We describe V G k for general groups in §2. These formulas were first written down by E. Verlinde [23] in the context of Conformal Field theory. The interest towards them in Algebraic Geometry stems form the fact that they give the Hilbert function of moduli spaces of principle bundles over projective curves. More precisely, let C be a smooth projective curve of genus g, and let MC be the modul...

We address the problem of the continuum limit for a system of Hausdorr lattices (namely lattices of isolated points) approximating a topological space M. The correct framework is that of projective systems. The projective limit is a universal space from which M can be recovered as a quotient. We dualize the construction to approximate the algebra C(M) of continuous functions on M. In a companio...

We display explicit half-conformally-flat metrics on the connected sum of any number of copies of the complex projective plane. These metrics are obtained from magnetic monopoles in hyperbolic 3-space by an analogue of the Gibbons-Hawking ansatz, and are conformal compactifications of asymptotically-flat, scalar-flat Kahler metrics on «-fold blow-ups of C . The corresponding twistor spaces are ...

We construct a noncommutative deformation of odd-dimensional spheres that preserves the natural partition of the (2N + 1)-dimensional sphere into (N + 1)many solid tori. This generalizes the case N = 1 referred to as the Heegaard quantum sphere. Our twisted odd-dimensional quantum sphere C∗-algebras are given as multipullback C∗-algebras. We prove that they are isomorphic to the universal C∗-al...

A theorem of A. Ostrowski describing meromorphic functions f such that the family {f(λz) : λ ∈ C∗} is normal, is generalized to holomorphic maps from C∗ to a projective space. MSC 2010: 30D45, 32A19.

Let C be a class of graphs closed under isomorphism and let ~ C be obtained from C by arc anchorage. A concept of ~ C-homogeneous graphs that include the C-ultrahomogeneous graphs is given, with the explicit construction of ~ C-homogeneous graphs that are not C-ultrahomogeneous, via ordered pencils of binary projective spaces.

We prove that there exists a positive integer νn depending only on n such that for every smooth projective n-fold of general type X defined over C, | mKX | gives a birational rational map from X into a projective space for every m ≥ νn. This theorem gives an affirmative answer to Severi’s conjecture. The key ingredients of the proof are the theory of AZD which was originated by the aurhor and t...

Generalising the notion of Galois corings, Galois comodules were introduced as comodules P over an A-coring C for which PA is finitely generated and projective and the evaluation map μC : Hom (P, C) ⊗S P → C is an isomorphism (of corings) where S = End(P ). It was observed that for such comodules the functors HomA(P,−)⊗S P and −⊗A C from the category of right A-modules to the category of right ...

Let C be a smooth projective curve defined over a number field k, A/k(C) an abelian variety and (τ, B) the k(C)/k-trace of A. We estimate how the rank of A(k(C))/τB(k) varies when we take a finite Galois k-cover π : C → C defined over k.

A cohomology class of a smooth complex variety dimension $n$ has coniveau $\geq c$ if it vanishes in the complement closed subvariety codimension c$, and strong comes by proper pushforward from $\leq n-c$. We show that these two notions differ general, both for integral classes on projective varieties rational open varieties.