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Andrei Moroianu works in differential geometry, including Kähler geometry and geometric differential operators. His authored lectures develop the interaction of a Riemannian metric, a complex structure $J$, and the associated Kähler form $\omega (X,Y)=g(JX,Y)$. For a Kähler manifold, the Levi-Civita connection preserves $J$, linking the real and complex descriptions of curvature.^[ref]

A concrete consequence is that the Ricci form represents the tangent First Chern class; prescribing Ricci curvature therefore has a topological constraint, not just a differential equation. His lectures explain the Kähler-Einstein metric equation and illustrate it on Complex projective space and Ricci-flat examples.^[ref] His personal page identifies him as a CNRS research director.^[ref]

Kähler geometry studies complex manifolds with a Hermitian metric $g$ whose associated real two-form $\omega (X,Y)=g(JX,Y)$ is closed. Here $J$ is the complex structure. Equivalently, the Levi-Civita connection preserves $J$. Thus the same metric simultaneously supports complex geometry, Riemannian geometry and symplectic geometry.^[ref]

On ${\mathbb{C}}^n$, the Euclidean metric gives $\omega ={\sum }_{j}d{x}_j\wedge d{y}_j$, which is closed and nondegenerate. Locally one can express a Kähler form as $\omega =i\partial \bar{\partial }\phi$ for a real Kähler potential $\phi$, using a stated normalization. On Complex projective space, $\phi =\log (1+{\sum }_j|z_j{|}^2)$ gives the Fubini-Study metric in an affine chart. The potentials need not glue as one global function; their forms glue. This is how locally elementary differentiation can define nontrivial global geometry.