##
**Complex manifolds.**
*(English)*
Zbl 0325.32001

Athena Series. Selected Topics in Mathematics. New York etc.: Holt, Rinehart and Winston, Inc. VII, 192 p. $ 10.00 (1971).

This book, based on lectures of Kodaira at Stanford University in 1965–1966, develops some of the most deep and fascinating aspects of the theory of the complex analytic manifolds, especially the deformation theory. Dealing with problems, to which Kodaira mainly contributed, and which involve complex analysis, differential geometry, as well as elliptic partial differential equations and algebraic topology, the bock remains self-contained (except for some standard results about elliptic partial differential equations, for which complete references are given) and the great number of examples permits a profound understanding of the subject. Examples taken from the Riemann surfaces case make the book very interesting also for the specialists in this field. The lines of development of the theory and its importance are pointed out by formulation of principal problems and succinct remarks. Chapter 1 introduces the complex (analytic) manifolds and the \(\Gamma\)-manifolds \((\Gamma-a\) pseudogroup of local diffeomorphisms of a domain of \(\mathbb{R}^n\) or \(\mathbb{C}^n)\). It contains examples and constructions of compact complex manifolds (submanifolds, quotient manifolds, complex tori and Hopf manifolds, surgeries, quadric transformations). Complex analytic families \(\{M_t|t\in B\}\) of compact complex manifolds \(M_t\) \((B-a\) connected complex manifold) and deformations of such a manifold \(M\) are defined, and numerous examples discussed.

Chapter 2 begins with the notions of germs of functions, sheaves and cohomology. Its main aim is to define the infinitesimal deformation \(\partial M_t/\partial t^\nu\) \((t=(t^1, \dots,t^n)\) a local coordinate on \(B)\) as a cohomology class of \(H^1(M_t, \Theta_t)\), where \(\Theta_t\) is the sheaf of germs of holomorphic vector fields on \(M_t\). Then the authors consider locally trivial complex analytic families, rigid compact complex manifolds and complete complex analytic families and give several theorems: for instance, those of Kodaira and Spencer on the local triviality and on the semicontinuity of \(\dim H^1(M_t,\Theta_t)\), the criterium \(H^1(M, \Theta)=0\) for the rigidity of \(M\), as well as Kodaira and Spencer completeness criterium. The cohomology sequence corresponding to a short exact sequence of sheaves, complex vector bundles (especially line bundles, tangent bundles and their duals, tensor bundles, differential forms as sections in tensor bundles), Poincaré’s lemma and de Rham’s theorem, the lemma and the theorem of Dolbeault complete this chapter.

In chapter 3 the authors develop the geometry of the complex manifolds, having as the main aim the Kähler ones, for which they give different characterisations. Let \(M\) be a compact complex manifold, \(A^{p,q}\) the sheaf over \(M\) of germs of \(C^\infty(p,q)\)-forms and \({\mathcal L}^q= \Gamma(A^{p,q})\) its sections; they introduce an Hermitian scalar product in \({\mathcal L}^q\), the operator, the adjoint operators of \(\overline \partial,\partial,d\), the Laplacians \(\square,\Delta\), and extension is done to the holomorphic vector bundles \(F\) over \(M\), with respect to a metric given by an Hermitian form on the fibres of \(F\). Using results on the elliptic operators concerning \({\mathcal L}^q\) and \({\mathcal H}^q=\{\varphi \in{\mathcal L}^q|\square\varphi= 0\}\), the covariant differentiation and curvature tensors on Kähler manifolds, they establish fundamental theorems on the cohomology groups: Kodeira’s theorem \((H^q(M, \Omega^p)\cong{\mathcal H}^q\), where \(\Omega^p\) is the sheaf of germs of holomorphic \(p\)-forms on \(M)\), Serre’s duality, Hodge-Kodaira-de Rham theorem \((H^q(M,\Omega^p)\simeq H^p(M,\Omega^q)\) and \(H^r(M,\mathbb{C})\simeq\oplus H^p(M, \Omega^q))\), results on Betti numbers and Chern classes, vanishing theorems for \(H^q(M,{\mathcal O} (F))\) (especially for positive and for sufficiently positive \(F)\) as well as for \(H^q(M,\Omega^p(F))\), (Nakano’s theorem). The next section on Hodge manifolds contains other Kodaira’s important theorems: the embedding theorem (every Hodge manifold is (projective) algebraic) with its original proof, and criteria for compact Kähler manifolds to be algebraic. In chapter 4 the theory of elliptic partial differential equations and methods developed in the previous chapters are applied to the study of the small deformations of compact complex manifolds. It is proved for instance, that under certain conditions the complex analytic structure on \(M_t\) is represented by a vector \((0,1)\)-form \(\varphi (t)\). Given a compact complex manifold \(M\) existence theorems for deformations are proved, first in the case \(H^2(M,\theta)=0\) (Kodaira, Nirenberg, Spencer), then in the general case (Kuranishi). Semicontinuity and stability theorems (small deformations of a Kähler manifold are also Kähler) end the book.

Chapter 2 begins with the notions of germs of functions, sheaves and cohomology. Its main aim is to define the infinitesimal deformation \(\partial M_t/\partial t^\nu\) \((t=(t^1, \dots,t^n)\) a local coordinate on \(B)\) as a cohomology class of \(H^1(M_t, \Theta_t)\), where \(\Theta_t\) is the sheaf of germs of holomorphic vector fields on \(M_t\). Then the authors consider locally trivial complex analytic families, rigid compact complex manifolds and complete complex analytic families and give several theorems: for instance, those of Kodaira and Spencer on the local triviality and on the semicontinuity of \(\dim H^1(M_t,\Theta_t)\), the criterium \(H^1(M, \Theta)=0\) for the rigidity of \(M\), as well as Kodaira and Spencer completeness criterium. The cohomology sequence corresponding to a short exact sequence of sheaves, complex vector bundles (especially line bundles, tangent bundles and their duals, tensor bundles, differential forms as sections in tensor bundles), Poincaré’s lemma and de Rham’s theorem, the lemma and the theorem of Dolbeault complete this chapter.

In chapter 3 the authors develop the geometry of the complex manifolds, having as the main aim the Kähler ones, for which they give different characterisations. Let \(M\) be a compact complex manifold, \(A^{p,q}\) the sheaf over \(M\) of germs of \(C^\infty(p,q)\)-forms and \({\mathcal L}^q= \Gamma(A^{p,q})\) its sections; they introduce an Hermitian scalar product in \({\mathcal L}^q\), the operator, the adjoint operators of \(\overline \partial,\partial,d\), the Laplacians \(\square,\Delta\), and extension is done to the holomorphic vector bundles \(F\) over \(M\), with respect to a metric given by an Hermitian form on the fibres of \(F\). Using results on the elliptic operators concerning \({\mathcal L}^q\) and \({\mathcal H}^q=\{\varphi \in{\mathcal L}^q|\square\varphi= 0\}\), the covariant differentiation and curvature tensors on Kähler manifolds, they establish fundamental theorems on the cohomology groups: Kodeira’s theorem \((H^q(M, \Omega^p)\cong{\mathcal H}^q\), where \(\Omega^p\) is the sheaf of germs of holomorphic \(p\)-forms on \(M)\), Serre’s duality, Hodge-Kodaira-de Rham theorem \((H^q(M,\Omega^p)\simeq H^p(M,\Omega^q)\) and \(H^r(M,\mathbb{C})\simeq\oplus H^p(M, \Omega^q))\), results on Betti numbers and Chern classes, vanishing theorems for \(H^q(M,{\mathcal O} (F))\) (especially for positive and for sufficiently positive \(F)\) as well as for \(H^q(M,\Omega^p(F))\), (Nakano’s theorem). The next section on Hodge manifolds contains other Kodaira’s important theorems: the embedding theorem (every Hodge manifold is (projective) algebraic) with its original proof, and criteria for compact Kähler manifolds to be algebraic. In chapter 4 the theory of elliptic partial differential equations and methods developed in the previous chapters are applied to the study of the small deformations of compact complex manifolds. It is proved for instance, that under certain conditions the complex analytic structure on \(M_t\) is represented by a vector \((0,1)\)-form \(\varphi (t)\). Given a compact complex manifold \(M\) existence theorems for deformations are proved, first in the case \(H^2(M,\theta)=0\) (Kodaira, Nirenberg, Spencer), then in the general case (Kuranishi). Semicontinuity and stability theorems (small deformations of a Kähler manifold are also Kähler) end the book.

Reviewer: C. Andreian Cazacu

### MSC:

32Cxx | Analytic spaces |

32-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces |

35J99 | Elliptic equations and elliptic systems |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

32Gxx | Deformations of analytic structures |

32J99 | Compact analytic spaces |