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Stein degree of proper morphisms

Caucher Birkar

Abstract

The notion of degree begins in field theory as the dimension of a field extension. In algebraic geometry, this idea reappears as the degree of a finite morphism, defined using the induced extension of function fields. For proper morphisms that are not necessarily finite, Stein factorization isolates the finite part of the map and leads to the notion of Stein degree. This invariant is especially useful in birational geometry, where it interacts naturally with singularities of pairs and the study of log Calabi-Yau fibrations. In this article we give an expository introduction to these ideas, discuss motivating examples, and explain a boundedness problem for Stein degree arising in recent work of the author and collaborators.

Stein degree of proper morphisms

Abstract

The notion of degree begins in field theory as the dimension of a field extension. In algebraic geometry, this idea reappears as the degree of a finite morphism, defined using the induced extension of function fields. For proper morphisms that are not necessarily finite, Stein factorization isolates the finite part of the map and leads to the notion of Stein degree. This invariant is especially useful in birational geometry, where it interacts naturally with singularities of pairs and the study of log Calabi-Yau fibrations. In this article we give an expository introduction to these ideas, discuss motivating examples, and explain a boundedness problem for Stein degree arising in recent work of the author and collaborators.
Paper Structure (17 sections, 3 theorems, 81 equations)

This paper contains 17 sections, 3 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Degree of field extensions
  3. Finite morphisms of varieties
  4. Further examples of finite morphisms
  5. Stein factorization
  6. Stein degree
  7. Examples of Stein degree
  8. Singularities of pairs
  9. The connectedness principle
  10. Log Calabi--Yau fibrations and Stein degree
  11. A general boundedness conjecture
  12. Recent progress
  13. Sketch of the proof strategy
  14. Horizontal and vertical cases
  15. From $H(1,t)$ to $V(1,t)$
  16. ...and 2 more sections

Key Result

Theorem 9.1

Let $(X,B)$ be a pair and let $f\colon X\to Y$ be a proper morphism with connected fibers. If is $f$-ample, then $\operatorname{Nklt}(X,B)$ is connected in a neighborhood of each fiber of $f$.

Theorems & Definitions (38)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Example 3.4
  • Example 4.1: Normalization of a cusp
  • ...and 28 more