Related papers: On Bott's residue formula for toric complete inter…

In this paper we shall give formulas for the pairings of intersection cohomology classes of complementary dimensions in the intersection cohomology of geometric invariant theoretic quotients for which semistability is not necessarily the…

We give complete geometric invariants of cobordisms of framed fold maps. These invariants consist of two types. We take the immersion of the fold singular set into the target manifold together with information about non-triviality of the…

A generic geodesic on a finite area, hyperbolic 2-orbifold exhibits an infinite sequence of penetrations into a neighborhood of a cone singularity, so that the sequence of depths of maximal penetration has a limiting distribution. The…

In this paper, we compute the number of self-intersections of a plane projection of a generic complete intersection curve defined by polynomials with the given support. Moreover, we discuss the tropical counterpart of this problem.

The set of volumes of stable surfaces does have accumulation points. In this paper, we study this phenomenon for surfaces with one cyclic quotient singularity, towards answering the question under which conditions we can still have…

We consider random Gaussian eigenfunctions of the Laplacian on the standard torus, and investigate the number of nodal intersections against a line segment. The expected intersection number, against any smooth curve, is universally…

We analyze the embedding dimension of a normal weighted homogeneous surface singularity, and more generally, the Poincar\'e series of the minimal set of generators of the graded algebra of regular functions, provided that the link of the…

In this paper we prove geometric residue theorems for bundle maps over a compact manifold. The theory developed associates residues to the singularity submanifolds of the map for any invariant polynomial. The theory is then applied to a…

In this short note, we provide a quantitative global Poincar\'e inequality for one forms on a closed Riemannian four manifold, in terms of an upper bound on the diameter, a positive lower bound on the volume, and a two-sided bound on Ricci…

The automorphism group of a projective bundle P(E) over a simplicial toric variety is described when the bundle E is a direct sum of line bundles. Applications to study of moduli of complete intersections on toric varieties, including…

Let $\mathcal{F}$ be a singular Riemann surface foliation on a complex manifold $M$, such that the singular set $E \subset M$ is non-discrete. We study the behavior of the foliation near the singular set $E$, particularly focusing on…

We explore systems of polynomial equations where we seek complex solutions with absolute value 1. Geometrically, this amounts to understanding intersections of algebraic varieties with tori -- Cartesian powers of the unit circle. We study…

We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…

An old conjecture of Durfee 1978 bounds the ratio of two basic invariants of complex isolated complete intersection surface singularities: the Milnor number and the singularity (or geometric) genus. We give a counterexample for the case of…

The purpose of this paper is to show how Rees algebras can be applied in the study of singularities embedded in smooth schemes over perfect fields. In particular, we will study situations in which the multiplicity of a hypersurface is a…

We consider a planar convex body $C$ and we prove several analogs of Roth's theorem on irregularities of distribution. When $\partial C$ is $\mathcal{C}% ^{2}$ regardless of curvature, we prove that for every set $\mathcal{P}_{N}$ of $N$…

We discuss various bifurcation problems in which two isolated periodic orbits exchange periodic ``bridge'' orbit(s) between two successive bifurcations. We propose normal forms which locally describe the corresponding fixed point scenarios…

We collect some classical results about holomorphic 1-forms of a reduced complex curve singularity, in particular of a complete intersection, and use them to compare the Milnor number, the Tjurina number and the dimension of the torsion…

This work is dedicated to studying holomorphic distributions on Grassmann manifolds and smooth quadric hypersurfaces. In special, we prove, under certain conditions, when the tangent and conormal sheaves of a distribution splits as a sum of…

For $G = \mathrm{GL}_2, \mathrm{SL}_2, \mathrm{PGL}_2$ we compute the intersection E-polynomials and the intersection Poincar\'e polynomials of the $G$-character variety of a compact Riemann surface $C$ and of the moduli space of $G$-Higgs…