abelian varieties
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2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Andreas Krug ◽  
Ciaran Meachan

AbstractWe consider certain universal functors on symmetric quotient stacks of Abelian varieties. In dimension two, we discover a family of $${{\mathbb {P}}}$$ P -functors which induce new derived autoequivalences of Hilbert schemes of points on Abelian surfaces; a set of braid relations on a holomorphic symplectic sixfold; and a pair of spherical functors on the Hilbert square of an Abelian surface, whose twists are related to the well-known Horja twist. In dimension one, our universal functors are fully faithful, giving rise to a semiorthogonal decomposition for the symmetric quotient stack of an elliptic curve (which we compare to the one discovered by Polishchuk–Van den Bergh), and they lift to spherical functors on the canonical cover, inducing twists which descend to give new derived autoequivalences here as well.


Author(s):  
Gian Paolo Grosselli ◽  
Abolfazl Mohajer

AbstractWe study Shimura (special) subvarieties in the moduli space $$A_{p,D}$$ A p , D of complex abelian varieties of dimension p and polarization type D. These subvarieties arise from families of covers compatible with a fixed group action on the base curve such that the quotient of the base curve by the group is isomorphic to $${{\mathbb {P}}}^1$$ P 1 . We give a criterion for the image of these families under the Prym map to be a special subvariety and, using computer algebra, obtain 210 Shimura subvarieties contained in the Prym locus.


Author(s):  
Paola Frediani ◽  
Gian Paolo Grosselli

We study Shimura curves of PEL type in the space of polarized abelian varieties [Formula: see text] generically contained in the ramified Prym locus. We generalize to ramified double covers, the construction done in [E. Colombo, P. Frediani, A. Ghigi and M. Penegini, Shimura curves in the Prym locus, Commun. Contemp. Math. 21(2) (2019) 1850009] in the unramified case and in the case of two ramification points. Namely, we construct families of double covers which are compatible with a fixed group action on the base curve. We only consider the case of one-dimensional families and where the quotient of the base curve by the group is [Formula: see text]. Using computer algebra we obtain 184 Shimura curves contained in the (ramified) Prym loci.


2021 ◽  
Vol 31 (1) ◽  
pp. 75-103
Author(s):  
Hsueh-Yung Lin

For every fibration f : X → B f : X \to B with X X a compact Kähler manifold, B B a smooth projective curve, and a general fiber of f f an abelian variety, we prove that f f has an algebraic approximation.


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