Scattering diagrams, stability conditions, and coherent sheaves on ℙ²

<p>We show that a purely algebraic structure, a two-dimensional scattering diagram, describes a large part of the wall-crossing behavior of moduli spaces of Bridgeland semistable objects in the derived category of coherent sheaves on <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P squared">
  <mml:semantics>
    <mml:msup>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mi mathvariant="double-struck">P</mml:mi>
      </mml:mrow>
      <mml:mn>2</mml:mn>
    </mml:msup>
    <mml:annotation encoding="application/x-tex">\mathbb {P}^2</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>. This gives a new algorithm computing the Hodge numbers of the intersection cohomology of the classical moduli spaces of Gieseker semistable sheaves on <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P squared">
  <mml:semantics>
    <mml:msup>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mi mathvariant="double-struck">P</mml:mi>
      </mml:mrow>
      <mml:mn>2</mml:mn>
    </mml:msup>
    <mml:annotation encoding="application/x-tex">\mathbb {P}^2</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>, or equivalently the refined Donaldson-Thomas invariants for compactly supported sheaves on local <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P squared">
  <mml:semantics>
    <mml:msup>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mi mathvariant="double-struck">P</mml:mi>
      </mml:mrow>
      <mml:mn>2</mml:mn>
    </mml:msup>
    <mml:annotation encoding="application/x-tex">\mathbb {P}^2</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>.</p>

<p>As applications, we prove that the intersection cohomology of moduli spaces of Gieseker semistable sheaves on <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P squared">
  <mml:semantics>
    <mml:msup>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mi mathvariant="double-struck">P</mml:mi>
      </mml:mrow>
      <mml:mn>2</mml:mn>
    </mml:msup>
    <mml:annotation encoding="application/x-tex">\mathbb {P}^2</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula> is Hodge-Tate, and we give the first non-trivial numerical checks of the general <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi">
  <mml:semantics>
    <mml:mi>χ<!-- χ --></mml:mi>
    <mml:annotation encoding="application/x-tex">\chi</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>-independence conjecture for refined Donaldson-Thomas invariants of one-dimensional sheaves on local <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P squared">
  <mml:semantics>
    <mml:msup>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mi mathvariant="double-struck">P</mml:mi>
      </mml:mrow>
      <mml:mn>2</mml:mn>
    </mml:msup>
    <mml:annotation encoding="application/x-tex">\mathbb {P}^2</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>.</p>