Picard group of isotropic realizations of twisted Poisson manifolds

AbstractWe establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in Bruzzo and Grassi (Commun Anal Geom 28: 1773–1786, 2020). We also show that the Hodge Conjecture holds asymptotically for suitable quasi-smooth hypersurface in the Noether–Lefschetz locus, where “asymptotically” means that the degree of the hypersurface is big enough, under the assumption that the ambient variety $${{\mathbb {P}}}_\Sigma ^{2k+1}$$ P Σ 2 k + 1 has Picard group $${\mathbb {Z}}$$ Z . This extends to a class of toric varieties Otwinowska’s result in Otwinowska (J Alg Geom 12: 307–320, 2003).

Download Full-text

Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

Journal of Algebraic Combinatorics ◽

10.1007/s10801-021-01025-x ◽

2021 ◽

Author(s):

Michele Rossi ◽

Lea Terracini

Keyword(s):

Free Group ◽

Toric Variety ◽

Class Group ◽

Abelian Groups ◽

Toric Varieties ◽

Picard Group ◽

Closed Field ◽

Finitely Generated ◽

Free Part ◽

Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

Download Full-text

Schottky groups acting on homogeneous rational manifolds

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2016-0065 ◽

2019 ◽

Vol 2019 (753) ◽

pp. 23-56 ◽

Cited By ~ 1

Author(s):

Christian Miebach ◽

Karl Oeljeklaus

Keyword(s):

Deformation Theory ◽

Group Actions ◽

Picard Group ◽

Schottky Group ◽

Fundamental Groups ◽

Complex Manifolds ◽

Schottky Groups ◽

Algebraic Dimension ◽

Geometric Invariants ◽

Compact Complex Manifolds

AbstractWe systematically study Schottky group actions on homogeneous rational manifolds and find two new families besides those given by Nori’s well-known construction. This yields new examples of non-Kähler compact complex manifolds having free fundamental groups. We then investigate their analytic and geometric invariants such as the Kodaira and algebraic dimension, the Picard group and the deformation theory, thus extending results due to Lárusson and to Seade and Verjovsky. As a byproduct, we see that the Schottky construction allows to recover examples of equivariant compactifications of {{\rm{SL}}(2,\mathbb{C})/\Gamma} for Γ a discrete free loxodromic subgroup of {{\rm{SL}}(2,\mathbb{C})}, previously obtained by A. Guillot.

Download Full-text