Duke Mathematical Journal

Symplectic invariance of uniruled affine varieties and log Kodaira dimension

Mark McLean

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We introduce some definitions of uniruledness for affine varieties and use these ideas to show symplectic invariance of various algebraic invariants of affine varieties. For instance we show that if A and B are symplectomorphic smooth affine varieties, then any compactification of A by a projective variety is uniruled if and only if any such compactification of B is uniruled. If A is acylic of dimension 2, then we show that B has the same log Kodaira dimension as A. If A has dimension 3, has log Kodaira dimension 2, and satisfies some other conditions, then B cannot be of log general type.

Article information

Source
Duke Math. J., Volume 163, Number 10 (2014), 1929-1964.

Dates
First available in Project Euclid: 8 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1404824305

Digital Object Identifier
doi:10.1215/00127094-2738748

Mathematical Reviews number (MathSciNet)
MR3229045

Zentralblatt MATH identifier
1312.53107

Subjects
Primary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx]
Secondary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35] 14R05: Classification of affine varieties

Citation

McLean, Mark. Symplectic invariance of uniruled affine varieties and log Kodaira dimension. Duke Math. J. 163 (2014), no. 10, 1929--1964. doi:10.1215/00127094-2738748. https://projecteuclid.org/euclid.dmj/1404824305


Export citation

References

  • [1] M. Abouzaid and P. Seidel, An open string analogue of Viterbo functoriality, Geom. Topol. 14 (2010), 627–718.
  • [2] K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), 601–617.
  • [3] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), 45–88.
  • [4] F. Bourgeois, T. Ekholm, and Y. Eliashberg, Effect of Legendrian surgery, with an appendix by S. Ganatra and M. Maydanskiy, Geom. Topol. 16 (2012), 301–389.
  • [5] K. Cieliebak and K. Mohnke, Symplectic hypersurfaces and transversality in Gromov-Witten theory, J. Symplectic Geom. 5 (2007), 281–356.
  • [6] Y. Eliashberg and M. Gromov, “Convex symplectic manifolds” in Several Complex Variables and Complex Geometry, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math. 52, Amer. Math. Soc., Providence, 1991, 135–162.
  • [7] J. W. Fish, Target-local Gromov compactness, Geom. Topol. 15 (2011), 765–826.
  • [8] H. Flenner and M. Zaidenberg, “$\mathbf{Q}$-acyclic surfaces and their deformations” in Classification of Algebraic Varieties (L’Aquila, 1992), Contemp. Math. 162, Amer. Math. Soc., Providence, 1994, 143–208.
  • [9] R. Friedman and J. W. Morgan, Algebraic surfaces and Seiberg-Witten invariants, J. Algebraic Geom. 6 (1997), 445–479.
  • [10] T. Fujita, On the topology of noncomplete algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), 503–566.
  • [11] K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant, Topology 38 (1999), 933–1048.
  • [12] R. V. Gurjar and M. Miyanishi, “Affine surfaces with $\overline{\kappa}\leq1$” in Algebraic Geometry and Commutative Algebra, Vol. I, Kinokuniya, Tokyo, 1988, 99–124.
  • [13] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, Ann. of Math. (2) 79 (1964), 109–203; II, 205–326.
  • [14] H. Hofer, K. Wysocki, and E. Zehnder, Applications of polyfold theory I: the polyfolds of Gromov-Witten theory, preprint, arXiv:1107.2097v2 [math.SG].
  • [15] J. Hu, T.-J. Li, and Y. Ruan, Birational cobordism invariance of uniruled symplectic manifolds, Invent. Math. 172 (2008), 231–275.
  • [16] S. Iitaka, “On logarithmic Kodaira dimension of algebraic varieties” in Complex Analysis and Algebraic Geometry, Iwanami Shoten, Tokyo, 1977, 175–189.
  • [17] S. Iitaka, Algebraic Geometry, Grad. Texts in Math. 76, Springer, New York, 1982.
  • [18] Y. Kawamata, “On the classification of noncomplete algebraic surfaces” in Algebraic Geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math. 732, Springer, Berlin, 1979, 215–232.
  • [19] T. Kishimoto, On the logarithmic Kodaira dimension of affine threefolds, Internat. J. Math. 17 (2006), 1–17.
  • [20] J. Kollár, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin, 1996.
  • [21] J. Kollár, “Low degree polynomial equations: arithmetic, geometry and topology” in European Congress of Mathematics, Vol. I (Budapest, 1996), Progr. Math. 168, Birkhäuser, Basel, 1998, 255–288.
  • [22] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), 119–174.
  • [23] J. Li and G. Tian, “Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds” in Topics in Symplectic $4$-Manifolds (Irvine, CA, 1996), First Int. Press Lect. Ser. 1, Int. Press, Cambridge, Mass., 1998, 47–83.
  • [24] T.-J. Li and W. Zhang, “Additivity and relative Kodaira dimensions” in Geometry and Analysis. No. 2, Adv. Lect. Math. (ALM) 18, Int. Press, Somerville, Mass., 2011, 103–135.
  • [25] D. McDuff, The structure of rational and ruled symplectic $4$-manifolds, J. Amer. Math. Soc. 3 (1990), 679–712.
  • [26] M. McLean, The growth rate of symplectic homology and affine varieties, Geom. Funct. Anal. 22 (2012), 369–442.
  • [27] M. Miyanishi, Open Algebraic Surfaces, CRM Monogr. Ser. 12, Amer. Math. Soc., Providence, 2001.
  • [28] M. Miyanishi and T. Sugie, Affine surfaces containing cylinderlike open sets, J. Math. Kyoto Univ. 20 (1980), 11–42.
  • [29] C. P. Ramanujam, A topological characterisation of the affine plane as an algebraic variety, Ann. of Math. (2) 94 (1971), 69–88.
  • [30] Y. Ruan, Symplectic topology on algebraic $3$-folds, J. Differential Geom. 39 (1994), 215–227.
  • [31] Y. Ruan, Topological sigma model and Donaldson-type invariants in Gromov theory, Duke Math. J. 83 (1996), 461–500.
  • [32] Y. Ruan, Virtual neighborhoods and pseudo-holomorphic curves, Turkish J. Math. 23 (1999), 161–231.
  • [33] Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), 259–367.
  • [34] P. Seidel, “A biased view of symplectic cohomology” in Current Developments in Mathematics, 2006, Int. Press, Somerville, Mass., 2008, 211–253.
  • [35] Z. Tian, Symplectic geometry of rationally connected threefolds, Duke Math. J. 161 (2012), 803–843.
  • [36] T. tom Dieck and T. Petrie, Contractible affine surfaces of Kodaira dimension one, Japan. J. Math. (N.S.) 16 (1990), 147–169.
  • [37] C. Voisin, Rationally connected $3$-folds and symplectic geometry, Astérisque 322 (2008), 1–21.
  • [38] E. Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), 769–796.
  • [39] M. Zaidenberg, Lectures on exotic algebraic structures on affine spaces, preprint, arXiv:math/9801075v2 [math.AG].