Lesson plan

Positive current

In mathematics, more particularly in complex geometry, algebraic geometry and complex analysis, a positive current is a positive (n-p,n-p)-form over an n-dimensional complex manifold, taking values in distributions. For a formal definition, consider a manifold M. Currents on M are (by definition) differential forms with coefficients in distributions; integrating over M, we may consider currents as "currents of integration", that is, functionals : \eta \mapsto \int_M \eta\wedge \rho on smooth forms with compact support. This way, currents are considered as elements in the dual space to the space \Lambda_c^*(M) of forms with compact support. Now, let M be a complex manifold. The Hodge decomposition \Lambda^i(M)=\bigoplus_{p+q=i}\Lambda^{p,q}(M) is defined on currents, in a natural way, the (p,q)-currents being functionals on \Lambda_c^{p, q}(M) . A positive current is defined as a real current of Hodge type (p,p), taking non-negative values on all positive (p,p)-forms.

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Article summary

In mathematics, more particularly in complex geometry, algebraic geometry and complex analysis, a positive current is a positive (n-p,n-p)-form over an n-dimensional complex manifold, taking values in distributions. For a formal definition, consider a manifold M. Currents on M are (by definition) differential forms with coefficients in distributions; integrating over M, we may consider currents as "currents of integration", that is, functionals : \eta \mapsto \int_M \eta\wedge \rho on smooth forms with compact support. This way, currents are considered as elements in the dual space to the space \Lambda_c^*(M) of forms with compact support. Now, let M be a complex manifold. The Hodge decomposition \Lambda^i(M)=\bigoplus_{p+q=i}\Lambda^{p,q}(M) is defined on currents, in a natural way, the (p,q)-currents being functionals on \Lambda_c^{p, q}(M) . A positive current is defined as a real current of Hodge type (p,p), taking non-negative values on all positive (p,p)-forms.

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