Related papers: A rigid analytic proof that the Abel-Jacobi map ex…

We investigate the "natural" locus of definition of Abel-Jacobi maps. In particular, we show that, for a proper, geometrically reduced curve C -- not necessarily smooth -- the Abel-Jacobi map from the smooth locus C^{sm} into the Jacobian…

For a complex projective manifold, Walker has defined a regular homomorphism lifting Griffiths' Abel-Jacobi map on algebraically trivial cycle classes to a complex abelian variety, which admits a finite homomorphism to the Griffiths…

Let $C$ be a nodal curve and $L$ be an invertible sheaf on $C$. Let $\alpha_{L}:C\dashrightarrow J_{C}$ be the degree-$1$ rational Abel map, which takes a smooth point $Q\in C$ to $\left[ m_{Q}\otimes L\right] $ in the Jacobian of $C$. In…

To a compact tropical variety of arbitrary dimension, we associate a collection of intermediate Jacobians defined in terms of tropical homology and tropical monodromy. We then develop an Abel-Jacobi theory in the tropical setting by…

We realize the infinitesimal Abel-Jacobi map as a morphism of formal deformation theories, realized as a morphism in the homotopy category of differential graded Lie algebras. The whole construction is carried out in a general setting, of…

Given a smooth curve with weighted marked points, the Abel-Jacboi map produces a line bundle on the curve. This map fails to extend to the full boundary of the moduli space of stable pointed curves. Using logarithmic and tropical geometry,…

In this article, we study the infinitemisal invariant of the relative higher Abel Jacobi map of a smooth open morphism. We give a generalization of a theorem of Voisin to open varieties and higher Chow groups and as a corollary a non…

We prove that Shimura varieties of abelian type satisfy a $p$-adic Borel-extension property over discretely valued fields. More precisely, let $\mathsf{D}$ denote the rigid-analytic closed unit disc and $\mathsf{D}^{\times} = \mathsf{D}…

We establish rigidity results for holomorphic mappings and plurisubharmonic functions in complex geometry. First, under mild conditions, we show that the gradient of a $\operatorname{U}(1)$-invariant strictly plurisubharmonic function in…

Let $f\col\C\ra B$ be a regular local smoothing of a nodal curve. In this paper, we find a modular description of the Abel--N\'eron map having values in Esteves's fine compactified Jacobian and extending the degree-2 Abel--Jacobi map of the…

The abelian Higgs model on a compact Riemann surface \Sigma supports vortex solutions for any positive vortex number d \in \ZZ. Moreover, the vortex moduli space for fixed d has long been known to be the symmetrized d-th power of \Sigma, in…

Let C be an integral projective curve in any characteristic. Given an invertible sheaf L on C of degree 1, form the associated Abel map A_L : C -> P, which maps C into its compactified Jacobian scheme P, and form its pullback map A_L^* :…

We show that the space of expanding maps contains an open and dense set where smooth conjugacy classes of expanding maps are determined by the values of the Jacobians of return maps at periodic points.

Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this paper the existence of families of…

Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this paper the existence of families of…

We study the interaction between the group law on an abelian variety and the additive structure induced on its image under a morphism to projective space. Let $A/F$ be a simple abelian variety, $f:A \rightarrow \mathbb{P}^n$ be a morphism…

Let $X$ be a compact connected Riemann surface of genus at least two. The Abel-Jacobi map $\varphi: {\rm Sym}^d(X) \rightarrow {\rm Pic}^d(X)$ is an embedding if $d$ is less than the gonality of $X$. We investigate the curvature of the…

The intermediate Jacobian map, which associates to a smooth cubic threefold its intermediate Jacobian, does not extend to the GIT compactification of the space of cubic threefolds, not even as a map to the Satake compactification of the…

The difference $[L_1]-[L_2]$ of a pair of skew lines on a cubic threefold defines a vanishing cycle on the cubic surface as the hyperplane section spanned by the two lines. By deforming the hyperplane, the flat translation of such vanishing…

We construct the Abel-Jacobi map for Mumford curves over any complete non-archimedean field, using multiplicative integrals and in the setting of Berkovich analytic geometry. Along the way, we proof some results concerning graphs and…