The metric geometry of singularity types

By Tam'as Darvas, E. Di Nezza, Chinh H. Lu. Journal für die Reine und Angewandte Mathematik, 2019

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<jats:p>Let <jats:italic>X</jats:italic> be a compact Kähler manifold. Given a big cohomology class <jats:inline-formula id="j_crelle-2020-0019_ineq_9999"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mi>θ</m:mi> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0019_eq_0709.png" /> <jats:tex-math>{\{\theta\}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, there is a natural equivalence relation on the space of θ-psh functions giving rise to <jats:inline-formula id="j_crelle-2020-0019_ineq_9998"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒮</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>θ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0019_eq_0556.png" /> <jats:tex-math>{\mathcal{S}(X,\theta)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, the space of singularity types of potentials. We introduce a natural pseudo-metric <jats:inline-formula id="j_crelle-2020-0019_ineq_9997"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>d</m:mi> <m:mi mathvariant="script">𝒮</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0019_eq_0791.png" /> <jats:tex-math>{d_{\mathcal{S}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> on <jats:inline-formula id="j_crelle-2020-0019_ineq_9996"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒮</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>θ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0019_eq_0556.png" /> <jats:tex-math>{\mathcal{S}(X,\theta)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> that is non-degenerate on the space of model singularity types and whose atoms are exactly the relative full mass classes. In the presence of positive mass we show that this metric space is complete. As applications, we show that solutions to a family of complex Monge–Ampère equations with varying singularity type converge as governed by the <jats:inline-formula id="j_crelle-2020-0019_ineq_9995"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>d</m:mi> <m:mi mathvariant="script">𝒮</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0019_eq_0791.png" /> <jats:tex-math>{d_{\mathcal{S}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-topology, and we obtain a semicontinuity result for multiplier ideal sheaves associated to singularity types, extending the scope of previous results from the local context.</jats:p>

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9a1420b470c8f22952c62590976fb0f2bb61775e