Elizabeth Gasparim

Holomorphic bundles on O(− k) are algebraic

We show that holomorphic bundles on O(−k) for k>0 are algebraic. We also show that holomorphic bundles on O(−1) are trivial outside the zero section. A corollary is that bundles on the blow-up of a surface at a point are trivial on a neighborhood of the exceptional divisor minus the exceptional divisor.

Chern classes of bundles on blown-up surfaces

Consider the blow upπ [Xtilde]→ Xof a complex surface Xat a point. Let Ẽ be a holomorphic bundle on [Xtilde]whose restriction to the exceptional divisor is O(j)n ⊗ O(-j), and define E+(π* E*)vv. Friedman and Morgan gave the following bounds for the second Chern classes j≤ c 2(Ẽ)- c 2(E) ≤ j 2We show that these bounds are sharp.

Numerical invariants for bundles on blow-ups

We suggest an effective procedure to calculate numerical invariants for rank two bundles over blown-up surfaces. We study the moduli spaces M j of rank two bundles on the blown-up plane splitting over the exceptional divisor as O(j)○+O(-j). We use the numerical invariants to give a topological decomposition of M j .

Vector Bundles Near Negative Curves: Moduli and Local Euler Characteristic

We study moduli of vector bundles on a two-dimensional neighbourhood Z k of an irreducible curve ℓ ≅ ℙ1 with ℓ2 = −k and give an explicit construction of their moduli stacks. For the case of instanton bundles, we stratify the stacks and construct moduli spaces. We give sharp bounds for the local holomorphic Euler characteristic of bundles on Z k and prove existence of families of bundles with prescribed numerical invariants. Our numerical calculations are performed using a Macaulay 2 algorithm, which is available for download at http://www.maths.ed.ac.uk/~s0571100/Instanton/.

Local moduli of holomorphic bundles

We study moduli of holomorphic vector bundles on non-compact varieties. We discuss filtrability and algebraicity of bundles and calculate dimensions of local moduli. As particularly interesting examples, we describe numerical invariants of bundles on some local Calabi�Yau threefolds.

Computing instanton numbers of curve singularities

We present an algorithm for computing instanton numbers of curve singularities. A comparison is made between these and some other invariants of curve singularities. The algorithm is implemented in Macaulay2, and can be downloaded from http://www.math.nmsu.edu/~iswanson/instanton.m2 or from http://emmy.nmsu.edu/~gasparim/m2code.

Adjoint Orbits of Semi-Simple Lie Groups and Lagrangian Submanifolds

We give various realizations of the adjoint orbits of a semi-simple Lie group and describe their symplectic geometry. We then use these realizations to identify a family of Lagrangean submanifolds of the orbits.

Moduli Stacks of Bundles on Local Surfaces

We give an explicit groupoid presentation of certain stacks of vector bundles on formal neighborhoods of rational curves inside algebraic surfaces. The presentation involves a Mobius type action of an automorphism group on a space of extensions.

Three Applications of Instanton Numbers

We use instanton numbers to: (i) stratify moduli of vector bundles, (ii) calculate relative homology of moduli spaces and (iii) distinguish curve singularities.

Classical deformations of noncompact surfaces and their moduli of instantons

Abstract We describe semiuniversal deformation spaces for the noncompact surfaces Z k : = Tot ( O P 1 ( − k ) ) and prove that any nontrivial deformation Z k ( τ ) of Z k is affine. It is known that the moduli spaces of instantons of charge j on Z k are quasi-projective varieties of dimension 2 j − k − 2 . In contrast, our results imply that the moduli spaces of instantons on any nontrivial deformation Z k ( τ ) are empty.