# zbMATH — the first resource for mathematics

Stacks of stable maps and Gromov-Witten invariants. (English) Zbl 0872.14019
From the authors’ summary: Let $$V$$ be a projective algebraic manifold. M. Kontsevich and Yu. Manin [Commun. Math. Phys. 164, No. 3, 525-562 (1994; Zbl 0853.14020)], described Gromov-Witten invariants of $$V$$ axiomatically as a collection of linear maps $I^V_{g,n,\beta} : H^*(V)^{\otimes n} \rightarrow H^*(\overline M_{g,n}, \mathbb Q), \quad \beta \in H_2(V,\mathbb Z),$ satisfying certain axioms, and suggested a program to construct them by algebro-geometric techniques. The program is based upon Kontsevich’s notion of a stable map $$(C,x_1,\dots,x_n,f), f: C \rightarrow V$$. These data consist of an algebraic curve $$C$$ with $$n$$ labeled points on it and a map $$f$$ such that if an irreducible component of $$C$$ is contracted by $$f$$ to a point, then this component together with its special points is Deligne-Mumford stable. The construction consists of three major steps.
$$A$$. Construct an orbispace (or rather a stack) of stable maps $$\overline M_{g,n}(V,\beta)$$ such that $$g=$$ genus of $$C$$, $$f_*([C])=\beta$$, and its two morphisms to $$V^n$$ and $$\overline M_{g,n}$$. On the level of points, these morphisms are given, respectively, by $p\colon(C,x_1,\dots,x_n,f)\mapsto (f(x_1),\dots,f(x_n)),\quad q\colon(C,x_1,\dots,x_n,f)\mapsto[(C,x_1,\dots,x_n)]^{stab},$ where the last expression means the stabilization of $$(C,x_1,\dots,x_n)$$.
$$B$$. Construct a “virtual fundamental class” $$[\overline M_{g,n}(V,\beta)]_{virt}$$ or “orientation”, and use it to define a correspondence in the Chow ring $$C^V_{g,n,\beta} \in A(V^n \times \overline M_{g,n})$$.
$$C$$. Use $$C^V_{g,n,\beta}$$ to construct the induced maps $$I^V_{g,n,\beta}$$ on any cohomology satisfying some version of the standard properties making it functorial on the category of correspondences. – A neat way to organize this information is to introduce the category of marked stable modular graphs indexing degeneration types of stable maps and to treat various modular stacks $$\overline M_{g,n}(V,\beta)$$ as values of this modular functor on the simplest one-vertex graphs.
Then the check of the axioms in the paper cited above essentially boils down to a calculation of this functor on a family of generating morphisms and objects in the graph category. The degeneration type of $$(C,x_1,\dots,x_n,f)$$ is described by the graph whose vertices are the irreducible components of $$C$$, edges are singular points of $$C$$, and tails (“one-vertex edges”) are $$x_1,\dots, x_n$$. In addition, each vertex is marked by the homology class in $$V$$, which is the $$f$$-image of the fundamental class of the respective component of $$C$$, and by the genus of the normalization of this component.
In part I of this paper, the authors treat in this way step A. – Part II is devoted to steps B and C for $$g=0$$ and convex manifolds $$V$$.

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14F99 (Co)homology theory in algebraic geometry
Full Text:
##### References:
 [1] 1 M. Artin, A. Grothendieck, and J. L. Verdier, Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos , Lecture Notes in Mathematics, vol. 269, Springer-Verlag, Berlin, 1972. · Zbl 0234.00007 · doi:10.1007/BFb0081551 [2] 2 M. Artin, A. Grothendieck, and J. L. Verdier, Théorie des topos et cohomologie étale des schémas. Tome 2 , Lecture Notes in Mathematics, vol. 270, Springer-Verlag, Berlin, 1972. · Zbl 0237.00012 · doi:10.1007/BFb0061319 [3] 3 M. Artin, A. Grothendieck, and J. L. Verdier, Théorie des topos et cohomologie étale des schémas. Tome 3 , Lecture Notes in Mathematics, vol. 305, Springer-Verlag, Berlin, 1973. · Zbl 0245.00002 · doi:10.1007/BFb0070714 [4] K. Behrend, Gromov-Witten invariants in algebraic geometry , to appear in Invent. Math. · Zbl 0909.14007 · doi:10.1007/s002220050132 [5] K. Behrend and B. Fantechi, The intrinsic normal cone , to appear in Invent. Math. · Zbl 0909.14006 · doi:10.1007/s002220050136 [6] P. Deligne and J. Milne, Tannakian categories , Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Math., vol. 900, Springer-Verlag, New York, 1982, pp. 101-228. · Zbl 0465.00010 [7] W. Fulton and R. MacPherson, Categorical framework for the study of singular spaces , Mem. Amer. Math. Soc. 31 (1981), no. 243, vi+165. · Zbl 0467.55005 [8] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology , to appear in the proceedings of the Algebraic Geometry Conference in Santa Cruz, Calif., 1995. · Zbl 0898.14018 [9] E. Getzler and M. Kapranov, Cyclic operads and cyclic homology , Geometry, topology, & physics ed. S. T. Yau, Conf. Proc. Lecture Notes Geom. Topology, IV, Internat. Press, Cambridge, MA, 1995, pp. 167-201. · Zbl 0883.18013 [10] A. Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique IV: Les schémas de Hilbert , Séminaire Bourbaki, 13e année, 221, 1960-61. · Zbl 0236.14003 · numdam:SB_1960-1961__6__249_0 · eudml:109614 [11] A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes , Inst. Hautes Études Sci. Publ. Math. (1961), no. 8, 222. · Zbl 0118.36206 · doi:10.1007/BF02684778 · numdam:PMIHES_1960__4__5_0 · numdam:PMIHES_1961__8__5_0 · numdam:PMIHES_1961__11__5_0 [12] 1 A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I , Inst. Hautes Études Sci. Publ. Math. (1964), no. 20, 259. · Zbl 0136.15901 · doi:10.1007/BF02684747 · numdam:PMIHES_1964__20__5_0 [13] 2 A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II , Inst. Hautes Études Sci. Publ. Math. (1965), no. 24, 231. · Zbl 0135.39701 · numdam:PMIHES_1965__24__5_0 · eudml:103851 [14] 3 A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III , Inst. Hautes Études Sci. Publ. Math. (1966), no. 28, 255. · Zbl 0144.19904 · doi:10.1007/BF02684343 · numdam:PMIHES_1966__28__5_0 · eudml:103860 [15] 4 A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV , Inst. Hautes Études Sci. Publ. Math. (1967), no. 32, 361. · Zbl 0153.22301 · numdam:PMIHES_1967__32__5_0 · eudml:103873 [16] A. Grothendieck and M. Raynaud, Revêtements étales et groupe fondamental , Lecture Notes in Mathematics, vol. 224, Springer-Verlag, Berlin, 1971. · Zbl 0234.14002 [17] S. Kleiman, Motives , Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer-School in Math., Oslo, 1970), Wolters-Noordhoff, Groningen, 1972, pp. 53-82. · Zbl 0285.14005 [18] F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks $$M\sbg,n$$ , Math. Scand. 52 (1983), no. 2, 161-199. · Zbl 0544.14020 · eudml:166839 [19] M. Kontsevich, Enumeration of rational curves via torus actions , The moduli space of curves (Texel Island, 1994), Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 335-368. · Zbl 0885.14028 [20] M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry , Comm. Math. Phys. 164 (1994), no. 3, 525-562. · Zbl 0853.14020 · doi:10.1007/BF02101490 [21] M. Kontsevich and Yu. Manin, Quantum cohomology of a product , Invent. Math. 124 (1996), no. 1-3, 313-339, (appendix by R. Kaufmann). · Zbl 0853.14021 · doi:10.1007/s002220050055 [22] A. J. Scholl, Classical motives , Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 163-187. · Zbl 0814.14001 [23] A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces , Invent. Math. 97 (1989), no. 3, 613-670. · Zbl 0694.14001 · doi:10.1007/BF01388892 · eudml:143716
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.