From real affine geometry to complex geometry

From Real Affine Geometry to Complex Geometry Mark Gross and Bernd Siebert

Introduction. 2 1. Fundamentals 8 1.1. Discrete data 8 1.2. Algebraic data 19 1.3. Statement of the Main Theorem 28 2. Main objects of the construction 31 2.1. Exponents, orders, rings 31 2.2. Automorphism groups 39 2.3. Slabs, walls and structures 43 2.4. The gluing morphisms 46 2.5. Loops around joints and consistency 50 2.6. Construction of finite order deformation 53 2.7. The limit k → ∞ 60...

Computational Geometry from the Viewpoint of Affine Differential Geometry

Affine Geometry, Projective Geometry, and Non- Euclidean Geometry

1. Affine Geometry 1.1. Affine Space 1.2. Affine Lines 1.3. Affine transformations 1.4. Affine Collinearity 1.5. Conic Sections 2. Projective Geometry 2.1. Perspective 2.2. Projective Plane 2.3. Projective Transformations 2.4. Projective Collinearity 2.5. Conics 3. Geometries and Groups 3.1. Transformation Groups 3.2. Erlangen Program 4. Non-Euclidean Geometry 4.1. Elliptic Geometry 4.2. Hyperb...

Affine Integral Geometry from a Differentiable Viewpoint

The notion of a homogeneous contour integral is introduced and used to construct affine integral invariants of convex bodies. Earlier descriptions of this construction rely on a Euclidean structure on the ambient vector space, so the behavior under linear transformations is established after the fact. The description presented here relies only on the linear structure and a Lebesgue measure on t...

Real Solutions to Equations from Geometry

Understanding, finding, or even deciding on the existence of real solutions to a system of equations is a very difficult problem with many applications. While it is hopeless to expect much in general, we know a surprising amount about these questions for systems which possess additional structure. Particularly fruitful—both for information on real solutions and for applicability—are systems who...