Spring Semester 2018 Math 7721: Kähler Geometry

[KG] refers to the .pdf notes on Kähler geometry, [DG] to the  .pdf notes on differential geometry,         [FR] to  further references. [Zheng’s book] to Professor Zheng’s “Differential Geometry.”

1. JANUARY  8.   Almost-complex manifolds.  Examples: complex vector spaces, open submanifolds, Cartesian products. The covariant 2-tensors  aJ  and  Ja  arising from a given covariant 2-tensor  a  on an almost-complex manifold.  Hermitian (symmetric) 2-tensors  a  and skew-Hermitian 2-forms  a,  defined by requiring that
   
   (1.1) aJ  =  Ja.
   
   Hermitian metrics on an almost-complex manifold, characterized, equivalently, by being those metrics which make  J  skew-adjoint at each point, as well as by being the metrics turning  J,   at each point, into a linear isometry of the tangent space.  The fact that, when one uses a Hermitian metric  g  to identify any covariant 2-tensor  a  with the endomorphism  A  of the tangent bundle  TM  such that   a(v, ⋅ )   =  g(Av, ⋅ )   for all tangent vector fields  v,  the 2-tensors  aJ  and  Ja  correspond in the same way to the composite bundle endomorphisms  AJ  and  JA. The one-to-one  J-correspondence between Hermitian 2-tensors and skew-Hermitian 2-forms. The Kähler form  Ω = gJ  of the given Hermitian metric  g  on an almost-complex manifold.  The equality
   
   (1.2) Ω ^∧m  =  m! dg,     where  m = dim_CM.
   
   [FR].   [KG]:  Section 3 except formula (3.2), and Remark 4.1(iii) except the last sentence.   Homework #1.
   
   
2. JANUARY  10.   Holomorphic mappings, biholomorphisms. Integrability of  J.  Complex manifolds. Kähler manifolds, examples: open submanifolds, complex vector spaces with Hermitian inner products, oriented Riemannian surfaces (since an oriented Euclidean plane is, naturally, a complex line), products of Kähler manifolds. Almost complex submanifolds,  examples: open submanifolds and nonempty preimages of points under constant-rank holomorphic mappings (including holomorphic submersions). The Levi-Civita connection of a submanifold metric. The fact that almost complex submanifolds of Kähler manifolds become Kähler manifolds when endowed with the submanifold metric. Nonsingular projective algebraic varieties as examples of Kähler manifolds.   [FR].   [KG]:  Section 7 (the first four lines and the final paragraph), Section 4 (the first three paragraphs).   Homework #2.
   
   
3. JANUARY  12.   Almost-Kähler metrics, including Kähler metrics as a special case. Finite partitions of unity [DG  p. 121]. Oriented integration of compactly supported continuous top degree differential forms [DG  p. 165]. The Stokes theorem [DG  p. 166]. The Kähler form  Ω = gJ   and the Kähler class
   
   (3.1) [Ω] ∊ H²(M, R)
   
   of an almost-Kähler manifold  (M, g). Positive and negative cohomology classes in  H²(M, R)  on an almost complex manifold  M, and the fact that, by (1.2), positivity implies being nonzero. Mutual exclusiveness of positivity/negativity/vanishing in  H²(M, R)  for a compact almost complex manifold  M, leading to the conclusion that  H²(M, R) ≠ {0}  for a compact almost Kähler manifold  (M, g). An example of a compact complex manifold  M  with  H²(M, R) = {0},  provided by a Hopf manifold, that is,   S¹ × S^2m−1  with the atlas consisting of the local inverses of the following locally diffeomorphic surjective mapping (with any fixed real constant  θ ≠ 0):
   
   (3.2) C^m − {0} ∋ x   →   (e ^{iθ log |x|} ,  x ⁄ |x|) ∊ S¹ × S^2m−1 .
   
   [FR].   [KG]:  Section 5 (the first paragraph, Remark 5.1, the 4-line paragraph preceding Theorem 5.3, and the text following the proof of Theorem 5.3).  Homework #3.
   
   
4. JANUARY  17.   The Betti numbers of spheres and the Künneth formula for   S¹ × M   [DG  p. 164], derived from the Mayer-Vietoris sequence [DG  p. 162]. The curvature   R   of a connection in a real/vomplex vector bundle [DG  p. 77]. Existence of connections (and fibre metrics) via partitions of unity [DG  p. 121]. The first Chern class of a complex line bundle defined with the aid of a connection and its independence of the connection. The first Chern class for arbitrary complex vector bundles, and  c₁(M)   for an almost complex manifold  M. Locally symmetric Kähler manifolds, and a proof of the fact that they are actually Kähler manifolds.   [FR].   [KG]:  Section 2 and, in Section 7, the paragraph immediately following Remark 7.2.   Homework #4.
   
   
5. JANUARY  19.   Examples of locally symmetric Kähler manifolds: complex vector spaces with constant metrics, open submanifolds, complex projective spaces with the Fubini-Study metrics, and complex hyperbolic spaces.   [FR].   [KG]:  Section 7 (the second paragraph following Remark 7.2).   Homework #5.
   
   
6. JANUARY  22.   The curvature tensor   R   and Ricci tensor   r   of a connection in the tangent bundle [DG  p. 80]. The first Bianchi identity [DG  p. 94]. The skew-adjointness property of the curvature for a connection in a vector bundle compatible with a fibre metric [DG  p. 103], symmetry of the Ricci tensor   r   of a Riemannian manifold [DG  p. 108]. Index raising and lowering [DG  p. 105]. The scalar curvature   s   [DG  p. 108]. The second Bianchi identity [DG  pp. 94-95]. The additional symmetry of the Riemannian curvature tensor due to the fact that the conditions
   
   (6.1) abcd = −bacd = −abdc    and    abcd + bcad + cabd = 0
   
   imply   abcd = cdab,  as one sees noting that   abcd = −abdc = badc, and hence [DG  p. 108], as illustrated by Milnor's octahedron,
   
           2cdba = (cdba + dcab) = −(bcda + dbca) − (cadb + adcb) = −(cbad + bdac) − (acbd + dabc)
           = −(cbad + acbd) − (bdac + dabc) = bacd + abdc = 2bacd.
   
   [FR].   Homework #6.
   
   
7. JANUARY  24.   The relations [DG  p. 110]
   
   (7.1) R(u, v)w  =   K [g(u, w)v − g(v, w)u]    and   r  =  Kg       in dimension  2,   where   K  =  s ⁄ 2.
   
   Einstein metrics and metrics of constant (sectional) curvature [DG  p. 130]. Complex-linearity of   R(v,w)  for Kähler manifolds. The equality
   
   (7.2) tr_R J[R(v,w)]  =  −2 ρ(v,w)  
   
   in a Kähler manifold, implying, for the Ricci tensor   r   of any Kähler manifold,
   
   (7.3) Hermitian symmetry of   r   and closedness  (dρ = 0) of the Ricci form  ρ = rJ.
   
   Proof of (7.2) based on the identity
   
   (7.4) ρ_kl  =  R_skl ^qJ_q^s.
   
   The conclusion that
   
   (7.5) tr_C[R(v,w)]  =   iρ(v,w),
   
   since  2 tr_C iA =  i tr_R A  for any self-adjoint endomorphism  A  of a finite-dimensional complex inner-product space (as one sees using a basis formed by eigenvectors of  A). The cohomology relation
   
   (7.6) [ρ] = 2π c₁(M)
   
   for the Ricci form   ρ   of any Kähler metric.   [KG]:  Section 4 (the two lines preceding formula (4.1) plus the 13 lines following it, and Remark 4.1); Section 6 (parts (i), (ii) of Remark 6.1).   [FR].   Homework #7.
   
   
8. JANUARY  26.   Examples of Einstein manifolds: vector spaces with constant metrics (which are flat, hence Ricci-flat); suitable products; Riemannian surfaces of constant Gaussian curvature. The Einstein condition as a consequence of irreducibility of the isotropy representation, further examples of Einstein manifolds (complex projective spaces with the Fubini-Study metrics, complex hyperbolic spaces, and standard spheres). Kähler-Einstein metrics and consequences for the first Chern class, including, in the compact case, nonnegativity of the integral of its top cup power when the complex dimension is even. Certain products of closed oriented Riemannian surfaces as examples showing that a Kähler-Einstein metric may fail to exist despite the existence of a Kähler metric. The Calabi conjecture (for   c₁ < 0,  proved independently by Aubin and Yau):
   
           ‘ every compact almost complex manifold   M   with   c₁(M)  <  0,
           admitting a Kähler metric, also admits a Kähler-Einstein metric ’.
   
   [FR].   [KG]:  Section 7 (the third paragraph following Remark 7.2).   Homework #8.
   
   
9. JANUARY  29.   The Calabi conjecture (for   c₁ = 0,  proved by Yau):
   
           ‘ if  ρ  is a closed skew-Hermitian 2-form on a compact almost complex mani-
           fold  M  admitting a Kähler metric, and  [ρ] = 2π c₁(M),  then every positive 
           cohomology class in  H²(M, R)  contains the Kähler form of a Kähler metric
           for which  ρ  is the Ricci form ’.
   
   The Goldberg conjecture (still open):
   
           ‘ a compact almost-Kähler Einstein manifold
           is necessarily a Kähler manifold ’.
   
   The contraction-derived consequences of the second Bianchi identity [DG  pp. 126-127]:
   
   (9.1) δR  =  − dr,          2δr  =  ds,
   
   the latter known as the Bianchi identity for the Ricci tensor.  Schur's lemma [DG  p. 131]:
   
           ‘ if   r = sg ⁄ n   on a connected Riemannian manifold  (M, g)   of dimen-
            sion   n > 2,    then   s   is constant,  that is,   g   is an Einstein metric ’.
   
   The second covariant derivative  ∇df  of a function  f  on a manifold with a connection  ∇,  also known as the Hessian of  f  [DG  p. 89], the coordinate description
   
   (9.2) f_, jk =  ∂_k ∂_j f  −  Γ_kj^l ∂_l f
   
   of  ∇df  and the formula
   
   (9.3) [∇_u(df )]v  =  d_u d_v f  −  d_w f   with    w  =  ∇_uv,
   
   for any vector fields  u  and  v, where  [∇df ](u,v) = [∇_u(df )](v).  Symmetry of the Hessian if  ∇  is torsionfree. The gradient of a smooth function on a Riemannian manifold [DG  p. 106]. A description of the Hessian in terms of gradient (namely,  ∇df  corresponds via  g  to  ∇w  for  w = ∇f ). The Laplacian and its representation as a composition: gradient followed by divergence [DG  p. 125]. The observation that, for any smooth function  f  on a Riemannian manifold,
   
   (9.4) ∇df  and  Δ f  are nonnegative / nonpositive at any local minimum / maximum of   f.
   
   The operator   i∂∂    on an almost complex manifold  M,  sending smooth functions  f : M  →  R  to exact 2-forms, and given by
   
   (9.5) 2i∂∂f  =  − d [(df )J].
   
   The fact (see Problem 1 in  Homework #9) that for any fixed torsionfree connection  ∇  in  TM  (not assumed to be in any way related to  J ), and any vector fields  u, v, one then has
   
   (9.6) 2 [i∂∂f ](Ju,v)  =   −  [∇df ]( Ju, Jv)  −  [∇df ](v, u)  −  d_w f,   
   
   where, this time,  w = [∇_Ju J]v + J [∇_v J]u.  The conclusion that, if  ∇J = 0  (for instance, if the almost complex manifold  M  admits a Kähler metric), then, for every smooth function  f,  the 2-form  2i∂∂f  is skew-Hermitian (as  [i∂∂f ](Ju,v)  is symmetric in  u  and  v). The equality (immediate when one replaces  u  in (9.6) with −Ju):
   
   (9.7) 2i∂∂f  =  (∇df )J  +  J(∇df ),
   
   for smooth functions  f  on Kähler manifolds. [FR].   [KG]:  formula (3.2) in Section 3 and formula (4.3.i) in Section 4.   Homework #9.
   
   
10. JANUARY  31.   The Ricci identity   (∇_u[∇ψ])w  −  (∇_w[∇ψ])u  =  R^∇(w,u) ψ,   that is,
    
    (10.1) ψ^a_, jk  −  ψ^a_, kj  =  R_jkc^a ψ^c,
    
    for sections  ψ  of a vector bundle with an arbitrary connection  ∇  over a manifold  M  and vector fields  u, w, where any fixed torsionfree connection  ∇  in  TM  is used to form the second covariant derivative. The divergence  δA  of a bundle morphism  A:TM  → TM. The formula
    
    (10.2) J_q^pw_k ,^q_p  =  ρ_lkw^l,     that is,     δ[J(∇w)^*]  =  ρ(w, ⋅ ),
    
    easily obtained from (10.1) and (7.2), and valid for all vector fields  w  in all Kähler manifolds. The identities, with  A = ∇w : TM  → TM,
    
    (10.3) tr JAJA =  (tr JA)² −  r(w,w) + δ[JAJw − (tr JA)Jw]
    
    and
    
    (10.4) tr JAJA^*  =  δ(JA^*Jw)  −  r(w,w),
    
    satisfied by any vector field  w  in any Kähler manifold, and derived from (10.1) - (10.2). The  μ-divergence operator  δ  acting on vector fields, associated with a fixed positive volume form  μ  on an oriented n-dimensional manifold, and given by the local-coordinate formula
    
    (10.5) μ_{1. . . n} δw   =   ∂_j (w^jμ_{1. . . n}).
    
    The fact that, if  μ = dg  is the volume form of a Riemannian metric  g,  then [DG  p. 125, Theorem 38.1] for the Levi-Civita connection  ∇  of  g  and any vector field  w,
    
    (10.6) δw   =   tr ∇w.
    
    The equalities [DG  pp. 124-125], used to establish (10.6):
    
    (10.7) μ_{1. . . n}  =  √ det g ,       g^kl ∂_j g_kl  =  ∂_j det g,        Γ_kj^k  =  ∂_j log√ det g 
    
    for  det g = det [g_kl]  and  μ = dg.   [FR].   [KG]:  formulae (4.1.b) and (4.2) in Section 4, plus formula (6.1) in Section 6.   Homework #10.
    
    
11. FEBRUARY  2.    The divergence theorem [DG  p. 127] and integration by parts. The fact that, for smooth functions  θ, f  on a compact oriented Riemannian manifold,
    
    (11.1) (f, Δθ)  =  − (∇f, ∇θ)  =  (Δf, θ),         (θ, Δθ)  =  − ||∇θ||²,
    
    where  ( , )  and  || ||  denote the  L²  inner product and  L²  norm, leading to Bochner’s lemma [DG  p. 128]:
    
    (11.2) if  Δθ  ≥  0,   then  θ  must be constant.
    
    The kernel and image of the Laplacian of a compact oriented Riemannian manifold. The relation
    
    (11.3)  tr_g [i∂∂θ]J   =   −Δθ 
    
    for any smooth function  θ  on a Kähler manifold. The   ∂∂  Lemma:
    
            ‘ any exact skew-Hermitian 2-form on a compact connected Kähler manifold
            equals   i∂∂θ   for some   θ : M  →  R,  unique up to an additive constant ’.
    
    [FR].   [KG]:   formulae (4.3.ii), (4.4) and Lemma 4.2 in Section 4.   Homework #11.
    
    
12. FEBRUARY  5.   The equality of  L²  norms:
    
    (12.1) √2 || ζ ||    =    || tr_g ζJ ||
    
    for any exact skew-Hermitian 2-form  ζ  on a compact Kähler manifold, and the resulting injectivity of the assignment sending  ζ  to  tr_g ζJ.  Proof of the   ∂∂  Lemma based on (12.1) combined with the fact that, due to (11.3), the assignment just mentioned sends  ζ − i∂∂θ  to  0  for a function  θ  with  Δθ = − tr_g ζJ.  The ratio  γ : M  →  R  of the volume forms of two Riemannian metrics  g  and  g′ on an oriented manifold  M,   defined by the relation  dg′ =  γ dg.   The equality
    
    (12.2) ρ′  =   ρ   −   i∂∂ log γ
    
    for the Ricci forms of two Kähler metrics on an almost complex manifold. The conclusion that the Ricci form (and Ricci tensor) of a Kähler metric is uniquely determined by its volume form. First part of the proof of (12.2).   [FR].   [KG]:   Section 4 (Lemma 4.2 and its proof), Section 6 (Remark 6.1(iii)).   Homework #12.
    
    
13. FEBRUARY  7.   The remainder of the proof of (12.2), based on the coordinate-free version
    
    (13.1) det_g g′  =   γ²
    
    of the coordinate formula  det g′ =  γ² det g.  The uniqueness assertions in the Calabi conjectures, both due to Calabi:
    
            ‘ one necessarily has  g = g′  whenever  g, g′  are two Kähler metrics
            on a compact almost complex manifold, such that
                    ρ  =  −Ω    and    ρ′  =  −Ω ′,
            or
                    ρ  =  ρ′    and    [Ω]  =  [Ω ′] ’.
    
    Proof of the first uniqueness assertion.   [FR].   [KG]:   Section 6 (formula (6.2.a) and Remark 6.1(iii), plus their justifications, and Theorem 6.2). w  Homework #13.
    
    
14. FEBRUARY  9.   Proof of the second uniqueness assertion. Tensor fields  Z  on a manifold  M,  including sections  θ, f  of  M × R  (that is, functions on  M), sections  w,u  of  TM  (vector fields),  ξ  of  T*M  (1-forms),  A, B  of  Hom(TM, TM)  (bundle endomorphisms of  TM) and, finally, sections  a  of  Hom(TM, T*M)  (2-tensors, also called twice-covariant tensors). Natural multiplications of tensor fields. The push-forward  [dF]Z  of a tensor field  Z  on  M  under a diffeomorphism  F:M  → N  between two manifolds, and the equality  [dF](ZZ′)  = ([dF]Z)([dF]Z′) ,  where  ZZ′  stands for any natural multiplication. The (local) flow  e^tw  of a vector field  w  on a manifold  M  [DG  pp. 219-221]. Completeness of a vector field. The Lie derivative  £_w Z  of a tensor field  Z  in the direction of a vector field  w  on a manifold, defined to be the derivative with respect to the real variable  t,  at  t = 0,  of  [de^−tw]Z.  The immediate conclusion that  [de^−sw]£_w Z  equals the derivative of  [de^−tw]Z  at  t = s,  and so
    
    (14.1) £_w Z  =  0      if and only if the local flow of  w  leaves  Z  invariant.
    
    The Leibniz rule
    
    (14.2) £_w (ZZ′)   =  (£_w Z)Z′  +  Z(£_w Z′).
    
    The relation
    
    (14.3) £_w θ  =  d_w θ
    
    for any smooth function  θ. Proofs of (14.2) - (14.3).   [FR].   [KG]:   Section 6 (proof of Theorem 6.2).   Homework #14.
    
    
15. FEBRUARY  12.   The identities stating that £_w u,   (£_w A)u,   (£_w a)(u, v)  are, respectively, equal to
    
    (15.1) [w, u],       [w, Au] − A[w, u],       d_w [a(u, v)] − a([w, u], v) − a(u, [w, v]),
    
    while  (£_w ξ)(u) = d_w [ξ(u)] − ξ([w, u])  for vector fields  u,v,w, endomorphisms  A  of the tangent bundle, twice-covariant tensor fields  a, and 1-forms  ξ.  Proof of (15.1). Holomorphic vector fields on an almost complex manifold, defined to be those smooth vector fields  w  for which  £_w J = 0, that is, the operators  J  and  [w, ⋅ ]  commute when both treated as endomorphisms of the space of all smooth vector fields. The equality
    
    (15.2) £_wJ  =  [J,∇w]
    
    (commutator of bundle morphisms  TM  → TM), for any smooth vector field on a Kähler manifold, and the resulting the characterization of holomorphic vector fields  w  on a Kähler manifold by the condition  [J,∇w] = 0.  Killing vector fields  w  on a Riemannian manifold  (M, g),  defined by requiring that  £_wg = 0. The formula
    
    (15.3) (£_wg)(u, v)  =  g(Bu, v),        where      B  =  ∇w  +  (∇w)^*
    
    and  u,v,w  are any smooth vector fields on a Riemannian manifold  (M, g).  The characterization of Killing fields  w  by skew-adjointness of  ∇w.   [FR].   [KG]:   Section 7 (the three lines immediately preceding Remark 7.1, parts (a) - (c) of Remark 7.1, and Remark 7.2).   Homework #15.
    
    
16. FEBRUARY  14.   The relation, satisfied by any smooth vector fields  w, u  on any manifold:
    
    (16.1) £_[w,u]  =  £_w£_u  −  £_u£_w.
    
    The proof of (16.1) based on the fact that, for  Ψ = £_[w,u] − £_w£_u + £_u£_w, and any tensor fields  Z, Z′,  one has the Leibniz rule  Ψ(ZZ′) = (ΨZ)Z′ + Z(ΨZ′.  The (real) Lie algebra of vector field on a given manifold whose flows preserve a given tensor field. The real Lie algebra  i(M, g)  of Killing vector fields on a Riemannian manifold  (M, g). The complex Lie algebra  h(M)  of holomorphic vector fields on a Kähler manifold  (M, g).  The Bochner identity [DG  p. 126], immediate from (10.1):
    
    (16.2) w^k_, jk −  w^k_, kj  =  R_jk w^k,     that is,     δ∇w  −  dδw  =  r( ⋅ , w),
    
    for vector fields  w  on any manifold with a fixed connection  ∇.   The Laplacian  Δ  acting on sections of a vector bundle with a connection over a Riemannian manifold  (M, g),  with  [Δψ]^a = g^jkψ^a_, jk .  The symbol  aw  for a twice-covariant tensor field  a  and a vector field  w  on a Riemannian manifold  (M, g),  denoting the unique vector field with  g(aw, ⋅ ) = a(w, ⋅ ).  The linear differential operator  D  in a Riemannian manifold  (M, g),  sending vector fields to vector fields, and given by
    
    (16.3) Dw  =  − Δw  −  rw
    
    or, equivalently,
    
    (16.4) g(Dw, ⋅ )  =  − δ£_wg   +  dδw,
    
    the equivalence of the two descriptions being due to the coordinate expression  [Dw]_j  =  − w_j, k ^k −  R_jk w^k,  the Bochner formula (16.2), and (15.3). The fact that, as an obvious consequence of (16.4),
    
    (16.5) Dw  =  0      for any Killing vector field   w   on a Riemannian manifold.
    
    The identity, satisfied by any smooth vector field  w  on a Kähler manifold:
    
    (16.6) δ[J,∇w]^*  =  − g(JDw, ⋅ ).
    
    Proof of (16.6) in local coordinates: the  kth  component of the left-hand side is  g_kl [J_p^lw^p_, q − w^l_, p J_q^p ]^, q = g_kl J_p^lw^p_, q^q − J_q^pw_k, p^q while (10.2) combined with skew-adjointness of  J  gives  J_q^pw_k, p^q = ρ_kl w^l.  The resulting trivial conclusion that
    
    (16.7) Dw  =  0     whenever   w   is a holomorphic vector field on a Kähler manifold.
    
    The obvious equality
    
    (16.8) |[J, A]|²  =  tr [J, A][J, A]^*  =  2 tr JAJA^*  +  2 tr AA^*,
    
    for any (real) linear endomorphism of the tangent bundle of an almost complex manifold. The integral formula for smooth vector fields  w  on a compact oriented Riemannian manifold  (M, g),  obvious from (16.3):
    
    (16.9) (Dw, w)   =   ||∇w||²  −  ∫ _M r(w,w) dg        and,  consequently,      Ker D   =  {0}   if   r < 0,
    
    the inequality  r < 0 meaning that the Ricci tensor  r  is negative definite at each point.   [FR].   [KG]:   parts of Section 9 (formulae (9.1) - (9.2) and (9.5), assertions (i) and (iii) in Remark 9.1).   Homework #16.
    
    
17. FEBRUARY  16.   The relation, satisfied by any compactly supported smooth vector field  w  on a Kähler manifold:
    
    (17.1) 2(Dw, w)   =   ||B||²     for     B = [J,∇w],     that is,     B = £_wJ,   cf.  (15.3),
    
    and obtained by combining (10.4) with (16.8) applied to  A = ∇w. The conclusion that in any compact Kähler manifold  (M, g)  the operator  D  is nonnegative, while its kernel consists precisely of all of holomorphic vector fields:
    
    (17.2) Ker D   =   h(M).
    
    An obvious consequence of (17.2) and (16.5):
    
    (17.3) ‘ on a compact Kähler manifold every Killing field is holomorphic ’.
    
    The inner product of twice-covariant tensor fields  a, b  on a Riemannian manifold  (M, g),  given by
    
    (17.4) < a, b >  =  a_j^kb^j_k,     that is,     < a, b >  =  tr AB^*
    
    A, B  being the bundle endomorphisms of  TM  characterized by  g(Aw, ⋅ ) = a(w, ⋅ )  and  g(Bw, ⋅ ) = b(w, ⋅ )  for all vector fields  w. The observation that
    
    (17.5) < a, b >  =  0    if   a   is symmetric and   b   skew-symmetric.
    
    The integral formula [DG  pp. 128-129] for compactly-supported smooth vector fields  w  and functions  f  on an oriented Riemannian manifold  (M, g),  also due to Bochner, and immediate from (16.2):
    
    (17.6) ∫ _M r(w,w) dg = ||δw||² − ∫ _M tr (∇w)² dg,     so  that    ∫ _M r(∇f, ∇f ) dg  =  ||Δf ||² − ||∇df ||².
    
    The fact that the first equality in (17.6) implies nonexistence of nontrivial Killing fields (or, harmonic 1-forms) on a compact Riemannian manifold having negative (or, respectively, positive) Ricci curvature, in the sense that its Ricci tensor  r  is negative/positive definite at each point. The Lichnerowicz inequality [DG  p. 128]:
    
    (17.7) (n − 1)τ  ≥  nλ
    
    for any positive eigenvalue  τ  of  −Δ  on a compact oriented  n-dimensional Riemannian manifold satisfying the lower bound  r ≥ λg  on the Ricci curvature with a constant  λ.  The conclusion that, by (16.9) and the second equality in (17.6), for a positive eigenvalue  τ  of  −Δ  on a compact oriented Riemannian manifold  (M, g),
    
    (17.8) τ ||w||²   =   (Dw, w)  +  2 ∫ _M r(w,w) dg      whenever    w = ∇f   and   Δf  =  − τf.
    
    The stronger version of the Lichnerowicz inequality (17.7):
    
    (17.9) τ  ≥  2λ,
    
    obvious from (17.8) along with nonnegativity of  D  (see (17.1)), and valid for any positive eigenvalue  τ  of  −Δ  on a compact Kähler manifold such that  r  ≥  λg  for a constant  λ. [FR].   [KG]:   Section 9 (Lemma 9.2, formula (9.9), and the first part of Theorem 9.4).   Homework #17.
    
    
18. FEBRUARY  19.   The theorem stating that
    
            ‘ in a compact oriented Einstein manifold  (M, g)  with nonzero Einstein
             constant  λ,  the assignment which sends  f  to  ∇f  is a linear isomorphism
             of the space of all functions  f : M  →  R  with  Δf = − 2λf  and the space
             p of all gradient vector fields  w  on  M  having  Dw = 0 ’,
    
    where the fact that the assignment takes values in the required space and is surjective follows from the following identity, valid for any smooth function  f  on any Riemannian manifold, and immediate from (16.2) - (16.3):
    
    (18.1) D∇f  =  − ∇Δf  − 2r∇f,
    
    so that
    
    (18.2) D∇f  =  − ∇(Δf  + 2λf )      when     r = λg.
    
    The version of the above theorem for Kähler manifolds, which, as a consequence of (17.2), reads
    
            ‘ in a compact Kähler-Einstein manifold  (M, g)  with nonzero Einstein
             constant  λ,  the assignment sending  f  to  ∇f  is a linear isomorphism
             of the space  Ker (Δ + 2λ)  of all functions  f : M → R  with  Δf  = − 2λf 
             and the space  p  of all holomorphic gradient vector fields on  M ’.
    
    The fact that, for any smooth differential 2-form  ζ  on a Riemannian manifold, one has
    
    (18.3) δδζ   =   0,                w_j, k ^jk  =  w_j, k ^kj,
    
    where  δζ  denotes the 1-form  ξ  with the components  ξ_j = ζ_jk^,k  and the divergence  δξ  of a 1-form  ξ  is, by definition, equal to  δw  for the vector field  w  with  g(w, ⋅ ) = ξ.  A proof of (18.3) based on the following equalities of  L² inner products, resulting from (17.5) via integration by parts, with any smooth function  f  having a (small) compact support:  0 = (ζ, ∇df ) = −(δζ, df ) = (δδζ, f ).  Also,
    
    (18.4) δδζ   =   0,                w_j, k ^jk  =  w_j, k ^kj,
    
    for smooth vector fields  w  on a Riemannian manifold, which is a special case of (18.3), with  ζ_jk = w_j, k − w_k, j.  Consequently,
    
    (18.5) δ[Dw  +  2rw]  =  − Δδw,
    
    since adding  2R_jk w^k  to the coordinate expression  [Dw]_j  =  − w_j, k ^k −  R_jk w^k,  following (16.4), we get  [Dw  +  2rw]_j  =  R_jk w^k − w_j, k ^k, so that the Bochner formula  R_jk w^k  =  w^k_, jk −  w^k_, kj  in (16.2) gives  [Dw  +  2rw]_j  =  w_k, j ^k − w_j, k ^k −  w^k_, kj,  and, therefore,  [Dw  +  2rw]_j^, j  =  −w^k_, kj^j  by (18.4), as required in (18.5). In particular, with a constant  λ,  (18.5) yields
    
    (18.6) δw ∊ Ker (Δ + 2λ)   and    D∇δw  =  0     whenever   Dw  =  0   and    r  =  λg,
    
    the second conclusion being derived from the first via (18.2) for  f  = δw.  The equality
    
    (18.7) 2(Dw, w)   =   ||£_wg||²  −  ||δw||²
    
    for compactly supported smooth vector fields  w  on oriented Riemannian manifolds, with  ( , )  and  || ||  denoting, as usual, the  L²  inner product and  L²  norm. The proof of (18.7) based on (16.4) and trivial integration by parts, along with the trivial observation that  |£_wg|² = 2w^j, k[w_j, k + w_k, j].  Matsushima’s theorem (the general Riemannian version):
    
            ‘ for any compact oriented Einstein manifold,   Ker D  is the  L²-orthogonal direct sum
             of the Lie algebra  k  of all Killing fields, and the space  p  of all gradients in  Ker D ’.
    
    The formulae for the  k-component  u  and  p-component  v  of any  w  in   Ker D  in the case where the Einstein constant  λ  is positive:
    
    (18.8) u  =  w  +  ∇φ,          v  =  −∇φ,       with    φ    given by    2λφ  =  δw.
    
    Proof of Matsushima’s theorem. First, by (16.9),   Ker D  =  k  =  p  = {0}  if  λ < 0,  while  p  = {0}  and   Ker D  =  k  is the space of all parallel vector fields if  λ = 0. Now let  λ > 0.  Then  u  and  v  in (18.8) lie in  Ker D,  which is clear for  v  in view of the second conclusion in (18.6), and hence obvious for  u.  Finally, due to (18.7) with  w  replaced by  u  (and  Du = 0),  u  in (18.8) is a Killing field, as  δu = δw  + δ∇φ = 2λφ + Δφ,  which vanishes since (18.8) and (18.6) give  φ ∊ Ker (Δ + 2λ).  [Another argument for  λ > 0: as we saw,  v  defined by (18.8) lies in  Ker D  whenever  w ∊ Ker D.  Thus, assigning  v  (18.8) to  w  we obtain a linear operator  Π : Ker D  →  p.  The restriction of  Π  to  p  is the identity: if  w = ∇f ∊ Ker D,  (18.2) allows us to assume, by adding a constant to  f,  that  f  lies in  Ker (Δ + 2λ),  and so  δw = Δf = −2λf,  giving  φ = −f  and  v = w  in (18.8). Hence  Π  is a direct-sum projection onto  p,  while, by the divergence theorem, (18.7), and (16.5),  Ker Π = i(M, g).]   [FR].   [KG]:   Section 9 (formula (9.3), Remark 9.1(iv), formula (9.4), the second part of Theorem 9.4, formula (9.8), and Theorem 9.6).   Homework #18.
    
    
19. FEBRUARY  21.   The Kähler case of Matsushima’s theorem, in which   Ker D  =  h(M)  by (17.2) and, in addition,
    
    (19.1) Jk  =  p       unless    λ = 0,
    
    the equality in (19.1) being immediate if one notes that, for any smooth vector field  w  on a Kähler manifold,
    
    (19.2) ∇(Jw)  =  J∇w.
    
    In fact, given a bundle morphism  A : TM  → TM, it is obvious that
    
    (19.3) [JA]^*  =  −JA^*       if   A   and   J    commute.
    
    Thus, by (19.2), for a holomorphic vector field  w,  self-adjointness of  ∇(Jw)  is equivalent to skew-adjointness of  ∇w,  and vice versa.  Complexifications and real forms of Lie algebras. The conclusion that, in any compact Kähler-Einstein manifold  (M, g)  with nonzero Einstein constant, the Lie algebra  k = i(M, g)  of Killing vector fields is a real form of  h(M),  and hence
    
    (19.4) dim_R i(M, g)  =  dim_C h(M).
    
    Compact Lie algebras, defined by requiring the existence of a Euclidean inner product  < , >  that makes  < [u,v],w >  skew-symmetric in  u,v,w  (or, equivalently, has  < [u,v],v > = 0  for all  u,v,  which is further equivalent to skew-adjointness of the operators  Ad u = [u, ⋅ ]  for all  u).  Compactness of  i(M, g)  for a compact oriented Riemannian manifold  (M, g),  with  < , >  chosen to be the  L²  inner product  ( , ).   Specifically,  ([u,v],v) = 0  for all Killing fields  u,v  since
    
    (19.5) g([u,v],v)  =  δw,        where      w  =  −[g(u,v)]v.
    
    The corollary that  h(M)  has a compact real form whenever the compact almost complex manifold  M  admits a Kähler-Einstein metric. The obvious fact that - by (19.2) and (19.3) - if the endomorphisms  J  and  ∇w  of the tangent bundle of a Kähler manifold commute, then  [∇w + (∇w)^*] J = ∇u − (∇u)^*  for  u = Jw.  Hence, in view of (15.2),
    
    (19.6) (£_wg) J  =  d [g(Jw, ⋅ )]     for any holomorphic vector field  w  on a Kähler manifold.
    
    Covariant differentiation of “smooth sections defined along a curve in the base manifold” of a real or complex vector bundle endowed with a connection. Geodesics of a connection in  TM,  for a manifold  M.  The equality, satisfied by every Killing field  u  and any smooth vector field  w  in any Riemannian manifold:
    
    (19.7) u^q_, jk   =   u^pR_pkj ^q,      that is,    ∇_w[∇u]  =  R(u,w).
    
    Relation (19.7) as a trivial consequence of the identity
    
    (19.8) 2u_q, jk   =   2R_qjk ^pu_p  +  a_qj, k  +  a_qk, j  −  a_jk, q,      with    a_jk  =  u_j, k  +  u_k, j,
    
    valid for any smooth vector field  u  on a Riemannian manifold.   [FR].   [KG]:   Section 9 (part (d) of Theorem 9.6).   Homework #19.
    
    
20. FEBRUARY  23.   Proof of (19.8), using the Ricci identity
    
    (20.1) u_j, kl  −  u_j, lk   =   R_lkj ^pu_p,
    
    immediate from (10.1), gives  2u_q, jk − a_qj, k − a_qk, j + a_jk, q = (R_qkj ^p + R_kjq ^p + R_qjk ^p)u_p,  which equals  2R_qjk ^pu_p  due to the first Bianchi identity. The first-order linear ordinary differential equation satisfied along any geodesic, in view of (19.7), by the pair  (u, ∇u),  for any Killing field  u  in a Riemannian manifold. The resulting injectivity of the linear operator sending  u ∊ i(M, g)  to  (u_x, (∇u)_x),  at any fixed point  x. (Namely, if  (u_x, (∇u)_x) = (0,0),  then  (u, ∇u)  vanishes along every broken geodesic emanating from  x,  while such broken geodesics reach every point in  M  due to connectivity.) The obvious consequence in the form of the upper bounds
    
    (20.2) dim_R i(M, g)   ≤  n(n+1) ⁄ 2,          dim_R k  ≤  m(m+2)
    
    for the Lie algebra  i(M, g)  of all Killing vector fields on an  n-dimensional Riemannian manifold  (M, g)  and, respectively, the Lie algebra  k  of holomorphic Killing vector fields on any Kähler manifold of complex dimension  m.  Linear vector fields on finite-dimensional real/complex vector spaces. Linearity of the flow transformations  e^tA  of a linear vector field  A  on a vector space, completeness of  A,  and the commutation relation  Ae^tA = e^tAA,  all derived from the existence and uniqueness of global solutions to first-order linear ordinary differential equations [DG  p. 208], applied to the equation  dΨ ⁄ dt = AΨ  satisfied by  Ψ  equal to  e^tAx,   or  e^tA(x + y) − e^tAx − e^tAy,  or  e^tA(cx) − ce^tAy,  or  Ae^tA − e^tAA,  where  x, y ∊ V  and  c ∊ R.   [FR].   Homework #20.
    
    
21. FEBRUARY  26.   Projectability of vector fields under smooth mappings between manifolds, with uniqueness and linearity of the push-forward operation when the mapping is surjective [DG  p. 23]. The fact that all linear vector fields on a finite-dimensional complex vector space  V  are projectable onto  P(V )  and their projected images are holomorphic, leading to a complex vector space of dimension  m(m+2)  consisting of holomorphic vector fields on  CP^m  and, consequently, the inequality  dim_C h(CP^m)  ≥  m(m+2).  The conclusion (from (17.3), (19.4), (20.2) and the last inequality) that
    
    (21.1) dim_C h(CP^m)  =  m(m+2)
    
    and the holomorphic vector fields on  P(V),  for any finite-dimensional complex vector space  V,  are precisely the projected images of linear vector fields on  V.  The equality
    
    (21.2) 2∇_wζ   =   ζ_w  +  Jζ_w J,        where      ζ  =  hJ        and      ζ_w  =  (dζ )(w, ⋅ , ⋅ ),
    
    for an almost complex manifold  M,  any torsion-free connection  ∇  in  TM,  and any Hermitian tensor field  h  on  M  with  ∇h = 0,  valid under the assumption that  J  is parallel relative to some torsion-free connection in  TM.  The special case of (21.2) in which  h  is a Hermitian metric  g,  while  ∇  and  ζ = Ω  denote the Levi-Civita connection and Kähler form of  g. The conclusion that on any fixed almost complex manifold the set of all Kähler metrics is empty or coincides with the set of all almost-Kähler metrics, while the latter set is a convex cone in a vector space.   [FR].   [KG]:  Section 5 (the statement of Lemma 5.2, and Theorem 5.3).   Homework #21.
    
    
22. FEBRUARY  28.   Proof of (21.2). The line segment of Kähler metrics joining two given Kähler metrics on a fixed almost complex manifold. The fact that for any Kähler metric  g  and any compactly supported smooth function  f  on an almost complex manifold  M  the tensor  g + ε(i∂∂f )J  is a Kähler metric as long as  ε  is sufficiently close to  0  in  R.  Time-dependent Riemannian metrics on a manifold. Relations satisfied by the derivatives with respect to time - here denoted by  ( )′ = d ⁄ dt  - with the time-dependent function  φ  defined by  dg′ = φ dg  whenever  g  is a time-dependent Riemannian metric:
    
    (22.1) δ′  =  dφ,               tr_g g′  =  2φ.
    
    [FR].   [KG]:  Section 5 (proof of Lemma 5.2), Section 8 (formula (8.10), the nine lines immediately preceding it, and the six lines following it).   Homework #22.
    
    
23. MARCH  2.   The existence, on any compact oriented Riemannian manifold  (M, g),  of a smooth function  f : M  →  R,  unique up to the addition of a constant, such that
    
    (23.1) Δ f   +   s     is constant
    
    (the latter constant being the average scalar curvature  s_avg).  The resulting differential operator  L,  sending smooth vector fields  v  to smooth functions, and given by
    
    (23.2) Lv  =  δv  −  d_v f      or,   equivalently,      Lv  =  e^fδ(e^− fv).
    
    The Futaki functional, assigning to any smooth vector field  v  on the given compact oriented Riemannian manifold  (M, g)  of dimension  n  the real number
    
    (23.3) Fv   =   (s_avg)^n ⁄ 2 ∫ _M d_v f dg,
    
    where  f  is chosen as in (23.1). The Futaki invariant  F : h(M)  →  R  of a compact almost complex manifold   M   having  c₁(M)  >  0  or  c₁(M)  <  0  and  admitting a Kähler metric, defined by (23.3) for a Kähler metric  g  on  M  satisfying the condition
    
    (23.4) i∂∂f   +   ρ   =   λΩ      with some constant   λ   and some smooth function   f : M  →  R,
    
    where  f  in (23.4) - necessarily, due to (11.3) - also satisfies (23.1), while  s_avg = 2mλ  for  m = dim_CM.  Futaki’s theorem stating that, if  c₁(M)  >  0  or  c₁(M)  <  0,  then
    
    (23.5) g   with  (23.4)  must exist,  and the Futaki invariant does not depend on its choice,
    
    the existence claim being obvious from the   ∂∂  Lemma in the lines following (11.3), and (12.2), since, by the   ∂∂  Lemma, (23.4) is equivalent to
    
    (23.6) [ρ]   =   λ[Ω]   in  H²(M, R),    for some real constant   λ.
    
    The fact that, whenever a Kähler manifold  (M, g)  satisfies (23.4), then, with  f  appearing in (23.4) used to define  L  as in (23.2), one has, for any smooth vector field  v  on  M,
    
    (23.7) ∇Lv  −  J∇LJv  =  −2λv  −  JEA,      where    A  =   £_vJ,
    
    E  being the linear differential operator that sends any smooth endomorphism  A  of  TM  to the vector field  EA  characterized by
    
    (23.8) g(EA, ⋅ )  =  δA  −  (df )A    or,  equivalently,    g(EA, ⋅ )  =  e^fδ(e^− fA).
    
    Proof of (23.7), based on the coordinate relation  (Jw)_k = −J_k^pw_p  for vector fields  w  (due to skew-adjointness of  J),  (9.7), (15.2) (in its coordinate form  J_k^pv^k_, q  =  A_q^p + J_q^kv^p_, k),  the Ricci identity
    
    (23.9) v^k_, pq  −  v^k_, qp  =  R_pql^k v^l,
    
    cf. (10.1), the Bochner formula (16.2), and the equality  J_l^qJ_k^pR_pqs^j = R_kls^j  (see Problem 2 in  Homework #6).   [FR].   [KG]:  Section 8 (from the beginning to the seventh line following formula (8.3) - which is the coordinate version of (23.7) above - except Theorem 8.2 and the last sentence of Theorem 8.1).   Homework #23.
    
    
24. MARCH  5.   The identity, using the same assumptions and notations as (23.7), which trivially follows from (23.7) (cf. Problem 1 in  Homework #23):
    
    (24.1) ΔLv  =  −2λ δv  −  δ(JEA),      where    A  =   £_vJ.
    
    The observation that, for a compact almost complex manifold  M,
    
    (24.2) the Futaki invariant vanishes if   M   admits a non-Ricci-flat Kähler-Einstein metric,
    
    as  (23.3)  yields  Fv = 0  for all  v  whenever  s  is constant. (Since (7.6) then gives  c₁(M)  >  0  or  c₁(M)  <  0,  the Futaki invariant is well defined.) Time-dependent Kähler metrics on a manifold with a fixed (time-independent) almost complex structure  J.  The case of a time-dependent Kähler metric with  Ω ′ = 2i∂∂χ  for some time-dependent function  χ,  in which, with the time-dependent function  φ  defined by  dg′ = φ dg,
    
    (24.3) φ  =  Δχ,                ρ′  =  − i∂∂ Δχ.
    
    (The first equality in (24.3) follows from the second part of (22.1) along with the relation  g = −ΩJ  and (11.3); the second - since (12.2) gives  ρ = ρ₀ − i∂∂ log γ  for  γ  with  dg = γ dg₀,   the subscript  ₀  standing for the value at a fixed time  t₀,  and so  φ = (log γ)′.)  The further conclusion that, if condition (23.4) holds just at one time  t₀,  and  Ω ′ = 2i∂∂χ  as before, then a time-dependent function  f  may be chosen so as to satisfy (23.4) at all times  t,  and, with  L  defined by (23.2),
    
    (24.4) L′  =  −2λdχ,                f ′  =  Δχ  +  2λχ,                λ′  =  0.
    
    (Namely,  f  is obtained by solving the equation  f ′ = Δχ + 2λχ  with prescribed  f₀,  and then (24.4) follows from (23.2), (22.1) and (24.3), while  (i∂∂f + ρ − λΩ )′ = 0  by the second equality in (24.3) with  Ω ′ = 2i∂∂χ.)  Proof of the “does not depend” clause in (23.5): both metrics may be assumed to satisfy (23.4) with the same  s_avg = 2mλ,  so that  Ω ′ = 2i∂∂χ  for the line segment joining them,  2i∂∂χ  being the difference of their Kähler forms, with a (time-independent) smooth function  χ  which exists in view of the   ∂∂  Lemma, and - letting  ~  mean ‘differs by a divergence’ - one has  d_v f ~ −Lv  by (23.2), while  (Lv dg)′ = [(Lv)Δχ − 2λd_v χ] dg  from the first equalities in (24.3) and (24.4); finally, (11.1) and (24.1) with  A = 0  give  (Lv, Δχ) = (ΔLv, χ) = −2λ(δv, χ)  and, at the same time,  −χδv ~ d_v χ.  Homotopy equivalence. Simple connectivity. The Poincaré conjecture.   [FR].   [KG]:  Section 8 (the last sentence of Theorem 8.1, and Lemma 8.7).   [FR].   [KG]:  Section 8 (Lemma 8.4(b), the last sentence of Theorem 8.1, and Lemma 8.7).   Homework #24.
    
    
25. MARCH  7.    The Ricci flow on a manifold  M,  the trajectories of which are defined to be time-dependent metrics  g  on  M  with  g′ = −2r.  Ricci solitons, that is, Riemannian manifolds  (M, g)  satisfying the condition
    
    (25.1) £_wg   +   r   =   λg        or,  equivalently,      w_j, k  +  w_k, j  +  R_jk   =   λg_jk,      with a constant  λ,
    
    for some smooth vector field  w.  Kähler-Ricci solitons, by which one means those Ricci solitons  (M, g)  which at the same time are Kähler manifolds for some almost complex structure  J  on  M.  The observation that  tr_g  applied to (25.1) yields
    
    (25.2) 2 δw   +   s   =   nλ,        where      n = dim M.
    
    The conclusion - obtained by integrating (25.2) and using the divergence formula - that when  M  is compact and oriented,  λ  is uniquely determined by  g  (being equal to  1/n  times the average scalar curvature), which in turn makes  w  unique up to the addition of a Killing field. Gradient Ricci solitons: the Ricci solitons with  w  in (25.1) which is a gradient,  w = ∇f ⁄ 2,  that is, the Riemannian manifolds  (M, g)  satisfying the condition
    
    (25.3) ∇df   +   r   =   λg        or,  equivalently,      f_, jk  +  R_jk   =   λg_jk,      with a constant  λ,
    
    for some smooth function  f.  The result of Perelman, which we will not prove or use:
    
            ‘ Every compact Ricci soliton is a gradient Ricci soliton ’.
    
    Einstein manifolds as the simplest examples of (gradient) Ricci solitons. A corollary:
    
    (25.4) Dw  =  0        whenever      £_wg   +   r   =   λg      with a constant  λ,
    
    which follows since applying  d tr_g − 2δ  to (25.1) we obtain  2g(Dw, ⋅ ) = 0  from (25.2), (16.4) and (9.1). The immediate consequence that, by (17.2),
    
    (25.5) w   in  (25.1)  is holomorphic for any compact Kähler-Ricci soliton  (M, g).
    
    The conclusion that, due to (25.5) and (19.6), in every compact Kähler-Ricci soliton, multiplying (25.1) from the right by  J  one obtains (23.6) or, equivalently, in view of the   ∂∂  Lemma, (23.4). Thus, just as in the Kähler-Einstein case,  c₁(M)  is positive, negative or zero for a compact almost complex manifold  M  admitting a Kähler-Ricci soliton metric, the sign being the same as that of the constant  λ  in (25.1). The fact that (23.1) follows both from (25.3) (by applying  tr_g) and from (23.4) (via (11.3)). More on complexifications of real vector spaces (example: the spaces of real-valued and complex-valued smooth functions on a manifold). The unique complex-linear and complex-bilinear/sesquilinear extensions to complexifications of real-linear and real-bilinear mappings (example: the real and complex L² inner products of compactly supported smooth functions on an oriented manifold with a fixed volume form). The real-part and conjugation operators. The fact that a complex-linear operator from a complex space into a complexification is uniquely determined by its real part (which may be any real-linear operator). The differential operator  P,  sending smooth vector fields  v  on a compact Kähler manifold to smooth complex-valued functions, and defined to be the unique complex-linear operator having the real part  L  defined by (23.2), so that
    
    (25.6) Pv  =  Lv  −  iLJv.
    
    The Tian-Zhu invariant  T : h(M)  →  C  of a compact almost complex manifold   M   having  c₁(M)  >  0  or  c₁(M)  <  0  and  admitting a Kähler metric, given by
    
    (25.7) Tv   =   (s_avg)^n ⁄ 2 ∫ _M e^Pv dg.
    
    for a Kähler metric  g  on  M  with (23.4). The fact that  T : h(M)  →  C  with (25.7) is holomorphic for any compact Kähler manifold  (M, g)  due to complex-linearity of  P  since, obviously,
    
    (25.8) dT_vu   =   (s_avg)^n ⁄ 2 ∫ _M e^PvPu dg
    
    whenever  v,u ∊ h(M).  The conclusion that, by (25.8) and (23.3),
    
    (25.9) the Futaki invariant equals   −Re dT₀.
    
    [FR].   [KG]:  Remark 9.1(ii) in Section 9.   Homework #25.
    
    
26. MARCH  9.   According to Problem 1 in  Homework #26, for a vector field  v  on a Kähler manifold  (M, g),
    
    (26.1) Jv ∊ i(M, g)      whenever    v    is a holomorphic gradient.
    
    On the other hand, given a Killing vector field  u  on a compact oriented Riemannian manifold  (M, g),  with  f  and  L  as in (23.1) and (23.2), due to uniqueness of  f  when it is normalized so that  f_avg = 0,  one has
    
    (26.2) d_u f  =  Lu  =  0.
    
    Next, if a Kähler manifold  (M, g)  satisfies (23.4) with  λ ≠ 0  then, by (23.7), for  L  and  P  defined by (23.2) and (25.6), where  f  is the function appearing in (23.4),
    
    (26.3) the restriction of   P   to   h(M)   is injective.
    
    The equality, valid for any  v,u ∊ h(M)  whenever  (M, g)  is a compact Kähler manifold,
    
    (26.4) (d_ud_uT)(v)   =   (s_avg)^n ⁄ 2 ∫ _M e^Pv(Pu)² dg,
    
    which is obvious from (25.8). Constancy of  Δf  − |∇f |² + 2λf  under the assumption (25.3) (see Problem 2 in Homework #25). The resulting fact that, by (25.8), (25.6) and the second part of (23.2), for any compact gradient Kähler-Ricci soliton  (M, g),  with  f  and  λ ≠ 0  satisfying (25.3),
    
    (26.5) v = ∇f ⁄ (2λ)  is a critical point of the Tian-Zhu invariant  T,
    
    where  T : h(M)  →  C  is given by (25.7), and  v ∊ h(M)  in view of (25.5). An immediate consequence of (26.2) - (26.4): if  k  denotes the Lie algebra of all Killing fields on  (M, g),  then the restriction of  T  to the space  Jk  is real-valued and, in view of (26.4), its Euclidean Hessian is positive-definite, so that  T : Jk  →  R  can have at most one critical point. The conclusion, from the last sentence, (26.1), (24.2), (25.9) and (26.5), that a compact almost complex manifold cannot simultaneously admit a Kähler-Einstein metric and a non-Einstein Kähler metric which is a gradient Ricci soliton, since if  λ = 0  we may use Problem 3 in  Homework #26, while for  λ ≠ 0 the existence of the latter metric implies nonzero Futaki invariant.   Homework #26.
    
    
27. MARCH  19.   The observation that, for any smooth vector field  v  on a Kähler manifold  (M, g)  with (23.4), defining  L, P  and  E  by (23.2), (25.6) and (23.8), applying  J  to both sides of (23.7), and then taking the divergence, one obtains
    
    (27.1) ∇LJv + J∇Lv = −2λJv + EA,     ΔLJv  =  −2λ δ(Jv)  +  δEA    with   A =  £_vJ.
    
    In addition, with the same assumptions and notations as in (27.1),
    
    (27.2) |∇Lv|²  +  2λd_vLv  +  g(∇Lv, JEA)  =  |∇LJv|²  +  2λd_JvLJv  −  g(∇LJv, EA)
    
    for any smooth vector field  v  on  M,  as well as
    
    (27.3) 2[g(∇Lv, ∇LJv)  +  λ(d_JvLv  + d_vLJv)]  =  g(∇Lv + J∇LJv, EA).
    
    In fact, the left-hand side of (27.2),  g(∇Lv, ∇Lv + 2λv + JEA), equals  g(∇Lv, J∇LJv),  by (23.7), so that it is invariant under the replacement of  v  with  Jv.  This invariance amounts to equality (27.2), since, whenever  v  is a smooth vector field on a Kähler manifold and  A  is any smooth endomorphism of its tangent bundle, (15.2), (19.2) and the definition (23.8) of  E  give
    
    (27.4) £_JvJ  =  −(£_vJ)J,          E(AJ)  =  −JEA.
    
    Similarly, the left-hand side of (27.3) is  g(∇LJv, ∇Lv + 2λv) +  g(∇Lv, ∇LJv + 2λJv),  so that (27.1) and (23.7) yield (27.3). The equality (see Problem 1 in  Homework #27), valid for smooth complex-valued functions  ψ  on any Riemannian manifold:
    
    (27.5) Δe^ψ  =  [Δψ  +  g(∇ψ, ∇ψ)] e^ψ,
    
    where  Δ  and  g  are extended complex-(bi)linearly, so that
    
    (27.6) Δψ  =  ΔReψ + iΔImψ   and   g(∇ψ, ∇ψ)  =  |∇Reψ|²  −  |∇Imψ|²  +  2ig(∇Reψ, ∇Imψ).
    
    Tian and Zhu’s theorem: if a compact almost complex manifold   M   having  c₁(M)  >  0  or  c₁(M)  <  0  admits a Kähler metric, then, for the Tian-Zhu invariant  T : h(M)  →  C  of  (M, g)  given by (25.7),
    
    (27.7) T   does not depend on the choice of a Kähler metric  g  with   (23.4).
    
    Note that such  g  must exist according to the lines following (23.5). Proof of (27.7):   Homework #27.
    
    
28. MARCH  21.   The spaces  XM  of all smooth vector fields and  F_cM  of all smooth complex-valued functions on a given manifold  M.  Three important complex-linear operators
    
    (28.1) Θ : F_cM  →  F_cM,        P : XM  →  F_cM,         ∂ : F_cM  →  XM
    
    in the case of a compact Kähler manifold  (M, g).  Namely, using a fixed smooth function  f : M  →  R  such that  Δ f  + s  is constant, we set
    
    (28.2) Θ   =   Δ  −  d_u  −  id_Ju          for     u = ∇f,
    
    while  P  and the complex gradient operator  ∂  are defined by (25.6) with (23.2) and, respectively
    
    (28.3) ∂ψ  =  ∇ Re ψ  +  J∇ Im ψ.
    
    Futaki’s theorem: for a compact Kähler manifold  (M, g)  satisfying (23.4) or,  equivalently, (23.6), with  λ ≠ 0,
    
    (28.4) ∂   maps   Ker (Θ + 2λ)   isomorphically onto   h(M),   and its inverse is   −P ⁄ (2λ).
    
    The Kähler version (19.1) of Matsushima’s theorem as a special case of (28.4). The first step in a proof of (28.4): on a Kähler manifold satisfying (23.4), with  A =  £_vJ  and  E  as in (23.8),
    
    (28.5) (Θ + 2λ)ψ  =  − iPEA     for    ψ  =  Pv
    
    whenever  v ∊ XM,  while, for any  ψ ∊ F_cM  and  v ∊ XM,
    
    (28.6) (i)    P∂ ψ  =  Θψ,        (ii)    ∂ Pv  =  −2λv  −  JEA.
    
    In fact, since  δ  =  L + df  by (23.2), relations (24.1), (23.7) and (27.1) imply (28.5). Similarly, (25.6), (23.2), (23.7) and (28.2) - (28.3) yield (28.6.i) and (28.6.ii).   Homework #28.
    
    
29. MARCH  23.   The  μ-adjoint  Π^*  of a real/complex linear operator  Π,  characterized by
    
    (29.1) (Πψ, θ)  =  (ψ, Π^*θ)      for all   ψ  and  θ,
    
    where  μ  is a fixed (positive) volume form on an oriented manifold  M  and  ( , )  denotes the L² inner product of compactly supported smooth sections of any given real/complex vector bundle over  M,  associated with  μ  and any fixed Riemannian/Hermitian fibre metric in the bundle, while  Π  sends compactly supported smooth sections of one such bundle to analogous sections of the other. Uniqueness of the  μ-adjoint  Π^*  (when it exists). Some trivial observations:  (Π Σ)^* =  Σ^*Π^*,  the composites  Π Π^*  and  Π^*Π  are  μ-self-adjoint,  Π^** = Π,
    
    (29.2) Π^*Π    is self-adjoint, nonnegative, and   Ker Π^*Π  =  Ker Π.
    
    The fact that, if  Π  has the  μ-adjoint  Π^*  and  f  is a smooth function on  M,  then   Π  has the  e^− fμ-adjoint  Π_f^*,  and
    
    (29.3) Π_f^*θ  =  e^fΠ^*(e^− fθ).
    
    The weighted  L²  inner products  ( , )_f,  related to  ( , ),  the original ones, by
    
    (29.4) ( ⋅ , θ)_f  =  ( ⋅ , e^− fθ),
    
    and the corresponding weighted  L²  norms  || ||_f,  so that, if  Π_f^*  exists,
    
    (29.5) (Πψ, θ)_f  =  (ψ, Π_f^*θ)_f      for all   ψ  and  θ.
    
    The Cauchy-Riemann operator  H  sending smooth vector fields on an almost complex manifold to endomorphisms of its tangent bundle, with
    
    (29.6) Hv  =  [J, ∇v],       that is,     Hv  =  £_vJ,
    
    cf.  (15.2). The  dg-adjoint of the Cauchy-Riemann operator  H  of a Kähler manifold  (M, g),  characterized by
    
    (29.7) g(H^*B, ⋅ )  =  δ[J, B^*].
    
    Justification of (29.7) based on the trivial observation that
    
    (29.8) [J, B]^*  =  [J, B^*],    for all endomorphisms   B   of the tangent bundle,
    
    and integration by parts of the inner product  tr A[J, ∇v]^* = tr A[J, (∇v)^*].  An obvious algebraic property of the commutator/anticommutator of  J  and an endomorphisms  A  of the tangent bundle of an almost complex manifold:
    
    (29.9) [J, A]  anticommutes,  JA + AJ  commutes with  J.
    
    Homework #29.
    
    
30. MARCH  26.   Given a smooth function  f  and a smooth vector field  v  on a Kähler manifold, one has
    
    (30.1) H_f^*Hv  =  2JEA^*,    where   A =  £_vJ
    
    and  E  is the operator defined by (23.8). In fact, for  B = ∇v  and  A = Hv  =  £_vJ = [J, B],  (29.3) with  Π = H  and (29.7) - (29.9) yield
    
      g(H_f^*Hv, ⋅ )  =  e^fg(H^*[J, e^− fB], ⋅ )  =  e^fδ[J, [J, e^− fB^*]]  
    
                           =  −2e^fδ(e^− f[J, B^*] J)  =  −2e^fδ(e^− fA^*J)
    
    which, due to the second relations in (23.8) and (27.4), equals  −2g(E(A^*J), ⋅ ) = 2g(JEA^*, ⋅ ),  proving (30.1). Furthermore, by (29.8), if  B  is self-adjoint, so must be both  [J, B]  and  [J, JB].  Thus, (28.3) gives, in an arbitrary Kähler manifold  (M, g),
    
    (30.2) (£_vJ)^*  =  £_vJ    whenever   v  =  ∂ψ   for any   ψ  ∊  F_cM.
    
    Let  f,  in addition, satisfy (23.4) with a constant  λ.  Then, for any smooth function  ψ,
    
    (30.3) H_f^*H∂ ψ  =  −2∂ [(Θ + 2λ)ψ],
    
    since, for  v = ∂ψ,  (30.2) gives (30.1) with  A^* = A,  so that, from (28.6.ii), the left-hand side of (30.2) equals  −2(∂ Pv  +  2λv),  and hence the equality follows from (28.6.i). We can now establish Futaki’s theorem (28.4): by and (30.3) and (29.2),   ∂   maps   Ker (Θ − 2λ)   into  h(M),  while (28.5) clearly shows that  P  maps  h(M)  into  Ker (Θ + 2λ) .  Since  λ ≠ 0,  (28.6) implies that the resulting operators  ∂ : Ker (Θ + 2λ)  →  h(M)  and  −P ⁄ (2λ) : h(M)  →  Ker (Θ + 2λ)   are each other’s inverses, which completes the proof of (28.4). Next, for the gradient operator  ∇  of any oriented Riemannian manifold  (M, g)  and any smooth function  f  on  M,  defining  L by (23.2), one has
    
    (30.4) δ  =  −∇^*,       L  =  −∇_f^*.
    
    In other words,  δ  is the  dg-adjoint, and  L  the  e^− fdg-adjoint of  −∇;  in fact, the first relation is obvious from integration by parts, and the second then follows via the last equality in (23.2) and (29.3). If, in addition,  (M, g)  is a Kähler manifold, then, according to Problem 1 in Homework #30, for  P  as in (25.6),
    
    (30.5) P  =  −∂_f^*.
    
    Another result of Futaki: in a compact Kähler manifold  (M, g)  with (23.4) (that is, having  [ρ] = λ[Ω]),  for a constant  λ ≠ 0,
    
    (30.6) Θ  =  −∂_f^*∂    is self-adjoint, nonpositive, and   Ker Θ  =  C,
    
    C  being the space of constant complex-valued functions. In fact, (30.6) is clear from (28.6.i), (30.5), (29.2) and Problem 2 in Homework #28.   Homework #30
    
    
31. MARCH  28.   The conclusion that, for any compact almost complex manifold,
    
    (31.1) h(M) = {0}   if   c₁(M) < 0   and   M   admits a Kähler metric.
    
    Proof of (31.1): nonnegativity of  −Θ  in (30.6) implies that, when  c₁(M) < 0  (and hence  λ < 0),  2λ  cannot be an eigenvalue of  −Θ.  Thus,  Ker (Θ + 2λ) = {0},  and we can use Futaki’s theorem (28.4). A generalization of Bochner’s integral formula (17.6): with  hv  defined by  g(hv, ⋅ ) = h(v, ⋅ ),
    
    (31.2) (hv, v)_f  =  || Lv ||²_f  −  (B, B^*)_f,    where   h = ∇df  +  r    and   B = ∇v,
    
    for any smooth compactly supported vector field  v  on an oriented Riemannian manifold  (M, g),  any smooth function  f : M  →  R,  and  L  defined as in (23.2), that is,  Lv = δv − d_v f.  The Kähler case, still with arbitrary  v  and  f,  in which, setting  ζ = i∂∂f + ρ,  one sees - from (9.7), (7.3) and (1.1) - that
    
      − 2ζJ  =  ∇df  −  J(∇df )J  +  r  −  JrJ,
    
    where  r − JrJ = 2r,   and so  − 2ζJ = h − JhJ.  Suppose now that one also has  i∂∂f +  ρ  =  λΩ  with a constant  λ,  as in (23.4) or, equivalently,
    
    (31.3) − 2ζJ  =  2λg.
    
    The resulting equality  h − JhJ = 2λg  combined with (31.2) and (25.6) gives, for any smooth compactly supported vector field  v,
    
    (31.4) 2λ||v||²_f  =  ||Pv||²_f  −  (B, B^*)_f  −  (JB, (JB)^*)_f,    with   B = ∇v.
    
    The observation (see Problem 2 in Homework #31) that, given any smooth function  ψ : M  →  C  on a Kähler manifold  (M, g),
    
    (31.5) 2 tr [B² + (JB)²]  =  |A|²     for     B  =  ∇v    and   A  =  £_vJ,     where     v  =  ∂ψ.
    
    The integration of (31.5) against the volume form  e^− fdg  yields  2(B, B^*)_f + 2(JB, (JB)^*)_f = ||£_vJ||²_f  whenever  f  or  ψ  is compactly supported (while one still sets  v = ∂ψ  and  B = ∇v).  Under the additional assumption (23.4), this allows us to rewrite (31.4), with the aid of (28.6.i), as
    
    (31.6) 4λ||v||²_f  =  2||Θψ||²_f  −  ||£_vJ||²_f     if     v  =  ∂ψ    for any  ψ  ∊  F_cM,
    
    Yet another theorem due to Futaki: in a compact Kähler manifold  (M, g)  satisfying  [ρ] = λ[Ω]  with a constant  λ ≠ 0,
    
    (31.7) τ  ≥  2λ         whenever   τ   is a nonzero eigenvalue of   −Θ.
    
    (Note that (17.9) is a special case.) Proof of (31.7): if  Θψ = −τψ,  then, by (30.6),
    
      ||Θψ||²_f  =  (Θψ, Θψ)_f  =  −τ(ψ, Θψ)_f  =  τ(ψ, ∂_f^*∂ψ)_f  =  τ(∂ψ, ∂ψ)_f  =  τ||v||²_f     for    v  =  ∂ψ,
    
    and so (31.6) with  v ≠ 0  trivially implies (31.7). Next: under the assumptions of Futaki’s theorem (28.4), for the isomorphism  ∂  in (28.4), with  u = ∇f,
    
    (31.8) ∂   maps   Ker (Θ + 2λ) ∩ Ker  d_Ju   isomorphically onto   k ⊕ p,    and   Jk  =  p.
    
    Here, as in (19.1),  k ⊕ p  is the  L²-orthogonal direct sum of the Lie algebra  k  of all Killing fields, and the space  p  of all holomorphic (real) gradients. In addition,
    
    (31.9) Ker (Θ + 2λ) ∩ Ker  d_Ju   =  {ψ ∊ F_cM : Re ψ,  Im ψ ∊ Ker (Θ + 2λ)}.
    
    In fact, (31.9) is obvious from (28.2), while (31.8) follows since an  L²-orthogonality argument (see the hint for Problem 1 in Homework #28) shows that, if  ∂ψ ∊ k ⊕ p,  then the projections of  ∂ψ  onto  k  and  p  must coincide with  ∇ Re ψ  and  J∇ Im ψ.   Homework #31.
    
    
32. MARCH  30.   For a compact oriented Riemannian manifold  (M, g)  and a smooth function  f : M  →  R,
    
    (32.1) d_w   is   ( , )_f-skew-adjoint if   w ∊ i(M, g)   and   d_wf  =  0.
    
    Let  (M, g)  now be a compact gradient Kähler-Ricci soliton with  f  and  λ ≠ 0  satisfying (25.3),  so that  u = ∇f  is holomorphic by (25.5), and  Ju  is a (holomorphic) Killing field according to Problem 1 in  Homework #26.  We proceed to discuss some observations made, in this case, by Tian and Zhu. As the Lie derivative  £_Ju   then commutes with both  Θ  and the complex gradient  ∂  (see Problem 2 in Homework #31),
    
    (32.2) the operator   −id_Ju   leaves   Ker (Θ + 2λ)   invariant
    
    (since so does  d_Ju).  Recall that  h(M)  is a complex Lie algebra, cf. (15.1) and Problem 2 in  Homework #16.  Under the complex-linear isomorphism  ∂  in (28.4),
    
    (32.3)   −id_Ju   corresponds to   Ad u = [u, ⋅ ] : h(M)  →  h(M),
    
    which is immediate from the fact that  d_Ju  corresponds to  Ad Ju.  From (28.2),
    
    (32.4) −id_Ju  =  −Δ  +  d_u  −  2λ     on     Ker (Θ + 2λ).
    
    On the other hand, by (32.1),
    
    (32.5) −id_Ju : Ker (Θ + 2λ)  →  Ker (Θ + 2λ)    is   ( , )_f-self-adjoint.
    
      Homework #32.
    
    
33. APRIL  2.   Again,  (M, g)  is a compact gradient Kähler-Ricci soliton, with  f  and  λ ≠ 0  satisfying (25.3). It follows from (32.3) and (32.5) that, as a vector space,  h(M)  is the direct sum of the eigenspaces  h_τ  corresponding to the eigenvalues  h_τ  of  Ad u = [u, ⋅ ].  More generally, let  h_τ = Ker ([u, ⋅ ] − τ)  whenever  τ ∊ R.  As  [u, ⋅ ]  always is, by the Jacobi identity, a derivation of the Lie algebra in question, we have, for any real  τ, σ,
    
    (33.1) [h_τ,  h_σ]  ⊆  h_τ + σ,      while   h₀   is the complex Lie subalgebra   k ⊕ p.
    
    Recall: holomorphic functions  φ : M  →  C  on an almost complex manifold  M  are characterized by  (dφ)J = idφ  (as they are nothing else than holomorphic mappings  M  →  C). The algebra of holomorphic functions. Holomorphicity of multiplicative inverses. Vector bundles over manifolds, local sections, local trivializations and their compatibility, in the sense of regularity of transition functions [DG  pp. 57-58]. Holomorphic complex vector bundles over almost complex manifolds, defined analogously to smooth real/complex vector bundles over smooth manifolds [DG  p. 58], just with holomorphicity of transition functions rather than their smoothness. Examples: tangent bundles of complex manifolds, and the tautological line bundle  T  over any complex projective space  P(V),  cf. [DG  p. 59]. The total space of a vector bundle [DG  p. 66]. The holomorphic complex line bundle  L  naturally associated with a complex submanifold  Q  of codimension one in a complex manifold  M,  and given by
    
    (33.2) L_x  =  (T_xM ⁄ T_xQ)^*    if    x ∊ Q,    and    L_x  =  C      otherwise.
    
    The atlas of local trivializations for this  L,  consisting of all  (U, φ)  such that
    
    (33.3) U ⊆ M   is open,   φ : U  →  C   is holomorphic,  and   φ⁻¹(0) = U ∩ Q,
    
    while  dφ ≠ 0  everywhere in  U ∩ Q.  Here  φ  is treated as a local section of  L  defined on  U,  with  φ_x = dφ_x  if  x ∊ Q  (cf. Problem 1 in  Homework #33)  and  φ_x = φ(x)  otherwise.   Homework #33.
    
    
34. APRIL  4.   The fact that an almost complex submanifold  Q  of a complex manifold  M  is itself a complex manifold as one sees, locally, using a mapping whose restriction to  Q,   due to the inverse mapping theorem, is a biholomorphisms. The fact, immediate from the holomorphic version of the rank theorem (see below) that locally, up to biholomorphisms, complex submanifolds appear as vector subspaces in a complex vector space. The conclusion from an integral version of the first-order Taylor formula (see Problem 1 in  Homework #34)  that a holomorphic function  φ  on an open set  U  in a complex vector space, vanishing on  U ∩ Ker ξ  for a nonzero linear functional  ξ,  is holomorphically divisible by  ξ,  with the quotient  φ ⁄ ξ  equal on  U ∩ Ker ξ  to the ratio  dφ ⁄ dξ  of differentials. The resulting mutual compatibility of the local trivializations  (U, φ)  in the line following formula (33.2). The equality  i∂∂f = 0  valid, in view of (9.5), whenever  f = Re φ  and  φ  is a holomorphic function. The fact that a holomorphic function on a compact connected Kähler manifold is necessarily constant (and, more generally, every holomorphic function on a Kähler manifold is harmonic). Holomorphic sections of a holomorphic vector bundle  B  over an almost complex manifold, and the space  H ^o(B)  of all such sections.   Homework #34.
    
    
35. APRIL  6.   The operators  ∂_v  and  ∂_v  acting on smooth functions  U  →  C,  where  U  is an open subset of an  m-dimensional complex vector space  V  and  v ∊ V,  and given by
    
    (35.1) 2∂_v  =  d_v  −  id_iv,           2∂_v  =  d_v  +  id_iv.
    
    The complex partial derivatives  ∂_j  and  ∂_j  relative to a fixed (complex) basis  e₁,..., e_m  of  V,  equal, respectively, to  ∂_v  and  ∂_v  for  v = e_j.  Holomorphicity of  φ ∊ F_cU  as equivalent to requiring that  ∂_vφ = 0,  or  ∂_jφ = 0,  or  ∂_vφ = d_vφ  for all  v  (or all  j).  The complex Jacobian matrix  [∂_jF^a],  representing the differential, at any point  x,  of a holomorphic mapping  F  from an open set  U  into a finite-dimensional complex vector space, where in both spaces fixed bases are used; in fact,  dF_xv = v^j(∂_jF^a)(x).  The holomorphic version of the rank theorem [DG  pp. 33-34].  The identities (see Problem 1 in  Homework #35):
    
    (35.2) [φv, w]  =  φ[v, w] − (d_wφ)v − (Im φ)(£_wJ)v,         (£_φvJ)u  =  φ(£_vJ)u − (d_Juφ − id_uφ)v,
    
    valid whenever  v,w,u ∊ XM  and  φ ∊ F_cM  in any almost complex manifold  M.  The relation
    
    (35.3) H ^o(TM)  =  h(M)    if    M    is a complex manifold.
    
    Homework #35.
    
    
36. APRIL  9.    Product bundles. Holomorphic vector-bundle morphisms and (holomorphically) trivial bundles. Operations on holomorphic vector bundles: direct sum,  Hom,  the dual, and the tensor product. Tensor powers of line bundles with integer exponents. The equalities
    
    (36.1) H ^o(T ^*)  =  V^*,        H ^o(T)  =  {0}
    
    for the tautological line bundle  T  over a complex projective space  P(V),  the first of which is a natural isomorphic identification associating with a linear functional on  V  the family of its restrictions to one-dimensional vector subspaces of  V.   Homework #36.
    
    
37. APRIL  11.   Proof of the remaining left-to-right inclusion in (36.1) (using only Liouville’s theorem, rather than the Hartogs extension theorem): holomorphic sections of  T ^*  are naturally identified with degree-one homogeneous holomorphic functions  φ : V − {0}  →  C,  while for such  φ  and  v,w ∊ V  the holomorphic functions  d_vφ  and  d_wd_vφ  are homogeneous of degree  0  and, respectively, of degree  −1,  so that the latter tends to  0  and infinity and hence, restricted to an complex affine line not containing  0,   must be identically zero by Liouville’s theorem. Holomorphic subbundles. Quotient bundles. The determinant bundle. The (anti)canonical bundle of an almost complex manifold. The kernel and image of a constant-rank holomorphic vector-bundle morphism. Pullbacks of holomorphic vector bundles under holomorphic mappings, including their restrictions to complex submanifolds. The normal bundle of a complex submanifold. The natural structure of a complex manifold on the total space of a holomorphic complex vector bundle over a complex manifold. Holomorphic sections as complex submanifolds of the total space, including the zero section, always identified with the base manifold. The tautological line bundle  T  over a complex projective space  P(V)  as a subbundle of the product bundle  P = P(V) × V,  for any finite-dimensional complex vector space  V.  The natural isomorphic identifications
    
    (37.1) T[P(V)]  =  Hom(T, P ⁄T),       T^⊗m  =  det T*[P(V)].
    
    Homework #37.
    
    
38. APRIL  13.   The Picard group  Pic(M)  of a complex manifold  M,  formed by the isomorphism classes of holomorphic line bundles over  M  with the Abelian-group operation induced by the tensor product. Riemann surfaces. Divisors on a closed Riemann surface  S,  defined to be integer-valued functions  D  on  S  such that the support  S − D⁻¹(0)  is finite. The additive group  Div(S)  of all divisors on  S,  the degree (sum of values) homomorphism  deg : Div(S)  →  Z,  and the subgroup  Pr(S)  of all principal divisors  D^F  on  S,  naturally associated with nonconstant holomorphic mappings  F :  S  →  CP¹.  The divisor class group  Cl(S) = Div(S) ⁄ Pr(S).  The convention that
    
    (38.1) C   is a subset of   CP¹ =  P(C²),    namely,   CP¹ − {∞},
    
    where  ∞ = C(1, 0)  (the span of  (1, 0))  and  z ∊ C  is identified with  C(z, 1).  The group homomorphism
    
    (38.2) Div(S)  →  Pic(S)
    
    uniquely determined by the requirement that for each  x ∊ Q  it send the generator  D^[x] ∊ Div(S)  having the support  {x}  and  D^[x](x) = 1  to the isomorphism class of the holomorphic line bundle  D  over  S  defined as in (33.2) - (33.3) with  M = S  and  Q = {x}.  The fact that
    
    (38.3) Pr(S)    is contained in the kernel of    deg.
    
    Homework #38.
    
    
39. APRIL  16.   The degree of a nonconstant holomorphic mapping  F :  S  →  Q  between closed Riemann surfaces, defined to be the number of  F-preimages, counted with multiplicities, of any  y ∊ Q.  Assertion (38.3) as a trivial consequence of the fact that the degree does not dependent on the choice of  y.  The equality between the degree just defined and the ordinary mapping degree, which uses the oriented-integral isomorphism  Hⁿ(M, R)  →  R  for any (connected) compact oriented  n-dimensional manifold  M  (and a simple proof of the latter for a closed Riemann surface  M = S  is based on the Stokes theorem [DG  p. 166]:  if  μ  denotes the area (volume) form of a fixed Riemannian metric  g  on  S  and  f ∊ FS  is a smooth function  S  →  R  having  f_avg = 0,  then for  θ ∊ FS  with  Δθ = f  we obtain  2i∂∂θ = fΩ = fμ  due to (11.3) and two-dimensionality, so that  fμ  is exact; note that  Ω = μ  as  Ω_x(u, Ju) = g_x(Ju, Ju) = 1  for any  x ∊ S  and any unit vector  u  tangent to  S  at  x).  Holomorphic (local) sections of the line bundle over  S  arising from a divisor  D  as in (38.2), identified with (local) holomorphic mappings into  CP¹  with the order of zero, at each point  x,  not less than  D(x) = 1  (where poles are treated as zeros of negative order). Triviality of the line bundles associated with a principal divisor  D^F  on  S,  a global trivializing section being  F  itself.   Homework #39.
    
    
40. APRIL  18.   The natural biholomorphic identification
    
    (40.1) T ^* =  P(V × C) − {{0} × C}
    
    for the tautological line bundle  T  over a complex projective space  P(V),  obtained by assigning the graph of  ξ :  Λ  →  C  to any pair  (Λ, ξ) ∊ T ^*,  with  Λ ∊ P(V).  The holomorphic Gauss mapping  G : B  →  V  for a holomorphic vector subbundle of the product bundle  P = M × V  over a complex manifold  M,  where  V  is any finite-dimensional complex vector space. Example:  G : T  →  V  for  T, V  as above. The restriction biholomorphism  G : T − P(V)  →  V − {0}.  Holomorphic fibre bundles, including the projectivization  P(B),  with the fibres
    
    (40.2) [P(B)]_x  =  P(B_x)
    
    for  x ∊ M,  and
    
    (40.3) the projective compactification     P(B ⊕ [M × C])
    
    of a holomorphic vector bundle  B  over a complex manifold  M.  The biholomorphism  W − {0}  →  W* − {0}  for any one-dimensional complex vector space  W,  given by
    
    (40.4) w  →   w ⁻¹,      with    w ⁻¹(w)  =  1.
    
    Gluing of two complex manifolds with the aid of a biholomorphism between a pair of their open submanifolds, leading to a new complex manifold as long as the Hausdorff axiom is satisfied. The observation that, for the total space of the projective compactification of a given complex line bundle  L  over a complex manifold  M,
    
    (40.5) P(L ⊕ [M × C])   arises from the fibrewise gluing of   L   and   L^*,
    
    via (40.4), along the complements of the zero section  M.  The resulting biholomorphic identification
    
    (40.6) P(L ⊕ [M × C])  =  P(L^*⊕ [M × C])
    
    between the projective compactifications of  L  and  L^*.   Homework #40.
    
    
41. APRIL  20.   The blow-up of a complex manifold  M  at a point  x,  defined to be the complex manifold  N  with the underlying set
    
    (41.1) N   =   (M − {x})  ∪  P(T_x M)
    
    and with the complex manifold structure obtained by choosing a biholomorphism  F : B  →  U  between a neighborhood  B  of  0  (a “ball”) in  T_x M,  and a neighborhood  U  of  x  in  M  such that  F(0) = x  and  dF₀ = Id,  and then gluing the total space  T  of the tautological line bundle over  P(T_x M)  to  M  using the following biholomorphism between open subsets:
    
    (41.2) F _º G : G⁻¹(B − {0})   →  U − {x}.
    
    Independence of the resulting complex structure of the choice of such  B, U  and  F.  The fact that, for the total space  T  of the tautological line bundle over  P(V),  where  V  is any finite-dimensional complex vector space,
    
    (41.3) T     is the blow-up of    V    at the point    0,
    
    since the gluing as above with  M = U = B = V = T_x M  and  F = Id  realizes  M − {x} = V − {0}  as a subset of  T  (namely,  T − P(V)).  The local nature of blow-ups:
    
    (41.4) the blow-up of   U   at   x   is open in the blow-up of   M   at   x
    
    whenever  U  is an open submanifold of a complex manifold  M  and  x ∊ U.  The natural biholomorphic identification
    
    (41.5) [the blow-up of  P(V × C)  at the point  {0} × C]   =   P(T ⊕ [M × C])
    
    Proof of (41.5) obtained by combining (40.5) for  L = T  with (40.1) and (41.3) - (41.4), where  V  is identified with the image of  V × {1}  under the holomorphic projection mapping  (V × C) − {(0, 0)}  →  P(V × C).   Homework #41.
    
    
42. APRIL  23.   Elliptic complexes and the Hodge-de Rham decomposition theorem. The case of differential forms. Hodge’s theorem and the Hodge decomposition for Kähler manifolds [Zheng’s book: Section 8.1]. Ampleness and very-ampleness of holomorphic line bundles over a complex manifolds [Zheng’s book: Section 5.4], and their generalizations to holomorphic vector bundles of any fibre dimension [Zheng’s book: Section 8.4]. The Chern connection of a Hermitian fibre metric on a holomorphic vector bundle [Zheng’s book: Section 7.3]. Positive line bundles and the Kodaira embedding theorem [Zheng’s book: Section 8.3].   Homework #42.