26 research outputs found
Enumeration of diagonally colored Young diagrams
In this note we give a new proof of a closed formula for the multivariable
generating series of diagonally colored Young diagrams. This series also
describes the Euler characteristics of certain Nakajima quiver varieties. Our
proof is a direct combinatorial argument, based on Andrews' work on generalized
Frobenius partitions. We also obtain representations of these series in some
particular cases as infinite products.Comment: Final version, 12 pages. To appear in Monatshefte f\"ur Mathemati
Recollement from DG quotients
We review a recollement for derived categories of DG categories arising from
Drinfeld quotients. As an application we show that an exact sequence well-known
in K-theory descends to numerical K-groups provided that either the quotient or
the category we take the quotient with has a Serre functor, and if either the
quotient functor preserves compactness or if the K-group of the quotient is
torsion-free.Comment: 7 pages, comments are welcome. arXiv admin note: text overlap with
arXiv:2105.1333
Slicing up multigraded linear series
Multigraded linear series generalize the classical morphism to the linear
series of a basepoint-free line bundle on a scheme. We investigate the
collection of the natural cornering morphisms into elementary bigraded linear
series obtained from direct summands of the original globally generated vector
bundle. Our main result is a condition on the injectivity of the product
morphism. We apply our result in three examples: modules over the
reconstruction algebra, equivariant Hilbert and Quot schemes of quotient stacks
and Kapranov's tilting bundle over the Grassmannian.Comment: 11 pages. Comments are welcom
Euler characteristics of Hilbert schemes of points on simple surface singularities
We study the geometry and topology of Hilbert schemes of points on the
orbifold surface [C^2/G], respectively the singular quotient surface C^2/G,
where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition
of the (equivariant) Hilbert scheme of the orbifold into affine space strata
indexed by a certain combinatorial set, the set of Young walls. The generating
series of Euler characteristics of Hilbert schemes of points of the singular
surface of type A or D is computed in terms of an explicit formula involving a
specialized character of the basic representation of the corresponding affine
Lie algebra; we conjecture that the same result holds also in type E. Our
results are consistent with known results in type A, and are new for type D.Comment: 57 pages, final version. To appear in European Journal of Mathematic
-fixed Hilbert schemes on surfaces, modular forms, and eta products
Let be a complex surface with an effective action of a group
which preserves the holomorphic symplectic form. Let
be the generating function for the Euler characteristics of the Hilbert schemes
of -invariant length subschemes. We show that its reciprocal,
is the Fourier expansion of a modular cusp form of weight
for the congruence subgroup . We give an
explicit formula for in terms of the Dedekind eta function for all 82
possible . The key intermediate result we prove is of independent
interest: it establishes an eta product identity for a certain shifted theta
function of the root lattice of a simply laced root system. We extend our
results to various refinements of the Euler characteristic, namely the Elliptic
genus, the Chi- genus, and the motivic class.Comment: Minor edits and reference update
Rigid ideal shaves and modular forms
Let X be a complex smooth quasi-projective surface acted upon by a finite group G such that the quotient X/G has singularities only of ADE type. We obtain an explicit expression for the generating series of the Euler characteristics of the zero-dimensional components in the moduli space of zero-dimensional subschemes of X invariant under the action of G. We show that this generating series (up to a suitable rational power of the formal variable) is a holomorphic modular form