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Izv. RAN. Ser. Mat., 2008, Volume 72, Issue 6, Pages 203–222 (Mi izv2664)  

This article is cited in 6 scientific papers (total in 6 papers)

Minimal Gromov–Witten rings

V. V. Przyjalkowski

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We construct an abstract theory of Gromov–Witten invariants of genus 0 for quantum minimal Fano varieties (a minimal class of varieties which is natural from the quantum cohomological viewpoint). Namely, we consider the minimal Gromov–Witten ring: a commutative algebra whose generators and relations are of the form used in the Gromov–Witten theory of Fano varieties (of unspecified dimension). The Gromov–Witten theory of any quantum minimal variety is a homomorphism from this ring to $\mathbb C$. We prove an abstract reconstruction theorem which says that this ring is isomorphic to the free commutative ring generated by ‘prime two-pointed invariants’. We also find solutions of the differential equation of type $DN$ for a Fano variety of dimension $N$ in terms of the generating series of one-pointed Gromov–Witten invariants.

DOI: https://doi.org/10.4213/im2664

Full text: PDF file (621 kB)
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English version:
Izvestiya: Mathematics, 2008, 72:6, 1253–1272

Bibliographic databases:

UDC: 512.772
MSC: 53D45, 14J45, 14N35
Received: 14.05.2007

Citation: V. V. Przyjalkowski, “Minimal Gromov–Witten rings”, Izv. RAN. Ser. Mat., 72:6 (2008), 203–222; Izv. Math., 72:6 (2008), 1253–1272

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Przyjalkowski V., “On Landau-Ginzburg models for Fano varieties”, Communications in Number Theory and Physics, 1:4 (2007), 713–728  crossref  mathscinet  isi  scopus
    2. Nathan Owen Ilten, Jacob Lewis, Victor Przyjalkowski, “Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models”, Journal of Algebra, 374 (2013), 104  crossref  mathscinet  zmath  isi  scopus
    3. V. V. Przyjalkowski, “Weak Landau–Ginzburg models for smooth Fano threefolds”, Izv. Math., 77:4 (2013), 772–794  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. V. V. Przyjalkowski, C. A. Shramov, “Laurent phenomenon for Landau–Ginzburg models of complete intersections in Grassmannians”, Proc. Steklov Inst. Math., 290:1 (2015), 91–102  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    5. V. V. Golyshev, D. Zagier, “Proof of the gamma conjecture for Fano 3-folds of Picard rank 1”, Izv. Math., 80:1 (2016), 24–49  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. V. V. Przyjalkowski, “Toric Landau–Ginzburg models”, Russian Math. Surveys, 73:6 (2018), 1033–1118  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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