Journal of the American Mathematical Society

Intersection theory on $\overline{\mathcal{M}}_{1,4}$ and elliptic Gromov-Witten invariants

Author(s): E. Getzler
Journal: J. Amer. Math. Soc. 10 (1997), 973-998.
MSC (1991): Primary 14H10, 14H52, 14N10, 81T40, 81T60
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Abstract: We find a new relation among codimension $2$

algebraic cycles in the moduli space $\overline{\mathcal{M}}_{1,4}$ , and use this to calculate the elliptic Gromov-Witten invariants of projective spaces $\mathbb{CP}^2$ and $\mathbb{CP}^3$ .

1.: D. Avritzer, Enumerative geometry of del Pezzo surfaces, ICTP preprint IC/93/53.
2.: K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), 601-617. CMP 97:07
3.: K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), 45-88. CMP 97:09
4.: K. Behrend and Yu. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke Math. J. 85 (1996), 1-60. CMP 97:02
5.: M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B405 (1993), 279-304. MR 94j:81254
6.: L. Caporaso and J. Harris, Counting plane curves of any genus, to appear, Invent. Math., alg-geom/9608025.
7.: T. Eguchi, K. Hori and C.-S. Xiong, Quantum cohomology and the Virasoro algebra, hep-th/9703086.
8.: C. Faber, Chow rings of moduli spaces of curves. I: The Chow ring of $\overline{\mathcal{M}}_3$ , Ann. Math. 132 (1990), 331-419. MR 91h:14009a
9.: W. Fulton, ``Intersection theory,'' Springer-Verlag, Berlin-Heidelberg-New York, 1984. MR 85k:14004
10.: W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, to appear in Proc. Symp. Pure Math., Amer. Math. Soc., Providence, R.I., alg-geom/9608011.
11.: E. Getzler, The semi-classical approximation for modular operads, Max-Planck-Institut preprint MPI-96-145, alg-geom/9612005.
12.: E. Getzler, Generators and relations for $H_\bullet (\overline{\mathcal{M}}_{1,n},\mathbb{Q})$ , in preparation.
13.: E. Getzler and R. Pandharipande, The number of elliptic space curves, in preparation.
14.: R. Hain and E. Looijenga, Mapping class groups and moduli spaces of curves, to appear in Proc. Symp. Pure Math., Amer. Math. Soc., Providence, R.I., alg-geom/9607004.
15.: L. Göttsche and R. Pandharipande, The quantum cohomology of blow-ups of $\mathbb{P}^2$ and enumerative geometry, alg-geom/9611012.
16.: J. Harris, On the Severi problem, Invent. Math. 84 (1986), 445-461. MR 87f:14012
17.: R. Kaufmann, The intersection form in the cohomology of the moduli space of genus -pointed curves $H^*(\overline{M}_{0,n})$ and the explicit Künneth formula in quantum cohomology, Max-Planck-Institut für Mathematik preprint MPI 96-106, alg-geom/9608026.
18.: S. Keel, Intersection theory of moduli spaces of stable -pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545-574. MR 92f:14003
19.: S. Kleiman, The transversality of a general translate, Compositio Math. 38 (1974), 287-297. MR 50:13063
20.: F.F. Knudsen, The projectivity of the moduli space of stable curves II. The stacks $\overline{\mathcal{M}}_{g,n}$ , Math. Scand. 52 (1983), 161-189. MR 85d:14038a
21.: J. Kollár, ``Rational curves on algebraic varieties,'' Springer-Verlag, Berlin-Heidelberg-New York, 1995. CMP 97:10
22.: M. Kontsevich, Intersection theory on moduli spaces of curves and the matrix Airy function, Commun. Math. Phys. 147 (1992), 1-23. MR 93e:32027
23.: M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology and enumerative geometry, Commun. Math. Phys. 164 (1994), 525-562. MR 95i:14049
24.: J. Li and G. Tian, Virtual moduli cycles and GW-invariants, alg-geom/9602007.
25.: J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, alg-geom/9608032.
26.: R. Pandharipande, A geometric realization of Getzler's relation, hep-th/9705016.
27.: Z. Ran, The degree of a Severi variety, Bull. Am. Math. Soc. 17 (1987), 125-128. MR 88i:14019
28.: I. Vainsencher and D. Avritzer, Compactifying the space of elliptic quartic curves, in ``Complex projective geometry,'' eds. G. Ellinsgrud et al., London Mathematical Society LNS 179, Cambridge Uni. Press, 1992, pp. 47-58. MR 94c:14027
29.: E. Witten, On the structure of the topological phase of two-dimensional gravity, Nucl. Phys. B340 (1990), 281-332. MR 91m:32020

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