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K3 surface - Definition

K3 surface - Definition: Encyclopedia II - K3 surface - Definition

There are many equivalent properties that can be used to characterize a K3 surface. The definition given depends on the context: In differential geometry, a typical definition is of "a compact, complex, simply connected surface with trivial canonical line bundle". In algebraic geometry, the definition "a surface, X, with trivial canonical class such that H1(X,OX) = 0." is preferred since it generalizes to more arbitrary base fields (not just the complex numbers). Here, See also:
K3 surface, K3 surface - Definition, K3 surface - Important Properties, K3 surface - Examples

K3 surface, K3 surface - Definition, K3 surface - Examples, K3 surface - Important Properties

K3 surface: Encyclopedia II - K3 surface - Definition

K3 surface - Definition

There are many equivalent properties that can be used to characterize a K3 surface. The definition given depends on the context:

In differential geometry, a typical definition is of "a compact, complex, simply connected surface with trivial canonical line bundle".

In algebraic geometry, the definition "a surface, X, with trivial canonical class such that H¹(X,O_X) = 0." is preferred since it generalizes to more arbitrary base fields (not just the complex numbers). Here, H¹(X,O_X) denotes the first sheaf cohomology group of O_X, the sheaf of regular functions on X.

Another characterization, sometimes found in physics literature, is that "a K3 manifold is a hyperkähler manifold of real dimension 4 with SU(2) holonomy".