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We formalize the concepts of holomorphic affine and projective structures along the leaves of holomorphic foliations by curves on complex manifolds. We show that many foliations admit such structures, we provide local normal forms for them at singular points of the foliation, and we prove some index formulae in the case where the ambient manifold is compact. As a consequence of these, we establish that a regular foliation of general type on a compact algebraic manifold of even dimension does not admit a foliated projective structure. Finally, we classify foliated affine and projective structures along regular foliations on compact complex surfaces.

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Significant investment in new capacities for experimental research at high temperatures and pressures have provided new levels of understanding about the physical properties of carbon in fluids and melts, including its viscosity, electrical conductivity, and density. This chapter reviews the physical properties of carbon-bearing melts and fluids at high temperatures and pressures and highlights remaining unknowns left to be explored. The chapter also reviews how the remote sensing of the inaccessible parts of the Earth via various geophysical techniques – seismic shear wave velocity, attenuation, and electromagnetic signals of mantle depths – can be reconciled with the potential presence of carbon-bearing melts or fluids.

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Foliated affine and projective structures

7 - The Link between the Physical and Chemical Properties of Carbon-Bearing Melts and Their Application for Geophysical Imaging of Earth’s Mantle

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Foliated affine and projective structures

7 - The Link between the Physical and Chemical Properties of Carbon-Bearing Melts and Their Application for Geophysical Imaging of Earth’s Mantle

Summary