<p>We prove a suite of results classifying holomorphic maps between configuration spaces of Riemann surfaces; we consider both the ordered and unordered setting as well as the cases of genus zero, one, and at least two. We give a complete classification of all holomorphic maps <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\operatorname {Conf}_n(\mathbb {C})\rightarrow \operatorname {Conf}_m(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>Conf</mo> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msub> <mo>Conf</mo> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> provided that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n\ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$m\le 2n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≤</mo> <mn>2</mn> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> extending the Tameness Theorem of Lin, which is the case <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$m = n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. We also give a complete classification of holomorphic maps between ordered configuration spaces of Riemann surfaces of genus at most one (answering a question of Farb), and show that the higher genus setting is closely linked to the still-mysterious “effective de Franchis problem”. The main technical theme of the paper is that holomorphicity allows one to promote group-theoretic rigidity results to the space level.</p>

Holomorphic maps between configuration spaces of Riemann surfaces

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2026年1月20日

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Lei Chen,
Nick Salter

摘要

We prove a suite of results classifying holomorphic maps between configuration spaces of Riemann surfaces; we consider both the ordered and unordered setting as well as the cases of genus zero, one, and at least two. We give a complete classification of all holomorphic maps $\operatorname {Conf}_n(\mathbb {C})\rightarrow \operatorname {Conf}_m(\mathbb {C})$ ${Conf}_{n} (C) \to {Conf}_{m} (C)$ provided that $n\ge 5$ $n \geq 5$ and $m\le 2n$ $m \leq 2 n$ extending the Tameness Theorem of Lin, which is the case $m = n$ $m = n$ . We also give a complete classification of holomorphic maps between ordered configuration spaces of Riemann surfaces of genus at most one (answering a question of Farb), and show that the higher genus setting is closely linked to the still-mysterious “effective de Franchis problem”. The main technical theme of the paper is that holomorphicity allows one to promote group-theoretic rigidity results to the space level.