arXiv subjects

Zsolt Szilágyi

Publications and source records attributed to Zsolt Szilágyi.

6 recordsLinked to original sources

arXiv2019

On Chern classes of tensor products of vector bundles

We present two formulas for Chern classes of the tensor product of two vector bundles. In the first formula we consider a matrix containing Chern classes of the first bundle and we take a polynomial of this matrix with Chern classes of the second bundle as coefficients. The determinant of this expression equals the Chern polynomial of the tensor product. In the second formula we express the total Chern class of the tensor product as resultant of two explicit polynomials with coefficient involving Chern classes of each vector bundles. This approach leads to determinantal formulas for the total Chern class of the second symmetric and alternating product of a vector bundle.

math.AT↗

arXiv2017

Némethi's division algorithm for zeta-functions of plumbed 3-manifolds

A polynomial counterpart of the Seiberg-Witten invariant associated with a negative definite plumbed 3-manifold has been proposed by earlier work of the authors. It is provided by a special decomposition of the zeta-function defined by the combinatorics of the manifold. In this article we give an algorithm, based on multivariable Euclidean division of the zeta-function, for the explicit calculation of the polynomial, in particular for the Seiberg--Witten invariant.

math.GT↗

arXiv2016

Non-normal affine monoids, modules and Poincaré series of plumbed 3-manifolds

We construct a non-normal affine monoid together with its modules associated with a negative definite plumbed $3$-manifold $M$. In terms of their structure, we describe the $H_1(M,\mathbb{Z})$-equivariant parts of the topological Poincaré series. In particular, we give combinatorial formulas for the Seiberg--Witten invariants of $M$ and for polynomial generalizations defined in a previous paper of the authors.

math.GT↗

arXiv2015

On Poincaré series associated with links of normal surface singularities

We study the counting function of topological Poincaré series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with some Taylor expansions. It is motivated by a theorem of Szenes and Vergne which expresses these coefficient functions in terms of Jeffrey--Kirwan residues. This is used to prove the uniqueness of the quasipolynomiality inside a special cone, the structure of the counting function in terms of the graph and construction for a polynomial generalization of the Seiberg--Witten invariant given by the Poincaré series. We also reprove and discuss surgery formulas of Némethi for the counting function, and of Braun and Némethi for the Seiberg--Witten invariant.

math.GT↗

arXiv2013

Computation of Jeffrey-Kirwan residue using Gröbner basis

The Jeffrey-Kirwan residue is a powerful tool for computation of intersection numbers or volume of symplectic quotients. In this article, we give an algorithm to compute it using Gröbner bases. Our result is parallel to that of Cattani-Dickenstein for Grothendieck residues.

math.AC↗

arXiv2013

Equivariant Jeffrey-Kirwan localization theorem in non-compact setting

We generalize the Jeffrey-Kirwan localization theorem for non-compact symplectic and hyperKahler quotients. Similarly to the circle compact integration of Hausel and Proudfoot we define equivariant integrals on non-compact manifolds using the Atiyah-Bott-Berline-Vergne localization formula as formal definition. We introduce a so called equivariant Jeffrey-Kirwan residue and we show that it shares similar properties as the usual one. Our localization formula has the same structure as the usual Jeffrey-Kirwan formula, but it uses formal integration and equivariant residue. We also give a version for hyperKahler quotients. Finally, we apply our formula to compute the equivariant cohomology ring of Hilbert scheme of points on the plane constructed as a hyperKahler quotient.

math.SG↗