρ 1, ρ 2 ) (ρ 1, ρ 2) E(ρ 1, ρ 2 ) E(ρ 1, ρ 2) biholomorphic Proof. biholomorphism Φ: E(ρ 1, ρ 2 ) E(ρ 1, ρ 2) ρ 1 = ρ 1, ρ 2 = ρ 2 Φ compact curve compact curve Φ(W ) = W elliptic curve modulus elliptic curve Φ W fiberwise biholomorphism base map (ρ 1 ) (ρ 1) identity ρ 1 = ρ 1 analytic continuation Φ E(ρ 1, ρ 2 ) fiberwise biholomorphism annulus fiber (1, ρ 2 ) annulus fiber (1, ρ 2) biholomorphic ρ 2 = ρ 2 Picard group Pic(E(ρ 1, ρ 2 )) O CP 1(k) CP 1 first Chern class k holomorphic line bundle, L k f E(ρ 1, ρ 2 ) line bundle f line bundle Picard group homomorphism f 8. f : Pic(CP 1 ) Pic(E(ρ 1, ρ 2 )) injective Pic(E(ρ 1, ρ 2 )) complex vector space Proof. L k O CP 1(k) L k nonvanishing holomorphic section τ O CP 1(k) (ρ 1 ), (ρ 1 0 ) σ 1, σ 2 nonvanishing holomorphic section f W 1 := W nonvanishing section f (σ 1 ) W 2 := (1, ρ 2 ) (ρ 1 0 ) nonvanishing section f (σ 2 ) W j holomorphic function τ j (j = 1, 2) τ Wj = τ j f (σ j ) W 1 = W compact fibers foliate τ 1 fiberwise constant (ρ 1 ) holomorphic function u 1 τ 1 = f (u 1 ) V = W 1 W 2 f (u 1 σ 1 ) = τ 2 f (σ 2 )

C E(ρ 1, ρ 2 ) CP 1 contractible space E(ρ 1, ρ 2 ) = R 4 null-homotopic branched covering map f i constant map compact holomorphic curve f 1 (w) (w (ρ 1 )) w 0 f 1 (w) modulus 1 log w elliptic curve

9 τ 2 V fiberwise constant analytic continuation τ 2 W 2 fiberwise constant, (ρ 1 0 ) holomorphic function u 2 τ 2 = f (u 2 ) u 1 σ 1 u 2 σ 2 O CP 1(k) nonvanishing holomorphic section O CP 1(k) f injectivity Pic(CP 1 ) = Z Pic(E(ρ 1, ρ 2 )) Z sheaf cohomology long exact sequence Pic(E(ρ 1, ρ 2 )) = H 1 (E(ρ 1, ρ 2 ), O ) = H 1 (E(ρ 1, ρ 2 ), O) H 1 (E(ρ 1, ρ 2 ), O) complex vector space Pic(E(ρ 1, ρ 2 )) complex vector space Z E(ρ 1, ρ 2 ) holomorphic vector bundle L k1 L k2 L kn (k 1 k 2 k n ) R 2n+4 biholomorphic non-kähler complex structure Calabi Eckmann complex structure compact holomorphic curve ([2], Theorem 4) 9. connected open orientable 4-manifold M 4 non-kähler complex structure Phillips [15] 10. M open manifold d: Sub(M, V ) Epi(T M, T V ); f df Sub(M, V ) M V submersion Epi(T M, T V ) T M T V surjective homomorphism M parallelizable M R n (n dim M) submersion M 4 parallelizable open 4- manifold (open spin 4-manifold ) C 2 immersion g : M 4 C 2 g C 2 complex structure M 4 Kähler complex structure M 4 connected open orientable 4-manifold CP 2 M 4 alomost complex structure Teichner-Vogt [17] Gompf [12] 9 Proof. M 4 spin ( paralleizable ) 10 M 4 E(ρ 1, ρ 2 ) = R 4 immersion h: M 4 E(ρ 1, ρ 2 ) h(m 4 ) elliptic curve rescaling

Pic(E(ρ 1, ρ 2 )) complex vector space Z E(ρ 1, ρ 2 ) holomorphic vector bundle L k1 L k2 L kn (k 1 k 2 k n ) R 2n+4 biholomorphic non-kähler complex structure Calabi Eckmann complex structure

10 M 4 complex structure non-kähler h immersion 4-ball B M 4 embedding ρ 1 < ρ 1, ρ 2 < ρ 2 E(ρ 1, ρ 2) E(ρ 1, ρ 2) E(ρ 1, ρ 2 ) R 4 open 4-ball h(b) E(ρ 1, ρ 2 ) R 4 open 4-ball h(b) E(ρ 1, ρ 2) R 4 diffeomorphism diffeomorphism rescaling h(b) = E(ρ 1, ρ 2) h E(ρ 1, ρ 2 ) complex strcutre M 4 complex structure E(ρ 1, ρ 2) holomorphic M 4 non-kähler complex structure (ρ 1, ρ 2 ) (ρ 1, ρ 2) elliptic curve modulus M 4 non-spin E(ρ 1, ρ 2 ) 1 blow up E(ρ 1, ρ 2 ) 11. E(ρ 1, ρ 2 ) A (E(ρ 1, ρ 2 )\A) strictly pseudoconcave boundary negative overtwisted contact 3-sphere E(ρ 1, ρ 2 ) A overtwisted contact 3-sphere concave holomorphic filling contact structure negative Hopf band negative contact structure, S 3 standard contact structure Hopf fiber half Lutz twist negative orientation E(ρ 1, ρ 2 ) X negative Lefschetz singularity standard model strictly pseudoconcavity strictly pseudoconcave boundary complex tangency negative contact structure d 3 -invariant contact structure E(ρ 1, ρ 2 ) S 3 concave hypersurface [1] A. J. Di Scala, N. Kasuya and D. Zuddas, Non-Kähler complex structures on R 4, arxiv: (2015). [2] A. J. Di Scala, N. Kasuya and D. Zuddas, Non-Kähler complex structures on R 4 II, arxiv: (2015). [3] K. Kodaira, On Compact Complex Analytic Surfaces: I, Ann. of Math. 71 (1960), [4] K. Kodaira, On Compact Analytic Surfaces: II, Ann. of Math. 77 (1963), [5] K. Kodaira, On Compact Analytic Surfaces: III, Ann. of Math. 78 (1963), 1 40.

$E(ρ 1, ρ 2 ) A (E(ρ 1, ρ 2 )\A) strictly pseudoconcave boundary negative overtwisted contact 3-sphere E(ρ 1, ρ 2 ) A overtwisted contact 3-sphere concave holomorphic filling contact structure$

11 [6] K. Kodaira, On the structures of compact complex analytic surfaces: I, Amer. J. Math. 86 (1964), [7] K. Kodaira, On the structures of compact complex analytic surfaces: II, Amer. J. Math. 88 (1966), [8] K. Kodaira, On the structures of compact complex analytic surfaces: III, Amer. J. Math. 90 (1968), [9] K. Kodaira, On the structures of compact complex analytic surfaces: IV, Amer. J. Math. 90 (1968), [10] Y. Matsumoto, On 4-manifolds fibered by tori, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 7, [11] Y. Miyaoka, Kähler metrics on elliptic surfaces, Proc. Japan Acad. Ser. A Math. Sci. 50 (1974), [12] R. E. Gompf, Spin c -structures and homotopy equivalences, Geom. Topol. 1 (1997), [13] R. E. Gompf and A. I. Stipcitz, 4-manifolds and Kirby calculus, Graduate Studies in Mathematics 20, American Mathematical Society (1999). [14] A. Phillips, Submersions of open manifolds, Topology 6 (1967), [15] Y. T. Siu, Every K3 surface is Kähler, Inv. Math. 73 (1983), [16] P. Teichner and E. Vogt, All oriented 4-manifolds have spin c -structures, preprint (1994).

[10] Y. Matsumoto, On 4-manifolds fibered by tori, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 7, 298 301. [11] Y. Miyaoka, Kähler metrics on elliptic surfaces, Proc. Japan Acad. Ser. A Math. Sci. 50 (1974), 533 536.

Akbulut-Karakurt diagram L = K 1 K 2 in S 3 Stein corks W = W (L). W the Mazur manifold K 1, K 2 are unknotted lk(k 1, K 2 ) = ±1 Involution τ : K 1 K

Akbulut-Karakurt diagram L = K 1 K 2 in S 3 Stein corks W = W (L) W the Mazur manifold K 1, K 2 are unknotted lk(k 1, K 2 ) = ±1 Involution τ : K 1 K 2 admits a Stein diagram h 0 h 1 h 2 h 2 is attached