Topological zeta functions of complex plane curve singularities (Q6046552)

With \(f\) a non-constant complex function on a smooth complex algebraic variety \(X\) with zero locus \(X_0\), Denef and Loeser have introduced the topological zeta function for \(f\). Let \(h:Y\rightarrow (X,X_0)\) be an embedded resolution of singularities of \(X_0\), i.e, a proper morphism \(h:Y\rightarrow X\) with \(Y\) smooth such that the restriction \(Y\setminus h^{-1}(X_0)\rightarrow X\setminus X_0\) is an isomorphism and \(h^{-1}(X_0)\) is a divisor with normal crossings. The exceptional divisors and irreducible components of the strict transform of \(h\) are denoted by \(E_i\), where \(i\) is in a finite set \(S\). For \(I\subseteq S\), one sets \(E_I:=\bigcap _{i\in I}E_i\) and \(E^{\circ}_I:=E_I\setminus \bigcup _{j\not\in I}E_j\). The topological zeta function is defined as \(Z_f^{\mathrm{top}}(s)=\sum _{I\subseteq S}\chi (E^{\circ}_I)\prod _{i\in I}1/(N_is+\nu _i)\), where \(N_i\) is the multiplicity of \(h^*f\) on \(E_i\) and \(\nu _i-1\) is the corresponding discrepancy of the Jacobian of \(h\). The authors study topological zeta functions of complex plane curve singularities using toric modifications and further developments. As applications of the research method, they prove that the topological zeta function is a topological invariant for complex plane curve singularities and give a short and new proof of the monodromy conjecture for plane curves.