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Gromov-Witten Invariants of Hypersurfaces Gromov-Witten invariants of hypersurfaces Dem Fachbereich Mathematik der Technischen Universitat¨ Kaiserslautern zur Erlangung der venia legendi vorgelegte Habilitationsschrift von Andreas Gathmann geboren am 9.4.1970 in Hannover 2003 Contents Preface 1 Chapter 1. Gromov-Witten invariants of projective spaces 9 1.1. Stable curves and stable maps 12 1.2. Gromov-Witten invariants 18 1.3. Basic relations among Gromov-Witten invariants 21 1.4. The Virasoro relations 28 1.5. Proof of the Virasoro algorithm 32 1.6. Numerical examples and applications 39 Chapter 2. Relative Gromov-Witten invariants in genus zero 45 2.1. Moduli spaces of stable relative maps 47 2.2. Raising the multiplicities 56 N 2.3. Proof of the main theorem for hyperplanes in P 59 2.4. Proof of the main theorem for very ample hypersurfaces 64 2.5. Enumerative applications 69 Chapter 3. The mirror theorem 81 3.1. The mirror transformation 85 3.2. Examples 94 3.3. Proof of the main technical lemmas 97 Chapter 4. The number of plane conics 5-fold tangent to a given curve 101 4.1. The Gromov-Witten approach 103 4.2. The components of M¯ (2;2;2;2;2) 108 4.3. Computation of the virtual fundamental classes 111 4.4. Application to rational curves on K3 surfaces 117 Chapter 5. Relative Gromov-Witten invariants in higher genus 121 5.1. Stable relative and non-rigid maps 123 5.2. The virtual push-forward theorem 132 5.3. Push-forwards in codimension 0 145 5.4. Push-forwards in codimension 1 155 5.5. Elliptic Gromov-Witten invariants of the quintic threefold 164 Bibliography 169 Preface Over the last decade Gromov-Witten invariants have become an invaluable tool in modern enumerative geometry. The goal of the theory is to count curves with some given conditions in a complex projective manifold, where the conditions are chosen so that the expected number of curves satisfying them is finite and non-zero. The most famous example of an enumerative problem has in fact been inspired by physics: let X be a quintic threefold, i.e. a smooth hypersurface of degree 5 in complex projective 4-space P4. For any g ≥ 0 and d > 0 a na¨ıve dimension count shows that one expects a finite number Ng;d of curves of genus g and degree d on X. In 1991 the string theorists Candelas et al. encountered this problem in their study of conformal field theories, and their computations led them to predictions for the numbers N0;d in genus zero [COGP]. Their main tool was a certain equivalence of topological field theories that is known to physicists as “mirror symmetry”. The discovery of this relation between physics and algebraic geometry was the starting point of modern enumerative geometry. The analysis of Candelas et al. showed already that in order to count curves embed- ded in X it is actually more natural to study maps from a (varying) source curve to the target X. These two notions are of course very much related as every map from a curve to X determines an image curve, and conversely every curve in X arises as the image of some map from a curve to X. There are subtle differences however: the map could e.g. be a k-fold cover onto its image, in which case the degree of the map is k times the degree of the set-theoretic image curve. In fact, the physicists’ mirror symmetry computations first led to a different set of numbers n0;d that are supposed to count maps of degree d from a source curve of genus g = 0 to X. These numbers are called the Gromov-Witten invariants. They are the coefficients of a generating function that can be obtained from a certain hypergeometric series by an explicit variable transformation. The enumerative numbers N0;d are then related to the Gromov-Witten invariants n0;d by a correction formula that takes care of the contributions from multiple covers. The precise form of this correction formula was simply guessed by the time the paper by Candelas et al. was written. 1 2 PREFACE In the following years physicists have vastly generalized the above results. Ber- shadsky et al. considered the case of higher genus. They conjectured a generating function for the Gromov-Witten invariants n1;d in genus 1 [BCOV1] and an al- gorithm that can in principle compute the invariants ng;d in any genus [BCOV2]. Gopakumar and Vafa have conjectured the general form of the transformation that relates the Gromov-Witten invariants ng;d to the enumerative numbers Ng;d for any g [GoV1, GoV2]. A non-trivial check for these conjectures is that the resulting numbers Ng;d actually turned out to be positive integers as far as they have been computed numerically — a statement that is far from being obvious from the arith- metic of the algorithms. In the meantime mathematicians were (and to a large extent still are) just puzzled by the physicists’ results. To start from the very basics of the problem, it has not even been proven yet that the enumerative number of genus-0 curves in a general quintic threefold X is at all finite. For higher genus the numbers are most definitely not finite since e.g. intersections of X with planes in P4 lead to infinite families of curves. Neither are the numbers of maps from curves to X of given genus and degree finite, not even in genus 0: multiple covers have ramification points whose position on the curve can vary continuously. So before mathematicians could try to compute the numbers ng;d or Ng;d their first problem was simply to define them rigorously. The problem of defining Gromov-Witten invariants mathematically has been solved around 1995 by the theory of stable maps initiated by Kontsevich [K2]. For any projective manifold X, genus g ≥ 0, and homology class b of a curve in X, Behrend and Manin have constructed a “nice” moduli space (read: separated and proper Deligne-Mumford stack) that parametrizes curves of genus g together with a map of class b to X [BM]. The dimension one would expect such a moduli space to have by a na¨ıve dimension count is called its virtual dimension; it is zero for the quintic threefold for any g and b. The actual dimension of the moduli space may in general well be bigger. But in any case one can construct a so-called virtual fundamental class of the moduli space: a cycle of the virtual dimension that replaces the ordi- nary fundamental class in homology [BF, B, LT1]. The virtual fundamental class thus makes the moduli space appear to have the virtual dimension for intersection- theoretic purposes. The idea is roughly the same as that of an excess intersection product: if Y and Z are subvarieties of X then the dimension of the intersection Y \ Z may be bigger than the expected (virtual) dimension dimY + dimZ − dimX, but in any case there is an intersection product Y ·Z: a homology class of the virtual dimension in Y \ Z. PREFACE 3 The Gromov-Witten invariants ng;d of a quintic threefold can now simply be defined to be the degrees of the (zero-dimensional) virtual fundamental classes of the mod- uli spaces of stable maps of genus g and degree d to the quintic. More generally, let X be any projective manifold and pick a genus g and homology class b such that the corresponding moduli space M of stable maps has non-negative virtual dimension. Moreover, choose a cohomology class on M whose codimension is equal to the vir- tual dimension of M. Usually this cohomology class is chosen so that it describes the condition that the curves in X (or more precisely: stable maps to X) pass through some fixed given subvarieties of X. Evaluating it on the virtual fundamental class gives a zero-dimensional cycle whose degree we call again a Gromov-Witten in- variant. It can be interpreted geometrically as the number of curves in X satisfying the given conditions. (Note that this count may be virtual though in the cases when the virtual fundamental class of M is not equal to the ordinary one.) Having defined the Gromov-Witten invariants of a projective manifold X the next questions are of course what their structure is and how they can be computed. In general this question is very difficult however and far from being solved. The goal of this thesis is to attack this problem in the case of hypersurfaces. More precisely, our main question can be posed as follows: Let Y be a smooth hypersurface of a complex projective manifold X. Can we compute the Gromov-Witten invariants of Y from those of X? This will mean two things: in a first step, we would like to find an algorithm that can be programmed on a computer and that enables us to compute every possible Gromov-Witten invariant of Y numerically if we know the invariants of X. A sec- ond question is whether the relation between the invariants of X and Y can then be phrased in some “nice” way, e.g. in the form of an explicit relation between the generating functions of the invariants for the two spaces. Moreover, we will com- pute all Gromov-Witten invariants of projective spaces directly. This way every relation between the invariants of a hypersurface and the ambient space that we get can be applied immediately to compute Gromov-Witten invariants of hypersurfaces in projective spaces, including the most interesting case of the quintic threefold.
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