We prove that every injective endomorphism of an affine algebraic variety over an algebraically closed field of characteristic zero is an automorphism. We also construct an analytic curve in ℂ⁶ and its holomorphic bijection which is not a biholomorphism.
We consider 2-dimensional semialgebraic topological manifolds from the differentialgeometric point of view. Curvatures at singularities are defined and a Gauss-Bonnet formula holds. Moreover, Aleksandrov's axioms for an intrinsic geometry of surfaces are fulfilled.
We introduce the notion of an instanton bundle on a Fano threefold of index 2. For such bundles we give an analogue of a monadic description and discuss the curve of jumping lines. The cases of threefolds of degree 5 and 4 are considered in a greater detail.
We study a class of torsion-free sheaves on complex projective spaces which generalize the much studied mathematical instanton bundles. Instanton sheaves can be obtained as cohomologies of linear monads and are shown to be semistable if its rank is not too large, while semistable torsion-free sheaves satisfying certain cohomological conditions are instanton. We also study a few examples of moduli spaces of instanton sheaves.
A polynomial map F = (P,Q) ∈ ℤ[x,y]² with Jacobian has a polynomial inverse with integer coefficients if the complex plane curve P = 0 has infinitely many integer points.
We use the methods that were developed by Adler and van Moerbeke to determine explicit equations for a certain moduli space, that was studied by Narasimhan and Ramanan. Stated briefly it is, for a fixed non-hyperelliptic Riemann surface of genus , the moduli space of semi-stable rank two bundles with trivial determinant on . They showed that it can be realized as a projective variety, more precisely as a quartic hypersurface of , whose singular locus is the Kummer variety of . We first construct...
The aim of these notes is to provide an introduction to the subject of integral canonical models of Shimura varieties, and then to sketch a proof of the existence of such models for Shimura varieties of Hodge and, more generally, abelian type. For full details the reader is refered to [Ki 3].