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**Tropical algebraic geometry.
2nd ed.**
*(English)*
Zbl 1165.14002

Oberwolfach Seminars 35. Basel: Birkhäuser (ISBN 978-3-0346-0047-7/pbk). viii, 104 p. (2009).

This textbook is one of the first self-contained introductions to tropical geometry, i.e. algebraic geometry over \((\max,+)\)-arithmetics. The book is mostly dedicated to recent applications of this emerging field in enumerative algebraic geometry. In the first chapter, basics of tropical geometry are presented: amoebas and non-archimedean amoebas (including the non-standard analysis approach), tropical curves and tropicalizations of algebraic hypersurfaces.

The third chapter contains applications of tropical geometry to enumerative algebraic geometry: Gromov-Witten and Welschinger invariants are introduced and studied by means of G. Mikhalkin’s correspondence theorem [J. Am. Math. Soc. 18, 313–377 (2005; Zbl 1092.14068)], which establishes a certain correspondence between algebraic and tropical singular curves. As an application, the asymptotics of Gromov-Witten and Welschinger invariants is studied [I. Itenberg, V. Kharlamov and E. Shustin, Russ. Math. Surv. 59, No. 6, 1093–1116 (2004); translation from Usp. Mat. Nauk 59, No. 6, 85-110 (2004; Zbl 1086.14047)].

The second chapter is dedicated to the proof of Mikhalkin’s correspondence theorem and contains all necessary preliminaries: toric varieties, patchworking (including the proof of Viro’s patchworking theorem and applications to real algebraic geometry), and singular patchworking. Proof of the correspondence theorem is reduced to a number of transversality problems, whose systematic treatment is outside the scope of the textbook (see the claim of subsection 2.4.4 and lemmas in subsection 2.5.10; the reader is then referred to the third author’s paper [E. Shustin, Topology 37, No. 1, 195–217 (1998; Zbl 0905.14008)]). Except for this reference, the proof is self-contained. Inevitably for a textbook, explanations of certain technical issues are somewhat intuitive, which therefore allows to make presentation very transparent and helpful for a beginner.

It was not an intention of the authors to overview other sides and applications of tropical geometry (multidimensional generalization, connections with combinatorics and Newton polyhedra, tropical intersection theory, etc.). The book features a collection of exercises and a representative list of references to other lecture notes on tropical geometry and related topics (some are added in the second edition).

The third chapter contains applications of tropical geometry to enumerative algebraic geometry: Gromov-Witten and Welschinger invariants are introduced and studied by means of G. Mikhalkin’s correspondence theorem [J. Am. Math. Soc. 18, 313–377 (2005; Zbl 1092.14068)], which establishes a certain correspondence between algebraic and tropical singular curves. As an application, the asymptotics of Gromov-Witten and Welschinger invariants is studied [I. Itenberg, V. Kharlamov and E. Shustin, Russ. Math. Surv. 59, No. 6, 1093–1116 (2004); translation from Usp. Mat. Nauk 59, No. 6, 85-110 (2004; Zbl 1086.14047)].

The second chapter is dedicated to the proof of Mikhalkin’s correspondence theorem and contains all necessary preliminaries: toric varieties, patchworking (including the proof of Viro’s patchworking theorem and applications to real algebraic geometry), and singular patchworking. Proof of the correspondence theorem is reduced to a number of transversality problems, whose systematic treatment is outside the scope of the textbook (see the claim of subsection 2.4.4 and lemmas in subsection 2.5.10; the reader is then referred to the third author’s paper [E. Shustin, Topology 37, No. 1, 195–217 (1998; Zbl 0905.14008)]). Except for this reference, the proof is self-contained. Inevitably for a textbook, explanations of certain technical issues are somewhat intuitive, which therefore allows to make presentation very transparent and helpful for a beginner.

It was not an intention of the authors to overview other sides and applications of tropical geometry (multidimensional generalization, connections with combinatorics and Newton polyhedra, tropical intersection theory, etc.). The book features a collection of exercises and a representative list of references to other lecture notes on tropical geometry and related topics (some are added in the second edition).

Reviewer: Alexander Esterov (Moscow)

### MSC:

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14T05 | Tropical geometry (MSC2010) |

15A80 | Max-plus and related algebras |

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |

14P25 | Topology of real algebraic varieties |

14H99 | Curves in algebraic geometry |