Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1215_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> be a holomorphic foliation at <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1215_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$p\in \mathbb {C}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, and let B be a separatrix of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1215_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation>. We prove the following upper bound <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1215_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="275" /> </InlineMediaObject> <EquationSource Format="TEX">$GSV_p(\mathcal {F},B)\le 4\tau _p(\mathcal {F},B)-3\mu _p(\mathcal {F},B)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mi>S</mi> <msub> <mi>V</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">F</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mn>4</mn> <msub> <mi>τ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">F</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mn>3</mn> <msub> <mi>μ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">F</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1215_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">$GSV_p(\mathcal {F},B)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mi>S</mi> <msub> <mi>V</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">F</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the Gómez-Mont-Seade-Verjovsky index of the foliation <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1215_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> with respect to B, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1215_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu _p(\mathcal {F},B)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>μ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">F</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the multiplicity of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1215_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> along B and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1215_Article_IEq9.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau _p(\mathcal {F},B)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>τ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">F</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the dimension of the quotient of <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1215_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {C}}\{x,y\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> by the ideal generated by the components of any 1-form defining <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1215_Article_IEq11.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> and any equation of B.

An upper bound for the GSV-index of a foliation

期刊论文
发表日期：

2025年3月22日

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Arturo Fernández-Pérez,
Evelia R. García Barroso,
Nancy Saravia-Molina

摘要

Let $\mathcal {F}$ $F$ be a holomorphic foliation at $p\in \mathbb {C}^2$ $p \in C^{2}$ , and let B be a separatrix of $\mathcal {F}$ $F$ . We prove the following upper bound $GSV_p(\mathcal {F},B)\le 4\tau _p(\mathcal {F},B)-3\mu _p(\mathcal {F},B)$ $G S V_{p} (F, B) \leq 4 τ_{p} (F, B) - 3 μ_{p} (F, B)$ , where $GSV_p(\mathcal {F},B)$ $G S V_{p} (F, B)$ is the Gómez-Mont-Seade-Verjovsky index of the foliation $\mathcal {F}$ $F$ with respect to B, $\mu _p(\mathcal {F},B)$ $μ_{p} (F, B)$ is the multiplicity of $\mathcal {F}$ $F$ along B and $\tau _p(\mathcal {F},B)$ $τ_{p} (F, B)$ is the dimension of the quotient of ${\mathbb {C}}\{x,y\}$ $C {x, y}$ by the ideal generated by the components of any 1-form defining $\mathcal {F}$ $F$ and any equation of B.