Tropical Geometry Grigory Mikhalkin, Johannes Rau November 16, 2018 Contents 1 Introduction5 1.1 Algebraic geometry over what? . .5 1.2 Tropical Numbers . .8 1.3 Tropical monomials and integer affine geometry . 11 1.4 Forgetting the phase leads to tropical numbers . 15 1.5 Amoebas of affine algebraic varieties and their limits . 17 1.6 Patchworking and tropical geometry . 22 1.7 Concluding Remarks . 28 2 Tropical hypersurfaces in Rn 29 2.1 Polyhedral geometry dictionary I . 30 2.2 Tropical Laurent polynomials and hypersurfaces . 34 2.3 The polyhedral structure of hypersurfaces . 40 2.4 The balancing condition . 47 2.5 Planar Curves . 56 2.6 Floor decompositions . 69 3 Projective space and other tropical toric varieties 76 3.1 Tropical affine space Tn ....................... 76 3.2 Tropical toric varieties . 80 3.3 Tropical projective space . 84 3.4 Projective hypersurfaces . 88 3.5 Projective curves . 95 4 Tropical cycles in Rn 100 4.1 Polyhedral geometry dictionary II . 100 4.2 Tropical cycles and subspaces in Rn ................ 102 4.3 Stable intersection . 106 4.4 The divisor of a piecewise affine function . 116 4.5 Tropical modifications . 127 2 Contents 4.6 The projection formula . 131 4.7 Examples and further topics . 136 5 Projective tropical geometry 145 5.1 Tropical cycles in toric varieties . 145 5.2 Projective degree . 149 5.3 Modifications in toric varieties . 153 5.4 Configurations of hyperplanes in projective space . 166 5.5 Equivalence of rational curves . 169 5.6 Equivalence of linear spaces . 170 6 Tropical cycles and the Chow group 172 6.1 Tropical cycles . 172 6.2 Push-forwards of tropical cycles . 173 6.3 Linear equivalence of cycles . 175 6.4 Algebraic equivalence of cycles . 176 6.5 Linear systems . 177 6.6 Recession Fans . 177 7 Tropical manifolds 179 7.1 Tropical spaces . 179 7.2 Regularity at Infinity . 187 7.3 Tropical morphisms and structure sheaves . 193 7.4 Smooth tropical spaces . 196 7.5 Rational functions, Cartier divisors and modifications on ab- stract spaces . 206 8 Tropical curves 210 8.1 Smooth tropical curves . 210 8.2 Divisors and linear systems . 213 8.3 Line bundles on curves . 215 8.4 Riemann-Roch theorem . 221 9 Tropical jacobians 229 9.1 (Coarse) differential forms, integration, and the Jacobian . 229 9.2 Abelian varieties, polarization, and theta functions . 234 9.3 The Abel-Jacobi theorem . 238 9.4 Riemann’s theorem . 250 3 Acknowledgements We benefited from numerous valuable conversations, in particular with Vic- tor Batyrev, Benoît Bertrand, Erwan Brugallé, Ana Cannas da Silva, Jim Carlson, Yakov Eliashberg, Sergey Fomin, Andreas Gathmann, Mark Gross, Amihay Hanany, Ilia Itenberg, Michael Kapovich, Mikhail Kapranov, Lud- mil Katzarkov, Yujiro Kawamata, Sean Keel, Viatscheslav Kharlamov, Askold Khovanskii, Barak Kol, Maxim Kontsevich, Lionel Lang, Conan Leung, An- drey Losev, Diane Maclagan, Hannah Markwig, Pierre Milman, Yong-Geun Oh, Mikael Passare, Jean-Jacques Risler, Hans Rullgård, Kristin Shaw, Eu- genii Shustin, Yan Soibelman, David Speyer, Bernd Sturmfels, Louis Felipe Tabera, Ilya Tyomkin, Israel Vainsencher, Oleg Viro, Ilia Zharkov and Günter Ziegler on tropical geometry, especially during early stages of its develop- ment. (To be continued.) We would like to acknowledge support and encouragement from the Clay Mathematical Institute (USA) at the initial stage of this project. The project was also supported in part by the European Research Council (TROPGEO project), the Swiss National Science Foundation (projects no. 125070 and 140666). We would also like to thank the Mathematical Sciences Research Institute in Berkeley and the Max-Planck-Institut für Mathematik in Bonn for hospitality at different times during work on this project. 4 1 Introduction 1.1 Algebraic geometry over what? The two words “Algebraic Geometry” may invoke different images for dif- ferent people today. For some it might be something very abstract, and even somewhat formalistic, adapted to work in the most general setting possible. For many others, including the authors of this book, Algebraic Geometry is a branch of geometry in the first place — geometry of spaces defined by polynomial equations. Note that (unlike many algebraic properties) the resulting geometry de- pends not only on the type of the defining equations, but also on the choice of the numbers where we look for solutions. The two most classical choices are the field R of real numbers and the field C of complex numbers. Both these fields come naturally enhanced with the so-called “Euclidean topol- ogy” induced by the metric x y between two points x, y R (or in C). Furthermore, real algebraicj varieties− j are differentiable manifolds2 (perhaps with singularities) from a topological viewpoint, and complex algebraic va- rieties are special kind of real algebraic varieties of twice the dimension. To illustrate the two parallel classical theories let us recall the classical example of the so-called elliptic curve. Namely, consider a cubic curve in the complex projective plane CP2. As long as the defining cubic polynomial is chosen generically, the resulting curve is topologically a 2-dimensional torus (see Figure 1.1). This torus is embedded in the complex projective plane CP2. As CP2 is 4-dimensional such embedding is beyond our imagination tools. Now consider the case of real coefficients. Even if the defining polynomial is chosen generically, the topological type of the curve is not fixed. But there are only two possible cases, see Figure 1.2. As the ambient real projective plane RP2 is indeed 2-dimensional, we can actually draw how the curve is embedded there. Recall that topologically 5 1 Introduction Figure 1.1: A complex elliptic curve. Figure 1.2: Two real elliptic curves. RP2 can be obtained from a disc D2 by identifying the antipodal points on its boundary circle, see Figure 1.3. antipodal points get identified Figure 1.3: Real projective plane. 6 1 Introduction oval non- non- trivial trivial comp. comp. Figure 1.4: Elliptic curves in the real projective plane. Figure 1.4 depicts embeddings of cubic curves in the real projective plane. Note that there might be different pictures inside D2 before the self-iden- tification of its boundary, but we get one of the two pictures above in 2 2 RP = D = for any smooth real curve. An example is given in Figure 1.5. ∼ non-trivial comp. oval Figure 1.5: An elliptic curve in another presentation of RP2 by a disk. Furthermore, the inclusion R C gives us an inclusion RP2 CP2 as well as an inclusion of the real curve⊂ into the corresponding complex⊂ curve (see Figure 1.6). We see that the same equation may yield quite different geometric spaces. At the same time, real and complex numbers may be the only examples of fields where algebraic geometry is that much geometric. There is also a continuation of this series with the quaternions H and octonions O which leads to very interesting geometry no longer based on fields as we loose 7 1 Introduction oval non-trivial component Figure 1.6: Complex elliptic curve with its real locus. commutativity in the case of H or even associativity in the case of O. In this book we study geometry based on a predecessor of the entire series R, C, H, O, called the tropical numbers T. 1.2 Tropical Numbers We consider the set T = [ , + ) = R −∞ 1 [ {−∞} enhanced with the arithmetic operations “x + y” = max x, y , (1.1) f g “x y” = x + y, (1.2) where we set “( )+ x” = “x +( )” = x and “( )x” = “x( )” = . These operations−∞ are called−∞tropical arithmetic−∞ operations.−∞ We use quotation−∞ marks to distinguish them from the usual operations on R. Definition 1.2.1 The set T enhanced with the arithmetic operations (1.1) and (1.2) is called the set of tropical numbers. 8 1 Introduction Remark 1.2.2 There are papers where min x, y is taken for tropical addition. In such case one has to modify the setf of tropicalg numbers to include + and ex- clude . It is hard to say which choice is better. The choice of1 max may be more−∞ natural from the mathematical viewpoint as we are more used to taking the logarithm whose base is greater than 1, cf. equation (1.4). Also when we add two numbers in this way the sum does not get smaller. How- ever in some considerations in Computer Science and Physics (cf. [IM12]) taking the minimum is more natural. Clearly it does not really matter as the two choices are isomorphic under x x. 7! − Remark 1.2.3 The term “tropical” was borrowed from Computer Science, where it was reportedly introduced to commemorate contributions of the Brazilian com- puter scientist Imre Simon. Simon introduced the semiring (N, min, +) which was later baptized “tropical” following a suggestion of Christian Chof- furt (see [Sim88; Cho92]). According to Jean-Eric Pin, the suggestion was (also) made by Dominique Perrin (see [Pin98]). Some years later, in the first days of what is now known as tropical geometry, the adjective “tropical” quickly outplayed other suggestions inspired by “non-archimedean amoe- bas”, “idempotent algebraic geometry” or “logarithmic limit sets”. The set T is a semigroup with respect to tropical addition. It is commu- tative, associative and admits the neutral element 0T = . Neverthe- less we do not get a group as there is no room for subtraction.−∞ Indeed, if “x + y” = 0T, then either x = 0T or y = 0T. Thus the only element ad- mitting an inverse with respect to tropical addition is the neutral element 0T = . We−∞ may note that tropical addition is idempotent, i.e. we have “x+x” = x for all x T.