Poincarétype

Poincaré type refers to a class of complete metrics used in complex geometry and geometric analysis on non-compact spaces, named for their cusp-like ends modeled on the Poincaré metric. The standard setting is a quasi-projective variety X = X̄ \ D, where X̄ is a compact complex manifold and D is a divisor with normal crossings. A Kähler metric on X is said to be of Poincaré type if, near each component of D, it is asymptotically like the product of Poincaré metrics in the transverse directions to D. In local coordinates, this means the metric grows like a constant multiple of dz d z̄ divided by |z|^2 log^2|z|^2 in the directions transverse to the divisor, producing cusp-like ends with finite volume.

The Poincaré type condition captures the geometry of the ends of X and provides a natural analytic framework for studying differential equations and cohomology on X. It is particularly important in the theory of complete Kähler–Einstein metrics with negative Ricci curvature on non-compact varieties. The existence of such metrics often relies on extending methods from the Calabi–Yau program to the non-compact setting, yielding metrics that are complete, have controlled curvature, and reflect the singularities of the divisor D.

Applications of Poincaré type metrics span several areas, including L2-cohomology, moduli problems, and the study of automorphic forms on non-compact spaces. The concept generalizes the classical Poincaré metric on the punctured disk to higher dimensions and more intricate divisor configurations, thereby providing a versatile tool in complex differential geometry. The term itself traces to cusp ends familiar from modular curves and the original Poincaré metric.