Related papers: Lazzeri's Jacobian of oriented compact riemannian …
Using divisors, an analog of the Jacobian for a compact connected nonorientable Klein surface $Y$ is constructed. The Jacobian is identified with the dual of the space of all harmonic real one-forms on $Y$ quotiented by the torsion-free…
The Torelli group of a compact non-orientable Klein surface is the subgroup of the modular group consisting of the mapping classes that act trivially on the first homology group of the surface. We prove that if a surface has genus at least…
In this paper, we study Jacobi operators associated to algebraic curvature maps (tensors) on lightlike submanifolds M. We investigate conditions for an induced Rie- mann curvature tensor to be an algebraic curvature tensor on M. We…
The notion of partial Jacobi manifold is introduced in the convenient ($c^\infty$-complete) framework of Fr\"olicher, Kriegl, and Michor. Explicit examples are provided in both finite and infinite dimensions, and the characteristic…
The geometrical representation of the Jacobian in the path integral reduction problem which describes a motion of the scalar particle on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple Lie…
We consider four-dimensional Riemannian manifolds with commuting higher order Jacobi operators defined on two-dimensional orthogonal subspaces (polygons) and on their orthogonal subspaces. More precisely, we discuss higher order Jacobi…
Jacobi algebroids, that is graded Lie brackets on the Grassmann algebra associated with a vector bundle which satisfy a property similar to that of the Jacobi brackets, are introduced. They turn out to be equivalent to generalized Lie…
By considering the projectivized spectrum of the Jacobi operator, we introduce the concept of projective Osserman manifold in both the affine and in the pseudo-Riemannian settings. If M is an affine projective Osserman manifold, then the…
We develop algebro-geometrical approach for the open Toda lattice. For a finite Jacobi matrix we introduce a singular reducible Riemann surface and associated Baker-Akhiezer functions. We provide new explicit solution of inverse spectral…
In this paper we define a Grassmann odd analogue of Jacobi structure on a supermanifold. The basic properties are explored. The construction of odd Jacobi manifolds is then used to reexamine the notion of a Jacobi algebroid. It is shown…
We generalize the property of Jacobi-orthogonality to indefinite scalar product spaces. We compare various principles and investigate relations between Osserman, Jacobi-dual, and Jacobi-orthogonal algebraic curvature tensors. We show that…
In this paper we describe compactified universal Jacobians, i.e. compactifications of the moduli space of line bundles on smooth curves obtained as moduli spaces of rank 1 torsion-free sheaves on stable curves, using an approach due to…
We give a general method for constructing compact K\"ahler manifolds $X_1$ and $X_2$ whose intermediate Jacobians $J^k(X_1)$ and $J^k(X_2)$ are isogenous for each $k$, and we exhibit some examples. The method is based upon the algebraic…
We investigate the differential equation for the Jacobi-type polynomials which are orthogonal on the interval $[-1,1]$ with respect to the classical Jacobi measure and an additional point mass at one endpoint. This scale of higher-order…
We reformulate the notion of a Jacobi algebroid in terms of weighted odd Jacobi brackets. We then show how a Jacobi algebroid can be understood in terms of a kind of curved Q-manifold. In particular the homological condition on the odd…
This manuscript is about abelian varieties that are Jacobians of curves. I started writing it for a lecture series at the Arizona Winter School in 2024 on abelian varieties. A longer more descriptive title might be: The Torelli locus in the…
The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the P\"oschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of…
We present a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier. We illustrate the easiness and the power of Jacobi's method by applying it to several…
We introduce and study the notion of contact dual pair adopting a line bundle approach to contact and Jacobi geometry. A contact dual pair is a pair of Jacobi morphisms defined on the same contact manifold and satisfying a certain…
Jacobi/Poisson algebras are algebraic counterparts of Jacobi/Poisson manifolds. We introduce representations of a Jacobi algebra $A$ and Frobenius Jacobi algebras as symmetric objects in the category. A characterization theorem for…