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Izv. RAN. Ser. Mat., 2001, Volume 65, Issue 5, Pages 3–32 (Mi izv355)  

This article is cited in 13 scientific papers (total in 13 papers)

Riemann–Roch variations

V. V. Golyshev


Abstract: We construct a mirror-type correspondence that assigns variations (that is, local systems, $D$-modules or $l$-adic sheaves) to pairs $(V,C)$, where $V$ is a variety and $C$ is a complex of densely filtered vector bundles over $V$. We consider Calabi–Yau complete intersections in projective spaces. In the particular case when the complex is quasi-isomorphic to the tangent bundle on a generic Calabi–Yau complete intersection, this construction yields the variation that arises in the relative cohomology of the mirror-dual pencil. We call it the Riemann–Roch variation. The Riemann–Roch data of the divisorial sublattice in the $K$-group can be read off the Riemann–Roch local system since it encodes the information about the Euler characteristics of all $\mathscr O(i)$ sheaves (in an essentially non-commutative way).

DOI: https://doi.org/10.4213/im355

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English version:
Izvestiya: Mathematics, 2001, 65:5, 853–881

Bibliographic databases:

MSC: 14J32, 18F20, 14N10
Received: 12.10.2000

Citation: V. V. Golyshev, “Riemann–Roch variations”, Izv. RAN. Ser. Mat., 65:5 (2001), 3–32; Izv. Math., 65:5 (2001), 853–881

Citation in format AMSBIB
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\paper Riemann--Roch variations
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\yr 2001
\vol 65
\issue 5
\pages 3--32
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\transl
\jour Izv. Math.
\yr 2001
\vol 65
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\crossref{https://doi.org/10.1070/IM2001v065n05ABEH000355}
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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. Golyshev V.V., “The geometricity problem and modularity of certain Riemann-Roch variations”, Doklady Mathematics, 66:2 (2002), 231–236  mathscinet  zmath  isi  elib
    2. F. M. Mukhamedov, “On expansion of quantum quadratic stochastic processes into fibrewise Markov processes defined on von Neumann algebras”, Izv. Math., 68:5 (2004), 1009–1024  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. A. L. Gorodentsev, S. A. Kuleshov, “Helix theory”, Mosc. Math. J., 4:2 (2004), 377–440  mathnet  mathscinet  zmath
    4. Tanabé S., “Invariant of the hypergeometric group associated to the quantum cohomology of the projective space”, Bull. Sci. Math., 128:10 (2004), 811–827  crossref  mathscinet  zmath  isi  elib  scopus
    5. V. V. Przyjalkowski, “Quantum cohomology of smooth complete intersections in weighted projective spaces and in singular toric varieties”, Sb. Math., 198:9 (2007), 1325–1340  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. Golyshev V., Stienstra J., “Fuchsian equations of type DN”, Communications in Number Theory and Physics, 1:2 (2007), 323–346  crossref  mathscinet  zmath  isi  scopus
    7. Katzarkov L., Kontsevich M., Pantev T., “Hodge theoretic aspects of mirror symmetry”, From Hodge Theory To Integrability and TQFT: TT*- Geometry, Proceedings of Symposia in Pure Mathematics, 78, 2008, 87–174  crossref  mathscinet  zmath  isi
    8. Proc. Steklov Inst. Math., 264 (2009), 62–69  mathnet  crossref  mathscinet  isi  elib  elib
    9. Golyshev V.V., “Minimal Fano threefolds: Exceptional sets and vanishing cycles”, Dokl. Math., 79:1 (2009), 16–20  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    10. Simpson C., “Katz's middle convolution algorithm”, Pure Appl. Math. Q., 5:2, Special Issue: In honor of Friedrich Hirzebruch, Part 1 (2009), 781–852  crossref  mathscinet  zmath  isi  elib  scopus
    11. Corti A., Golyshev V., “Hypergeometric equations and weighted projective spaces”, Science China-Mathematics, 54:8 (2011), 1577–1590  crossref  mathscinet  zmath  adsnasa  isi  scopus
    12. Galkin S. Golyshev V. Iritani H., “Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures”, Duke Math. J., 165:11 (2016), 2005–2077  crossref  mathscinet  zmath  isi  elib  scopus
    13. Tanabe S., “On Monodromy Representation of Period Integrals Associated to An Algebraic Curve With Bi-Degree (2,2)”, Analele Stiint. Univ. Ovidius C., 25:1 (2017), 207–231  crossref  mathscinet  zmath  isi  scopus
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