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We discuss the history of the monodromy theorem, starting from Weierstrass, and the concept of monodromy group. From this viewpoint we compare then the Weierstrass , the Legendre and other normal forms for elliptic curves, explaining their geometric meaning and distinguishing them by their stabilizer in P SL(2,Z) and their monodromy. Then we focus on the birth of the concept of the Jacobian variety, and the geometrization of the theory of Abelian functions and integrals. We end illustrating the methods of complex analysis in the simplest issue, the difference equation $f(z) = g(z+1) - g(z)$ on $mathbb C$.
A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincare showed that theta surfaces are precisely the surface
Let $f:mathcal{X}to S$ be a proper holomorphic submersion of complex manifolds and $G$ a complex reductive linear algebraic group with Lie algebra $mathfrak{g}$. Assume also given a holomorphic principal $G$-bundle $mathcal{P}$ over $mathcal{X}$ whic
For every integer $g ,geq, 2$ we show the existence of a compact Riemann surface $Sigma$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${mathcal O}^{oplus 2}_{Sigma}$ admits holomorphic connections with $text{SL}(2,{mathbb R})
We introduce a notion of normal form for transversely projective structures of singular foliations on complex manifolds. Our first main result says that this normal form exists and is unique when ambient space is two-dimensional. From this result o
We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. Normal form algorithms provide an algebraic approach to solve this problem. The frame