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Indecomposable cycles and the case of the twisted Hodge D conjecture

Karim Mansour

Abstract

We will introduce twisted cycles and their associated regulators to cohomology. We prove the conjecture that this regulator is surjective for a general smooth projective surface. We construct indecomposable twisted cycles on elliptic fourfolds, and use that to show that this map is non-zero.

Indecomposable cycles and the case of the twisted Hodge D conjecture

Abstract

We will introduce twisted cycles and their associated regulators to cohomology. We prove the conjecture that this regulator is surjective for a general smooth projective surface. We construct indecomposable twisted cycles on elliptic fourfolds, and use that to show that this map is non-zero.
Paper Structure (10 sections, 30 theorems, 170 equations)

This paper contains 10 sections, 30 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Introduction to flat line bundles and twisted cycles
  3. Some Hodge theory
  4. A Tame symbol
  5. A Milnor complex
  6. Some examples
  7. General smooth projective surfaces
  8. Higher dimension twisted cycles
  9. Hypersurface arrangements
  10. Main result

Key Result

Lemma 1.6

Let us assume given a line bundle $L$ as above on a smooth projective $Z$, with first Chern class $c_1(L) = 0$. Then one can find a corresponding metric for which the curvature form $\nu_L = 0$.

Theorems & Definitions (70)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • Lemma 1.6
  • proof
  • Definition 1.7
  • Remark 1.8
  • Definition 1.9
  • ...and 60 more