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HEP-TH-9405035 Maxim Kontsevich Introduction CORE Metadata, citation and similar papers at core.ac.uk Provided by CERN Document Server ENUMERATION OF RATIONAL CURVES VIA TORUS ACTIONS Maxim Kontsevich Max-Planck-Institut fur Mathematik, Bonn and University of California, Berkeley Introduction This pap er contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry. In a sense, we almost solved b oth problems. There are still certain gaps in foundations. Nevertheless, we obtain \closed" formulas for generating functions in top ological sigma-mo del for a wide class of manifolds, covering many Calabi-Yau and Fano varieties. We reduced Mirror Symmetry in a basic example to certain complicated but explicit identity. Wehave made several computer checks. All results were as exp ected. In particular, we computed the \physical" number of rational curves of degree 4 on a quintic 3-folds (during 5 minutes on Sun), which was out of reach of previuos algebro-geometric metho ds. The text consists of 5 parts. The rst part contains the de nition of stable maps used through all the pap er. We establish several basic prop erties of mo duli spaces of stable maps. Also, we give an outline of a contsruction of Gromov-Witten invariants for all algebraic pro jective or closed symplectic manifolds. For reader who is interested mainly in computations it is enough to lo ok through 1.1 and to the statements of theorems in 1.3.1-1.3.2. In section 2 we describ e few examples of counting problems in enumerative ge- ometry of curves. One of examples is rational curves on quintics. We give a simple algebro-geometric de nition for the numb er of curves without assuming the validity of the Clemens conjecture or using symplectic metho ds. The main b o dy of computations is contained in section 3. Our strategy here is quite standard: we reduce problems to questions concerning Chern classes on a space of rational curves lying in pro jective spaces (A. Altman - S. Kleiman, S. Katz), and then use Bott's residue formula for the action of the group of diagonal matrices (G. Ellingsrud and S. A. Strmme). As a result we get in all our examples certain sums over trees. HEP-TH-9405035 In section 4 we develop a general scheme for summation over trees. By Feyn- man rules we know that such a sum should b e equal to the critical value of some functional. Using a trickwe obtain an equivalent functional which is a quadratic p olynomial in in nitely manyvariables with co ecients dep ending on a nite num- ber of variables. Thus, all our counting problems are reduced to the inversion of certain explicit square matrices with co e cients of hyp ergeometric kind. This last Typ eset by A S-T X M E 2 MAXIM KONTSEVICH step wewere not able to accomplish. Presumably, there is here a hidden structure of an integrable system and Sato's grassmanians. In section 5 we describ e extensions of our computation scheme to other enumer- ative problems, including Calabi-Yau and Fano complete intersections (of arbitrary dimension) in pro jective spaces, toric varieties and generalized ag varieties. 1. Stable maps. 1.1. De nition. Let V beascheme of nite typ e over a eld (or a smo oth scheme, or a complex manifold, or an almost complex manifold). De nition. Stable map is a structure (C ; x ;:::;x ;f) consisting of a connected 1 k compact reduced curve C with k 0 pairwise distinct marked non-singular points x and at most ordinary double singular points, and a map f : C ! V having no i non-trivial rst order in nitesimal automorphisms, identical on V and x ;:::;x 1 k (stability). The condition of stability means that every irreducible comp onentof C of genus 0 (resp. 1) which maps to a p ointmust have at least 3 (resp. 1) sp ecial (i.e. marked or singular) p oints on its normalization. Also, it means that the automorphism group of (C ; x ;:::;x ;f) is nite. 1 k For a curve C with at most ordinary double singular p oints its arithmetic genus 1 p (C ):=dim H (C; O) can b e computed from the formula a sing 2 2p (C )=(CnC ) : a 2 Let 2 H (V; Z) b e a homology class. (In algebro-geometric situation should b e an element of the group of 1-dimensional cycles mo dulo homological equivalence). Notation. M (V; ) denotes the moduli stack of stable maps to V of curves of g;k arithmetic genus g 0 with k 0 markedpoints such that f [C ]= . More precisely, in algebro-geometric setting one can de ne a family of at maps to V as a at prop er morphism C!Stoascheme S of nite typ e over the ground eld and a map f : C!V such that its restriction to each geometric b er of C over S is a stable map. M (V; ) as a set of In the setting of almost complex manifolds we consider g;k equivalence classes of stable maps endowed with a natural top ology (see [P]) and an orbispace structure (see the next subsection). Remark.For many reasons one has to consider curves not in a xed manifold V but in manifolds V varying in families. Wehave not develop ed the corresp onding formalism yet. In subsection 5.4 we describ e a simple example of algebraic K 3- surfaces which shows the necessity of families. It is also clear from our example that one can consider non-compact V as well. 1.2. Orbispaces. The notion of orbispace intro duced here is a top ological counterpart of (1) algebraic stacks (from algebraic geometry), and (2) orbifolds, or V-manifolds (from di erential top ology). ENUMERATION OF RATIONAL CURVES VIA TORUS ACTIONS 3 We de ne orbispace as a small top ological category C (i.e. a category for which Ob C and Mor C carry top ological structures) satisfying following axioms. A.1. C is a group oid (every morphism is invertible). A.2. For each X; Y 2 Ob C the set of morphisms Mor (X; Y ) is nite. C A.3. Two maps from Mor C to Ob C , assigning to a morphism its source and its target resp ectively, are lo cally homeomorphisms (etale maps). Functors b etween orbispaces which are continous, lo cally homeomorphisms and induce equivalence of categories we can call equivalences b etween orbispaces. The set jS j of equivalence classes of ob jects of C has natural induced top ology. We can asso ciate with each element[X]2jSjan equivalence class (mo dulo interior automorphisms) of nite groups, Aut(X ). 1.3. Prop erties of mo duli spaces of stable maps. The notion of a stable map is a mixture of the notion of a stable curve from algebraic geometry and of the notion of a cusp-curve from symplectic top ology. e By de nition from [P], cusp-curve is a holomorphic map f from a compact (not e necessarily connected) smo oth complex curve C to an almost-complex manifold V e and a nite collection S of non-intersecting 2-element subsets of C such that, for e each S 2S, its image f (S ) is a 1-element set. Glueing p oints from pairs S 2S together we obtain a curve C with at most ordinary double singular p oints and a map f : C ! V .P.Pansu claimed in [P] that if V is compact and endowed with a riemannian metric, then the space of equivalence classes of cusp-curves of b ounded genus and area is compact and Hausdor . His claim is wrong, exactly b ecause the condition of stability on comp onents which are mapp ed to a p ointwas forgotten! It seems that, after appropriate corrections, the pro of from [P] shows that the mo duli space of stable maps of b ounded genus and area is compact and Hausdor . Recall that in symplectic top ology one considers usually almost-complex struc- tures on symplectic manifolds compatible in an evident sense with the symplectic form. Such a structure de nes a Riemannian metric on the underlying manifold, and the riemannian area of each holomorphic curve coincides with its symplectic area. The latter is a pure homological invariant. Hence M (V; ) is compact and g;k Hausdor in such a situation. In the next subsection, we prove analogous prop erties of M (V; ) in algebro- g;k geometric setting. In 1.3.2, we describ e a situation in which the mo duli space of stable maps is smo oth (as a stack). 1.3.1. Algebraicity and prop erness. Theorem. Let V bea projective scheme of nite type over a eld. Then M (V; ) g;k is an algebraic proper stack of nite type. The pro of uses results from [DM]. We refere to [DM] for de nitions concerning prop erties of stacks, and for other technical details as well. M (V; ) as a quotient stackofascheme of nite typ e Wewant to realize g;n mo dulo etale equivalence relation. From the b oundness of the Hilb ert scheme of 1-dimensional subschemes of V it follows that for (C ; x ; ) with xed p (C ) and , the numb er of singular p oints on a C and the numb er of irreducible comp onents of C are b ounded. 4 MAXIM KONTSEVICH In the next step, we will realize M (V; ) as a quotient space of a space of maps g;n of stable curves into V .For this we can cho ose a nite collection of hyp ersurfaces D in V such that each non-stable comp onentofany curve C from M (V; ) i g;n intersects transversally some of D at least at three non-sp ecial p oints.
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