<p>We derive a canonical symmetry reduction associated to a compact non-Kähler Bismut-Hermitian-Einstein manifold. In real dimension 6, the transverse geometry is conformally Kähler, and we give a complete description in terms of a single scalar PDE for the underlying Kähler structure. In the case when the soliton potential is constant, we show that that the Bott–Chern number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$h^{1,1}_{BC} \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>h</mi> <mrow> <mi mathvariant="italic">BC</mi> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msubsup> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and that equality holds if and only if the metric is Bismut-flat, and hence a quotient of either <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${{\,\textrm{SU}\,}}(2) \times {\mathbb {R}} \times {\mathbb {C}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mspace width="0.166667em" /> <mtext>SU</mtext> <mspace width="0.166667em" /> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mi mathvariant="double-struck">R</mi> <mo>×</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${{\,\textrm{SU}\,}}(2) \times {{\,\textrm{SU}\,}}(2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mspace width="0.166667em" /> <mtext>SU</mtext> <mspace width="0.166667em" /> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mrow> <mspace width="0.166667em" /> <mtext>SU</mtext> <mspace width="0.166667em" /> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Rigidity results for non-Kähler Calabi–Yau geometries on threefolds

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2025年10月24日

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Vestislav Apostolov,
Giuseppe Barbaro,
Kuan-Hui Lee,
Jeffrey Streets

摘要

We derive a canonical symmetry reduction associated to a compact non-Kähler Bismut-Hermitian-Einstein manifold. In real dimension 6, the transverse geometry is conformally Kähler, and we give a complete description in terms of a single scalar PDE for the underlying Kähler structure. In the case when the soliton potential is constant, we show that that the Bott–Chern number $h^{1,1}_{BC} \ge 2$ $h_{BC}^{1, 1} \geq 2$ , and that equality holds if and only if the metric is Bismut-flat, and hence a quotient of either ${{\,\textrm{SU}\,}}(2) \times {\mathbb {R}} \times {\mathbb {C}}$ $SU (2) \times R \times C$ or ${{\,\textrm{SU}\,}}(2) \times {{\,\textrm{SU}\,}}(2)$ $SU (2) \times SU (2)$ .