Positivity for quantum cluster algebras

Abstract

Building on work by Kontsevich, Soibelman, Nagao and Efimov, we prove the positivity of quantum cluster coefficients for all skew-symmetric quantum cluster algebras, via a proof of a conjecture first suggested by Kontsevich on the purity of mixed Hodge structures arising in the theory of cluster mutation of spherical collections in 3-Calabi–Yau categories. The result implies positivity, as well as the stronger Lefschetz property conjectured by Efimov, and also the classical positivity conjecture of Fomin and Zelevinsky, recently proved by Lee and Schiffler. Closely related to these results is a categorified “no exotics” type theorem for cohomological Donaldson–Thomas invariants, which we discuss and prove in the appendix.

Note: To view the article, click on the URL link for the DOI number.

  • [BBD] A. A. Beuilinson, J. Bernstein, and P. Deligne, "Faisceaux pervers," in Analysis and Topology on Singular Spaces, I, Soc. Math. France, Paris, 1982, vol. 100, pp. 5-171.
    @INCOLLECTION{BBD,
      author = {Beĭlinson, A. A. and Bernstein, J. and Deligne, P.},
      title = {Faisceaux pervers},
      booktitle = {Analysis and Topology on Singular Spaces, {I}},
      venue = {{L}uminy, 1981},
      series = {Astérisque},
      volume = {100},
      pages = {5--171},
      publisher = {Soc. Math. France, Paris},
      year = {1982},
      mrclass = {32C38},
      mrnumber = {0751966},
      mrreviewer = {Zoghman Mebkhout},
      zblnumber = {0536.14011},
      }
  • [Br12] Go to document C. Brav, V. Bussi, D. Dupont, D. Joyce, and B. SzendrHoi, "Symmetries and stabilization for sheaves of vanishing cycles," J. Singul., vol. 11, pp. 85-151, 2015.
    @ARTICLE{Br12,
      author = {Brav, C. and Bussi, V. and Dupont, D. and Joyce, D. and Szendrői, B.},
      title = {Symmetries and stabilization for sheaves of vanishing cycles},
      note = {With an appendix by Jörg Schürmann},
      journal = {J. Singul.},
      fjournal = {Journal of Singularities},
      volume = {11},
      year = {2015},
      pages = {85--151},
      issn = {1949-2006},
      mrclass = {14F05 (14C25)},
      mrnumber = {3353002},
      mrreviewer = {Shintarou Yanagida},
      zblnumber = {1325.14057},
      doi = {10.5427/jsing.2015.11e},
      }
  • [Bon10] Go to document M. V. Bondarko, "Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)," J. K-Theory, vol. 6, iss. 3, pp. 387-504, 2010.
    @ARTICLE{Bon10,
      author = {Bondarko, M. V.},
      title = {Weight structures vs. {$t$}-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)},
      journal = {J. K-Theory},
      fjournal = {Journal of K-Theory. K-Theory and its Applications in Algebra, Geometry, Analysis \& Topology},
      volume = {6},
      year = {2010},
      number = {3},
      pages = {387--504},
      issn = {1865-2433},
      mrclass = {18E30 (14C15 18G40 19E08 19E15)},
      mrnumber = {2746283},
      mrreviewer = {Florence Lecomte},
      doi = {10.1017/is010012005jkt083},
      url = {http://dx.doi.org/10.1017/is010012005jkt083},
      zblnumber = {1303.18019},
      }
  • [LBP90] Go to document L. Le Bruyn and C. Procesi, "Semisimple representations of quivers," Trans. Amer. Math. Soc., vol. 317, iss. 2, pp. 585-598, 1990.
    @ARTICLE{LBP90,
      author = {Le Bruyn, Lieven and Procesi, Claudio},
      title = {Semisimple representations of quivers},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the American Mathematical Society},
      volume = {317},
      year = {1990},
      number = {2},
      pages = {585--598},
      issn = {0002-9947},
      mrclass = {16A64 (14D25 14L30)},
      mrnumber = {0958897},
      mrreviewer = {Alfred G. Wiedemann},
      doi = {10.2307/2001477},
      url = {http://dx.doi.org/10.2307/2001477},
      zblnumber = {0693.16018},
      }
  • [BZ05] Go to document A. Berenstein and A. Zelevinsky, "Quantum cluster algebras," Adv. Math., vol. 195, iss. 2, pp. 405-455, 2005.
    @ARTICLE{BZ05,
      author = {Berenstein, Arkady and Zelevinsky, Andrei},
      title = {Quantum cluster algebras},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {195},
      year = {2005},
      number = {2},
      pages = {405--455},
      issn = {0001-8708},
      mrclass = {20G42 (14M17 22E46)},
      mrnumber = {2146350},
      mrreviewer = {Oleg V. Ogievetsky},
      doi = {10.1016/j.aim.2004.08.003},
      url = {http://dx.doi.org/10.1016/j.aim.2004.08.003},
      zblnumber = {1124.20028},
      }
  • [GeomEng] Go to document W. Chuang, D. Diaconescu, J. Manschot, G. W. Moore, and Y. Soibelman, "Geometric engineering of (framed) BPS states," Adv. Theor. Math. Phys., vol. 18, iss. 5, pp. 1063-1231, 2014.
    @ARTICLE{GeomEng,
      author = {Chuang, Wu-yen and Diaconescu, Duiliu-Emanuel and Manschot, Jan and Moore, Gregory W. and Soibelman, Yan},
      title = {Geometric engineering of (framed) {BPS} states},
      journal = {Adv. Theor. Math. Phys.},
      fjournal = {Advances in Theoretical and Mathematical Physics},
      volume = {18},
      year = {2014},
      number = {5},
      pages = {1063--1231},
      issn = {1095-0761},
      mrclass = {81T13 (81P16 81T30)},
      mrnumber = {3281276},
      mrreviewer = {Matthew B. Young},
      url = {http://projecteuclid.org/euclid.atmp/1416929531},
      zblnumber = {1365.81092},
      }
  • [DaMe4] Go to document B. Davison and S. Meinhardt, "Motivic Donaldson-Thomas invariants for the one-loop quiver with potential," Geom. Topol., vol. 19, iss. 5, pp. 2535-2555, 2015.
    @ARTICLE{DaMe4,
      author = {Davison, Ben and Meinhardt, Sven},
      title = {Motivic {D}onaldson-{T}homas invariants for the one-loop quiver with potential},
      journal = {Geom. Topol.},
      fjournal = {Geometry \& Topology},
      volume = {19},
      year = {2015},
      number = {5},
      pages = {2535--2555},
      issn = {1465-3060},
      mrclass = {14N35 (14D23)},
      mrnumber = {3416109},
      mrreviewer = {Victor Przyjalkowski},
      doi = {10.2140/gt.2015.19.2535},
      url = {http://dx.doi.org/10.2140/gt.2015.19.2535},
      zblnumber = {06503548},
      }
  • [DaMe15b] B. Davison and S. Meinhardt, Cohomological Donaldson–Thomas theory of a quiver with potential and quantum enveloping algebras, 2016.
    @MISC{DaMe15b,
      author = {Davison, Ben and Meinhardt, Sven},
      title = {Cohomological {D}onaldson--{T}homas theory of a quiver with potential and quantum enveloping algebras},
      arxiv = {1601.02479},
      year = {2016},
      zblnumber = {},
      }
  • [DMSS13] Go to document B. Davison, D. Maulik, J. Schürmann, and B. SzendrHoi, "Purity for graded potentials and quantum cluster positivity," Compos. Math., vol. 151, iss. 10, pp. 1913-1944, 2015.
    @ARTICLE{DMSS13,
      author = {Davison, Ben and Maulik, Davesh and Schürmann, Jörg and Szendrői, Balázs},
      title = {Purity for graded potentials and quantum cluster positivity},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {151},
      year = {2015},
      number = {10},
      pages = {1913--1944},
      issn = {0010-437X},
      mrclass = {14C30 (13F60 14N35 32S35)},
      mrnumber = {3414389},
      mrreviewer = {Ruifang Song},
      doi = {10.1112/S0010437X15007332},
      url = {http://dx.doi.org/10.1112/S0010437X15007332},
      zblnumber = {1345.14014},
      }
  • [DWZ08] Go to document H. Derksen, J. Weyman, and A. Zelevinsky, "Quivers with potentials and their representations. I. Mutations," Selecta Math. (N.S.), vol. 14, iss. 1, pp. 59-119, 2008.
    @ARTICLE{DWZ08,
      author = {Derksen, Harm and Weyman, Jerzy and Zelevinsky, Andrei},
      title = {Quivers with potentials and their representations. {I}. {M}utations},
      journal = {Selecta Math. (N.S.)},
      fjournal = {Selecta Mathematica. New Series},
      volume = {14},
      year = {2008},
      number = {1},
      pages = {59--119},
      issn = {1022-1824},
      mrclass = {16G10 (13F60 16G20 16S38)},
      mrnumber = {2480710},
      mrreviewer = {Mátyás Domokos},
      doi = {10.1007/s00029-008-0057-9},
      url = {http://dx.doi.org/10.1007/s00029-008-0057-9},
      zblnumber = {1204.16008},
      }
  • [DWZ10] Go to document H. Derksen, J. Weyman, and A. Zelevinsky, "Quivers with potentials and their representations II: applications to cluster algebras," J. Amer. Math. Soc., vol. 23, iss. 3, pp. 749-790, 2010.
    @ARTICLE{DWZ10,
      author = {Derksen, Harm and Weyman, Jerzy and Zelevinsky, Andrei},
      title = {Quivers with potentials and their representations {II}: applications to cluster algebras},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {23},
      year = {2010},
      number = {3},
      pages = {749--790},
      issn = {0894-0347},
      mrclass = {16G20 (13F60)},
      mrnumber = {2629987},
      mrreviewer = {Mátyás Domokos},
      doi = {10.1090/S0894-0347-10-00662-4},
      url = {http://dx.doi.org/10.1090/S0894-0347-10-00662-4},
      zblnumber = {1208.16017},
      }
  • [Efi11] A. Efimov, Quantum cluster variables via vanishing cycles, 2011.
    @MISC{Efi11,
      author = {Efimov, A.},
      title = {Quantum cluster variables via vanishing cycles},
      arxiv = {1112.3601},
      year = {2011},
      zblnumber = {},
      }
  • [EdGr98] Go to document D. Edidin and W. Graham, "Equivariant intersection theory," Invent. Math., vol. 131, iss. 3, pp. 595-634, 1998.
    @ARTICLE{EdGr98,
      author = {Edidin, Dan and Graham, William},
      title = {Equivariant intersection theory},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {131},
      year = {1998},
      number = {3},
      pages = {595--634},
      issn = {0020-9910},
      mrclass = {14C17 (14F99)},
      mrnumber = {1614555},
      mrreviewer = {Burt Totaro},
      doi = {10.1007/s002220050214},
      url = {http://dx.doi.org/10.1007/s002220050214},
      zblnumber = {0940.14003},
      }
  • [FZ02] Go to document S. Fomin and A. Zelevinsky, "Cluster algebras. I. Foundations," J. Amer. Math. Soc., vol. 15, iss. 2, pp. 497-529, 2002.
    @ARTICLE{FZ02,
      author = {Fomin, Sergey and Zelevinsky, Andrei},
      title = {Cluster algebras. {I}. {F}oundations},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {15},
      year = {2002},
      number = {2},
      pages = {497--529},
      issn = {0894-0347},
      mrclass = {16S99 (14M99 17B99)},
      mrnumber = {1887642},
      mrreviewer = {Eric N. Sommers},
      doi = {10.1090/S0894-0347-01-00385-X},
      url = {http://dx.doi.org/10.1090/S0894-0347-01-00385-X},
      zblnumber = {1021.16017},
      }
  • [GHKK14] M. Gross, P. Hacking, S. Keel, and M. Kontsevich, Canonical bases for cluster algebras, 2014.
    @MISC{GHKK14,
      author = {Gross, M. and Hacking, P. and Keel, S. and Kontsevich, M.},
      title = {Canonical bases for cluster algebras},
      arxiv = {1411.1394},
      year = {2014},
      zblnumber = {},
      }
  • [ginz] V. Ginzburg, Calabi-Yau algebras, 2006.
    @MISC{ginz,
      author = {Ginzburg, V.},
      title = {{C}alabi-{Y}au algebras},
      arxiv = {math/0612139},
      year = {2006},
      zblnumber = {},
      }
  • [Ke375] B. Keller, "Cluster algebras, quiver representations and triangulated categories," in Triangulated Categories, Cambridge Univ. Press, Cambridge, 2010, vol. 375, pp. 76-160.
    @INCOLLECTION{Ke375,
      author = {Keller, Bernhard},
      title = {Cluster algebras, quiver representations and triangulated categories},
      booktitle = {Triangulated Categories},
      series = {London Math. Soc. Lecture Note Ser.},
      volume = {375},
      pages = {76--160},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {2010},
      mrclass = {13F60 (16G20 18E30)},
      mrnumber = {2681708},
      mrreviewer = {Gregoire Dupont},
      zblnumber = {1215.16012},
      }
  • [CYTC] Go to document B. Keller, "Calabi-Yau triangulated categories," in Trends in Representation Theory of Algebras and Related Topics, Eur. Math. Soc., Zürich, 2008, pp. 467-489.
    @INCOLLECTION{CYTC,
      author = {Keller, Bernhard},
      title = {Calabi-{Y}au triangulated categories},
      booktitle = {Trends in Representation Theory of Algebras and Related Topics},
      series = {EMS Ser. Congr. Rep.},
      pages = {467--489},
      publisher = {Eur. Math. Soc., Zürich},
      year = {2008},
      mrclass = {18E30},
      mrnumber = {2484733},
      mrreviewer = {Xueqing Chen},
      doi = {10.4171/062-1/11},
      url = {http://dx.doi.org/10.4171/062-1/11},
      zblnumber = {1202.16014},
      }
  • [Kel09] Go to document B. Keller, "Deformed Calabi-Yau completions," J. Reine Angew. Math., vol. 654, pp. 125-180, 2011.
    @ARTICLE{Kel09,
      author = {Keller, Bernhard},
      title = {Deformed {C}alabi-{Y}au completions},
      note = {With an appendix by Michel Van den Bergh},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {654},
      year = {2011},
      pages = {125--180},
      issn = {0075-4102},
      mrclass = {18E30 (13F60 16E35 16E45 18E35 18G10)},
      mrnumber = {2795754},
      mrreviewer = {Gregoire Dupont},
      doi = {10.1515/CRELLE.2011.031},
      url = {http://dx.doi.org/10.1515/CRELLE.2011.031},
      zblnumber = {1220.18012},
      }
  • [Keller10] Go to document B. Keller, "On cluster theory and quantum dilogarithm identities," in Representations of Algebras and Related Topics, Eur. Math. Soc., Zürich, 2011, pp. 85-116.
    @INCOLLECTION{Keller10,
      author = {Keller, Bernhard},
      title = {On cluster theory and quantum dilogarithm identities},
      booktitle = {Representations of Algebras and Related Topics},
      series = {EMS Ser. Congr. Rep.},
      pages = {85--116},
      publisher = {Eur. Math. Soc., Zürich},
      year = {2011},
      mrclass = {13F60 (16G10 33D05)},
      mrnumber = {2931896},
      mrreviewer = {Xueqing Chen},
      doi = {10.4171/101-1/3},
      url = {http://dx.doi.org/10.4171/101-1/3},
      zblnumber = {1307.13028},
      }
  • [King94] Go to document A. D. King, "Moduli of representations of finite-dimensional algebras," Quart. J. Math. Oxford Ser. (2), vol. 45, iss. 180, pp. 515-530, 1994.
    @ARTICLE{King94,
      author = {King, A. D.},
      title = {Moduli of representations of finite-dimensional algebras},
      journal = {Quart. J. Math. Oxford Ser. (2)},
      fjournal = {The Quarterly Journal of Mathematics. Oxford. Second Series},
      volume = {45},
      year = {1994},
      number = {180},
      pages = {515--530},
      issn = {0033-5606},
      mrclass = {16G10 (14D25)},
      mrnumber = {1315461},
      doi = {10.1093/qmath/45.4.515},
      url = {http://dx.doi.org/10.1093/qmath/45.4.515},
      zblnumber = {0837.16005},
      }
  • [KQ14] Go to document Y. Kimura and F. Qin, "Graded quiver varieties, quantum cluster algebras and dual canonical basis," Adv. Math., vol. 262, pp. 261-312, 2014.
    @ARTICLE{KQ14,
      author = {Kimura, Yoshiyuki and Qin, Fan},
      title = {Graded quiver varieties, quantum cluster algebras and dual canonical basis},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {262},
      year = {2014},
      pages = {261--312},
      issn = {0001-8708},
      mrclass = {13F60 (16G20 18D10 20G42)},
      mrnumber = {3228430},
      mrreviewer = {Xueqing Chen},
      doi = {10.1016/j.aim.2014.05.014},
      url = {http://dx.doi.org/10.1016/j.aim.2014.05.014},
      zblnumber = {1331.13016},
      }
  • [KS] Go to document M. Kontsevich and Y. Soibelman, "Motivic Donaldson-Thomas invariants: summary of results," in Mirror Symmetry and Tropical Geometry, Amer. Math. Soc., Providence, RI, 2010, vol. 527, pp. 55-89.
    @INCOLLECTION{KS,
      author = {Kontsevich, Maxim and Soibelman, Yan},
      title = {Motivic {D}onaldson-{T}homas invariants: summary of results},
      booktitle = {Mirror Symmetry and Tropical Geometry},
      series = {Contemp. Math.},
      volume = {527},
      pages = {55--89},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2010},
      mrclass = {14N35 (14C15 14F05 14J32 18E30)},
      mrnumber = {2681792},
      mrreviewer = {Balázs Szendrői},
      doi = {10.1090/conm/527/10400},
      url = {http://dx.doi.org/10.1090/conm/527/10400},
      zblnumber = {1214.14014},
      }
  • [COHA] Go to document M. Kontsevich and Y. Soibelman, "Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants," Commun. Number Theory Phys., vol. 5, iss. 2, pp. 231-352, 2011.
    @ARTICLE{COHA,
      author = {Kontsevich, Maxim and Soibelman, Yan},
      title = {Cohomological {H}all algebra, exponential {H}odge structures and motivic {D}onaldson-{T}homas invariants},
      journal = {Commun. Number Theory Phys.},
      fjournal = {Communications in Number Theory and Physics},
      volume = {5},
      year = {2011},
      number = {2},
      pages = {231--352},
      issn = {1931-4523},
      mrclass = {14N35 (14F43 16G20)},
      mrnumber = {2851153},
      mrreviewer = {Mark Gross},
      doi = {10.4310/CNTP.2011.v5.n2.a1},
      url = {http://dx.doi.org/10.4310/CNTP.2011.v5.n2.a1},
      zblnumber = {1248.14060},
      }
  • [KellerMutations] Go to document B. Keller and D. Yang, "Derived equivalences from mutations of quivers with potential," Adv. Math., vol. 226, iss. 3, pp. 2118-2168, 2011.
    @ARTICLE{KellerMutations,
      author = {Keller, Bernhard and Yang, Dong},
      title = {Derived equivalences from mutations of quivers with potential},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {226},
      year = {2011},
      number = {3},
      pages = {2118--2168},
      issn = {0001-8708},
      mrclass = {16G20 (18E30)},
      mrnumber = {2739775},
      mrreviewer = {Alex Martsinkovsky},
      doi = {10.1016/j.aim.2010.09.019},
      url = {http://dx.doi.org/10.1016/j.aim.2010.09.019},
      zblnumber = {1272.13021},
      }
  • [Koszul] Go to document D. M. Lu, J. H. Palmieri, Q. S. Wu, and J. J. Zhang, "Koszul equivalences in $A_\infty$-algebras," New York J. Math., vol. 14, pp. 325-378, 2008.
    @ARTICLE{Koszul,
      author = {Lu, Di Ming and Palmieri, John H. and Wu, Quan Shui and Zhang, James J.},
      title = {Koszul equivalences in {$A_\infty$}-algebras},
      journal = {New York J. Math.},
      fjournal = {New York Journal of Mathematics},
      volume = {14},
      year = {2008},
      pages = {325--378},
      issn = {1076-9803},
      mrclass = {16E45 (16E35 16S37)},
      mrnumber = {2430869},
      mrreviewer = {Mikael Vejdemo Johansson},
      url = {http://nyjm.albany.edu:8000/j/2008/14_325.html},
      zblnumber = {1191.16011},
      }
  • [LS15] Go to document K. Lee and R. Schiffler, "Positivity for cluster algebras," Ann. of Math. (2), vol. 182, iss. 1, pp. 73-125, 2015.
    @ARTICLE{LS15,
      author = {Lee, Kyungyong and Schiffler, Ralf},
      title = {Positivity for cluster algebras},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {182},
      year = {2015},
      number = {1},
      pages = {73--125},
      issn = {0003-486X},
      mrclass = {13F60 (05E15)},
      mrnumber = {3374957},
      mrreviewer = {Alfredo Nájera Chávez},
      doi = {10.4007/annals.2015.182.1.2},
      url = {http://dx.doi.org/10.4007/annals.2015.182.1.2},
      zblnumber = {1350.13024},
      }
  • [Ma01] Go to document D. B. Massey, "The Sebastiani-Thom isomorphism in the derived category," Compositio Math., vol. 125, iss. 3, pp. 353-362, 2001.
    @ARTICLE{Ma01,
      author = {Massey, David B.},
      title = {The {S}ebastiani-{T}hom isomorphism in the derived category},
      journal = {Compositio Math.},
      fjournal = {Compositio Mathematica},
      volume = {125},
      year = {2001},
      number = {3},
      pages = {353--362},
      issn = {0010-437X},
      mrclass = {32S20 (18E30 32S55 32S60)},
      mrnumber = {1818986},
      mrreviewer = {José Ignacio Burgos Gil},
      doi = {10.1023/A:1002608716514},
      url = {http://dx.doi.org/10.1023/A:1002608716514},
      zblnumber = {0986.32004},
      }
  • [MFK94] D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, Third ed., Springer-Verlag, Berlin, 1994, vol. 34.
    @BOOK{MFK94,
      author = {Mumford, D. and Fogarty, J. and Kirwan, F.},
      title = {Geometric Invariant Theory},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {34},
      edition = {Third},
      publisher = {Springer-Verlag, Berlin},
      year = {1994},
      pages = {xiv+292},
      isbn = {3-540-56963-4},
      mrclass = {14D25 (58E05 58F05)},
      mrnumber = {1304906},
      mrreviewer = {Yi Hu},
      zblnumber = {0797.14004},
      }
  • [Mi63] J. Milnor, Morse Theory, Princeton Univ. Press, Princeton, N.J., 1963, vol. 51.
    @BOOK{Mi63,
      author = {Milnor, J.},
      title = {Morse Theory},
      note = {{B}ased on lecture notes by M. Spivak and R. Wells},
      series = {Ann. of Math. Stud.},
      volume = {51},
      publisher = {Princeton Univ. Press, Princeton, N.J.},
      year = {1963},
      pages = {vi+153},
      mrclass = {57.50 (53.72)},
      mrnumber = {0163331},
      mrreviewer = {H. I. Levine},
      zblnumber = {0108.10401},
      }
  • [MeRe14] Go to document M. Reineke, "Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants," Compos. Math., vol. 147, iss. 3, pp. 943-964, 2011.
    @ARTICLE{MeRe14,
      author = {Reineke, Markus},
      title = {Cohomology of quiver moduli, functional equations, and integrality of {D}onaldson-{T}homas type invariants},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {147},
      year = {2011},
      number = {3},
      pages = {943--964},
      issn = {0010-437X},
      mrclass = {16G20 (14D20)},
      mrnumber = {2801406},
      mrreviewer = {Ada Boralevi},
      doi = {10.1112/S0010437X1000521X},
      url = {http://dx.doi.org/10.1112/S0010437X1000521X},
      zblnumber = {1266.16013},
      }
  • [Na13] Go to document K. Nagao, "Donaldson-Thomas theory and cluster algebras," Duke Math. J., vol. 162, iss. 7, pp. 1313-1367, 2013.
    @ARTICLE{Na13,
      author = {Nagao, Kentaro},
      title = {Donaldson-{T}homas theory and cluster algebras},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {162},
      year = {2013},
      number = {7},
      pages = {1313--1367},
      issn = {0012-7094},
      mrclass = {14N35 (13F60)},
      mrnumber = {3079250},
      mrreviewer = {Wan Keng Cheong},
      doi = {10.1215/00127094-2142753},
      url = {http://dx.doi.org/10.1215/00127094-2142753},
      zblnumber = {06180315},
      }
  • [Na11] Go to document H. Nakajima, "Quiver varieties and cluster algebras," Kyoto J. Math., vol. 51, iss. 1, pp. 71-126, 2011.
    @ARTICLE{Na11,
      author = {Nakajima, Hiraku},
      title = {Quiver varieties and cluster algebras},
      journal = {Kyoto J. Math.},
      fjournal = {Kyoto Journal of Mathematics},
      volume = {51},
      year = {2011},
      number = {1},
      pages = {71--126},
      issn = {2156-2261},
      mrclass = {13F60 (14D21 16G20 17B37)},
      mrnumber = {2784748},
      mrreviewer = {Kyungyong Lee},
      doi = {10.1215/0023608X-2010-021},
      url = {http://dx.doi.org/10.1215/0023608X-2010-021},
      zblnumber = {1223.13013},
      }
  • [Pl11] Go to document P. Plamondon, "Cluster characters for cluster categories with infinite-dimensional morphism spaces," Adv. Math., vol. 227, iss. 1, pp. 1-39, 2011.
    @ARTICLE{Pl11,
      author = {Plamondon, Pierre-Guy},
      title = {Cluster characters for cluster categories with infinite-dimensional morphism spaces},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {227},
      year = {2011},
      number = {1},
      pages = {1--39},
      issn = {0001-8708},
      mrclass = {13F60 (16E35 16G20 18E30)},
      mrnumber = {2782186},
      mrreviewer = {Gregoire Dupont},
      doi = {10.1016/j.aim.2010.12.010},
      url = {http://dx.doi.org/10.1016/j.aim.2010.12.010},
      zblnumber = {1288.13016},
      }
  • [Reineke_HN] Go to document M. Reineke, "The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli," Invent. Math., vol. 152, iss. 2, pp. 349-368, 2003.
    @ARTICLE{Reineke_HN,
      author = {Reineke, Markus},
      title = {The {H}arder-{N}arasimhan system in quantum groups and cohomology of quiver moduli},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {152},
      year = {2003},
      number = {2},
      pages = {349--368},
      issn = {0020-9910},
      mrclass = {16G20 (14L24 17B37)},
      mrnumber = {1974891},
      mrreviewer = {Michel Van den Bergh},
      doi = {10.1007/s00222-002-0273-4},
      url = {http://dx.doi.org/10.1007/s00222-002-0273-4},
      zblnumber = {1043.17010},
      }
  • [Re05] Go to document M. Reineke, "Cohomology of noncommutative Hilbert schemes," Algebr. Represent. Theory, vol. 8, iss. 4, pp. 541-561, 2005.
    @ARTICLE{Re05,
      author = {Reineke, Markus},
      title = {Cohomology of noncommutative {H}ilbert schemes},
      journal = {Algebr. Represent. Theory},
      fjournal = {Algebras and Representation Theory},
      volume = {8},
      year = {2005},
      number = {4},
      pages = {541--561},
      issn = {1386-923X},
      mrclass = {16G20 (14D20 16S38)},
      mrnumber = {2199209},
      mrreviewer = {Jon Eivind Vatne},
      doi = {10.1007/s10468-005-8762-y},
      url = {http://dx.doi.org/10.1007/s10468-005-8762-y},
      zblnumber = {1125.14006},
      }
  • [Saito10] Go to document M. Saito, "Mixed Hodge modules," Publ. Res. Inst. Math. Sci., vol. 26, iss. 2, pp. 221-333, 1990.
    @ARTICLE{Saito10,
      author = {Saito, Morihiko},
      title = {Mixed {H}odge modules},
      journal = {Publ. Res. Inst. Math. Sci.},
      fjournal = {Kyoto University. Research Institute for Mathematical Sciences. Publications},
      volume = {26},
      year = {1990},
      number = {2},
      pages = {221--333},
      issn = {0034-5318},
      mrclass = {14D07 (14C30 32J25)},
      mrnumber = {1047415},
      mrreviewer = {Min Ho Lee},
      doi = {10.2977/prims/1195171082},
      url = {http://dx.doi.org/10.2977/prims/1195171082},
      zblnumber = {0727.14004},
      }
  • [Sa88] Go to document M. Saito, "Modules de Hodge polarisables," Publ. Res. Inst. Math. Sci., vol. 24, iss. 6, pp. 849-995 (1989), 1988.
    @ARTICLE{Sa88,
      author = {Saito, Morihiko},
      title = {Modules de {H}odge polarisables},
      journal = {Publ. Res. Inst. Math. Sci.},
      fjournal = {Kyoto University. Research Institute for Mathematical Sciences. Publications},
      volume = {24},
      year = {1988},
      number = {6},
      pages = {849--995 (1989)},
      issn = {0034-5318},
      mrclass = {32C35 (14C30 32C38 32C42 32G99)},
      mrnumber = {1000123},
      mrreviewer = {J. H. M. Steenbrink},
      doi = {10.2977/prims/1195173930},
      url = {http://dx.doi.org/10.2977/prims/1195173930},
      zblnumber = {0691.14007},
      }
  • [Sai89duality] Go to document M. Saito, "Duality for vanishing cycle functors," Publ. Res. Inst. Math. Sci., vol. 25, iss. 6, pp. 889-921, 1989.
    @ARTICLE{Sai89duality,
      author = {Saito, Morihiko},
      title = {Duality for vanishing cycle functors},
      journal = {Publ. Res. Inst. Math. Sci.},
      fjournal = {Kyoto University. Research Institute for Mathematical Sciences. Publications},
      volume = {25},
      year = {1989},
      number = {6},
      pages = {889--921},
      issn = {0034-5318},
      mrclass = {32C38 (32C37)},
      mrnumber = {1045997},
      mrreviewer = {J.-E. Björk},
      doi = {10.2977/prims/1195172510},
      url = {http://dx.doi.org/10.2977/prims/1195172510},
      zblnumber = {0712.32011},
      }
  • [Sai89] M. Saito, "Introduction to mixed Hodge modules," Astérisque, iss. 179-180, p. 10, 145-162, 1989.
    @ARTICLE{Sai89,
      author = {Saito, Morihiko},
      title = {Introduction to mixed {H}odge modules},
      note = {Actes du Colloque de Théorie de Hodge (Luminy, 1987)},
      journal = {Astérisque},
      fjournal = {Astérisque},
      number = {179-180},
      year = {1989},
      pages = {10, 145--162},
      issn = {0303-1179},
      mrclass = {32S35 (14C05 14C30 32J25)},
      mrnumber = {1042805},
      zblnumber = {0753.32004},
      }
  • [Saito89] M. Saito, "Mixed Hodge modules and admissible variations," C. R. Acad. Sci. Paris Sér. I Math., vol. 309, iss. 6, pp. 351-356, 1989.
    @ARTICLE{Saito89,
      author = {Saito, Morihiko},
      title = {Mixed {H}odge modules and admissible variations},
      journal = {C. R. Acad. Sci. Paris Sér. I Math.},
      fjournal = {Comptes Rendus de l'Académie des Sciences. Série I. Mathématique},
      volume = {309},
      year = {1989},
      number = {6},
      pages = {351--356},
      issn = {0764-4442},
      mrclass = {32J25 (14C30 32S35)},
      mrnumber = {1054250},
      mrreviewer = {Aleksandr G. Aleksandrov},
      zblnumber = {0765.14006},
      }
  • [Sai90] Go to document M. Saito, "Mixed Hodge modules," Publ. Res. Inst. Math. Sci., vol. 26, iss. 2, pp. 221-333, 1990.
    @ARTICLE{Sai90,
      author = {Saito, Morihiko},
      title = {Mixed {H}odge modules},
      journal = {Publ. Res. Inst. Math. Sci.},
      fjournal = {Kyoto University. Research Institute for Mathematical Sciences. Publications},
      volume = {26},
      year = {1990},
      number = {2},
      pages = {221--333},
      issn = {0034-5318},
      mrclass = {14D07 (14C30 32J25)},
      mrnumber = {1047415},
      mrreviewer = {Min Ho Lee},
      doi = {10.2977/prims/1195171082},
      url = {http://dx.doi.org/10.2977/prims/1195171082},
      zblnumber = {0727.14004},
      }
  • [StZu85] Go to document J. Steenbrink and S. Zucker, "Variation of mixed Hodge structure. I," Invent. Math., vol. 80, iss. 3, pp. 489-542, 1985.
    @ARTICLE{StZu85,
      author = {Steenbrink, Joseph and Zucker, Steven},
      title = {Variation of mixed {H}odge structure. {I}},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {80},
      year = {1985},
      number = {3},
      pages = {489--542},
      issn = {0020-9910},
      mrclass = {32G20 (14C30 32J25)},
      mrnumber = {0791673},
      mrreviewer = {Sampei Usui},
      doi = {10.1007/BF01388729},
      url = {http://dx.doi.org/10.1007/BF01388729},
      zblnumber = {0626.14007},
      }

Authors

Ben Davison

School of Mathematics and Statistics University of Glasgow, University Place Glasgow, United Kingdom