Symplectic Geometry, Lecture 14

SYMPLECTIC GEOMETRY, LECTURE 14 1. Kähler Geometry Let (M, ω, J) be a Kähler manifold, with ω a symplectic form and J an integrable complex structure com­ patible with ω. � • Compatibility ω(Ju, Jv) = ω(u, v): note that, for a (2, 0)-form γ = � ai,j dzi ∧ dzj , we have γ(Ju, Jv) = −γ(u, v), and similarly for a (0, 2)-form. For a (1, 1)-form γ = ai,j dzi ∧ dz j , we have γ(Ju, Jv) = γ(u, v), implying that ω ∈ Ω1,1 . • Closedness dω = 0 ⇔ ∂ω = 0, ∂ω = 0: in particular, [ω] ∈ H∂1,1 (M ) lives in the Dolbeault cohomology �n of M . Moreover ω is real (i.e. ω = ω). Writing ω locally as 2i j,k=1 hjk dzj ∧ dz k , so ω= (1) n i � hjk dzk ∧ dz j 2 j,k=1 we have that hjk = hkj , and (hjk ) must be a Hermitian matrix. • Nondegeneracy ω n �= 0 ⇔ (hjk ) is invertible, since (2) i ω n = ±( )n n!(det (hjk ))dz1 ∧ · · · ∧ dzn ∧ dz 1 ∧ · · · ∧ dz n 2 � • Positivity ω(v, Jv) > 0 ⇔ positivity of g(·, ·) = ω(·, J ·) = hjk dzj dz k ⇔ (hjk ) is a positive definite Hermitian matrix. 1,1 Thus, we � find that, given a complex manifold (M, J), ω is a Kähler form ⇔ ω ∈ ΩR , ∂ω = 0, and locally 1 ω = 2 hjk dzj ∧ dz k for (hjk ) a positive definite Hermitian matrix. Moreover, since these properties are preserved by convex linear combinations, any two Kähler forms for the same complex structure J are deformation equivalent and isotopic if [ω] is fixed. 1.1. Kähler potential. Definition 1. For M a complex manifold, φ ∈ C ∞ (M, R) is strictly plurisubharmonic (spsh) if on each complex chart (U, zj ), the matrix ( ∂z∂φ ) is positive definite at every point. j ∂z k 2 Recall that J integrable, d2 = 0 =⇒ ∂ 2 = 0, ∂∂ + ∂∂ = 0, ∂ = 0. Proposition 1. φ spsh ⇔ 2i ∂∂φ is Kähler. � � 2 |zj | = Example. On Cn , φ = zj zj is strictly plurisubharmonic since ( ∂z∂φ ) is the identity matrix, and j ∂z k � i the corresponding symplectic form ω = 2 dzj ∧ dz j is the standard one. We have the following converse. Theorem 1. For ω a closed, real-valued (1, 1)-form on p ∈ M , ∃ a neighborhood U � p, φ ∈ C ∞ (U, R) s.t. ω = 2i ∂∂φ. This φ is called a local Kähler potential for ω. 1 2 1.2. Examples of Kähler Manifolds. Example. Any complex submanifold of (Cn , ω) is Kähler, with the inherited complex and symplectic structures. Example. Complex projective space CPn = Cn+1 � {0}/C∗ is Kähler: letting (3) Ui = {(z0 : · · · : zi−1 : 1 : zi+1 : · · · : zn )} be the standard charts ∼ = Cn , we have the Fubini-Study K¨ahler form i 2 (4) ωF S = ∂∂ log(1 + |z| ) 2 (since f (z) = log(1 + |z |)2 is spsh). Explicitly, � � � 2 � zj dz j (1 + |z| ) dzj ∧ dz j − ( z j dzj ) ∧ ( zj dz j ) (5) ∂∂f = ∂ 2 = 2 1 + |z| (1 + |z| )2 Applying this to v ∈ T 1,0 , v ∈ T 0,1 , we obtain 2 (6) 2 2 (1 + |z| ) |v| − |�z, v�| 2 (1 + |z| )2 2 ≥ |v| 2 (1 + |z| )2 Since 2i ∂∂f (u, iu) = ∂∂f (u1,0 , u1,0 ), we have the desired positivity. Moreover, for φ a transition map (WLOG 2 2 between U0 and U1 ), we have that φ∗ f = log(1 + |z| ) − log |z1 | =⇒ ∂∂(φ∗ f ) = ∂∂f since z1 dz 1 d z1 2 ∂∂ log |z1 | = ∂ =0 (7) 2 =∂ z 1 |z1 | Finally, recall that H 2 (CPn , R) = R, and H2 (CPn ) is generated by [CP1 ]. The class of [ω] is thus defined by the value of � 1 [ω] · [CP ] = ωF S = Area(CP1 , ωF S ) (8) CP1 Example. Any complex submanifold of CPn (i.e. complex projective variety) is Kähler. Theorem 2 (Kodaira Embedding). Let (X, ω, J) be a compact Kähler manifold, with [ω] ∈ H 2 (X, R) an integral class. Then ∃ a holomorphic embedding X �→ CPn making it a complex projective variety, with ω differing from ωF S by a scaling factor. Theorem 3 (Hodge). For (M, ω) a compact Kähler manifold, the Dolbeault cohomology groups H∂p,q (M ) satisfy � H k (M, C) = p+q=k H∂p,q (M ) and H p,q ∼ = H q,p . Corollary 1. dim H k (M ) is even for odd k. Example. In the 70’s, Kodaira and Thurston independently studied a closed 4-manifold which carries both a complex structure and a symplectic structure but which is not Kähler.