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Special representatives of complexified Kähler classes

  • Carlo Scarpa [1] ; Jacopo Stoppa [2]
    1. [1] University of Quebec

      University of Quebec

      Canadá

    2. [2] SISSA, Italy Institute for Geometry and Physics (IGAP), Italy
  • Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 4, 2024, págs. 1-45
  • Idioma: inglés
  • DOI: 10.1007/s00029-024-00955-1
  • Enlaces
  • Resumen

    • Motivated by constructions appearing in mirror symmetry, we study special representatives of complexified Kähler classes, which extend the notions of constant scalar curvature and extremal representatives for usual Kähler classes. In particular, we provide a moment map interpretation, discuss a possible correspondence with compactified Landau–Ginzburg models, and prove existence results for such special complexified Kähler forms and their large volume limits in certain toric cases.

  • Referencias bibliográficas
    • Álvarez Cónsul, L., García-Fernández, M., García-Prada, O.: Coupled equations for Kähler metrics and Yang-Mills connections. Geom. Topol....
    • Álvarez Cónsul, L., Garcia-Fernandez, M., García-Prada, O.: On the Kähler-Yang-Mills-Higgs equations. Pure Appl. Math. Q. 15(4), 1181–1217...
    • Apostolov, V.: The Kähler geometry of toric manifolds. Lecture Notes of CIRM winter school (2019).
    • Aspinwall, P., Bridgeland, T., Craw, A., Douglas, M., Gross, M., Kapustin, A., Moore, G., Segal, G., Szendrói, B., Wilson, P.: Dirichlet branes...
    • Bayer, A.: Polynomial Bridgeland stability conditions and the large volume limit. Geom. Topol. 13(4), 2389–2425 (2009).
    • Biquard, O., Rollin, Y.: Smoothing singular constant scalar curvature Kähler surfaces and minimal Lagrangians. Adv. Math. 285, 980–1024 (2015).
    • Chen, G.: The J-equation and the supercritical deformed Hermitian-Yang-Mills equation. Invent. Math. 225(2), 529–602 (2021).
    • Chen, X., Cheng, J.: On the constant scalar curvature Kähler metrics (I)–a priori estimates. J. Am. Math. Soc. 34(4), 909–936 (2021).
    • Chen, X., Cheng, J.: On the constant scalar curvature Kähler metrics (II)–existence results. J. Am. Math. Soc. 34(4), 937–1009 (2021).
    • Chu, J., Lee, M.-C., Takahashi, R.: A Nakai-Moishezon type criterion for supercritical deformed Hermitian-Yang-Mills equation. To appear in...
    • Collins, T., Jacob, A., Lin, Y.-S.: The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces. arXiv:2012.05416...
    • Collins, T., Jacob, A., Yau, S.-T.: (1, 1) forms with specified Lagrangian phase: a priori estimates and algebraic obstructions. Camb. J....
    • Collins, T.C., Yau, S.-T.: Moment maps, nonlinear PDE, and stability in mirror symmetry. arXiv:1811.04824 [math.DG].
    • Collins, T.C., Yau, S.-T.: Moment maps, nonlinear PDE and stability in mirror symmetry, I: geodesics. Ann. PDE 7(1), 11 (2021).
    • Datar, V.V., Pingali, V.P.: On coupled constant scalar curvature Kähler metrics. J. Symplectic Geom. 18(4), 961–994 (2020).
    • Datar, V.V., Pingali, V.P.: A numerical criterion for generalised Monge-Ampère equations on projective manifolds. Geom. Funct. Anal. 31(4),...
    • Dervan, R.: Stability conditions for polarised varieties. Forum Math. Sigma 11, e104 (2023).
    • Dervan, R., McCarthy, J., Sektnan, L.: Z-critical connections and Bridgeland stability conditions. To appear in Camb. J. Math. arXiv:2012.10426...
    • Dervan, R., Naumann, P.: Moduli of polarised manifolds via canonical Kähler metrics. arXiv:1810.02576 [math.AG].
    • Dervan, R., Ross, J.: Stable maps in higher dimensions. Math. Ann. 374(3–4), 1033–1073 (2019).
    • Donaldson, S.K.: Symmetric spaces, Kähler geometry and Hamiltonian dynamics. In: Northern California Symplectic Geometry Seminar, volume 196...
    • Donaldson, S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 62(2), 289–349 (2002).
    • Donaldson, S.K.: Interior estimates for solutions of Abreu’s equation. Collect. Math. 56(2), 103–142 (2005).
    • Donaldson, S.K.: Constant scalar curvature metrics on toric surfaces. Geom. Funct. Anal. 19(1), 83–136 (2009).
    • Fujiki, A., Schumacher, G.: The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics. Publ. Res. Inst....
    • Futaki, A.: An obstruction to the existence of Kähler-Einstein metrics. Invent. Math. 73, 437–443 (1983).
    • Garcia-Fernandez, M.: Coupled equations for Kähler metrics and Yang-Mills connections. PhD thesis, Universidad Autónoma de Madrid. arXiv:1102.0985...
    • Gross, M., Hacking, P., Keel, S.: Mirror symmetry for log Calabi-Yau surfaces I. Publ. Math. Inst. Hautes Études Sci. 122, 65–168 (2015).
    • Gross, M., Katzarkov, L., Ruddat, H.: Towards mirror symmetry for varieties of general type. Adv. Math. 308, 208–275 (2017).
    • Hattori, M.: On K-stability of Calabi-Yau fibrations. arXiv:2203.11460 [math.AG].
    • He, W.: On Calabi’s extremal metric and properness. Trans. Am. Math. Soc. 372(8), 5595–5619 (2019).
    • Hultgren, J., Witt Nyström, D.: Coupled Kähler-Einstein metrics. Int. Math. Res. Not. IMRN 21, 6765–6796 (2019).
    • Jacob, A., Yau, S.-T.: A special Lagrangian type equation for holomorphic line bundles. Math. Ann. 369(1–2), 869–898 (2017).
    • Katzarkov, L., Kontsevich, M., Pantev, T.: Bogomolov-Tian-Todorov theorems for Landau-Ginzburg models. J. Differ. Geom. 105(1), 55–117 (2017).
    • Keller, J., Tønnesen-Friedman, C.W.: Nontrivial examples of coupled equations for Kähler metrics and Yang-Mills connections. Cent. Eur. J....
    • Leung, N.C., Yau, S.-T., Zaslow, E.: From special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai. Adv. Theor. Math. Phys. 4(6), 1319–1341...
    • Li, A.-M., Lian, Z., Sheng, L.: Extremal metrics on toric manifolds and homogeneous toric bundles. J. Reine Angew. Math. (Crelles J.) 2023(798),...
    • Lin, C.-M.: The deformed Hermitian-Yang-Mills equation, the Positivstellensatz, and the solvability. Adv. Math. 433, 109312 (2023).
    • Przyjalkowski, V.: Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds. Sb. Math. 7(208), 992–1013 (2017).
    • Scarpa, C.: A K-energy functional for complexified Kähler classes. arXiv:2307.09904 [math.DG].
    • Schlitzer, E., Stoppa, J.: Deformed Hermitian Yang-Mills connections, extended gauge group and scalar curvature. J. Lond. Math. Soc. (2) 104(2),...
    • Schlitzer, E., Stoppa, J.: Examples of dHYM connections in a variable background. Can. J. Math. 2023, 1–29 (2023). https://doi.org/10.4153/S0008414X23000561.
    • Schoen, R., Wolfson, J.: Minimizing area among Lagrangian surfaces: the mapping problem. J. Differ. Geom. 58, 1–86 (2001).
    • Sheridan, N.: Versality in mirror symmetry. In: Current Developments in Mathematics 2017, pp. 37–86. International Press, Somerville, MA (2019).
    • Stoppa, J.: Twisted constant scalar curvature Kähler metrics and Kähler slope stability. J. Differ. Geom. 83(3), 663–691 (2009).
    • Thomas, R.P., Yau, S.-T.: Special Lagrangians, stable bundles and mean curvature flow. Commun. Anal. Geom. 10(5), 1075–1113 (2002).
    • Schumacher, G.: Construction of the coarse moduli space of compact polarized Kähler manifolds with c1 = 0. Math. Ann. 1(264), 81–90 (1983)
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