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Oberwolfach Seminars Volume 35 IElilali oIttte nHb. eLrigeb GRoribgeorrty SMeiirkinhgalekrin EJaung ePnhiii liSph Suostloinvej Jakob Yngvason TTrhoep iMcaalthematics of Athlgee Bboraseic Gas Ganedo mites tCryondensation Birkhäuser Verlag Basel Boston Berlin • • Authors: EIllilaio Ittte nHb. eLrigeb ERuogbeenrti iS Sehiruinstgienr DIReMpaAr,t Umneinvtesr soitfé M Loauthise mPaasttiecus rand Physics SDcehpoaortlm oef nMt aotfh Pehmyasticicsal Sciences P7r rinucee Rtoenné U Dneivscearsritteys RPariynmceotnodn aUnndiv Beersvietryly Sackler Faculty J6a7d0w84in S Htraalsl,b Po.uOr. gB Coexd 7e0x8 oJaf dEwxainc tH Saclile, nP.cOe.s Box 708 PFrrainncceeton, NJ 08544 TPerli nAcveitvo Un,n NivJe r0s8it5y44 Ue-SmAail: [email protected] RUaSmAat Aviv, 69978 Tel Aviv [email protected] [email protected] e-mail: [email protected] GJarnig Pohryil iMp iSkohlaolvkeinj Jakob Yngvason DDeeppaarrttmmeenntt ooff MMaatthheemmaattiiccss Institut für Theoretische Physik UUnniivveerrssiittyy ooff TCooropnetnohagen Universität Wien TUonroivnetrosi, tOetnstp. aMrk5eSn 25E4 Boltzmanngasse 5 C21a0n0a dCaopenhagen 1090 Wien eD-emnmaila:r [email protected] Austria [email protected] [email protected] 22000000 MMaatthheemmaattiiccaall SSuubbjjeecctt CCllaassssiiffiiccaattiioonn 1842MBx2x5, 14N35, 14N10, 52B20, 14P25, 14H99 Library of Congress Control Number: 2007920172 A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliografische Information Der Deutschen Bibliothek Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über <http://dnb.ddb.de> abrufbar. ISBN 937-786-34-37-674333-68-390 B9i-r1kh Bäiurksehrä uVseerrla Vge, rBlaags,e lB –as Belo –st oBno s–to Bne –rl iBnerlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. ©2005 by Elliott H. Lieb, Robert Seiringer, Jan Philip Solovej, Jakob Yngvason Published by Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland ©Pa r2t0 o0f7 SBpirriknhgäeurs eSrc iVeenrclea+g BAuGs,i nP.eOs.s B Moxe d1i3a3, CH-4010 Basel, Switzerland PCaorvte or fd Sepsirginng: eMr iScchiean Lcoet+roBvusskiyn,e CssH M-4e1d0i6a Therwil, Switzerland PPrriinntteedd oonn aacciidd--ffrreeee ppaappeerr pprroodduucceedd offro cmhl ocrhilnoer-infree-ef rpeue lpp.u TlpC.F T(cid:102)C(cid:3)F (cid:39) PPrriinntteedd iinn GGeerrmmaannyy IISSBBNN--1100:: 33--77664433--878333030969---797 e-ISBN-10: 3-7643-8310-0 IISSBBNN--1133:: 997788--33--77664433--878333030969---181 e-ISBN-13: 978-3-7643-8310-7 99 88 77 66 55 44 33 22 11 www.birkhauser.cwhww.birkhauser.ch Contents Preface vii 1 Introduction to tropical geometry 1 1.1 Images under the logarithm . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Families of amoebas . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Non-Archimedean amoebas . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Non-standard complex numbers . . . . . . . . . . . . . . . . . . . . 7 1.5 The tropical semifield T . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Tropical curves and integer affine structure . . . . . . . . . . . . . 11 1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Patchworking of algebraic varieties 17 2.1 Introduction: A general idea of the patchworking construction . . . 17 2.2 Elements of toric geometry . . . . . . . . . . . . . . . . . . . . . . 19 2.2.1 Construction of toric varieties . . . . . . . . . . . . . . . . . 19 2.2.2 A toric variety associated with a fan . . . . . . . . . . . . . 20 2.2.3 A toric variety associated with a convex lattice polyhedron 21 2.2.4 Embedding of Tor(∆) into a projective space . . . . . . . . 22 2.2.5 The real part of a toric variety and the moment map . . . . 22 2.2.6 Hypersurfaces in toric varieties . . . . . . . . . . . . . . . . 24 2.3 Viro’s patchworking method . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 Chart of a real polynomial. . . . . . . . . . . . . . . . . . . 25 2.3.2 Patchworkingof real nonsingular hypersurfaces . . . . . . . 27 2.3.3 Combinatorialpatchworking. . . . . . . . . . . . . . . . . . 30 2.3.4 A tropical point of view on the combinatorial Viro patchworking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.5 Patchworkingof pseudo-holomorphic curves on ruled surfaces 37 2.4 Patchworkingof singular algebraic hypersurfaces . . . . . . . . . . 45 2.4.1 Initial data . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.4.2 Transversalityconditions . . . . . . . . . . . . . . . . . . . 46 2.4.3 The patchworking theorem . . . . . . . . . . . . . . . . . . 47 2.4.4 Some S-transversality criteria . . . . . . . . . . . . . . . . . 50 vi Contents 2.5 Tropicalizationand patchworking in the enumerationof nodal curves 51 2.5.1 Plane tropical curves . . . . . . . . . . . . . . . . . . . . . . 52 2.5.2 Algebraic enumerative problem and its tropical analogue. . 54 2.5.3 Tropical formulas for the Gromov–Witten and Welschinger invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.5.4 Tropical limit . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.5.5 Tropicalization of nodal curves . . . . . . . . . . . . . . . . 57 2.5.6 Reconstruction of a simple tropical curve . . . . . . . . . . 61 2.5.7 Reconstruction of the limit curve C(0) . . . . . . . . . . . . 63 2.5.8 Refinement of a tropical limit . . . . . . . . . . . . . . . . . 64 2.5.9 Refinement of the condition to pass through a fixed point . 67 2.5.10 Refined patchworking theorem . . . . . . . . . . . . . . . . 69 2.5.11 The real case: Welschinger invariants . . . . . . . . . . . . . 73 2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3 Applications of tropical geometry to enumerative geometry 77 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2 Tropical hypersurfaces in Rn . . . . . . . . . . . . . . . . . . . . . 78 3.3 Geometric description of plane tropical curves . . . . . . . . . . . . 81 3.4 Count of complex nodal curves . . . . . . . . . . . . . . . . . . . . 83 3.5 Correspondence theorem . . . . . . . . . . . . . . . . . . . . . . . . 84 3.6 Mikhalkin’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.7 Welschinger invariants . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.8 Welschinger invariants W for small m. . . . . . . . . . . . . . . . 88 m 3.9 Tropical calculation of Welschinger invariants . . . . . . . . . . . . 89 3.10 Asymptotic enumeration of real rational curves . . . . . . . . . . . 90 3.11 Recurrence formula for Welschinger invariants . . . . . . . . . . . . 92 3.12 Welschinger invariants W . . . . . . . . . . . . . . . . . . . . . . 93 m,i 3.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 List of Figures 97 Bibliography 99 Preface This book is based on the lectures given at the Oberwolfach Seminar on Tropical Algebraic Geometry in October 2004. Tropical Geometry first appeared as a subject of its own in 2002, while its roots can be traced back at least to Bergman’s work [1] on logarithmic limit sets. Tropical Geometry is now a rapidly developing area of mathematics. It is inter- twined with algebraic and symplectic geometry, geometric combinatorics, inte- grablesystems,andstatisticalphysics.TropicalGeometrycanbeviewedasasort of algebraic geometry with the underlying algebra based on the so-called tropical numbers.Thetropicalnumbers(theterm“tropical”comesfromcomputerscience and commemorates Brazil, in particular a contribution of the Brazilian school to the language recognition problem) are the real numbers enhanced with negative infinity and equipped with two arithmetic operations called tropicaladdition and tropical multiplication. The tropical addition is the operation of taking the max- imum. The tropical multiplication is the conventional addition. These operations are commutative, associative and satisfy the distribution law. It turns out that such tropical algebra describes some meaningful geometric objects, namely, the Tropical Varieties. From the topological point of view the tropical varieties are piecewise-linearpolyhedralcomplexesequippedwithaparticulargeometricstruc- ture coming from tropical algebra. From the point of view of complex geometry this geometric structure is the worst possible degeneration of complex structure ona manifold.Fromthe point ofview of symplectic geometrythe tropicalvariety is the result of the Lagrangiancollapse of a symplectic manifold (along a singular fibration by Lagrangiantori). The targetaudience ofthe Oberwolfachseminarwasgraduatestudents.The seminar was designed to introduce young mathematicians to this perspective re- search field, including presentation of basic notions and motivations for tropical algebraic geometry as well as demonstration of some of its striking applications. Duringthepreparationoftheselecturenotesforpublication,weadaptedthenotes to a wider audience, both beginners and established researchers. As a result, the discussionsinthis bookaremoredetailedandcontainsomenew resultsthatwere obtained after the seminar itself. Besidesageneralintroductiontotropicalgeometry,wediscusstheconceptsof complexandnon-Archimedeanamoebas,aswellasthepatchworkingconstruction viii Preface andenumerativetropicalgeometry.Foramoreadvancedstudyofthesetopics,we recommend the articles [7, 21, 26, 37, 38, 39, 40, 45, 56]. We do not in this book attempt to cover all facets of tropical geometry. For instance,we donotdiscuss the combinatorialaspects oftropicalvarieties (see,for example, [29, 51, 59, 61]), or abstract tropical varieties of dimension greater than 1 [17, 32, 33]. Furthermore, we do not touch various other branches of tropical mathematics, but only recommend some references: [34, 61] (computational as- pects), [4, 14, 50] (max-plus algebra), [8, 30, 35, 47, 49] (tropical mathematics in applied problems). Thebookconsistsofthreechapters.Thefirstchapter,“Introductiontotropi- calgeometry”byG.Mikhalkin,isabasic treatmentoftropicalvarietiesandtheir relation to classical geometry, in particular the theory of amoebae. Special em- phasis is put on tropical curves. The second chapter, “Patchworking of algebraic varieties” by E. Shustin, deals with the patchworking construction in algebraic geometry, the link between real algebraic geometry and tropical geometry. The chapterstartswiththeoriginalViromethodofgluingrealalgebraichypersurfaces, then goes through various modifications and generalizations of the Viro method. InthefinalsectionthepatchworkingconstructionisusedtoproveMikhalkin’scor- respondence theorem between real and complex algebraiccurves on toric surfaces on one side and plane tropical curves on the other side. The third chapter, “Ap- plications of tropical geometry to enumerative geometry” by I. Itenberg, presents various applications, based on Mikhalkin’s correspondence theorem, of tropical geometry in real and complex enumerative geometry. These applications mostly concerncalculationofGromov–WitteninvariantsandWelschingerinvariants(the latter invariants can be seen as real counterparts of genus zero Gromov–Witten invariants). Each chapter is supplemented by exercises, most of which were proposed to and discussed by the participants of the seminar. Acknowledgements. We are grateful to Mathematisches Forschungsinstitut Oberwolfach for a unique opportunity to run a seminar on tropical algebraic geometry. Our special thanks go to Oliver Wienand. We are very grateful to him for taking notes of our lectures and helping in expanding them for publication. His role in the work on this book is hard to overestimate. The first author was partially supported by the ANR-05-0053-01 grant of Agence Nationale de la Recherche and a grantof Université Louis Pasteur,Stras- bourg. The second author is supported in part by NSERC. The third author was supported by the Hermann-Minkowski-Minerva center for Geometry at the Tel Aviv University and by the grant no. 465/04 from the Israel Science Foundation. The first and the third authors were partially supported by a grant of the Min- istèredesAffairesEtrangères,FranceandtheMinistryofScienceandTechnology, Israel. Chapter 1 Introduction to tropical geometry Inthis sectionthe notionofanamoebaofavarietywillbe introducedandseveral examples of such amoebas are given. Then we consider a degenerations process where an amoeba becomes a piecewise-linear object. 1.1 Images under the logarithm We start with an algebraic variety over (C∗)n. Namely, let I be an ideal in the ring of polynomials in n variables over C. Then the variety is given by V ={x∈(C∗)n | f(x)=0 for all f ∈I}. We define the map Log:(C∗)n −→Rn, (z ,z ,...,z )(cid:3)−→(log|z |,log|z |,...,log|z |). 1 2 n 1 2 n Definition 1.1. Let V ⊂(C∗)n be an algebraic variety. Then we define its amoeba as A(V)=Log(V). This is a subset of Rn: A(V)=LogV ⊂Rn . 0-dimensional amoebas If V is 0-dimensional, then it is just a collection of points and so is Log(V). Amoeba of a line in P2 For our first example of an amoeba of a 1-dimensional variety, consider the case when V ⊂(C∗)2 ⊂CP2 is a line given by equation z+w+1=0. (1.1)