*(English)*Zbl 0685.32001

The author’s book [1st ed. (1966; Zbl 0138.062), 2nd. ed. (1973; Zbl 0271.32001)] is known to deal with topics in complex analysis in several variables based on the theory of partial differential equations, more exactly on the solution of the $\overline{\partial}$ Neumann problem as the main tool. In this third edition several additions are made where these methods are particularly important: The new section 7.7 is devoted to the study of modules with polynomial generators in order to prove the full “fundamental principle” of Ehrenpreis. For this reason the earlier section 7.6 on “Cohomology on bounds” is modified.

A new section 6.5 on analytic sets is added. Finally, a discussion of the theorem of Siu on Lelong numbers of plurisubharmonic functions is included.

##### MSC:

32-02 | Research monographs (several complex variables) |

32U05 | Plurisubharmonic functions and generalizations |

32D15 | Continuation of analytic objects (several variables) |

32E05 | Holomorphically convex complex spaces, reduction theory |

32E20 | Polynomial convexity |

32E10 | Stein spaces, Stein manifolds |

32W05 | $\overline{\partial}$ and $\overline{\partial}$-Neumann operators |

32B05 | Analytic algebras and generalizations, preparation theorems |