impanga 2024/25

We show that for every $k\ge 3$ there exist complex algebraic cones of dimension $k$ with isolated singularities, which are bi-Lipschitz and semi-algebraically equivalent but have different degrees. We also prove that homeomorphic projective hypersurfaces with dimension greater than 2 have the same degree. In the final part of the paper, we classify links of real cones with base $\mathbb{P}^1\times \mathbb{P}^2.$ As an application we give an example of three four-dimensional real algebraic cones in $\mathbb{R}^8$ with isolated singularity which are semi-algebraically and bi-Lipschitz equivalent but have non-homeomorphic bases. We discover also some new tools to study the links of real algebraic varieties. Moreover, we give examples of real manifolds, which are not diffeomorphic to projective manifolds of odd degree.

I will explain in detail the construction of the A'Campo space. It plays a crucial role in the proof of Zariski Multiplicity Conjecture, as it provides a representative of monodromy with good dynamics. In this lecture, I will focus on other applications of this construction. For example, in the context of maximal Calabi-Yau degenerations, it provides a family of Kahler metrics with properties similar to those expected from the Ricci flat ones. The key idea is to perform most computations at the boundary of the A'Campo space, and then push the result back to the original degeneration using the symplectic connection.

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Stated in 1971, the Zariski multiplicity conjecture asserts that the multiplicity of a reduced complex hypersurface singularity is an embedded topological invariant. More than 50 years later, this problem is still open in general. Several partial answers have been given over the years, with a major breakthrough by J. Fernández de Bobadilla and T. Pełka who recently proved a version of the conjecture for 1-parameter families of isolated singularities. In this talk, I will give an overview of the most significant results that have been obtained.

This is the first of two lectures concerning my joint work with J. F. de Bobadilla, proving Zariski multiplicity conjecture for μ-constant families of isolated hypersurface singularities. In this lecture I will explain the key ingredient of the proof, namely the construction of a representative of monodromy which is an exact symplectomorphism with very good dynamic properties. Given a family of Kähler manifolds over a punctured disk, together with a model with snc central fiber, I will show how to extend it to a symplectic fibration over an annulus, such that over the inner circle ("at radius zero") most choices become irrelevant, in particular the symplectic monodromy can be read off from the dual complex of the central fiber. The resulting fibration, called the A'Campo space, can be applied to many degeneration problems in algebraic geometry. If time permits, I will show how to use it to produce Lagrangian tori approximating those predicted by Mirror Symmetry.

In this introductory lecture, I will describe basics of Heegaard Floer homology (construction and properties) focusing on examples of special importance to experts in singularity theory and low-dimensional algebraic geometry.

In the second lecture, I will explain how to use the A'Campo space, to prove the Zariski conjecture. Just like the classical topological A'Campo construction allows to compute Lefschetz number of monodromy's iterates, our symplectic version yields a spectral sequence converging to its Floer homology. This proof is a slight generalization of the previous work by McLean, in particular it recovers his characterization of multiplicity as the smallest integer m such that the m-th iterate of monodromy has nonvanishing Floer homology. Our more general setting allows us to apply it to each member of a μ-constant family, and infer that the resulting Floer homology - hence multiplicity - stays the same.