On the ampleness of positive CR line bundles over Levi-flat manifolds

.. On the ampleness of positive CR line bundles over Levi-flat manifolds . Masanori ADACHI Nagoya U.. September 12, 2013. Geometry and Foliations 2013 at Tokyo. Masanori ADACHI (Nagoya U.). Ampleness of positive bundles over Levi-flats. September 12, 2013. 1 / 10. Definition Levi-flat 3-manifold := 3-manifold with C ∞ non-singular foliation by Riemann surfaces {Σα } (Levi foliation). Levi-flat real hypersurface := Levi-flat 3-manifold realized in a complex surface. leafwise holomorphic function := function holomorphic on each leaf Σα . (No transverse regularity is assumed a priori.). Masanori ADACHI (Nagoya U.). Ampleness of positive bundles over Levi-flats. September 12, 2013. 2 / 10. Motivation The starting point is Inaba’s Theorem (1992): Continuous leafwise holomorphic functions on compact Levi-flat manifolds are leafwise constant. In particular, they are constant if a dense leaf exists. It permits us to study function theory on Levi-flat 3-manifolds as (1 + 1) dimensional generalization of (meromorphic) function theory on compact Riemann surfaces. We expect to see the “complexity” of Levi foliation affecting the transverse regularity of leafwise meromorphic functions.. Masanori ADACHI (Nagoya U.). Ampleness of positive bundles over Levi-flats. September 12, 2013. 3 / 10. The existence problem of leafwise meromorphic functions CR line bundle L → M over Levi-flat manifold M := C ∞ C-line bundle over M with C ∞ leafwise holomorphic transition functions. How many leafwise holomorphic sections can a CR line bundle L → M possess? Question When is a CR line bundle over a compact Levi-flat manifold C k -ample? i.e., whether we can make an embedding (s0 : · · · : sN ) : M ,→ CPN where s0 , · · · , sN are appropriately chosen C k leafwise holomorphic sections of L⊗n (n = n(k) ≫ 1).. Masanori ADACHI (Nagoya U.). Ampleness of positive bundles over Levi-flats. September 12, 2013. 4 / 10. A necessary condition for L to be C k -ample L → M is positive : ⇐⇒ ∃C ∞ hermitian metric s.t. its Chern curvature along each leaf Σα is positive. A Levi-flat 3-manifold M posesses a positive CR line bundle ⇐⇒ its Levi foliation is taut / (2-)calibrated. Ohsawa-Sibony’s embedding theorem (2000) L → M is positive =⇒ C k -ample for any k ∈ N. Question What about k = ∞?. Masanori ADACHI (Nagoya U.). Ampleness of positive bundles over Levi-flats. September 12, 2013. 5 / 10. Main result The positivity of L → M cannot imply C ∞ -ampleness in general though it implies C k -ampleness for any k ∈ N. A counterexample Σ: a compact Riemann surface of genus ≥ 2. L([p]) → Σ: a holomorphic line bundle defined by an one point divisor. ρ : π1 (Σ) ,→ PSL(2, R) ≃ Aut(D): a non-trivial quasiconformal deformation (or more generally we suppose the action of Image(ρ) on CP1 has no finite orbit & no ±holomorphic section in its associated disc bundle.) π : Mρ := Σ ×ρ S 1 → Σ: the suspension. L := π ∗ L([p]) → Mρ gives a counterexample. Masanori ADACHI (Nagoya U.). Ampleness of positive bundles over Levi-flats. September 12, 2013. 6 / 10. Why isn’t this L → Mρ C ∞ -ample? . Mρ is a Levi-flat real hypersuface in Xρ := Σ ×ρ CP1 ⊃ Mρ := Σ ×ρ S 1 .. 1. L is a restriction of a holomorphic line bundle over Xρ : L = L([π −1 (p)])|Mρ . . The complement Xρ \ Mρ is “Takeuchi 1-complete” by using the ρ-equivariant harmonic diffeo. Σ → D/Image(ρ).. 2. . For any n ∈ N, we can find k = k(n) ∈ N s.t. any C k leafwise holomorphic section of L⊗n → Mρ extends to a holomorphic section of L([π −1 (p)])⊗n → Xρ .. 3. . But, L([π −1 (p)]) is not ample and their sections cannot distinguish the points in the same fiber.. 4. Masanori ADACHI (Nagoya U.). Ampleness of positive bundles over Levi-flats. September 12, 2013. 7 / 10. The space of C k leafwise holomorphic sections of L⊗n (in the counterexample L → Mρ ). k. differentiability of CR sections not very ample. k = k(n). finite dimensional. n = n(k). Ohsawa-Sibony. very ample infinite dimensional A.. n. degree (the order of allowed poles) Masanori ADACHI (Nagoya U.). Ampleness of positive bundles over Levi-flats. September 12, 2013. 8 / 10. Takeuchi 1-completeness X : a complex manifold, D ⋐ X : a rel. cpt. domain, ∂D ∈ C 2 . D is Takeuchi 1-complete : ⇐⇒ ∃ r : a C 2 defining function of D s.t. (Any eigenvalue of i∂∂(− log |r |) w.r.t. ω) > ε > 0 on D where ω is an arbitrary hermitian metric on X . A Bochner-Hartogs type Theorem X : a compact complex surface, M: a compact Levi-flat real hypersurface, L: a holomorphic line bundle over X . Suppose X \ D: disjoint union of Takeuchi 1-complete domains. Then, ∃κ ∈ N s.t. any C κ leafwise holomorphic section of L|M extends to a holomorphic section of L.. Masanori ADACHI (Nagoya U.). Ampleness of positive bundles over Levi-flats. September 12, 2013. 9 / 10. Remarks Akira Takeuchi (1964): X = CPn , D: proper pseudoconvex. The level sets {r = Cst.} give a “uniform” contact deformation. Takeuchi 1-convexity (i.e. the eigenvalues ≥ 0 and > ϵ > 0 except a cpt. K ⊂ D) ⇐⇒ the positivity of the normal bundle of the Levi foliation if M is C 3 and saturated by a holomorphic foliation. (cf. the works of Brunella (2008)) Thank you for your attention.. Masanori ADACHI (Nagoya U.). Ampleness of positive bundles over Levi-flats. September 12, 2013. 10 / 10.