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Trudy Mat. Inst. Steklova, 2009, Volume 264, Pages 69–76 (Mi tm804)  

This article is cited in 15 scientific papers (total in 15 papers)

An Update on Semisimple Quantum Cohomology and $F$-Manifolds

C. Hertlinga, Yu. I. Maninbc, C. Telemande

a Institut für Mathematik, Universität Mannheim, Mannheim, Germany
b Max-Planck-Institut für Mathematik, Bonn, Germany
c Northwestern University, Evanston, USA
d University of Edinburgh, UK
e University of California, Berkeley, USA

Abstract: In the first section of this note, we show that Theorem 1.8.1 of Bayer–Manin can be strengthened in the following way: If the even quantum cohomology of a projective algebraic manifold $V$ is generically semisimple, then $V$ has no odd cohomology and is of Hodge–Tate type. In particular, this answers a question discussed by G. Ciolli. In the second section, we prove that an analytic (or formal) supermanifold $M$ with a given supercommutative associative $\mathcal O_M$-bilinear multiplication on its tangent sheaf $\mathcal T_M$ is an $F$-manifold in the sense of Hertling–Manin if and only if its spectral cover, as an analytic subspace of the cotangent bundle $T^*_M,$ is coisotropic of maximal dimension. This answers a question of V. Ginzburg. Finally, we discuss these results in the context of mirror symmetry and Landau–Ginzburg models for Fano varieties.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2009, 264, 62–69

Bibliographic databases:

UDC: 514.743.2
Received in July 2008
Language:

Citation: C. Hertling, Yu. I. Manin, C. Teleman, “An Update on Semisimple Quantum Cohomology and $F$-Manifolds”, Multidimensional algebraic geometry, Collected papers. Dedicated to the Memory of Vasilii Alekseevich Iskovskikh, Corresponding Member of the Russian Academy of Sciences, Trudy Mat. Inst. Steklova, 264, MAIK Nauka/Interperiodica, Moscow, 2009, 69–76; Proc. Steklov Inst. Math., 264 (2009), 62–69

Citation in format AMSBIB
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\by C.~Hertling, Yu.~I.~Manin, C.~Teleman
\paper An Update on Semisimple Quantum Cohomology and $F$-Manifolds
\inbook Multidimensional algebraic geometry
\bookinfo Collected papers. Dedicated to the Memory of Vasilii Alekseevich Iskovskikh, Corresponding Member of the Russian Academy of Sciences
\serial Trudy Mat. Inst. Steklova
\yr 2009
\vol 264
\pages 69--76
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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\jour Proc. Steklov Inst. Math.
\yr 2009
\vol 264
\pages 62--69
\crossref{https://doi.org/10.1134/S0081543809010088}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. David L., Hertling C., “(T)-Structures Over Two-Dimensional F-Manifolds: Formal Classification”, Ann. Mat. Pura Appl.  crossref  mathscinet  isi
    2. Merkulov S.A., “Wheeled Props in Algebra, Geometry and Quantization”, European Congress of Mathematics 2008, eds. Ran A., Riele H., Wiegerinck J., Eur. Math. Soc., 2010, 83–114  crossref  mathscinet  zmath  isi
    3. Teleman C., “Topological Field Theories in 2 Dimensions”, European Congress of Mathematics 2008, eds. Ran A., Riele H., Wiegerinck J., Eur. Math. Soc., 2010, 197–210  crossref  mathscinet  zmath  isi
    4. Usher M., “Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms”, Geom. Topol., 15:3 (2011), 1313–1417  crossref  mathscinet  zmath  isi  scopus
    5. Teleman C., “The Structure of 2D Semi-Simple Field Theories”, Invent. Math., 188:3 (2012), 525–588  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Marcolli M., Tabuada G., “From Exceptional Collections To Motivic Decompositions Via Noncommutative Motives”, J. Reine Angew. Math., 701 (2014), 153–167  crossref  mathscinet  isi  scopus
    7. Galkin S., Mellit A., Smirnov M., “Dubrovin'S Conjecture For Ig(2,6)”, Int. Math. Res. Notices, 2015, no. 18, 8847–8859  crossref  mathscinet  zmath  isi  elib  scopus
    8. Izv. Math., 81:4 (2017), 818–826  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. David L., Hertling C., “Regular F-Manifolds: Initial Conditions and Frobenius Metrics”, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 17:3 (2017), 1121–1152  mathscinet  zmath  isi
    10. Plaza Martin F.J., Tejero Prieto C., “Virasoro and KdV”, Lett. Math. Phys., 107:5 (2017), 963–994  crossref  mathscinet  zmath  isi  scopus
    11. Cruz Morales J.A., Mellit A., Perrin N., Smirnov M., Kuznetsov A., “On Quantum Cohomology of Grassmannians of Isotropic Lines, Unfoldings of a(N)-Singularities, and Lefschetz Exceptional Collections”, Ann. Inst. Fourier, 69:3 (2019), 955–991  crossref  isi
    12. Ke H.-Zh., “On Semisimplicity of Quantum Cohomology of P-1-Orbifolds”, J. Geom. Phys., 144 (2019), 1–14  crossref  mathscinet  isi
    13. Giordano Cotti, Boris Dubrovin, Davide Guzzetti, “Local Moduli of Semisimple Frobenius Coalescent Structures”, SIGMA, 16 (2020), 040, 105 pp.  mathnet  crossref
    14. David L., Hertling C., “(Te)-Structures Over the Irreducible 2-Dimensional Globally Nilpotent F-Manifold Germ”, Rev. Roum. Math. Pures Appl., 65:3, SI (2020), 235–284  mathscinet  isi
    15. Sanda F., Shamoto Y., “An Analogue of Dubrovin'S Conjecture”, Ann. Inst. Fourier, 70:2 (2020), 621–682  crossref  mathscinet  isi
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