<p>In this paper, the singularity structure in moduli spaces of <i>G</i>-Higgs bundles arising from degenerations of Riemann surfaces to nodal curves is examined. Specifically, for a complex reductive Lie group <i>G</i> and a standard degeneration <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\pi : {\mathcal {X}} \rightarrow \Delta \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo>:</mo> <mi mathvariant="script">X</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">Δ</mi> </mrow> </math></EquationSource> </InlineEquation> with central fiber <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(X_0 = Y_1 \cup Y_2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>Y</mi> <mn>1</mn> </msub> <mo>∪</mo> <msub> <mi>Y</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, we prove the existence of a flat family of schemes <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {M}_G \rightarrow \Delta \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mi>G</mi> </msub> <mo stretchy="false">→</mo> <mi mathvariant="normal">Δ</mi> </mrow> </math></EquationSource> </InlineEquation> whose general fiber is the moduli space of semistable <i>G</i>-Higgs bundles. The central fiber <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal {M}_G(0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> admits a stratification indexed by conjugacy classes of parabolic subgroups <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(P \subset G\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo>⊂</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation>, where each stratum <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathcal {M}_G^P(0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="script">M</mi> <mi>G</mi> <mi>P</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is isomorphic to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathcal {M}_L(Y_1, p) \times _{H^0(p, \mathfrak {l}/\mathfrak {z})} \mathcal {M}_L(Y_2, p)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mi>L</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mo>×</mo> <mrow> <msup> <mi>H</mi> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi mathvariant="fraktur">l</mi> <mo stretchy="false">/</mo> <mi mathvariant="fraktur">z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <msub> <mi mathvariant="script">M</mi> <mi>L</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for the Levi factor <i>L</i> of <i>P</i>. We demonstrate that the formal neighborhood of a stable point in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\mathcal {M}_G^P(0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="script">M</mi> <mi>G</mi> <mi>P</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is analytically equivalent to the cone on <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(G/P \times G/P^{\textrm{op}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">/</mo> <mi>P</mi> <mo>×</mo> <mi>G</mi> <mo stretchy="false">/</mo> <msup> <mi>P</mi> <mtext>op</mtext> </msup> </mrow> </math></EquationSource> </InlineEquation> with its Plücker embedding, and construct a canonical resolution <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\tilde{\mathcal {M}}_G(0) \rightarrow \mathcal {M}_G(0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover accent="true"> <mi mathvariant="script">M</mi> <mo stretchy="false">~</mo> </mover> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msub> <mi mathvariant="script">M</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> whose exceptional locus over <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(\mathcal {M}_G^P(0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="script">M</mi> <mi>G</mi> <mi>P</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(\mathbb {P}^{\dim (G/P)-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mrow> <mo>dim</mo> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">/</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-bundle. Furthermore, we establish that <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(\mathcal {M}_G(0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> carries a natural Poisson structure whose symplectic leaves are precisely the strata <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(\mathcal {M}_G^P(0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="script">M</mi> <mi>G</mi> <mi>P</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Finally, as an application, we derive explicit formulas for Gromov-Witten invariants of associated symplectic fibrations in terms of relative invariants on <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\(Y_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Y</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq16"> <EquationSource Format="TEX">\(Y_2.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Y</mi> <mn>2</mn> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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Singularities and canonical resolutions of moduli spaces of G-Higgs bundles on nodal curves

  • Álvaro Antón-Sancho

摘要

In this paper, the singularity structure in moduli spaces of G-Higgs bundles arising from degenerations of Riemann surfaces to nodal curves is examined. Specifically, for a complex reductive Lie group G and a standard degeneration \(\pi : {\mathcal {X}} \rightarrow \Delta \) π : X Δ with central fiber \(X_0 = Y_1 \cup Y_2\) X 0 = Y 1 Y 2 , we prove the existence of a flat family of schemes \(\mathcal {M}_G \rightarrow \Delta \) M G Δ whose general fiber is the moduli space of semistable G-Higgs bundles. The central fiber \(\mathcal {M}_G(0)\) M G ( 0 ) admits a stratification indexed by conjugacy classes of parabolic subgroups \(P \subset G\) P G , where each stratum \(\mathcal {M}_G^P(0)\) M G P ( 0 ) is isomorphic to \(\mathcal {M}_L(Y_1, p) \times _{H^0(p, \mathfrak {l}/\mathfrak {z})} \mathcal {M}_L(Y_2, p)\) M L ( Y 1 , p ) × H 0 ( p , l / z ) M L ( Y 2 , p ) for the Levi factor L of P. We demonstrate that the formal neighborhood of a stable point in \(\mathcal {M}_G^P(0)\) M G P ( 0 ) is analytically equivalent to the cone on \(G/P \times G/P^{\textrm{op}}\) G / P × G / P op with its Plücker embedding, and construct a canonical resolution \(\tilde{\mathcal {M}}_G(0) \rightarrow \mathcal {M}_G(0)\) M ~ G ( 0 ) M G ( 0 ) whose exceptional locus over \(\mathcal {M}_G^P(0)\) M G P ( 0 ) is a \(\mathbb {P}^{\dim (G/P)-1}\) P dim ( G / P ) - 1 -bundle. Furthermore, we establish that \(\mathcal {M}_G(0)\) M G ( 0 ) carries a natural Poisson structure whose symplectic leaves are precisely the strata \(\mathcal {M}_G^P(0)\) M G P ( 0 ) . Finally, as an application, we derive explicit formulas for Gromov-Witten invariants of associated symplectic fibrations in terms of relative invariants on \(Y_1\) Y 1 and \(Y_2.\) Y 2 .