Convergence of the mirror to a rational elliptic surface

Transcription

1 Convergence of the mirror to a rational elliptic surface Trinity College, University of Cambridge Submitted for the degree of Doctor of Philosophy Lawrence Jack Barrott July 12, 2018

2 To my family, the outdoors and the good people of the CUMC.

3 Declaration I declare that all material unless explicitly marked is my own. It is not substantially the same as any that I have submitted, or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution. I further state that no substantial part of my dissertation has already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification at the University of Cambridge or any other University or similar institution.

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5 Acknowledgements My deepest thanks go to all the people who have supported me through my continuing mathematical education. Firstly to my family, my father James, my mother Lisa and my sister Isobel, who always come first in everything. Also to my girlfriend Corlijn Reijgwart for her passion, food and love supporting me through the writing up period Mathematically I must begin by thanking my supervisor Mark Gross. His continuous advice, both mathematical and grammatical, has helped mould this thesis from my original drafts to the current readable version you now hold. His humour and kindness have encouraged me to pursue ideas that at first seemed outlandish and he has often introduced novel insights in my own ideas. Second I owe a debt of gratitude to Pelham Wilson, my departmental advisor who originally suggested I apply to Mark Gross. On numerous occasions he has acted on my behalf with both the department and my college. I hope that he enjoys reading this thesis. Thirdly to Tyler Kelly. During the four years I have worked on this thesis he has been there, supporting me through difficult periods when I have felt unsure of myself. When I doubted myself he was always willing to offer advice, though conversation and inspirational quotes. I must thank a series of my supervisor s collaborators, Paul Hacking, Sean Keel and Bernd Siebert. Their work over these years created the program that I am now contributing one small piece towards. Their works is embedded in the wider world of mathematics but I am very happy to call this place home. In London Tom Coates and Alessio Corti have provided me with many new ideas and directions of study, through informal

6 conversations, their choices of graduate students and the conferences they have organised. Within the department here there are numerous people with whom I have had important and revealing conversations. Roberto Svaldi, Joe Waldron and Julius Ross mathematically and our lunch group of Benjamin Barrett, Tom Berrett, Nigel Burke, James Kilbane, Jo Evans and Nicolas Dupré and James Munro for their friendship. I owe the staff of the CMS greatly for all the times I have been late with forms or needed advice with applications, to Vivienne, Julia and Anita. Outside of Cambridge I must thank Alexander Betts for his humour and sharp mathematical mind, Josh Jackson for being willing to explain his thoughts on geometry. To Alex Torzewski for being himself and being enthusiastic about any part of mathematics. In Cambridge itself I owe Chris Hands who always caught me when I fell and to Veronika Siska for her support, friendship and tea. Finally to Amanda Sondergard Schiller for not fitting in, even to this list.

7 The mountains are calling and I must go. - John Muir

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11 Abstract The construction introduced by Gross, Hacking and Keel in [28] allows one to construct a mirror family to (S, D) where S is a smooth rational projective surface and D a certain type of Weil divisor supporting an ample or anti-ample class. To do so one constructs a formal smoothing of a singular variety they call the n-vertex. By arguments of Gross, Hacking and Keel one knows that this construction can be lifted to an algebraic family if the intersection matrix for D is not negative semi-definite. In the case where the intersection matrix is negative definite the smoothing exists in a formal neighbourhood of a union of analytic strata. A proof of both of these is found in [28]. In our first project we use these ideas to find explicit formulae for the mirror families to low degree del Pezzo surfaces. In the second project we treat the remaining case of a negative semi-definite intersection matrix, corresponding to S being a rational elliptic surface and D a rational fibre. Using intuition from the first project we prove in the second project that in this case the formal family of their construction lifts to an analytic family.

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13 Contents 1 The Saga of Mirror Symmetry The mathematical framework Constructing mirror pairs The Gross-Siebert program Log Geometry Outline of this thesis Conventions Mirror families to low degree del Pezzo surfaces Looijenga Pairs The Mumford degeneration Scattering diagrams Broken lines The combinatorics of scattering diagrams Explicit Gromov-Witten counts and Monodromy Explicit formulae The mirror family to a rational elliptic surfaces Counting sections of rational elliptic and K3 surfaces The unravelled threefold The ramified fibres The unravelled threefold Our approach Givental s techniques Formulae for the invariants

14 3.3.3 Log Geometry and Algebraic Geometry A geometric interpretation Some painful analysis Multivariate holomorphic functions Bounding Gromov-Witten invariants Convergence of the mirror Reflections Future questions The moduli space of surfaces Understanding the mirror family Mirrors to low degree del Pezzo surfaces

15 Chapter 1 The Saga of Mirror Symmetry Algebraic geometers have long been interested in enumerative questions on varieties. These problems have led to the development of techniques in intersection theory and deformation theory which have then been applied to solve many other problems. As an example let us consider the following problem How many degree one curves are there on a smooth cubic surface S in P 3? To understand when we expect these problems to have a finite number of solutions we must turn to deformation theory and the Riemann-Roch theorem. By [34] the deformations of a smooth subvariety Z of a smooth variety V are controlled by the normal bundle N Z/V to Z in V. The global sections of this sheaf correspond to first order deformations of the embedding whilst elements of the module H 1 (Z, N Z/V ) correspond to obstructions to lifting these deformations to order two. By the Riemann-Roch theorem applied to the deformations of a smooth curve C inside a smooth variety V the difference of the ranks of these two modules, or expected dimension, is given by (1 g(c))(dim V 3) K V.[C] In the above problem this expected dimension is zero. One can show that the normal bundle N Z/V has no higher cohomology, so the solution space, in this case the moduli space of maps from P 1 to S, genuinely is zero di- 1

16 mensional and has length 27, corresponding to the 27 lines on the cubic surface. Now looking at the above formula one notices that on a Calabi- Yau threefold, that is to say a threefold with trivial canonical class, any class β ought to support a finite number of rational curves, although the actual picture is more subtle. Throughout the history of algebraic geometry many mathematcians for many reasons have attempted to calculate these invariants for low degree rational curves lying on a smooth quintic in P 4. This question rapidly becomes more intractable as the degree grows, degree one was known to mathematicians in the 19 th century, degree two was calculated in [38] by Sheldon Katz and degree three by Ellingsrud and Strømme in [21]. This question however was catapulted to fame by the work of physicists in [11]. Using this work a general formula was given by Givental in [24]. 1.1 The mathematical framework During the 80 s and 90 s physicists had explored a great number of topological conformal field theories, string theories built from maps from a Riemann surface into a target space. Of these two, the A and B-models are twists of the same theory, depending on symplectic and complex data respectively. Therefore all the physical data, boundary conditions, partition functions and so on, should be equal up to a change of basis. The driving motivation for Mirror Symmetry is that to any target space X there should be another target space ˇX, called the mirror, interchanging these two models. We will not formally attempt to describe these theories as this is purely motivational. Instead we will talk about what data goes into building them. Let us begin this section by recalling some motivation from symplectic geometry. This is relevant to us as the Gross-Siebert program is an attempt to recreate algebro-geometrically this symplectic data. Since this relies on the reader being familiar with symplectic geometry we will not attempt to build examples, rather we jump in with the definitions. 2

17 Definition 1. A symplectic manifold is a 2n-dimensional smooth manifold M together with a choice ω of closed non-degenerate two-form on M. An almost complex structure on M is a choice of automorphism J of the tangent bundle T M such that J 2 = 1. Such a structure is called compatible if ω(v, Jv) defines a metric on M. Given (M, ω) a symplectic manifold a submanifold N M is a symplectic submanifold if ω N is closed and non-degenerate. It is isotropic if ω N is zero and it is Lagrangian if it is both isotropic and of dimension n. Examples of symplectic manifolds include complex Kähler manifolds, so include smooth complex projective varieties with the restriction of the Fubini-Study symplectic form on P n. Let (M, ω) be such a variety. The Lagrangian submanifolds of (M, ω) naturally arise in the construction of the A-model as the boundary conditions of strings, called D-branes. Mathematically these boundary conditions should form a category, in this case the Fukaya category. There are many different constructions of this category dependent on what properties one wants of it, see [23] for the initial definitions. In broad brush-strokes the objects of this category are Lagrangian submanifolds with a choice of additional data, in our later example a flat U(1) bundle over that Lagrangian. Morphisms L 1 L 2 in this category correspond to elements of a homology theory built from intersection points of L 1 and L 2. In the philosophy of Kontsevich and Soibelman there should be a whole system of operations defining an A structure hidden behind these definitions as described in [39]. Associated to this k-linear category is its Hochschild cohomology. This is a canonical construction of a ring from a k-linear category given by the bar complex and in many cases the Hochschild cohomology of the (wrapped) Fukaya category of a symplectic manifold (V, ω) is isomorphic to the symplectic cohomology, written SH (V, ω). We won t precisely define the symplectic cohomology here, but will introduce part of the definition later as motivation. The properties of this construction are not geometrically clear, but following ideas of Seidel in [53] there is a morphism from this to the quantum cohomology of the space. This morphism 3

18 is conjecturally an isomorphism following promising results of Bourgeois, Ekholm and Eliashberg in [9], although the technical difficulties of calculating the Hochschild cohomology makes proving this hard. We will follow this conjecture and define the quantum product. The (small) quantum cohomology of a smooth variety X is a deformation of the singular cohomology incorporating the three point functions arising in the physics. Letting H i be cohomology classes, the product H 1 H 2 is defined by H 1 H 2, 3 = I 0,3 (X, α, H 1, H 2, H 3 )z α H α H 2 (X) where I 0,3 (X, α, H 1, H 2, H 3 ) counts rational curves in class α with three marked points lying on classes Poincaré dual to H 1, H 2 and H 3. This produces a product on the cohomology of X with coefficients in C deforming the classical product over a formal neighbourhood of the origin in k[h 2 (X)]. In fact this product remains associative, and the details of this construction are described in [16] Theorem On the complex geometric side the D-branes correspond to elements in the derived category of the target and strings correspond to morphisms between them. Here the A structure suggested by Kontsevich can be seen more explicitly as in [12]. The equivalence between these two constructions is made explicit by the following famous conjecture of Kontsevich as described in [40] Conjecture 1. Let X and ˇX be a mirror pair. After taking split completions there is an isomorphism Fuk(X) = D b (Coh( ˇX)) Split completions preserve the Hochschild cohomology, so there is an induced isomorphism of Hochschild cohomology rings. In general the Hochschild cohomology of a derived category is hard to compute. Fortunately when working with the category of derived category of coherent sheaves on an affine scheme X we can say much more. 4

19 Lemma 1. Let Spec A be an affine variety, then there is an natural isomorphism of rings A = HH 0 (D b Coh(Spec A)) Proof. The Hochschild cohomology of a category is described in terms of the endomorphism ring of the identity functor, E which is canonically isomorphic to A via its action on A. To be precise it is the cohomology of the bar complex applied this functor with multiplication given by cup product. Taking the dual of the bar complex 0 E Hom (E, E) Hom (E, E E)... The first differential is given by : e 1 (e 2 e 1 e 2 e 2 e 1 ). Since E is a commutative ring this differential is zero and we obtain the claimed isomorphism. One prediction of mirror symmetry therefore would be that the two Hochschild cohomologies are isomorphic. 1.2 Constructing mirror pairs There are many flavours of mirror symmetry constructions, work of Dolgachev on mirror symmetry for K3 surfaces [19], of Borcea and Voisin [56], on LG/CY mirror symmetry of Berglund, Hübsch and Krawitz for hypersurfaces in weighted projective spaces [42], and this list is far from exhaustive. For our interests we will follow Batyrev and Borisov, who studied toric varieties associated to polytopes. The best reference to learn about toric geometry is [17]. Recall that from a lattice M with dual lattice N and a rational polytope M R one can construct a dual polytope inside N R. This is constructed as = {n N m, n 1 m } 5

20 and the polytope is called reflexive if is a lattice polytope in N. From any rational reflexive polytope one can construct a toric variety P and inside this a canonical family of Calabi-Yau hypersurfaces CY ( ). Entries in the Hodge diamond of a generic hypersurface can be read off from the combinatorics of the polytopes. Batyrev exploited this in [6] to prove that one has an equality h 1,1 (CY ( )) = h 2,1 (CY ( )) for nice families of anti-canonical hypersurfaces inside a large class of projective toric varieties. Together with Borisov he extended this result to complete intersections in Gorenstein Fano toric varieties. This approach is relevant to us in that it is the natural testing ground for the following conjecture of [55] Conjecture 2 (The SYZ conjecture). Let X, ˇX be a mirror pair of Calabi- Yau varieties. Then there is an affine manifold with singularities B and maps φ : X B, ˇφ : ˇX B which are dual special Lagrangian fibrations. This is shown schematically below T T X ˇX φ φ Let us use this to explore fully mirror symmetry for the most simple example, an elliptic curve. 6

21 Example 1. Let E = T 2 be an elliptic curve over C with projections π L and π R. Appropriately identified the projection to the first factor π L : E S 1 gives a special Lagrangian fibration. Therefore the mirror manifold should be fibred by the dual circles, i.e., it is the dual elliptic curve Ě and has two basic types of Lagrangians, Lagrangian sections and Lagrangian fibres. Now the derived category of quasi-coherent shaves on E is generated by the structure sheaf O E and skyscraper sheaves, so it is enough to find the Lagrangians mirror to these objects. Conjecturally a line bundle should be mirror to a Lagrangian section of the fibration Ě S1. Under this the structure sheaf is dual to a Lagrangian section with winding number zero under the projection π R to S 1. The associated U(1) bundle is the trivial bundle. The skyscraper sheaves should be mirror to a special Lagrangian fibre of this fibration, together with a choice of flat U(1) bundle. Again conjecturally the mirror to a skyscraper sheaf is a fibre of the special Lagrangian fibration. There is a natural candidate for the fibre mirror to k P, the fibre over the image of P. A choice of flat U(1) bundle on S 1 is equivalent to a choice of phase in C. This choice of phase is the value of P under π R. This correspondence extends to the entirety of both categories by taking exact sequences. To test that this correspondence is correct we consider the homomorphisms between objects, which also should be preserved. Given two distinct points P and Q of E lying on distinct Lagrangian fibres it is immediate that Hom (k P, k Q ) is trivial. On the symplectic side the homomorphisms between transverse Lagrangians L 1 and L 2 are given by elements of the vector space C {L 1 L 2 }. In this case the mirrors to k P and k Q are disjoint and so there is only the zero homomorphism. Similarly the space Hom (O E, k P ) is one dimensional and the corresponding Lagrangians meet transversely at a single point. Finally if the two Lagrangians are identical one needs to apply a Hamiltonian isotopy to one of them so that the Lagrangians become transverse. In this case one can check that the Hom space between the two Lagrangians is dimension one in degrees 0 and 1, which agrees with the derived Hom spaces between O E and O E or between 7

22 k P and k P. In the case of toric varieties there is a natural choice of fibration given by the action coordinates. One can show that this conjecture is in fact realised for toric varieties under the above construction, with the fibres degenerating over the boundary of the polytope. It was this understanding of mirror symmetry for toric varieties that is at the core of the Gross-Siebert program. In its full generality the Gross-Siebert program is believed to be a mirror construction for a pair (V, D) where V is any smooth variety and D is a divisor supporting an ample or anti-ample class. Let us review the philosophy of the program for Fano varieties since this will occupy us later. 1.3 The Gross-Siebert program According to Givental s 1994 ICM lecture mirror symmetry predicts that associated to a toric Fano variety V there should exist a mirror Landau- Ginzburg model. This is a pair ( ˇV, ˇW ) where ˇV is a smooth variety and ˇW : ˇV C the super-potential, a regular function. This was extended in work of Cho and Oh, who in [15] realised that a choice of effective anti-canonical divisor played a role in determining the Landau-Ginzburg model. To construct this we work in two stages, Gross, Hacking and Keel suggest the mirror to the complement V \ D should be the underlying scheme ˇV, which under the conditions on D is affine. The intuition for this is not clear and indeed is false in higher dimensions. Following our discussion about Hochschild cohomology if we could describe algebraically the symplectic cohomology of V \ D we could recover ˇV as being the spectrum of the ring SH 0 (V, ω). A technical difficulty that we will gloss over is that V \ D is not proper, so that one should work with the wrapped Fukaya category. The super-potential is then defined in terms of a canonical basis of the ring SH 0 (V, ω), the ϑ-functions. The mirror category to the Fukaya category of (V, D) should be the category of matrix factorisations of W. The corresponding Hochschild cohomology ring (in the case that ˇW has 8

23 isolated singularities) is given by the Jacobian ring Jac( ˇV, ˇW ) of the singularity. One interpretation of the statement that these two are mirror pairs is the statement that there is an isomorphism of rings QH(V ) = Jac( ˇV, ˇW ) but any proof of this requires a stronger understanding of the glueing properties of Gromov Witten invariants. We expect a proof of this to follow from future work of Gross and Siebert but for now we leave this as a conjecture. So how does one calculate algebraically the symplectic cohomology? Here the SYZ heuristic enters the picture. If we suppose the existence of an SYZ fibration, the image of a stable map under this fibration is called a tropical amoeba. This is a space retracting to a piecewise linear graph, as described in [48]. As one moves closer and closer to a large complex limit point of a variety V this space collapses onto its skeleton, a tropical curve in the base. Taking this as a stand point we should count all possible tropical curves on the base of the SYZ fibration and then following [26] reconstruct the original invariants using log geometry. Due to the presence of singularities of the SYZ fibration these piecewise linear objects can bend in a highly controlled manner, corresponding to glueing Maslov index zero disks. Gross, Hacking and Keel encode this symplectic data in the definition of the canonical scattering diagram, and from this algebraic variant of the symplectic cohomology of X they attempt to construct the underlying scheme of the mirror family. The techniques themselves were introduced by Kontsevich and Soibelman in their paper [41] to study K3 surfaces. These were applied by Gross in [26] to study the prototype example of the construction, mirror symmetry for P 2, and by Gross and Siebert in [31] to construct mirror partners to Calabi-Yau threefolds. It was expanded upon in work of both [30] and [13] to construct mirrors to Fano varieties. This just leaves how to find the base B of the SYZ fibration. It should be invariant topologically under small deformations of X and ˇX. This inspired Mark Gross and Bernd Siebert to study toric degenerations of 9

24 varieties as described in [31]. The idea is to deform to a degenerate central fibre, construct the base there and attempt to deform the algebra structure to incorporate the missing singular fibres. This construction depends on a choice of degeneration to a toric variety. Let us define the objects of study. Definition 2. A toric degeneration is a flat family X C with C a nonsingular curve with a point 0 C such that the fibre X 0 is a union of toric varieties glued along toric strata and étale locally at zero-dimensional strata of the central fibre, the morphism X C is given by a monomial. These were initially studied by Mumford in [50]. Of the many different degenerations possible the philosophy of Gross and Siebert is that we should study those with maximally unipotent monodromy as described in [44]. In fact, instead of working with toric degenerations, Gross, Hacking and Keel work with pairs (S, D) where S is a smooth Fano surface and D a particular type of anti-canonical divisor on S. From this data we construct a base B from the dual intersection complex of D. Given this we describe the mirror family via the data of a scattering diagram on the base B. This encodes data necessary to construct a smoothing of a union of affine planes. The scattering diagram involves relative Gromov-Witten invariants by assigning to each rational line in B a generating power series for these invariants. In the symplectic cohomology the product comes from counts of Maslov index two disks. The tropical analogue of these are broken lines, decorated balanced piecewise linear graphs and Gross, Hacking and Keel define the product analogously. The super-potential has a simple form in terms of these theta functions: it is a sum of the theta functions attached to each one-cell of B. Furthermore the theta functions should induce an embedding of the mirror into affine space. Therefore if we have a generating basis then we obtain a formal family smoothing a central fibre, the union of affine planes. By controlling the combinatorics of the products of these theta functions we will show that this family is in fact algebraic or analytic dependent on the surface. 10

25 1.4 Log Geometry A key technical tool in mirror symmetry is log geometry, which formalises the open sector of the Gromov-Witten theory and systematises the study of toric degenerations. These were first introduced by Kato in [35], and were studied by Deligne and Illusie in [36]. The foundational properties were explored in [51] which will form a basis for the entire theory. In [26] Mark Gross used them to more strongly relate the tropical and algebraic pictures of [48]. The philosophy is that log geometry provides a connecting link between algebraic and tropical geometry. We will introduce these log schemes. Definition 3. A pre-log scheme X is a pair (X, α X : M X O X ) of a scheme X (the underlying scheme), a sheaf of monoids on X in the étale topology M X and a homomorphism α X of sheaves of monoids from M X to the multiplicative monoid O X. Saying that this is a log scheme means that the restriction of α X to the inverse image of O X is an isomorphism. To any pre-log scheme there is an associated log structure given by taking the amalgamated sum M X α 1 O X O X A morphism of log schemes φ : X Y is a pair (φ, φ # ) with φ : X Y a morphism of schemes and φ # fitting into a commutative diagram φ 1 M Y φ # M X φ 1 O Y φ O X Given a morphism of schemes φ : X Y and the structure of a log scheme on Y the inverse image sheaf φ 1 (M Y ) on X is naturally a pre-log structure. We call the associated log structure the pull-back log structure, written φ M Y. A morphism of log schemes φ : X Y is strict if the natural map φ # : φ M Y M X is an isomorphism. 11

26 There is an interesting invariant of a log scheme which contains the combinatorial data not seen by classical geometry. The ghost or characteristic sheaf is the quotient M X /O X. We introduced log structures to study degenerations of pairs (X, D), so we now explain how to construct a log structure from such a pair. Example 2. Let X be a smooth scheme and D be a simple normal crossings divisor. Write U for the complement of D in X and let j be the inclusion of U into X. Let M be the sheaf j OU O X, and α X the canonical inclusion. This defines a log scheme X, the divisorial log structure on X. General log schemes exhibit pretty wild behaviour so we restrict those we consider. The objects we want to study are localled modelled on toric varieties together with their toric boundary, which has another intrinsic definition which we give below. Example 3. Let P be a finitely generated integral monoid. Take P to be the constant sheaf on Spec k[p ], which maps to O Spec k[p ] via p z p. Let Spec log k[p ] denote the corresponding log scheme, the toric log scheme, whose log structure is refered to as the toric log structure. If T is any toric variety then each toric Zariski open affine subset carries a log structure as just described. These turn out to be canonically isomorphic on intersections of these Zariski open subsets, yielding a log structure on T. The corresponding characteristic sheaf is only trivial over the big torus G n m T. Let us describe the ghost sheaf on A 2 = Spec k[n 2 ] with its toric log structure. On the big torus the toric monomials are certainly invertible and so the ghost sheaf has trivial stalks at these points. Now along the two toric strata z (1,0) = 0 and z (0,1) = 0 those same monomials are not invertible. Since an an element of M A 2 in a neighbourhood of one of these strata takes the form z (a,b) f for f invertible and either b = 0 or a = 0 in these two cases. Hence each contribute an N to the ghost sheaf. Drawing the stalks of the ghost sheaf we obtain the following picture: 12

27 N N 2 N Now the actual objects we want are locally modelled on this construction Definition 4. A log scheme X is called fine saturated (abbreviated as fs) if étale locally on X there are strict morphisms to toric log schemes. A surprisingly important example is the standard log point, Spec k = (Spec k, α : k N k) where α sends (a, 0) to a and (a, 1) to 0. This is isomorphic to the origin inside A 1 with its toric log structure. Now we will only work with fs log schemes and so it is worth a little revision of history: from now on whenever we refer to log schemes we mean fs log schemes. The category of fs log schemes has many fantastic properties Theorem 1 (Log schemes have fibre products). The category of fs log schemes possesses fibre products. Given a diagram X Z Y there is a canonical morphism X Z Y X Z Y which is the composition of a finite surjective map followed by a closed immersion. If the map X Z is strict then the above canonical morphism is an isomorphism and the induced map X Z Y Y is strict. Proof. This is Corollary of [51]. Let us give an example where the above map is not an isomorphism Example 4. Let A 2 denote the plane with its toric log structure. There is a toric blowup of this, blowing up the origin which we denote π : Bl A 2. 13

28 Choose two different points of the exceptional curve P = Q = (Spec k, k N). We claim that the product P A 2 Q is empty, but the product of the underlying schemes is a point. To show that the product is empty note that it would be enough to prove that it doesn t have any k-points. Now any fs k-valued point would have log structure k Q, which by choosing an element we can reduce to having log structure k N. This reduces the problem to one of algebra, what is the push-out of the diagram N 2 N N This push-out is zero, so the sections coming from N 2 map to zero. But then the maps to k do not commute and there is no such point. The geometry of log schemes is highly related to the combinatorial data of the log structure and hence to the combinatorics of toric varieties. One way to organise the data of toric varieties and their morphisms is through fans and refinements of fans. Suppose we have lattices M = Z n and N = M contained inside vector spaces M R = M Z R and N R = MR. A cone σ inside N is a subset consisting of positive linear combinations of a finite collection of rational vectors φ 1,..., φ n. Such a cone is convex and rational polyhedral and has an associated dual cone given by σ = {v M R φ(v) 0 φ N R } Associated to σ is the monoid σ M and hence a monoid ring k[σ M]. After passing to Spec we obtain a scheme and thus a covariant functor from cones and inclusions of cones to toric varieties and morphisms of toric varieties. This is not an equivalence as the image consists only of affine toric varieties. To include the glueing conditions we introduce the notion of a fan: Definition 5. A fan Σ inside N is a simplicial complex of cones. It is 14

29 complete if the collection covers all of N. A morphism of fans f : Σ 1 Σ 2 is a linear map on underlying vector spaces mapping each cone of Σ 1 into a cone of Σ 2. According to [17] this produces an equivalence of categories between the category of toric varieties and toric morphisms and the category of fans and their morphisms. Log geometry was introduced to make sense of degeneration and moduli problems, which are two sides of the same coin. Classically the local deformations of a scheme are given by the tangent bundle, or perhaps normal bundles constructed from this. To a morphism of log schemes f : X Y there is an induced sheaf of log differentials whose sections correspond to infinitesimal deformations. I learnt the definition from [37] and we repeat this here. This plays the role of the usual tangent sheaf, for instance a morphism is log smooth just when this sheaf is locally free. Definition 6. Let f : X Y be a morphism of log schemes. A log derivation is a pair of maps D, Dlog from O X or M X respectively to a sheaf E satisfying: 1. D is a derivation in the classical sense. 2. Dlog vanishes on sections of f 1 (M Y ). 3. For every element m M X (U) one has an equality D(α X (m)) = α X (m)dlog(m). There is a universal sheaf classifying these, the logarithmic cotangent bundle Ω 1, X/Y. There is a natural pullback map, given f : X Y and g : Y Z there is a canonical morphism f Ω 1, Y/Z Ω1, X/Z. We write ω X/Y for the top tensor power in the case where Ω 1, X/Y is a vector bundle. We will need more results on log geometry in our second project and we will introduce more results as we need them. 15

30 1.5 Outline of this thesis In the first project we study the mirror construction of [28] for low degree del Pezzo surfaces. These were studied symplectically in [5] and predictions made for the structure of the mirror family. We construct an algebraic family smoothing the central fibre using the combinatorics of the scattering diagram. We will see that these agree with the predictions of [5], and obtain descriptions of the mirror family. This is novel for the case of del Pezzo surfaces of degree two. In the second project we again study this construction, this time using the intuition from the first project to prove that the formal smoothing to the mirror family of a rational elliptic surface lifts to an analytic smoothing. To do this we identify an infinite set of coefficients which appear in the relations between the theta functions. Whereas in the previous cases we could explicitly calculate the finite set of relevant coefficients here we attempt to bound those coefficients. To do this we construct a larger target space, an unravelled threefold, which lifts the desired classes. This space occurs as a nef complete intersection inside a toric variety so we can apply the techniques of Givental [25] and calculate the invariants on the threefold. Careful analysis relates these invariants to the original desired invariants, producing the convergence result. 1.6 Conventions Throughout this paper we will work over C, although it is clear that the first project makes sense over arbitrary base fields. Now let us collect some terminology here, so that one can check here rather than searching through the bulk of the text. Recall the notion of a virtual fundamental class from [8]. In full generality given a morphism f : X Y this associates to an embedding C E of a cone stack into an abelian cone stack E over X (an obstruction theory) a class in the bivariant Chow theory of f : X Y of dimension rk(e). We supress the choice of embedding and write [E, C] for 16

31 the associated class. Given a variety V the moduli space of stable maps of genus g and class β and with n marked points sits over the moduli stack M g,n. If V is smooth then there is a canonical obstruction theory with target h 0 /h 1 (Rπ f Θ X/Spec k ). We write I g,n,β (...) for the associated Gromov-Witten invariant. If there is a divisorial log structure on V, D = D 1 D 2... D n we will write I 0,1,β (D i) for log Gromov Witten invariant tangent to full order to the component D i not meeting any other component of D. Often we will be interested in connected components of a given stack of stable maps. To describe this situation we introduce the following terminology Definition 7. Let f : X Y be a morphism of stacks. We say that it is an inclusion of components if it is an isomorphism between X and the union of a set of connected components of Y. In the second project we will study rational elliptic surfaces. Here ρ : S P 1 is a rational elliptic surface and [F ] is the class of a fibre. We will write F 0 for a fixed rational fibre, of type I 4, over a point 0 P 1. This says that F 0 is a cycle of four rational curves. We write S for the log structure on S given by F 0. We will study the moduli stack of genus zero log stable maps to S in a class β in our second project. We write M (S, β) for this moduli space, requiring that the curve meets F 0 at a single point to maximal tangency order. This space carries a universal family which we denote by C, it comes with two maps, π : C M (S, β) and f : C S. This stack admits a map down to the moduli space of stable maps with basic log structures constructed in [32], we denote this space M 0,0,S. 17

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33 Chapter 2 Mirror families to low degree del Pezzo surfaces Mathematicians have over many centuries come to understand the geometry of algebraic surfaces in great detail. Of the many classification results known we recall the following Theorem 2 (Classification of del Pezzo surfaces). Let S be a smooth rational Fano surface, that is to say a smooth projective surface S with ω S ample. Then S is isomorphic to one of the following surfaces P 1 P 1. The blow-up of P 2 in zero through eight points in general position. In the second case the surface is described by the degree of the anticanonical class, also called the degree of S and we write dp d for a generic degree d surface. The respective Chow groups A 1 (S) are generated by The pullback of a point under the two projection maps. A hyperplane class H and the exceptional curves E 1,..., E 9 d. whilst the groups A 0 (S) and A 2 (S) are isomorphic to Z generated by a point and a fundamental class respectively. 19

34 Proof. See [52]. Of these the surfaces P 1 P 1, P 2, dp 8, dp 7 and dp 6 are all toric and mirror families can be constructed using techniques we will introduce to motivate the Gross-Siebert construction. The next three surfaces were studied in [5]. To build up our intuition with the Gross-Siebert program we will show that these agree with the predictions there. We will then handle one of the remaining two cases, the blow up in seven points. Explicit formulae for these families are new. Precisely the same techniques can be applied in the case of dp 1 but given the frankly atrocious nature of the equations for dp 2 we do not pursue this. First we introduce a method to systematically study toric degenerations. 2.1 Looijenga Pairs To study moduli problems in geometry one often needs a method to rigidify the problem and remove automorphisms. In studying the moduli space of Fano varieties this is most naturally done by considering a pair (X, D) of a variety X and an anti-canonical divisor D. This produces a moduli space with a boundary component consisting of degenerate divisors D. One locus within this boundary is particularly well suited to our needs. A Looijenga pair is a pair (X, D) where X is a smooth rational surface and D is an anti-canonical divisor consisting of a cycle of 1-rational curves. Such a pair can be viewed in many different ways. The most enlightening for us is that these are log Calabi-Yau varieties, log varieties with trivial log canonical sheaf. This is calculation is a specific case of the following general calculation Lemma 2. Let (X, D) be a pair with X smooth and D having simple normal crossings. Write X for the divisorial log structure on X. Then there is an isomorphism ω X = ω X O(D). When X is a toric surface and D the union of toric strata this structure allows us to recover the fan for X, at least up to a choice of embedding 20

35 into R 2. This is done using the dual intersection complex (S,D) of D, an abstract fan, which we now construct. Construction 1. Let D = D 1 D 2... D n be a cycle of rational curves on a smooth surface S such that D i D j is a single point just when i and j differ by 1 mod n and otherwise is empty (in the case n = 2 we relax this to saying that there are two points in the unique intersection D 1 D 2 ). Then (S,D) contains precisely one zero-dimensional cell {0} corresponding to the interior S \D. For each component D i, (S,D) contains a one-dimensional cone with v i its primitive generator. Attach the zero-dimensional cone as 0 inside each of these rays. Now introduce a two-dimensional cone in (S,D) for each intersection point of D i D j, spanned by v i and v j. This produces the dual intersection complex (S,D) as a cone complex but it carries more structure. We write B to be the underlying topological space of the cone complex (S,D). It is homeomorphic to R 2. We give B \ {0} an affine manifold structure by defining an affine coordinate chart by embedding the union of the cones R 0 v i+1 R 0 v i and R 0 v i R 0 v i 1 into R 2 via the relations v i 1 (1, 0), v i (0, 1) and v i+1 ( 1, D 2 i ). This expresses (S,D) not just as a complex of sets but an affine manifold with singularities (indeed a single singularity at the origin). An integral affine manifold M carries a sheaf of integral vector fields, which we write Λ M. On a chart this is isomorphic to the constant sheaf with coefficients Z n. The transition functions naturally give identifications of these constant sheaves. In addition, we define M(Z) to be the set of points of M with integer coordinates in some, hence all, integral affine coordinate chart. This construction is part of a general philosophy: the base of the SYZ fibration should be an affine manifold with singularities, a topological manifold with charts such that the transition functions lie in the affine transformation group R n Sl(Z n ). Of course the fan for a toric surface is a subdivision of R 2 whereas the dual intersection complex is just an abstract fan. The following lemma makes sense of the situation. Lemma 3 (Toric reconstruction). Let S be a compact toric surface and D 21

36 the toric boundary. Then (S, D) is a Looijenga pair, (S,D) embeds into R 2 and any choice of embedding gives the fan for a toric variety isomorphic to S. Proof. See [17]. Given a choice of ample divisor H on S toric, the fan (S,D) carries a natural choice of piecewise linear function up to a choice of global linear function and we denote this φ. Such a function is defined by how it changes where it is non-linear. If φ vi,v i+1 denotes the linear extension of φ restricted to R 0 v i +R 0 v i+1, then we determine φ by insisting that φ vi,v i+1 φ vi 1,v i = (H.D i )n i, where n i is a primitive element of the dual space vanishing on v i and positive on v i R 0 + v i+1 R 0. Now we ask how we can reconstruct the mirror from this data? This is surprisingly easy, for each point P of the integral structure introduce a variable ϑ P together with the multiplication rule that for points P and Q the product ϑ P ϑ Q is z φ(p +Q) φ(p ) φ(q) ϑ P +Q. This recreates the counts of holomorphic triangles as suggested by Auroux; in this toric case there are no additional singularities to account for. For non-zero choices of coefficients z φ(p ) the spectrum ˇW of the ring spanned by the ϑ is then an algebraic torus and the mirror should be the pair ( ˇW, ϑ vi ). Let us perform this in the case of P 2. Example 5. Our first goal is to find the fan. Fortunately we learnt a fan for P 2 back in infancy: it has one cells generated by (1, 0), (0, 1) and ( 1, 1). (1, 1) Calculating the product defined above, using the ample divisor H being the class of a line in P 2, one finds ϑ (1,0) ϑ (0,1) = zϑ ( 1, 1). As in the basis 22

37 constructions of [14] these functions morally ought to produce an embedding of the mirror into affine space and their sum should be the superpotential. Doing this we obtain the mirror as being G 2 m with superpotential x + y + z/xy. As expected this is the mirror predicted by other constructions. Calculating the Jacobian ring of the singularity we find that it is Z[X, z]/ X 3 z, isomorphic to the quantum cohomology of P 2. There were several steps involved in the above construction which we need to generalise. In order we 1. Constructed a base B. 2. Found a canonical piecewise linear function φ on B. 3. Introduced functions coming from integral points of B and a multiplication rule on these functions. 4. Used these functions to embed the mirror into affine space. Table 4 We will now set about generalising these in order to the case of a Looijenga pair (S, D). We will want to keep track of the data of the pair (S,D) we have chosen later, so let us record here a choice of boundary divisor and affine manifold for all the non-toric del Pezzo surfaces. Here we record the boundary divisor in P 2 before blowing up. The circles represent points to be blown up, with the exceptional divisor included in the boundary if the circle is red. Since we can t embed the dual complex as an affine manifold in R 2, we instead provide a list of cones in R 2 and an integral affine isomorphism between two of these cones. The dual intersection complex is then obtained by identifying these two cones using this isomorphism. In the pictures below the cone shaded grey is identified integral linearly with the standard first quadrant, whilst the dotted region is removed entirely. 23

38 Surface Boundary Class of boundary Dual complex dp 5 D i = E 1 (H E 1 E 2 ) E 2 (H E 2 E 3 ) (H E 1 E 4 ) dp 4 D i = E 1 (H E 1 E 2 ) (H E 3 E 4 ) (H E 1 E 5 ) dp 3 D i = (H E 1 E 2 ) (H E 3 E 4 ) (H E 5 E 6 ) dp 2 D i = (H E 1 E 2 ) (2H E 3 E 4 E 5 E 6 E 7 ) The Mumford degeneration Here we attempt to generalise bullet points 1 and 2 of Table 4. The construction of the mirror family was inspired by the Mumford degeneration of an Abelian variety to the union of toric varieties. Let us recall the construction of [50] Example 6. Let B N R be a lattice polyhedron, P a lattice polyhedral decomposition of B and φ : B R a strictly convex piecewise linear integral function. One takes the graph over φ to produce a new polyhedron Γ(B, φ) := {(n, r) N R R n B, r φ(n)} This produces a lattice polyhedron unbounded in the positive direction on R. To construct a family from this we perform a cone construction, let 24

39 C(Γ) be the closure of the cone over Γ(B, φ): C(γ) = {(n, r 1, r 2 ) N R R R (n, r 1 ) r 2 Γ(B, φ)}. This carries an action of R + given by translating the second component of Γ(B, φ). The integral points of C(Γ) form a graded monoid and we can take P roj of this to produce a variety projective over A 1. We write this P Γ(B,φ). The general fibre is a toric variety, whilst the fibre over the origin is a union of toric varieties. The construction of [28] mimics this in reverse: it claims that the central fibre should be the n-vertex, which for n > 2 is a cycle of n copies of A 2 glued adjacently along their axes, and then attempts to smooth. We cannot hope to have global coordinates as in the toric Mumford degeneration, only local coordinates. To incorporate this data the authors of [28] define a twist of the tangent sheaf which has enough global sections, in particular the function φ lifts to a section of this bundle. Definition 8. Let (S, D) be a Looijenga pair with associated dual intersection complex (S,D) and let η : NE(S) M be a homomorphism of monoids. We want to construct a multi-valued piecewise linear function on (S,D) which will bend only along the one-cells. This will be a collection of piecewise linear functions defined on open subsets of B which differ by linear functions on overlaps. Such functions are determined by their bends at one-cells, which are encoded as follows. For a one-cell τ = R 0 v i, choose an orientation σ + and σ of the two two-cells separated by τ and let n τ be the unique primitive linear function positive on σ + and annihilating τ. We want to construct a representative φ i for φ on σ + σ. Writing φ + and φ for the linear function defined by φ i on σ + and σ, the function φ i is then defined up to a linear function by the requirement that φ + φ = n τ η([d i ]) Such a function is convex in the sense of [28] Definition 1.11, and one says that it is strictly convex if η([d i ]) is not invertible for any i. 25

40 This function determines a M gp R-torsor P as defined in [33], Construction 1.14, on B \ {0} which is trivial on each (σ + σ ) \ {0}, i.e., is given by ((σ + σ ) \ {0}) (M gp R). These trivial torsors are glued on the overlap of two adjacent such sets, namely on R 0 v i R 0 v i+1, via the map (x, p) (x, p + φ i+1 (x) φ i (x)) which induce isomorphisms on the monoid of points lying above φ. This construction is designed to allow us to run the Mumford construction locally, even if we cannot run it globally. Write π : P B \ {0}, and we write M for the sheaf π (Λ P ), bearing in mind that P also has the structure of an affine manifold. Then there is a canonical exact sequence 0 M gp M r Λ B 0 We write r for the second map in this sequence. This will not be mentioned again until we define the canonical scattering diagram so keep this in mind until then. Furthermore, [27], Definition 1.16 gives a subsheaf of monoids M + M. If one performs this construction in the case of a toric variety with the function φ and pairs it with an ample class one obtains the height function we used in the Mumford degeneration. After introducing this Gross, Hacking and Keel continue to explain how such a φ gives rise to a canonical deformation of the n-vertex. This concerns their proof that the construction smooths the n-vertex and we wait until later on to introduce this. 2.2 Scattering diagrams The next step to move from the toric world to the non-toric world is bullet point 3 of Table 4 is to modify the construction of the ϑ-functions. To do so in a systematic way we must introduce scattering diagrams. In Mumford s degeneration of an Abelian variety above all the fibres of 26

41 the SYZ torus fibration on a generic member of the family are smooth. This can not be the case for a special Lagrangian fibration of a general variety as there must be some singular fibres and around these singular fibres there will be some monodromy action. This monodromy is an obstruction to a naive definition of theta functions being well-defined on the mirror. Thus one needs a corrected notion of theta functions, which from symplectic geometry can be done by counting so-called Maslov index two disks as described in [4]. These can be generated by gluing on Maslov index zero disks onto standard Maslov index two disks. Rather than trying to make this symplectic heuristic precise, we instead are motivated by this picture as follows. If we stand well away from the singularities and tropicalise these Maslov index zero disks they appear to be a collection of lines passing out from the origin together with the information of their class. This data is the inspiration for the definition below of a scattering diagram on the base B. Definition 9. Let B be an affine manifold with a single singularity, with B homeomorphic to R 2 with singularity at the origin, so that B := B \{0} is an affine manifold. Let M be a locally constant sheaf of abelian groups on B with a subsheaf of monoids M + M and equipped with a map r : M Λ B. Let J be a sheaf of ideals in M + with stalk J x maximal in M + x for all x B. Let R denote the sheaf of rings locally given by the completion of k[m + ] at J. A scattering diagram with values in the pair (M, J ) on B is a function f which assigns to each rational ray from the origin an section of the restriction of R to the ray. We require the following properties of this function: For each d one has f(d) = 1 mod J d. For each n there are only finitely many d for which f(d) is not congruent to 1 mod J n d. These d are called walls. For each ray d and for each monomial z p appearing in f(d) one has r(p) tangent to d. A line for which r(p) is a positive generator of d for all p with c p 0 is called an incoming ray. If instead r(p) is a 27