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Numerical inequalities for quasi-projective surfaces

Rita Pardini, Sofia Tirabassi

Abstract

Let $V$ be a smooth quasi-projective complex surface with compactification $(X,D)$ and set $\overline P_1(V):=h^0(X,K_X+D)$, $\overline q(V):=h^0(X,Ω^1_X(\log D))$. We prove that $\overline P_1(V)\ge \overline q(V)-1$ if $V$ has maximal Albanese dimension and $\overline P_1(V)\ge\frac 16( \overline q(V)-5)$ otherwise. Both bounds are sharp.

Numerical inequalities for quasi-projective surfaces

Abstract

Let be a smooth quasi-projective complex surface with compactification and set , . We prove that if has maximal Albanese dimension and otherwise. Both bounds are sharp.

Paper Structure

This paper contains 10 sections, 7 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Logarithmic invariants
  4. (Quasi)-Albanese varieties
  5. Irrational pencils
  6. WWPB equivalence
  7. The inequality ${\overline P}_1\ge \overline q-1$ for surfaces of maximal Albanese dimension
  8. Surfaces of maximal Albanese dimension and log general type with ${\overline P}_1(V)=1$, $\overline q(V)=2$
  9. $w$-fibers
  10. The inequality $\overline P_1\ge\frac{1}{6}( \overline q-5)$ for surfaces of Albanese dimension 1

Key Result

Theorem 2.1

Let $f \colon V \rightarrow W$ be a dominant morphism of complex algebraic varieties. If the general fiber of $f$ is an irreducible curve $F$, then we have the following inequality for logarithmic Kodaira dimensions:

Theorems & Definitions (21)

  • Theorem 2.1: Kawamata subadditivity formula
  • Theorem 2.2: ingrid, Thm. 1.1
  • Example 2.3
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Example 3.5
  • Example 3.6: $X$ abelian
  • ...and 11 more