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Monodromy conjecture and the Hessian differential form. (English) Zbl 1388.14050

The author considers a germ of an analytic nonconstant function \(f:(\mathbb{C}^{n+1},0)\rightarrow (\mathbb{C},0)\). The topological zeta function is an analytic, but not a topological invariant associated to this germ and a germ of an \((n+1)\)-differential form living in \((\mathbb{C}^{n+1},0)\). For plane curves, i.e. for \(f\in \mathbb{C}\{ x,y\}\), one defines the Hessian as \(\mathrm{Hess}(f):=f_{xx}f_{yy}-f_{xy}^2\). For the differential form \(\mathrm{Hess}(f)dx\wedge dy\) the author shows that it does not satisfy the monodromy conjecture, i.e., the Hessian form is not an allowed differential 2-form. The result is illustrated by the example of \(f(x,y)=y^5-2x^3y^7+x^4y^3+x^6\).

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
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