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  • 1.
    Nill, Benjamin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Paffenholz, Andreas
    On the equality case in Ehrhart's volume conjecture2014In: Advances in Geometry, ISSN 1615-715X, E-ISSN 1615-7168, Vol. 14, no 4, p. 579-586Article in journal (Refereed)
    Abstract [en]

    Ehrhart's conjecture proposes a sharp upper bound on the volume of a convex body whose barycenter is its only interior lattice point. Recently, Berman and Berndtsson proved this conjecture for a class of rational polytopes including reflexive polytopes. In particular, they showed that the complex projective space has the maximal anticanonical degree among all toric Kahler-Einstein Fano manifolds. In this note, we prove that projective space is the only such toric manifold with maximal degree by proving the corresponding convex-geometric statement. We also discuss a generalized version of Ehrhart's conjecture involving an invariant corresponding to the so-called greatest lower bound on the Ricci curvature.

  • 2.
    Oneto, Alessandro
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Petracci, Andrea
    On the quantum periods of del Pezzo surfaces with 1/3 (1,1) singularities2018In: Advances in Geometry, ISSN 1615-715X, E-ISSN 1615-7168, Vol. 18, no 3, p. 303-336Article in journal (Refereed)
    Abstract [en]

    In earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces X with isolated cyclic quotient singularities such that X admits a Q-Gorenstein toric degeneration correspond via Mirror Symmetry to maximally mutable Laurent polynomials f in two variables, and that the quantum period of such a surface X, which is a generating function for Gromov-Witten invariants of X, coincides with the classical period of its mirror partner f. In this paper we give strong evidence for this conjecture. Contingent on conjectural generalisations of the Quantum Lefschetz theorem and the Abelian/non-Abelian correspondence, we compute many quantum periods for del Pezzo surfaces with 1/3 (1, 1) singularities. Our computations also give strong evidence for the extension of these two principles to the orbifold setting.

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