Related papers: Multiple-point residue formulas for holomorphic ma…

Let $S$ be the affine plane regarded as a toric variety with an action of the 2-dimensional torus $T$. We study the equivariant Chow ring $A_{K}^*(Hilb^n(S))$ of the punctual Hilbert scheme $Hilb^n(S)$ with equivariant coefficients…

This article provides a summary of arXiv:1701.08899 and arXiv:1701.08902 where the authors studied the enumerative geometry of nested Hilbert schemes of points and curves on algebraic surfaces and their connections to threefold theories,…

A continuous map $\mathbb{C}^n\to Gr(\tau, N)$ is $k$-regular if the $\tau$-dimensional subspaces corresponding to images of any $k$ distinct points span a $\tau k$-dimensional space. For $\tau = 1$ this essentially recovers the classical…

This paper seeks to advance the theory of nonexpansive mappings by introducing and exploring a novel class of nonexpansive type mappings, which we aptly designate as perimetric nonexpansive mappings. We establish that the collection of…

We describe the ring structure of the cohomology of the Hilbert scheme of points for a smooth surface X. When the canonical class K_X = 0, this was done by Lehn and Sorger, extending earlier work when X = C^2. Their approach does not…

In this paper, using the formula for the integrals of the $\psi$-classes over the double ramification cycles found by S. Shadrin, L. Spitz, D. Zvonkine and the author, we derive a new explicit formula for the $n$-point function of the…

This is the first part in a series of papers on counting surfaces on Calabi-Yau 4-folds. Besides the Hilbert scheme of 2-dimensional subschemes, we introduce \emph{two} types of moduli spaces of stable pairs. We show that all three moduli…

We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients.…

We review some constructions and properties of complex manifolds admitting pluriclosed and balanced metrics. We prove that for a 6-dimensional solvmanifold endowed with an invariant complex structure J having holomorphically trivial…

Constructive methods for matrices of multihomogeneous (or multigraded) resultants for unmixed systems have been studied by Weyman, Zelevinsky, Sturmfels, Dickenstein and Emiris. We generalize these constructions to mixed systems, whose…

We express nested Hilbert schemes of points and curves on a smooth projective surface as "virtual resolutions" of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories…

Let $C$ be a hyperelliptic curve embedded in its Jacobian $J$ via an Abel-Jacobi map. We compute the scheme structure of the Hilbert scheme component of $\textrm{Hilb}_J$ containing the Abel-Jacobi curve as a point. We relate the result to…

The goal of this thesis is to study the singularities of the exponential map of Riemannian and Finsler manifolds (a concept related to caustics and catastrophes), and the object known as the cut locus (aka ridge, medial axis or skeleton),…

We seek to characterize homology classes of Lagrangian projective spaces embedded in irreducible holomorphic-symplectic manifolds, up to the action of the monodromy group. This paper addresses the case of manifolds deformation-equivalent to…

The multiple root loci among univariate polynomials of degree $n$ are indexed by partitions of $n$. We study these loci and their conormal varieties. The projectively dual varieties are joins of such loci where the partitions are hooks. Our…

We expand on a result of Barvinok and Hartigan to derive asymptotic formulas for the number of integer and binary integer points in a wide class of multi-index $k_1\times k_2\times \ldots \times k_{\nu}$ transportation polytopes. A simple…

Thom polynomial describes the cohomology class Poincar\'e dual to the locus of particular singularity of a generic holomorphic map. In this paper we derive a closed formula for the generating function of its coefficients. The method is…

We explicitly study Kato's residue homomorphisms in Milnor $K$-theory, which are closely related to Contou-Carr\`ere symbols. As applications we establish several reciprocity laws for certain locally defined maps on $K$-groups that are…

Various equivariant intersection numbers on Hilbert schemes of points on the affine plane are computed, some of which are organized into tau-functions of 2-Toda hierarchies. A correspondence between the equivariant intersection on Hilbert…

The cohomology jump loci of a space $X$ are of two basic types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems, and the resonance varieties, constructed from information encoded in…