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In a recent paper, M. Green and P. Griffiths used R. Thomas' works on nodal hypersurfaces to establish the equivalence of the Hodge conjecture and the existence of certain singular admissible normal functions. Inspired by their work, we…
We classify the irreducible Hermitian real variations of Hodge structure admitting an infinitesimal normal function, and draw conclusions for cycle-class maps on families of abelian varieties with a given Mumford-Tate group.
Given a family of smooth complex projective varieties, the Hodge conjecture predicts the algebraicity of the locus of Hodge classes. This was proven unconditionnally by Cattani, Deligne and Kaplan in 1995. In a similar way, conjectures on…
If there is a topologically locally constant family of smooth algebraic varieties together with an admissible normal function on the total space, then the latter is constant on any fiber if this holds on some fiber. Combined with spreading…
Two simple observations are made: (1) If the normal function associated to a Hodge class has a zero locus of positive dimension, then it has a singularity. (2) The intersection cohomology of the dual variety contains the cohomology of the…
Entire functions in one complex variable are extremely relevant in several areas ranging from the study of convolution equations to special functions. An analog of entire functions in the quaternionic setting can be defined in the slice…
Consider the one-parameter generalizations of the logarithmic and exponential functions which are obtained from the integration of non-symmetrical hyperboles. These generalizations coincide to the one obtained in the context of…
We show that the zero locus of a normal function on a smooth complex algebraic variety S is algebraic provided that the normal function extends to a admissible normal function on a smooth compactification of S with torsion singularity. This…
Expanding upon recent work, a new class of $A$-functions is introduced that can be viewed as an appropriate generalization of the class of regular $A$-functions, the class of structured $A$-functions, and the class of perfect $A$-functions.…
Let $\phi$ be a normalized convex function defined on open unit disk $\mathbb{D}$. For a unified class of normalized analytic functions which satisfy the second order differential subordination $f'(z)+ \alpha z f''(z) \prec \phi(z)$ for all…
We determine the possible functions that can occur, up to asymptotic equivalence, as growth functions of semigroups, hereditary languages, and algebras.
We give a complete description of zero sets for some well-known subclasses of entire functions of exponential growth (bounded on real axis, Cartwright's class)
For families of smooth complex projective varieties we show that normal functions arising from algebraically trivial cycle classes are algebraic, and defined over the field of definition of the family. In particular, the zero loci of those…
Given a family of intermediate Jacobians (for a polarized variation of Hodge structure of weight -1) on a Zariski-open subset of a complex manifold, we construct an analytic space that naturally extends the family. Its two main properties…
We show that the zero locus of an admissible normal function on a smooth complex algebraic variety is algebraic.
We study two canonically defined admissible normal functions on the moduli space of smooth genus 4 algebraic curves including the Ceresa normal function. In particular, we study the vanishing criteria for the Griffiths infinitesimal…
In this paper we will develop a systematic method to answer the questions $(Q1)(Q2)(Q3)(Q4)$ (stated in Section 1) with complete generality. As a result, we can solve the difficulties $(D1)(D2)$ (discussed in Section 1) without uncertainty.…
We present two new classes of orthogonal functions, log orthogonal functions (LOFs) and generalized log orthogonal functions (GLOFs), which are constructed by applying a $\log$ mapping to Laguerre polynomials. We develop basic approximation…
Prompted by an observation about the integral of exponential functions of the form $f(x)=\lambda e^{\alpha x}$, we investigate the possibility to exactly integrate families of functions generated from a given function by scaling or by…
The theory of quaternionic slice regular functions was introduced in 2006 and successfully developed for about a decade over symmetric slice domains, which appeared to be the natural setting for their study. Some recent articles paved the…