scholarly journals Atiyah’s work on holomorphic vector bundles and gauge theories

2021 ◽  
Vol 58 (4) ◽  
pp. 567-610
Author(s):  
Simon Donaldson
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Vector Bundles ◽  
Gauge Theories ◽  
Holomorphic Vector Bundles ◽  
And Topology ◽  
The 1950S

The first part of the article surveys Atiyah’s work in algebraic geometry during the 1950s, mainly on holomorphic vector bundles over curves. In the second part we discuss his work from the late 1970s on mathematical aspects of gauge theories, involving differential geometry, algebraic geometry, and topology.

scholarly journals Vector-valued modular forms and the modular orbifold of elliptic curves

2016 ◽  
Vol 13 (01) ◽  
pp. 39-63 ◽  
Author(s):  
Luca Candelori ◽  
Cameron Franc
Keyword(s):  
Algebraic Geometry ◽  
Elliptic Curves ◽  
Modular Forms ◽  
Vector Bundles ◽  
Projective Line ◽  
Free Module ◽  
Weighted Projective Line ◽  
Holomorphic Vector Bundles ◽  
Vector Valued ◽  
Standard Techniques

This paper presents the theory of holomorphic vector-valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are vector-valued modular forms. This perspective simplifies the theory, and it clarifies the role that exponents of representations of [Formula: see text] play in the holomorphic theory of vector-valued modular forms. Further, it allows one to use standard techniques in algebraic geometry to deduce free-module theorems and dimension formulae (deduced previously by other authors using different techniques), by identifying the modular orbifold with the weighted projective line [Formula: see text].

Hermitian–Einstein metrics and jumping lines forSp(n+1)-homogeneous bundles over ℙ2n+1

1993 ◽  
Vol 114 (3) ◽  
pp. 443-451
Author(s):  
Al Vitter
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Vector Bundles ◽  
Einstein Metrics ◽  
Einstein Metric ◽  
Holomorphic Vector Bundles ◽  
Kähler Form ◽  
Points Of View ◽  
The Stability ◽  
Complex Projective

Stable holomorphic vector bundles over complex projective space ℙnhave been studied from both the differential-geometric and the algebraic-geometric points of view.On the differential-geometric side, the stability ofE-→ ℙncan be characterized by the existence of a unique hermitian–Einstein metric onE, i.e. a metric whose curvature matrix has trace-free part orthogonal to the Fubini–Study Kähler form of ℙn(see [6], [7], and [13]). Very little is known about this metric in general and the only explicit examples are the metrics on the tangent bundle of ℙnand the nullcorrelation bundle (see [9] and [10]).

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