Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
aa r X i v : . [ m a t h . AG ] S e p INTRODUCTION TO TROPICAL GEOMETRY,NOTES FROM THE IMPA LECTURES, SUMMER 2007
GRIGORY MIKHALKIN Tropical varieties
Generalities.A.
In our viewpoint an algebraic variety is a pair ( X, O ) consisting ofa topological space X and a sheaf of function (to the base field) called regular functions subject to some nice geometric properties. B. In tropical geometry the base field is not strictly speaking a field,but only a semifield T . It consists of the real numbers augmented by −∞ and enhanced by two arithmetic operation: “ x + y ” = max { x, y } and “ xy ” = x + y . These operations are commutative associate andsatisfy to the distribution law. We have the additive zero 0 T = −∞ and the multiplicative unit 1 T = 0. The multiplicative group T × = T r {−∞} is an honest Abelian group with a division, but there is nosubtraction whatsoever. The tropical addition is idempotent, “ x + x ” = x . C. A tropical Laurent polynomial in n variables is a function f ( x , . . . , x n ) = “ X a j ...j n x j . . . x j n n ” = max j j x + · · · + j n x n , where the sum is finite and the indices j l are integers. The space( T × ) a pprox R n equipped with the Euclidean topology and consideredwith the sheaf of tropical Laurent polynomial is an example of a tropicalvariety (analogous to the torus ( C ∗ ) n ). D. One can evaluate a Laurent polynomial at any point of R n andat some points of T n ⊃ R n . Namely, a Laurent polynomial is regularat a point of T n if it has a limit at this point bounded from above(possibly −∞ but not + ∞ ). This defines a regular sheaf on the tropicalaffine space T n . The only functions regular on the entire T n are honestpolynomials (such that all the powers j l are non-negative). The author is partially supported by the Canada Research Chair program,NSERC and PREA. E. Starting from R n we can get compact tropical varieties by taking aquotient under an arbitrary discrete lattice Λ ≈ Z n embedded to R n .Later we’ll see that some of these tropical tori are algebraic while someare not (as it is the case for abelian varieties and other complex tori).This gives us some examples of tropical varieties modeled on manifolds,but the underlying topological space for most tropical varieties is nota manifold but only a polyhedral complex.1.2. Polyhedral complexes and tropical varieties.A.
A convex polyhedron P with integer slope in R N is an intersectionof half-spaces in R N defined by equations ax ≥ b . Here x ∈ R N , a ∈ Z N , b ∈ R . A convex polyhedron in T N is a closure of P in T N ⊃ R N . A dimension of such a polyhedron is its usual topologicaldimension. B. An (integer-slope n -dimensional) polyhedral complex P in R N is aunion of a collection of convex polyhedra with integer slope of dimen-sion n such that the intersection of any subcollection is a common face.Again, to get a polyhedral complex in the tropical affine space T N wesimply take the closure. C. Sedentarity of a point in T N is the number of coordinates equal to −∞ . The sedentarity of a face of P is the sedentarity of a generic pointthere. We call points and faces of zero sedentarity inner . D. Consider an ( n − E of P . Let F , . . . , F k bethe facets ( n -dimensional faces) adjacent to E . Lel λ : R N → R N − n +1 be the projection whose kernel is parallel to E . The facets F , . . . , F k project to rays adjacent to λ ( E ). Clearly, these rays have integer slopein R N − n +1 . Let v , . . . , v k ∈ Z N − n +1 be the primitive integer vectorsparallel to λ ( F ) , . . . , λ ( F k ). We say that P is balanced if k P j =1 v j = 0. E. An n -dimensional tropical variety is a pair ( X, O ) if X is a Haus-dorff space and can be covered by finitely many open sets V α ⊂ X suchthat for each α there exists an open set U α ⊃ V α in X , a balancedpolyhedral complex P α ⊂ T N α and an embedding ι α : U α → P α withthe property that the closure of ι α ( V α ) is contained in ι α ( U α ) and thatthe restriction of the sheaf O to U α coincides with the restriction of O T Nα under ι α . NTRODUCTION TO TROPICAL GEOMETRY 3 F. It is easy to see that the product of a finite number of balancedpolyhedral complexes is again a balanced polyhedral complex and thusthe product of a finite number of tropical varieties is again a tropicalvariety. G. Remark.
It is also possible to define a smooth tropical variety bymodeling it on a simply balanced polyhedral complex . A slightly differ-ent class of tropical varieties is formed by images of smooth tropicalvarieties under tropical morphisms. This leads to a more proper defi-nition of tropical varieties. However, for the sake of simplicity in theselectures we use the definition given above.1.3.
Hypersurfaces and complete intersections.A.
Let f : T N → T be a regular function (tropical polynomial). The hypersurface V f defined by f is the set of points in T n where “ T f ” = − f is not regular. In other words, V f is the corner locus of the convexpiecewise-linear function f . B. The hypersurface V f is an ( n − T N . However, in general it is NOT balanced. Nevertheless, onemay associate a natural number (called the weight ) that makes it intoa weighted balanced polyhedral complex (so that we get zero when wemodify the equality from 1.2.D by weighting all the primitive vectors. C. Similarly, one can define a hypersurface associated to a regularfunction f on a tropical variety X (or an open set U ⊂ X ). D. The weights on the facets play the role of the orders of zeroes of f .Alternatively they may be defined recursively by saying that that theorder of “ f g ” at a facet F is a + b if a is the weight of f at F and b isthe weight of g at F . If all the weights of the facets of V f are 1 then V f is a tropical subvariety. Otherwise it is only a tropical (Cartier)divisor . E. More generally, one can define a tropical k -cycle A ⊂ X by defin-ing them as (locally) weighted balanced k -dimensional polyhedral sub-complex inside the tropical variety. It is convenient to allow not onlynatural, but integer weights. F. Warning.
It may happen that A is an ( n − n -dimensional variety X , but not a hypersurface for any regular function f (even locally and even in the case when X is smooth). In other wordsin tropical geometry there exist non-Cartier divisors even for smoothvarieties. GRIGORY MIKHALKIN G. A tropical subvariety Y ⊂ X is a balanced unweighted subcomplex(i.e. such a divisor that all the weights are 1). H. Example.
A tropical cubic polynomial in 3 variables defines asmooth hypersurface in T if the (partially defined) coefficient func-tion is convex. We may find generic cubics with 27 distinct lines as1-cycles. ( Warning: for some other generic choices we have positive-dimensional families of lines, so there have infinitely many lines, unlikethe complex case.)1.4.
Projective spaces and cycles there.A.
Let φ : T × → T × be defined by z “ z ”. The result of gluing oftwo copies of T ⊃ T × along φ is called the tropical projective line andis denoted with TP . This is the tropical counterpart of the Riemannsphere. B. As a topological space it is just a closed interval. Geometricallythe tropical structure on TP may be reformulated as the integer affinestructure on R = T × ⊂ TP , the stratum of sedentarity 0. Indeed, tobe able to define tropical polynomials we only need to know what isan affine-linear function with an integer slope. The boundary divisor TP r R n consists of two components, each isomorphic to the single-point space TP . C. In a similar way we may glue n + 1 copies of T n to get the tropicalprojective space TP n . The sedentarity 0 stratum is R n . Again its (tau-tological) integer affine-linear structure encodes the tropical structureon TP n . The complement TP n r R n consists of ( n + 1) divisors calledthe boundary divisors and isomorphic to TP n − . D. A point has sedentarity k if and only if it is contained in exactly k different boundary divisors. Points of sedentarity k are thus dividedinto (cid:18) n + 1 k (cid:19) strata that correspond to the k -dimensional faces of the(topological) n -simplex TP n . Each individual stratum is isomorphic to TP k . E. We may define the projective degree of a k -cycle A ⊂ TP n by com-puting its intersection number with one of the boundary TP n − k . If weequip each intersection point with the proper multiplicity then (thanksto the balancing condition) it does not depend on the choice of TP n − k . NTRODUCTION TO TROPICAL GEOMETRY 5 F. Instead of giving the definition of the intersection number in thisboundary case (which is also possible, but more complicated) we maychoose yet another (non-boundary and easier to use) representative of TP n − k . Take the negative parts of the n coordinate axes in R n andthe ray ( t, . . . , t ) , t ≥
0. Alltogether these are n + 1 rays emanatingfrom the origin. For each ( n − k ) out of these n + 1 rays we take theirconvex hull. This gives us (cid:18) n + 1 n − k (cid:19) ( n − k )-dimensional quadrants(each isomorphic to R n − k ≥ . We denote the union of all the closure ofthese quadrants in with L n − k ⊂ TP n . It is a tropical variety. G. Note that any translation in R n may be extended to TP n by takingthe closure. We may move L n − k by such translations τ . Note thatfor a generic choice of τ the polyhedral complexes A and τ L n − k aretransverse, i.e. they intersect in a finitely many points of sedentarity0 and these points are in the interior of the facets of A and τ L n − k respectively. H. At a point of a transverse intersection at interior points of a facetof dimension k and and a facet of dimension ( n − k ) the definition ofthe local intersection number is especially easy. Indeed, locally such anintersection is given by two transverse affine-linear subspaces. Let Λ ⊂ Z n (resp. Λ ) be the integer vectors parallel to the first (resp. second)subspace. We define the local intersection number as the index of thesubgroup Λ + Λ ⊂ Z n multiplied by the weights of the correspondingfacets. The balancing condition ensures that the intersection number isinvariant with respect to the deformations, in particular to translationsin R n of one of the cycles. I. Note that the subspace L k is not even homeomorphic to TP n . Never-theless, it can be considered as a result of deformation of the boundary TP k . The tropical varieties L k and TP k are not isomorphic but theyare equivalent .1.5. Equivalence of tropical varieties.A.
Let f : T n → T be a regular function. It is easy to see that theset-theoretical graph of f is NOT a hypersurface in T n +1 = T n × T .Namely, the balancing condition fails at the corner locus of f as therethe function is strictly convex. GRIGORY MIKHALKIN B. In the same time the set-theoretical graph of f is contained in thehypersurface Γ f defined by y + f ( x ) (where x ∈ T n , y ∈ T ). Thishypersurface can be considered as the “complete tropical graph” of f . It is different from the set-theoretical graph by attaching a ray { x } × [ −∞ , f ( x )] for each x ∈ V f . Furthermore, the weights of thecorresponding facets in V f and Γ f coincide. We note that Γ f is NOTisomorphic to T n . We postulate that Γ f and T n are equivalent. C. More generally, the same situation holds for a regular function on atropical variety X or an open set U ⊂ X . Recall that the hypersurface V f ⊂ U is still well-defined as the set of points where “ T f ” is not regularand we still have weights at the facets of V f . Thus we can still define Γ f by attaching { x } × [ −∞ , f ( x )] for each x ∈ V f and enhancing the newfacets with the weights coinciding with the weights of the correspondingfacets in V f . Alternatively one can define the weights of the new facetsof Γ f so as to make Γ f ⊂ T n +1 to be a weighted balanced polyhedralcomplex. D. Remark.
Once we’ll develop the tropical intersection theory belowwe’ll have yet another alternative way of definition of Γ f (together withthe weights on its facets). Namely, X locally coincides with a balancedpolyhedral complex P ⊂ T N and f comes as a restriction of a tropicalLaurent polynomial F : T N → T . We can define Γ f locally as the stableintersection of P × T and the hypersurface of y + F ( x ) in T N +1 . E. Definition.
We say that a map Y → X between tropical varieties isa tropical modification if locally it coincides with the projection Γ f → U for some regular f : U → T . Two tropical varieties are called equivalent if they can be connected with a sequence of tropical modifications (antheir inverse operations). F. Remark. If X is a tropical variety then the tropical structure on Y is defined by f if all the weights of Γ f are equal to 1. We may alterna-tively define SMOOTH n -dimensional tropical varieties as topologicalspaces with sheaves of functions locally equivalent to T n after addingsome finiteness and separability conditions. Note that this definition isanalogous to analytic definition of complex manifolds: a complex man-ifold is a topological space with a sheaf of functions locally isomorphicto C n . Tropical morphisms.
NTRODUCTION TO TROPICAL GEOMETRY 7 A. A map φ : Y → X is called a tropical morphism if for every y ∈ Y there exists a neighborhood y ∈ U ⊂ Y and an affine-linear map Φ : R N → R M such that the linear part of Φ is defined over Z and suchthat under the charts ι : U ⊂ T N and φ ( U ) ⊂ T M the map φ is givenby the closure of Φ to a partially defined map from T N to T M . Herewe demand that whenever z ∈ ι ( U ) the closure of Φ is actually definedat z . B. Note that if X and ˜ X are equivalent tropical varieties then thereexists their “common resolution” ¯ X , i.e. a tropical variety equivalentto X and ˜ X with tropical morphisms φ : ¯ X → X and ˜ φ : ¯ X → ˜ X thatare both composition of tropical modifications. Indeed, if ψ : V → U isa tropical modification corresponding to a regular function g : U → T and f : U → T is another regular function then f ◦ ψ is tropical on ψ − ( U ) and we can make consider a tropical modification W → V corresponding to f ◦ ψ . Furthermore, we can obtain the same variety W by making a tropical modification along f first and then by thepull-back of g . In any case W is obtained from U by attaching raysalong the hypersurfaces of g and f and by attaching a positive quadrantalong their stable intersection (see below) which is symmetric. C. Two tropical morphisms φ : Y → X and ˜ φ : ˜ Y → ˜ X are calledequivalent if ˜ Y is equivalent to Y , ˜ X is equivalent to X and ˜ φ cor-responds to φ under these equivalences (in other words, they can belifted to the same morphism from the common resolution of Y and ˜ Y to the common resolution of X and ˜ X .2. Curves
Tropical curves as metric graphs.A.
Zero-dimensional tropical varieties are just disjoint union of points.The first interesting example of tropical varieties appears in dimension1. Clearly, any tropical curve is a graph. A compact tropical curve is afinite graph ¯Γ. Let us assume in addition that ¯Γ is connected. Denotewith Γ the complement of the set of 1-valent vertices in ¯Γ. B. The graph ¯Γ has an additional geometric structure that carriesthe same information as enhancing it with a structure sheaf of regulartropical functions. This structure is the complete inner metric on Γ.Recall that a leaf is an edge adjacent to a 1-valent vertex. Completenessof the metric ensures that all leaves have infinite length. The otheredges (called inner edges ) have finite length.
GRIGORY MIKHALKIN C. Once Γ is equipped with a complete inner metric we can reconstructthe structure sheaf O ¯Γ in the following way. For each point inside anedge a metric gives a well-defined (up to translation) embedding to T × ≈ R and we pull back the regular functions (the tropical Laurentpolynomials). Near a 1-valent vertex this embedding can be extendedto an embedding to T . Finally, for a vertex of valence k + 1 ≥ C k ⊂ R k obtained as a union of thenegative halves of k coordinate axes with the diagonal ( t, . . . , t ), t ≥ k variables to C k locally form the structure sheaf. D. Conversely, if Γ is a tropical curve then we have the notion of a primitive tangent vector to Γ. At a point inside an edge such vectoris well-defined up to the sign. The metric on Γ is defined by makingthese vectors unitary. The finiteness condition ensures that every leafhas an infinite length. E. Remark
More generally, a tropical structure on a tropical varietycan be rephrased in terms of the so-called integer affine structure .While traditionally such structure is considered only on smooth man-ifolds we can express the tropical structure as its extension to certainpolyhedral complexes.2.2.
Tropical modifications for curves.A.
Tropical modifications for curves are especially easy since hypersur-faces in curves are disjoint unions of points (with some natural weights).We restrict our attention to the case of weight 1, i.e. a modificationalong a simple point. In the language of metric graph such a modifi-cation results in attaching an interval of infinite length to this point.We easily see that all compact metric trees are equivalent. From nowon we restrict our consideration to the case of compact and connectedtropical curves. B. If we have a finite collection of marked points of sedentarity 0 thenwe can make tropical modifications at them to replace the curve withan equivalent model so that all the marked points have sedentarity 1,i.e. are the 1-valent vertices (ends) of the curve. Furthermore, if thenumber of the marked points is at least 2 then we can do the inversetropical modifications so that the ends are precisely the marked points.2.3.
Differential forms and their integrals.
NTRODUCTION TO TROPICAL GEOMETRY 9 A. A constant differential k -form α on R N is just an element of Λ k (( R N ) ∗ ).We say that it extends to a point x ∈ T N r R N if for any infinite coor-dinate of x the corresponding basis vector is in the kernel of α . B. A regular differential k -form on a tropical variety X is a differen-tial form that can be locally obtained as a restriction of a constantdifferential form in T N defined at all points of X . C. We may integrate differential forms against respective chains. E.g.let α be a 1-form on a tropical curve C and let γ : [0 , → C be a path.Note that because of the extension condition any regular 1-form on aleaf of a compact tropical curve must vanish. Thus the integral R γ α isa real number.2.4. The genus of tropical curve and its Jacobian.A.
All regular 1-forms on a compact connected tropical curve C forma finite-dimensional (real) vector space Ω( C ). Its dimension is equal to b ( C ) = dim H ( C ). This number is called the genus of C . B. Integration along 1-cycles gives a map H ( C ; Z ) → Ω ∗ ( C ), this mapis an embedding and its image is called the periods . The quotientJac( C ) = Ω ∗ ( C ) /H ( C ; Z )is called the Jacobian of C . Topologically it is a torus, but it is naturallyequipped with a tropical structure coming from the tautological integeraffine structure on Ω( C ) (its integer affine structure is coming from theregular forms that take integer values on integer tangent vectors to C ).2.5. The divisor group and the Picard group.A.
The Jacobian Jac( C ) can also be interpreted as the Picard group ofdegree 0. Let us recall its definition. The divisor group Div is definedas all formal (finite) linear combinations D = P a j x j , where a j ∈ Z and x j ∈ C . The degree of the divisor D is defined as P a j ∈ Z . Thedivisors of degree 0 form a group Div . B. Note that there are no non-constant globally defined regular func-tions on compact tropical curves. As in the complex case the reason isthe maximum principle: at the point of its maximum a regular func-tion must be locally constant. Nevertheless, the function which atevery point can be presented as the locally tropical quotient of tworegular functions can be defined globally. Such functions are called ra-tional functions , they are piecewise-linear, though no longer necessarilyconvex. C. A point x ∈ C is called a zero of order m of a rational function φ : C → T if near x the function φ coincides with a regular functionwith a zero of order m . Similarly, x ∈ C is called a pole of order m ifnear x “ φ ” = − φ coincides with a regular function with a zero of order m . D. The divisor of a rational function φ is the linear combination ofall its zeroes taken with the coefficients equal to their order and allits poles taken with the coefficients equal to minus their order. Suchdivisors are called principal . Their degree is always zero. Two divisorsare called linearly equivalent if their difference is a principal divisor.The quotient of the divisor group by the linear equivalence is calledthe Picard group and is denoted with Pic( C ). The quotient of thedivisors of degree d by the linear equivalent is called Pic d ( C ).2.6. The Abel-Jacobi theorem.A.
We have a map from the product of k copies of the curve C (whichis a tropical variety) to Pic k ( C ). Indeed, any k points form a divisorof degree k and we can take its equivalence class in Pic( C ). Note thatPic ( C ) and Pic k ( C ) can be identified by an invertible map given byadding a copy of a fixed point in C k times. Thus Pic k ( C ) ≈ Pic ( C )as a set. B. Integration gives a natural map from Pic ( C ) to Jac( C ). Namely,any element of Pic ( C ) is a 0-dimensional cycle in C and thus it boundsa 1-chain γ . Integration along γ gives a linear R -valued functional onthe regular 1-forms Ω( C ), i.e. an element of Ω ∗ ( C ). The 1-chain γ isdefined up to 1-cycle in C . Thus the functional is defined up to theperiods, so we get a well-defined image in Jac( C ). C. The resulting map α : Pic ( C ) → Jac( C ) is called the Abel-Jacobimap . The Abel-Jacobi Theorem states that it is an isomorphism. Thistheorem holds in tropical geometry. It can be used to induce the trop-ical structure to Pic ( C ) and thus to all other Pic k ( C ). The map C × · · · × C → Pic k ( C ) is a tropical morphism.2.7. The Riemann-Roch Theorem.A.
A divisor D ∈ Div( C ) is called effective (we write D ≥
0) if itis presented as P a j x j with a j ≥
0. If two effective divisors
D, D ′ are linearly equivalent then they can be connected by a deformation.Indeed we can find a rational function φ such that D is the divisor of itszeroes and D ′ is the divisor of its poles. The full graph of φ is a tropical NTRODUCTION TO TROPICAL GEOMETRY 11 curve in C × TP (considered with ( x, y )-coordinates). Functions y + t , t ∈ T × , define the required family of divisors connecting D and D ′ . B. All effective divisor in a given equivalence class thus form a con-nected space. It is denoted with | D | . In complex geometry this space isthe complex projective space CP n , where n is dim H ( D, O ). In tropi-cal geometry the space | D | is also kind of projective, though usually itis not isomorphic to TP n for any n . C. There is a tropical vector space, i.e. a space that contains T andadmits addition and multiplication by scalars (elements of T ) such that | D | comes as its tropical projectivisation. Namely, choose D ∈ | D | andconsider the space V of all rational functions φ such that at every x ∈ D either φ is regular or has a pole of order not more than the multiplicityof D at x . Clearly, if φ, ψ ∈ V then “ aφ + bψ ” ∈ V for any a, b ∈ T × .Furthermore, each D ∈ | D | corresponds to a ray in V , i.e. an elementof V up to a multiplication by a scalar. D. In a contrast with the complex case there are many tropical vectorspace not isomorphic to T n . Nevertheless, we can define the dimensionof | D | in the following way: we say that dim | D | ≥ k if for any effectivedivisor D ′ of degree k the space | D − D ′ | is nonempty. The Riemann-Roch theorem states that dim | D | − dim | K − D | = d − g + 1. Here d = deg D , g is the genus of the curve and K is the divisor obtained bytaking every vertex of C with the coefficient equal to its valence minustwo. The equivalene class of the divisor K is called the canonical class .3. Tropical Gromov-Witten theory and enumerativegeometry
Moduli spaces of tropical curves.A.
Consider all tropical curves of a given genus g with k distinctmarked points up to equivalence generated by the tropical modifica-tions. If g = 0 we assume in addition that k ≥ g = 1 that k >
0. As above applying the tropical modifications and the operationsinverse to them we may assume that all marked points are at the endsof the curve. B. If g = 0 all such curves form a tropical variety of dimension k − M ,k . This tropical variety can be embedded to R N forsome N as a balanced complex by the so-called cross-ratio functions.A cross-ratio on C is defined by a choice of two ordered disjoint pairsof elements of the marking set { , . . . , k } . Namely, we connect thepoints in each pair with an embedded path (unique in the tree C ) and measure the length l of the intersection. The corresponding cross-ratiois l if the orientations of the paths agree and − l if they disagree. C. If g > M g,k is not a tropical variety. The reasonis possible symmetries of the corresponding curves. To exclude suchsymmetries one may introduce the homotopy marking , or a homotopyequivalence h : F g → C , where F g is the (topological) wedge of g copiesof the circle S . The tropical curves enhanced with such marking formthe universal covering ˜ M g,k of M g,k . The deck group is isomorphic tothe group of outer automorphisms of the free group on g generators. InGeometric Group Theory the space ˜ M g, is known as the Outer Space. D. The space ˜ M g,k has a natural local structure of a tropical varietyof dimension 3 g − k . To verify this we consider a curve C ∈ ˜ M g,k .Near any vertex of valence v ≥ C maybe perturbed andthe perturbation space is parameterized by the tropical variety M ,v .Furthermore for each inner edge we can locally deform its length. Thusnear C the space ˜ M g,k coincides with a product of tropical varieties.The dimension 3 g − k is the number of inner edges in the case whenall inner vertices of C are 3-valent. E. Note that ˜ M g,k is not compact and even does not have a finitetype. Nevertheless its quotient by the group of deck transformation M g,k does have a finite type (in some sense) and can be viewed as a tropical orbifold .3.2. Stable tropical curves.A.
As in the classical case we may compactify M g,k . To do this itis sufficient to allow the inner edges to assume infinite length. Theresult is denoted with M g,k . It is clearly compact. Furthermore, it isa tropical variety as the locally the compactification construction justcorresponds to passing from R N to T N by taking the closure. B. The elements of M g,k are called stable curves . To establish thecorrespondence with the classical construction of the Deligne-Mumfordcompactification by curves with several components it suffices to con-tract all edges of infinite length (note that then we may have severalintersecting and self-intersecting components). NTRODUCTION TO TROPICAL GEOMETRY 13 C. Let X be a tropical variety. We may consider the space M g,k ( X )of all tropical morphisms from C to X , where C ∈ M g,k . The imageof C is a 1-cycle in X . We may specify the homology class of such1-cycle. To simplify our considerations we assume for the rest of theselectures that X = TP n . Then specifying the homology class is the sameas fixing the degree of the curve. We denote the corresponding set oftropical morphisms with M g,k,d ( TP n ). D. Clearly, M g,k,d ( X ) is a polyhedral complex (though not necessarilyof a pure dimension and its absence does not allow us to check thebalancing condition in general). It follows from the tropical Riemann-Roch formula thatdim M g,k,d ( TP n ) ≥ ( n + 1) d + ( n − − g ) . E. Let us consider a the closure in M g,k,d ( TP n ) tropical morphismsthat are realizable by topological immersions that locally deform inexactly (( n + 1) d + ( n − − g ))-dimensional space. This sub-space is called the moduli space of regular curves , we denote it with M regg,k,d ( TP n ). It can be checked that it is a tropical variety (of di-mension ( n + 1) d + ( n − − g )). The elements of M g,k,d ( TP n ) r M regg,k,d ( TP n ) are called the superabundant curves. F. As in the complex case it is possible to compactify M regg,k,d ( TP n )to a compact tropical variety M regg,k,d ( TP n ). To do this we considerthe morphisms from stable curves with the Kontsevich condition oneach component (of the complement of infinite length edges) that iscontracted to zero. G. There are two special cases when there are no superabundant curves.Namely, for immersions in the case when n = 2 and in the case when g = 0.3.3. Tropical Gromov-Witten theory.A.
Suppose that M g,k,d ( X ) is a compact tropical variety of dimensionprescribed by the Riemann-Roch formula. Then we have the evaluationmap ev j : M g,k,d ( X ) → X for each of the k marked points. This map is a tropical morphism. B. Given a tropical cycle A ⊂ X we may pull it back by ev j to atropical cycle ev ∗ j ( A ) ⊂ M g,k,d ( X ). This pull-back is especially easy todefine in the case when A is a Cartier divisor or a transverse intersectionof such divisors as we may just pull-back the corresponding regularfunctions. C. This gives an alternative (with respect to the usual tropical stableintersection in X ) product operation for the cycles in X . First we maypull-back the cycles to M g,k,d ( X ) and then take their tropical (stable)intersection in M g,k,d ( X ). These products are known as the Gromov-Witten numbers. D. In the case of g = 0 the resulting product in the clsssical (complex)set-up is known as the quantum product. Its associativity is related tothe so-called WDVV equation. This equation holds in the tropical caseas well. As in the classical case it follows from the linear equivalenceof all points in M , and, in particular, its boundary (sedentarity 1)points. E. Geometrically, the Gromov-Witten numbers may be inerpreted asthe number of curves of genus g and degree d passing via the cor-responding k cycles. Clearly, to have the non-zero number a certaincondition on the sum of the codimensions of the cycles has to hold. F. It can be shown that in the regular case (in particular, in the case n = 2 or g = 0 there is a correspondence between tropical curvesand their classical (real and complex) counterparts. This gives a wayto reduce computation of classical invariants to finding the tropicalcurves and computing their real and complex multiplicities (which aredifferent, in general). There are different algorithms for doing that. Inthe lectures we consider many examples of tropical computations. G. One of the powerful technique to enumerate tropical curves in TP n is via the so-called floor decompositions . The technique is based on theobservation that each component of the complement of vertical edgesfor a tropical curve in R n is projected (vertically) to a tropical curve in R n − . This allows to find curves inductively by dimension. Each suchcomponent is called a floor . The floor diagram is the graph obtainedby contracting each floor to a point. The resulting computation isespecially simple in the case of n = 2. NTRODUCTION TO TROPICAL GEOMETRY 15 Historical remarks and credits
Passing to the equivalent of tropical limit for counting holomorphiccurves and holomorphic disks respectively was suggested by Kontse-vich and Fukaya around 2000. Formalization of the limiting objectsas tropical varieties and their geometry as Tropical Geometry wereintroduced by Mikhalkin and Sturmfels in 2002. Applications of Trop-ical Geometry in classical enumerative (complex and real) curves werestarted by Mikhalkin in 2002. These applications can be consideredas a build-up on patchworking (Viro, 1979) and amoebas (Gelfand,Kapranov, Zelevinski, 1994), in particular, non-Archimedean amoebas(Kapranov, 2000). A general understanding of passing from classical totropical Mathematics as a sort of dequantization goes back to Maslovand his school (Kolokoltsov, Litvinov et al.) in the early 90s. Beloware some references for the material of these lectures.
References [1] E. Brugall´e, G. Mikhalkin,
Enumeration of curves via floor diagrams ,http://arxiv.org/abs/0706.0083.[2] A. Gathmann, H. Markwig,
Kontsevich’s formula and the WDVV equationsin tropical geometry , http://arxiv.org/abs/math.AG/0509628.[3] G. Mikhalkin,
Amoebas of algebraic varieties and tropical geometry, http://arxiv.org/abs/math.AG/0403015, in
Different Faces of Geometry , In-ternational Mathematical Series , Vol. 3 (Donaldson, Simon; Eliashberg,Yakov; Gromov, Mikhael (Eds.)) 2004, 257–300.[4] G. Mikhalkin,
Tropical geometry and its application , Proceedings of the ICM2006 Madrid, 25 pages, http://arxiv.org/abs/math.AG/0601041.[5] G. Mikhalkin,
Tropical Geometry
Tropical curves, their Jacobians and theta-functions ,http://arxiv.org/abs/math.AG/0612267.[7] J. Richter-Gebert, B. Sturmfels, Th. Theobald,
First steps in tropical geometry ,http://arxiv.org/abs/math.AG/0306366.[8] M. Vidgeland,
Smooth tropical surfaces with infinitely many tropical lines ,http://arxiv.org/abs/math/0703682.[9] O. Viro,
Dequantization of real algebraic geometry on loga-rithmic paper , Proceedings of the 3d ECM, Barcelona 2000.http://arxiv.org/abs/math.AG/0005163.
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