In this paper, we obtain several characterizations of varieties of Fano type and projective spaces via absolute complexity. Also, we show that if the absolute complexity of a given pair ((X,Delta )) is negative, then the pair ((X,Delta )) does not admit any (-(K_X+Delta ))-minimal models.
We obtain upper bounds for the torsion in the K-groups of the ring of integers of imaginary quadratic number fields, in terms of their discriminants.
We give a precise estimate for the average scalar curvature of the Weil–Petersson metric on the moduli space ({overline{{{mathcal {M}}} }}_g) as (grightarrow infty ) up to the order (1/g^2).
We compute Hodge numbers and zeta function of a Kummer Calabi-Yau 3-folds introduced by M. Andreatta and J. Wiśniewski in [2] and investigated by M. Donten-Bury in [13].
The shifting numbers measure the asymptotic amount by which an endofunctor of a triangulated category translates inside the category, and are analogous to Poincaré translation numbers that are widely used in dynamical systems. Motivated by this analogy, Fan–Filip raised the following question: “Do the shifting numbers define a quasimorphism on the group of autoequivalences of a triangulated category?” An affirmative answer was given by Fan–Filip for the bounded derived category of coherent sheaves on an elliptic curve or an abelian surface, via properties of the spaces of Bridgeland stability conditions on these categories. We prove in this article that the question has an affirmative answer for abelian varieties of arbitrary dimensions, generalizing the result of Fan–Filip. One of the key steps is to establish an alternative definition of the shifting numbers via bounded t-structures on triangulated categories. In particular, the full package of a Bridgeland stability condition (a bounded t-structure, and a central charge on a charge lattice) is not necessary for the purpose of computing the shifting numbers.
We provide a closed form expression for linear Hodge integrals on the hyperelliptic locus. Specifically, we find a succinct combinatorial formula for all intersection numbers on the hyperelliptic locus with one (lambda )-class, and powers of a (psi )-class pulled back along the branch map. This is achieved by using Atiyah–Bott localization on a stack of stable maps into the orbifold (left[ {mathbb {P}}^1/{mathbb {Z}}_2right] ).
In this paper, we study the Dirichlet problem for p-convex hypersurfaces with prescribed curvature. We prove that there exists a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition. An interior curvature estimate is also obtained.
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