<p>There is a local-to-global <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1075_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{Ext}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>Ext</mtext> </math></EquationSource> </InlineEquation> spectral sequence <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1075_Article_IEq2.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="272" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{E}_{2}^{p,q}= \textrm{H}^{p}(\textrm{L},\Omega ^{q}_\textrm{L}) \Rightarrow \textrm{Ext}_{\mathscr {O}_{\textrm{X}}}^{p+q}(i_{}\mathscr {L},i_{}\mathscr {L})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mtext>E</mtext> <mrow> <mn>2</mn> </mrow> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msubsup> <mo>=</mo> <msup> <mtext>H</mtext> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mtext>L</mtext> <mo>,</mo> <msubsup> <mi mathvariant="normal">Ω</mi> <mtext>L</mtext> <mi>q</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">⇒</mo> <msubsup> <mtext>Ext</mtext> <mrow> <msub> <mi mathvariant="script">O</mi> <mtext>X</mtext> </msub> </mrow> <mrow> <mi>p</mi> <mo>+</mo> <mi>q</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>i</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> </msub> <mi mathvariant="script">L</mi> <mo>,</mo> <msub> <mi>i</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> </msub> <mi mathvariant="script">L</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for a line bundle <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1075_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathscr {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation> supported on a smooth Lagrangian subvariety <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1075_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{L}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>L</mtext> </math></EquationSource> </InlineEquation> in a hyperkähler variety <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1075_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{X}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>X</mtext> </math></EquationSource> </InlineEquation>. We prove its degeneration on <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1075_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{E}_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>E</mtext> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> in special cases and various generalisations thereof.</p>

Degeneration of spectral sequences and complex Lagrangian submanifolds

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2025年8月11日

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Borislav Mladenov

摘要

There is a local-to-global $\textrm{Ext}$ $Ext$ spectral sequence $\textrm{E}_{2}^{p,q}= \textrm{H}^{p}(\textrm{L},\Omega ^{q}_\textrm{L}) \Rightarrow \textrm{Ext}_{\mathscr {O}_{\textrm{X}}}^{p+q}(i_{*}\mathscr {L},i_{*}\mathscr {L})$ $E_{2}^{p, q} = H^{p} (L, Ω_{L}^{q}) \Rightarrow {Ext}_{O_{X}}^{p + q} (i_{*} L, i_{*} L)$ for a line bundle $\mathscr {L}$ $L$ supported on a smooth Lagrangian subvariety $\textrm{L}$ $L$ in a hyperkähler variety $\textrm{X}$ $X$ . We prove its degeneration on $\textrm{E}_{2}$ $E_{2}$ in special cases and various generalisations thereof.