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On the Geometry of a Homological Invariant of a Plane Curve Singularity

Xavier Gomez-Mont

Centro de Investigación en Matemáticas, México   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

A first invariant of an oriented manifold $M$ consists of its (co)homology groups (over $\mathbb{Z},\mathbb{Q}$), that comes provided with a bilinear form $<\ ,\ >$ induced from the intersection of cycles of complementary dimension. If part of $M$ contracts to a point, we obtain an isolated singularity. In case of a hypersurface $\{f=0\}$ in $\mathbb{C}^n$, going around the critical value gives rise to an automorphism $h$ of $M$ called the geometric monodromy, and its action on (co)homology is the algebraic monodromy $h_*$. The algebraic monodromy has a periodic and a non-periodic part. The non-periodic part is codified in $N$: the logarithm of the unipotent part of $h_*$. Hodge Theory distinguishes the bilinear forms $\langle N^j\,,\, \rangle$. The problem we address is to find the geometric content of these bilinear forms in terms of the geometric monodromy $h$.

We will describe for plane curves singularities the geometric content of the symmetric bilinear form $\langle N\ ,\ \rangle$ in terms of Dehn twists of $h$. The Theory of Resolution of Plane Curve Singularities provides a finite number of closed curves $ \{\gamma_j\}$ in the smooth oriented compact surface with boundary $S :=f^{-1}(\varepsilon)$ and positive integers $m$ and $m_j$, attached to each $\gamma_j$, so that $h^m$ is the identity outside of tubular neighbourhoods of $\gamma_j$ and it is a Dehn twist by $m_j$ on a tubular neigbourhood of $\gamma_j$ (an explicit Thurston-Nielsen representative of $h$). We show $$\langle N\alpha,\beta \rangle =\frac{1}{m} \sum_jm_j \langle \alpha ,\gamma_j \rangle \langle \beta ,\gamma_j \rangle \hskip 1cm,\hskip 1cm \alpha,\beta \in H_1(S,\mathbb{Z}).$$ The bilinear form is the sum with weights of the product of the common times that the closed curves $\alpha$ and $\beta$ intersect the curves $\gamma_j$, everything done in 1-homology $H_1(S,\mathbb{Z})$. This bilinear form induces a non-degenerate bilinear form on $W_1:=\frac{H_1(S,\mathbb{Z})}{Ker(N)}$, and the above expression shows that it is positive definite (i.e. Riemann-Hodge positivity). We identify $W_1$ as the 1-homology group of a graph $\tilde \Gamma$ (the graph of the semistable reduction of the singularity), providing explicit generators of the non-periodic part $W_1$ of $H_1(S,\mathbb{Z})$, since in $\tilde \Gamma$ there are no 2-chains (arXiv:2011.12332).

Joint work with Lily Alanis (Universidad Autónoma de Nuevo León, México), Enrique Artal (Universidad de Zaragoza, España), Christian Bonatti (Université de Bourgogne, Dijon, France), Manuel González-Villa (CIMAT, México) and Pablo Portilla (CIMAT, México).

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