2011, 18: 131-143. doi: 10.3934/era.2011.18.131

Semisimplicity of the quantum cohomology for smooth Fano toric varieties associated with facet symmetric polytopes

1. 

Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

2. 

Yale University, MathematicsDepartment, PO Box 208283, New Haven, Connecticut 06520, United States

Received  March 2011 Revised  July 2011 Published  September 2011

The degree zero part of the quantum cohomology algebra of a smooth Fano toric symplectic manifold is determined by the superpotential function, $W$, of its moment polytope. In particular, this algebra is semisimple, i.e. splits as a product of fields, if and only if all the critical points of $W$ are non-degenerate. In this paper, we prove that this non-degeneracy holds for all smooth Fano toric varieties with facet-symmetric duals to moment polytopes.
Citation: Maksim Maydanskiy, Benjamin P. Mirabelli. Semisimplicity of the quantum cohomology for smooth Fano toric varieties associated with facet symmetric polytopes. Electronic Research Announcements, 2011, 18: 131-143. doi: 10.3934/era.2011.18.131
References:
[1]

Matthew Strom Borman, Quasi-states, quasi-morphisms, and the moment map,, , ().   Google Scholar

[2]

T. Delzant, Hamiltoniens periodiques et image convexes de l'application moment, Bulletin de la Societe Mathmatique de France, 116 (1988), 315-339.  Google Scholar

[3]

David Cox, Minicourse on Toric Varieties, given at the University of Buenos Aires., Available from: , ().   Google Scholar

[4]

Y. Eliashberg and L. Polterovich, Symplectic quasi-states on the quadric surface and Lagrangian submanifolds,, , ().   Google Scholar

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M. Entov and L. Polterovich, Symplectic quasi-states and semi-simplicity of quantum homology, in "Toric Topology," Contemp. Math., 460, American. Math. Society, Providence, RI, (2008), 47-70.  Google Scholar

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Günter Ewald, On the classification of Toric Fano varieties, Discrete and Computational Goemetry, 3 (1988), 49-54. doi: 10.1007/BF02187895.  Google Scholar

[7]

K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Spectral invariants with bulk quasimorphisms and Lagrangian Floer theory,, , ().   Google Scholar

[8]

William Fulton, "Introduction to Toric Varieties," Annals of Mathematics Studies, 131, The William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[9]

W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, in "Algebraic Geometry-Santa Cruz 1995," Proc. of Symp. in Pure Math., 62 Part 2, Amer. Math. Soc., Providence, RI, (1997), 45-96.  Google Scholar

[10]

Alexander Givental, A tutorial on quantum cohomology, in "Symplectic Geometry and Topology" (Park City, UT, 1997), 231-264, IAS/Park City Math. Ser., 7, Amer. Math. Soc., Providence, RI, 1999.  Google Scholar

[11]

Robin Hartshorne, "Algebraic Geometry," Graduate Texts in Math., 52, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[12]

J. Lagarias and G. Ziegler, Bounds for lattice polytopes containing a fixed number of interior points in a sublattice, Can. J. Math., 43 (1991), 1022-1035. doi: 10.4153/CJM-1991-058-4.  Google Scholar

[13]

Wilhelm Ljunggren, On the irreducibility of certain trinomials and quadrinomials, Math. Scand., 8 (1960), 65-70.  Google Scholar

[14]

D. McDuff and D. Salamon, "J-holomorphic Curves and Symplectic Topology," American Math. Society Collaquium Publications, 52, American Mathematical Society, Providence, RI, 2004.  Google Scholar

[15]

Benjamin Nill, "Reflexive Polytopes - Combinatorics and Convex Goemetry.", Available from: , ().   Google Scholar

[16]

Mikkel Obro, "Classification of smooth Fano polytopes,'' Dissertation, Univ. of Aarhus, 2007., Available from: , ().   Google Scholar

[17]

Yaron Ostrover, Calabi quasi-morphisms for some non-monotone symplectic manifolds, Algebr. Geom. Topol., 6 (2006), 405-434. doi: 10.2140/agt.2006.6.405.  Google Scholar

[18]

Y. Ostrover and I. Tyomkin, On the quantum homology algebra of toric Fano manifolds, Selecta Math. (N.S.), 15 (2009), 121-149.  Google Scholar

[19]

Michael Usher, Deformed Hamiltonian Floer theory, capacity estimates, and Calabi quasimorphisms,, , ().   Google Scholar

show all references

References:
[1]

Matthew Strom Borman, Quasi-states, quasi-morphisms, and the moment map,, , ().   Google Scholar

[2]

T. Delzant, Hamiltoniens periodiques et image convexes de l'application moment, Bulletin de la Societe Mathmatique de France, 116 (1988), 315-339.  Google Scholar

[3]

David Cox, Minicourse on Toric Varieties, given at the University of Buenos Aires., Available from: , ().   Google Scholar

[4]

Y. Eliashberg and L. Polterovich, Symplectic quasi-states on the quadric surface and Lagrangian submanifolds,, , ().   Google Scholar

[5]

M. Entov and L. Polterovich, Symplectic quasi-states and semi-simplicity of quantum homology, in "Toric Topology," Contemp. Math., 460, American. Math. Society, Providence, RI, (2008), 47-70.  Google Scholar

[6]

Günter Ewald, On the classification of Toric Fano varieties, Discrete and Computational Goemetry, 3 (1988), 49-54. doi: 10.1007/BF02187895.  Google Scholar

[7]

K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Spectral invariants with bulk quasimorphisms and Lagrangian Floer theory,, , ().   Google Scholar

[8]

William Fulton, "Introduction to Toric Varieties," Annals of Mathematics Studies, 131, The William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[9]

W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, in "Algebraic Geometry-Santa Cruz 1995," Proc. of Symp. in Pure Math., 62 Part 2, Amer. Math. Soc., Providence, RI, (1997), 45-96.  Google Scholar

[10]

Alexander Givental, A tutorial on quantum cohomology, in "Symplectic Geometry and Topology" (Park City, UT, 1997), 231-264, IAS/Park City Math. Ser., 7, Amer. Math. Soc., Providence, RI, 1999.  Google Scholar

[11]

Robin Hartshorne, "Algebraic Geometry," Graduate Texts in Math., 52, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[12]

J. Lagarias and G. Ziegler, Bounds for lattice polytopes containing a fixed number of interior points in a sublattice, Can. J. Math., 43 (1991), 1022-1035. doi: 10.4153/CJM-1991-058-4.  Google Scholar

[13]

Wilhelm Ljunggren, On the irreducibility of certain trinomials and quadrinomials, Math. Scand., 8 (1960), 65-70.  Google Scholar

[14]

D. McDuff and D. Salamon, "J-holomorphic Curves and Symplectic Topology," American Math. Society Collaquium Publications, 52, American Mathematical Society, Providence, RI, 2004.  Google Scholar

[15]

Benjamin Nill, "Reflexive Polytopes - Combinatorics and Convex Goemetry.", Available from: , ().   Google Scholar

[16]

Mikkel Obro, "Classification of smooth Fano polytopes,'' Dissertation, Univ. of Aarhus, 2007., Available from: , ().   Google Scholar

[17]

Yaron Ostrover, Calabi quasi-morphisms for some non-monotone symplectic manifolds, Algebr. Geom. Topol., 6 (2006), 405-434. doi: 10.2140/agt.2006.6.405.  Google Scholar

[18]

Y. Ostrover and I. Tyomkin, On the quantum homology algebra of toric Fano manifolds, Selecta Math. (N.S.), 15 (2009), 121-149.  Google Scholar

[19]

Michael Usher, Deformed Hamiltonian Floer theory, capacity estimates, and Calabi quasimorphisms,, , ().   Google Scholar

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