synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
A complex manifold is a manifold holomorphically modeled on polydiscs in (complexified -dimensional cartesian space):
a smooth manifold locally isomorphic to whose transition functions are holomorphic functions;
equivalently: a smooth manifold equipped with an integrable almost complex structure;
equivalently a smooth complex analytic space.
Every complex manifold admits a good open cover in .
For instance (Maddock, lemma 3.2.8).
A complex manifold of complex dimension 1 is called a Riemann surface.
A complex manifold whose canonical bundle is trivializable is a Calabi-Yau manifold. In complex dimension 2 this is a K3 surface.
Textbook accounts:
Claire Voisin, section 2 of Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3
Franc Forstnerič, Section 1.5 in: Stein manifolds and holomorphic mappings – The homotopy principle in complex analysis, Springer 2011 (doi:10.1007/978-3-642-22250-4)
(in the context of the Oka principle)
See also:
With an eye towards application in mathematical physics:
Lectures notes:
Last revised on July 19, 2021 at 09:12:15. See the history of this page for a list of all contributions to it.