Related papers: A generalized Quot scheme and meromorphic vortices

Using the idea of the degree of a smooth mapping between two manifolds of the same dimension we present here the topological (homotopical) classification of the mappings between spheres of the same dimension, vector fields, monopole and…

We use bounds of mixed character sums modulo a prime $p$ to estimate the density of integer points on the hypersurface $$ f_1(x_1) + \ldots + f_n(x_n) =a x_1^{k_1} \ldots x_n^{k_n} $$ for some polynomials $f_i \in {\mathbb Z}[X]$, nonzero…

For positive integers $m,n, d\geq 1$ with $(m,n)\not= (1,1)$ and a field $\Bbb F$ with its algebraic closure $\overline{\Bbb F}$, let $\text{Poly}^{d,m}_n(\Bbb F)$ denote the space of all $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \Bbb F [z]$…

Let $X, Y$ be smooth algebraic varieties of the same dimension. Let $f, g : X \to Y$ be finite polynomial mappings. We say that $f, g$ are equivalent if there exists a regular automorphism $\Phi \in Aut(X)$ such that $f = g\circ \Phi$. Of…

Generalized Heisenberg algebras $\H(f)$ for any polynomial $f(h)\in\C[h]$ have been used to explain various physical systems and many physical phenomena for the last 20 years. In this paper, we first obtain the center of $\H(f)$, and the…

For an arbitrary field K, let I be an ideal in the ring K[[x,y]] expressible as a polynomial in either the pair of ideals (x, y^4) and (x,y) or the pair (x,y^3) and (x^2, y). Let G be the group of automorphisms of K[[x,y]] sending the ideal…

We investigate the symplectic geometric and differential geometric aspects of the moduli space of connections on a compact Riemann surface $X$. Fix a theta characteristic $K^{1/2}_X$ on $X$; it defines a theta divisor on the moduli space…

Let G be a Lie goup, let M and N be smooth connected G-manifolds, let f be a smooth G-map from M to N, and let P denote the fiber of f. Given a closed and equivariantly closed relative 2-form for f with integral periods, we construct the…

Let (M,E) be a generalized polarized manifold, i.e., a pair of an n-dimensional smooth projective variety and an ample vector bundle E of rank r on M. Let t be the nef value of a polarized manifold (M, det E), i.e., the minimum of the set…

We consider the moduli space $\mathcal{R}_n$ of pairs of monic, degree $n$ polynomials whose resultant equals $1$. We relate the topology of these algebraic varieties to their geometry and arithmetic. In particular, we compute their…

Let G < SL(V) be a finite group, V is finite dimensional over a field F, p=char F and S(V) is the symmetric algebra of V. We determine when the subring of G-invariants S(V)^G is a polynomial ring. As a consequence, we classify, if F is…

We give some concrete examples of moduli spaces of connections. Precisely, we explain how to explicitly construct the moduli spaces of rank 2 fuchsian systems and logarithmic connections on the Riemann sphere with 4 poles. The former ones…

Let $X$ be a locally symmetric space $\Gamma\backslash G/K$ where $G$ is a connected non-compact semisimple real Lie group with trivial centre, $K$ is a maximal compact subgroup of $G$, and $\Gamma\subset G$ is a torsion-free irreducible…

In 2006, Kenyon and Okounkov computed the moduli space of Harnack curves of degree $d$ in $\mathbb{C}\mathbb{P}^2$. We generalize to any projective toric surface some of the techniques used there. More precisely, we show that the moduli…

Let f be a newform, as specified by its Hecke eigenvalues, on a Shimura curve X. We describe a method for evaluating f. The most interesting case is when X arises as a compact quotient of the hyperbolic plane, so that classical q-expansions…

Let $M$ be a $n$-dimensional complex manifold and $f,g:M\to M$ two distinct holomorphic self-maps. Suppose that $f$ and $g$ coincide on a globally irreducible compact hypersurface $S\subset M$. We show that if one of the two maps is a local…

Let $X$ be a compact Riemann surface. Let $f$ be a holomorphic self-correspondence of $X$ with dynamical degrees $d_1$ and $d_2$. Assume that $d_1\neq d_2$ or $f$ is non-weakly modular. We show that the graphs of the iterates $f^n$ of $f$…

Let $\Gamma \subset \mathbf{PU}(2,1)$ be a lattice which is not co-compact, of finite Bergman-covolume and acting freely on the open unit ball $\mathbf{B} \subset \mathbb{C}^2$. Then the compactification $X = \bar{\Gamma \setminus…

We study consequences and applications of the folklore statement that every double complex over a field decomposes into so-called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences…

We construct the geometric quantization of a compact surface using a singular real polarization coming from an integrable system. Such a polarization always has singularities, which we assume to be of nondegenerate type. In particular, we…