We study moduli of vector bundles on a two-dimensional neighbourhood Z k of an irreducible curve ℓ ∼ = P 1 with ℓ 2 = −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/.

We study the local holomorphic Euler characteristic χ x, F of sheaves near a surface singularity obtained from contracting a line ℓ inside a smooth surface Z. We prove non-existence of sheaves with certain prescribed numerical invariants. Non-existence of instantons on Z with certain charges follows, and we conclude that ℓ 2 poses an obstruction to instanton decay. A Macaulay 2 algorithm to compute χ is made available at Proposition 4.1. Let E 1 and E 2 be sl(2, C)-bundles over Z k with splitting types j 1 and j 2 , respectively. There exists an isomorphismIn particular, E 1 can decay totally over Z k if and only if j 1 ≡ 0 mod k.This paper consists of applications of the local holomorphic Euler characteristic to problems of existence and decay of instantons. We also discuss the Kobayashi-Hitchin correspondence over Z k . We obtain, via discussion of the physical consequences and an ad hoc definition of stability (Definition 5.2), the following conclusions:Proposition 5.4. There is a one-to-one correspondence between framed SU (2)-instantons on Z k with local charge n and framed-stable sl(2, C)-bundles on Z k with χ loc = n.Corollary 5.5. An sl(2, C)-bundle over Z k represents an instanton if and only if its splitting type is a multiple of k.Theorem 6.8. The minimal local charge of a nontrivial SU (2)-instanton on Z k is χ min k = k−1. The local moduli space of (unframed) instantons on Z k having fixed local charge χ min k has dimension k − 2.

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.Comment: 18 pages. Revision history: v1: As submitted for publication. v2: minor corrections, as publishe

We present DeepMind Lab2D, a scalable environment simulator for artificial intelligence research that facilitates researcher-led experimentation with environment design. DeepMind Lab2D was built with the specific needs of multi-agent deep reinforcement learning researchers in mind, but it may also be useful beyond that particular subfield.

We introduce a new notion of deformation of complex structure, which we use as an adaptation of Kodaira's theory of deformations, but that is better suited to the study of noncompact manifolds. We present several families of deformations illustrating this new approach. Our examples include toric Calabi-Yau threefolds, cotangent bundles of flag manifolds, and semisimple adjoint orbits, and we describe their Hodge theoretical invariants, depicting Hodge diamonds and KKP diamonds.