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2 results

  • 8 - Constant-Degree Polynomial Partitioning and Incidences in C2

  • Adam Sheffer, Bernard M. Baruch College, City University of New York
  • Book:
    Polynomial Methods and Incidence Theory
    Published online:
    17 March 2022
    Print publication:
    24 March 2022, pp 108-124

  • Summary

    It is usually easier to study problems over the complex than over the reals. Discrete geometry problems are an exception, often being significantly simpler over the reals. While there are several simple proofs of the Szemerédi–Trotter theorem over the reals, we only have rather involved proofs for the complex variant of the theorem. To avoid such involved proofs, we prove a slightly weaker variant of the complex Szemerédi–Trotter theorem. Our analysis is based on thinking of C^2 as R^4.

    In Chapter 7, we began to prove the distinct distances theorem by studying the ESGK framework. We complete this proof in Chapter 9, by relying on the constant-degree polynomial partitioning technique. In the current chapter we introduce this technique by studying incidences with lines in the complex plane. This is a warm-up towards Chapter 9, where we use constant-degree polynomial partitioning in more involved ways.

  • Rank 3 rigid representations of projective fundamental groups

  • Part of
  • Adrian Langer, Carlos Simpson
  • Journal:
    Compositio Mathematica / Volume 154 / Issue 7 / July 2018
    Published online by Cambridge University Press:
    30 May 2018, pp. 1534-1570
    Print publication:
    July 2018
    • Article
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  • Let $X$ be a smooth complex projective variety with basepoint $x$ . We prove that every rigid integral irreducible representation $\unicode[STIX]{x1D70B}_{1}(X\!,x)\rightarrow \operatorname{SL}(3,\mathbb{C})$ is of geometric origin, i.e., it comes from some family of smooth projective varieties. This partially generalizes an earlier result by Corlette and the second author in the rank 2 case and answers one of their questions.