complexgeometric

Complexgeometric is a field that studies geometric structures on complex manifolds and complex algebraic varieties. It sits at the crossroads of differential geometry, algebraic geometry, and several complex variables, and emphasizes holomorphic structure, metric properties, and intrinsic invariants such as Chern classes and Hodge numbers.

Key objects in complexgeometric include complex manifolds, complex projective varieties, and line bundles; geometric structures such as Hermitian and Kähler metrics; and curvature notions including Ricci and scalar curvature. Central concerns involve canonical metrics such as Kähler-Einstein and constant scalar curvature Kähler metrics, as well as the existence and uniqueness results tied to them. The field engages with the Calabi conjecture, various geometric flows like the Kähler-Ricci flow, and the study of deformation and moduli theory, Hodge theory, and mirror symmetry. In addition, it intersects with invariants from symplectic and enumerative geometry, including Gromov-Witten invariants.

Techniques in complexgeometric combine partial differential equations, complex analysis, and algebraic methods, employing analytic tools to solve complex Monge-Ampère equations and to study stability conditions. The interplay between differential geometry and algebraic stability yields results such as the link between K-stability and the existence of constant scalar curvature metrics. Complexgeometric methods inform and are informed by theoretical physics, notably string theory, where moduli of complex structures and Calabi-Yau manifolds are central, contributing to ideas like mirror symmetry and enumerative counts of curves.

Origins trace back to classical complex analysis and differential geometry, with foundational work by Riemann, Chern, and Kodaira, and later breakthroughs by Calabi, Yau, and Tian. Today the field continues to grow through cross-disciplinary collaborations and applications in geometry, topology, and mathematical physics.