Related papers: Refined Verlinde and Segre formula for Hilbert sch…

Using vertex algebra techniques, we determine a set of generators for the cohomology ring of the Hilbert schemes of points on an arbitrary smooth projective surface over the field of complex numbers.

We develop algorithms to compute two versions of the motivic Hilbert zeta function for curve singularities: the classical version, applicable to singularities with a monomial valuation semigroup or to singular curves defined by…

We show how the "finite Quot scheme method" applied to Le Potier's strange duality on del Pezzo surfaces leads to conjectures (valid for all smooth complex projective surfaces) relating two sets of universal power series on Hilbert schemes…

Given two sequences $\fa=(a_n)_{n\geq 0}$ and $\fb=(b_n)_{n\geq 0}$ of complex numbers such that their generating series are of the form $\sum_{n\geq 0}a_n t^n=\frac{\fh(\fa)(t)}{(1-t)^{d_{\fa}}}$ and $\sum_{n\geq 0}b_n…

In this paper, we give an extension of the classical story of the elliptic modular function to the Hilbert modular case for $\mathbb{Q}(\sqrt{5})$. We construct the period mapping for a family $\mathcal{F}=\{S(X,Y)\}$ of $K3 $ surfaces with…

We introduce spectral Hirzebruch-Milnor classes for singular hypersurfaces. These can be identified with Steenbrink spectra in the isolated singularity case, and may be viewed as their global analogues in general. Their definition uses…

Let M be a moduli space of stable sheaves on a K3 or Abelian surface S. We express the class of the diagonal in the cartesian square of M in terms of the Chern classes of a universal sheaf. Consequently, we obtain generators of the…

An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers $S$ for the reproducing kernel Hilbert space ${\mathcal H}(k_{d})$ on the unit ball ${\mathbb…

We give explicit formulae for the generators of H^2(Hol(\Sigma_r,{\cal F}_{\lambda}(\Sigma_r)) in terms of affine and projective connections. This is done using the cocycles of V. Ovsienko and C. Roger for the case of the circle and…

For every positive integer $N$ we determine the Enriques--Kodaira type of the Humbert surface of discriminant $N^2$ which parametrises principally polarised abelian surfaces that are $(N,N)$-isogenous to a product of elliptic curves. A key…

We study tautological vector bundles over the Hilbert scheme of points on surfaces. For each K-trivial surface, we write down a simple criterion ensuring that the tautological bundles are big and nef, and illustrate it by examples. In the…

In this paper we give a formula for the classes (in the Grothendieck ring of complex quasi-projective varieties) of irreducible components of $(1,k)$-quasi-homogeneous Hilbert schemes of points on the plane. We find a new simple geometric…

We construct a new autoequivalence of the derived category of the Hilbert scheme of n points on a K3 surface, and of the variety of lines on a smooth cubic 4-fold. For Hilb^2 and the variety of lines, we use the theory of spherical…

The Severi variety $V_{d,n}$ of plane curves of a given degree $d$ and exactly $n$ nodes admits a map to the Hilbert scheme $\mathbb{P}^{2[n]}$ of zero-dimensional subschemes of $\mathbb{P}^2$ of degree $n$. This map assigns to every curve…

The Beauville-Voisin conjecture for a hyperk\"ahler manifold X states that the subring of the Chow ring A^*(X) generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of X. We prove a weak…

Here we investigate meaningful families of vector bundles on a very general polarized $K3$ surface $(X,H)$ and on the corresponding Hyper--Kaehler variety given by the Hilbert scheme of points $X^{[k]}:= {\rm Hilb}^k(X)$, for any integer $k…

K-theoretic Donaldson invariants are holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 sheaves on surfaces. We develop an algorithm which determines the generating functions of K-theoretic Donaldson…

Given a finite, simple, vertex-weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to…

We propose a variation of the classical Hilbert scheme of points - the double nested Hilbert scheme of points - which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth…

Let $E$and $F$ be Hermitian vector bundles over a complex manifold $X$ and let $g\colon E\to F$ be a holomorphic morphism. We prove a Poincar\'e-Lelong type formula with a residue term $M^g$. The currents $M^g$ so obtained have an expected…