2000 character limit reached
Symplectic maps and hyperKähler moment map geometry
Published 6 Oct 2021 in math.SG and math.DG | (2110.02679v2)
Abstract: We obtain a correspondence between the group of symplectic diffeomorphisms of a 4-dimensional real torus and the vanishing locus of a certain hyperK\"ahler moment map. This observation gives rise to a new flow, called the modified moment map flow. The construction can be adapted to the polyhedral setting, for which we prove a Duistermaat type theorem. This paper lays out the ground work for some effective polyhedral symplectic geometry and for a potential Morse-Bott theory, with applications to the topology of the space of symplectic maps of the 4-torus.
- M. Abreu and D. McDuff. Topology of symplectomorphism groups of rational ruled surfaces. J. Am. Math. Soc., 13(4):971–1009, 2000.
- M. F. Atiyah and R. Bott. The Yang-Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond., Ser. A, 308:523–615, 1983.
- A. Banyaga. The structure of classical diffeomorphism groups, volume 400 of Math. Appl., Dordr. Dordrecht: Kluwer Academic Publishers, 1997.
- M. Bertelson and J. Distexhe. PL approximations of symplectic manifolds. J. Symplectic Geom., 2024. to appear.
- E. Bierstone and P. D. Milman. Semianalytic and subanalytic sets. Publ. Math., Inst. Hautes Étud. Sci., 67:5–42, 1988.
- J. Distexhe. Triangulating symplectic manifods. Université Libre de Bruxelles, PhD thesis, 2019.
- S. K. Donaldson. A new proof of a theorem of Narasimhan and Seshadri. J. Differ. Geom., 18:269–277, 1983.
- Y. M. Ehliashberg. A theorem on the structure of wave fronts and its applications in symplectic topology. Funct. Anal. Appl., 21(1-3):227–232, 1987.
- S. Etourneau. Approximation C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT d’immersions isotropes lisses par des immersions isotropes PL. Nantes University, PhD thesis, 2023.
- B. Gratza. Piecewise linear approximations in symplectic geometry. ETH Zurich, PhD thesis, 1998.
- M. Gromov. Pseudo holomorphic curves in symplectic manifolds. Invent. Math., 82:307–347, 1985.
- M. Gromov. Partial differential relations, volume 9 of Ergeb. Math. Grenzgeb., 3. Folge. Springer, Cham, 1986.
- F. Jauberteau and Y. Rollin. Numerical flow for polyhedral symplectic maps. In preparation, 2024.
- Discrete geometry and isotropic surfaces. Mém. Soc. Math. Fr., Nouv. Sér., 161:1–99, 2019.
- F. C. Kirwan. Cohomology of quotients in symplectic and algebraic geometry, volume 31 of Math. Notes (Princeton). Princeton University Press, Princeton, NJ, 1984.
- F. Lalonde and D. McDuff. The geometry of symplectic energy. Ann. Math. (2), 141(2):349–371, 1995.
- J. A. Lees. On the classification of Lagrange immersions. Duke Math. J., 43:217–224, 1976.
- E. Lerman. Gradient flow of the norm squared of a moment map. Enseign. Math. (2), 51(1-2):117–127, 2005.
- S. Łojasiewicz. Ensembles semi-analytiques. IHES preprint, 1965.
- D. McDuff and D. Salamon. Introduction to symplectic topology, volume 27 of Oxf. Grad. Texts Math. Oxford: Oxford University Press, 3rd edition edition, 2016.
- Geometric invariant theory., volume 34 of Ergeb. Math. Grenzgeb. Berlin: Springer-Verlag, 3rd enl. ed. edition, 1994.
- Y. Rollin. Polyhedral approximation by Lagrangian and isotropic tori. J. Symplectic Geom., 20(6):1349–1383, 2022.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.