Geometry and Topology of Singularities (Durfee Conjectures)

Ziel

"Much of geometry and topology of algebraic varieties is hidden in their singular points. To extract this information one typically compares invariants of a singular variety to those of its smoothing deformation or resolution. This gives two fundamental invariants of a singular point: the Milnor number and the singularity genus.

They were studied by many geometers and topologists. In particular, the relation between the Milnor number and singularity genus has been a hot topic since 50's. The case of curve singularities is elementary. Already for normal surface singularities these two invariants are related via combinatorial invariants of resolution graph and topological invariants of the link. Thus for a long time there has been a quest for a simple relation/inequality involving only Milnor number and singularity genus.

In 1978 A.Durfee stated a conjectural bound for isolated surface singularities that are complete intersections. After a long chain of partial confirmations and verifications the conjectural bound was stated for isolated complete intersections in arbitrary dimension.

Surprisingly, we found counterexamples to this conjecture. We succeeded to formulate the asymptotically sharp form of the bound and proved it for a large class of singularities.

In the current project I intend to prove the (corrected) Durfee bound for complete intersection surface singularities. In higher dimensions I hope to prove the bound for those hypersurface singularities that admit especially nice resolutions (by blowing up at centers of a particular type). This will be done by tracing the change of invariants at each blowup, reducing the problem to some particular singularities, ""building blocks"".

In parallel I intend to develop methods to trace the change-under-modification for other singularity invariants, such as the signature of the singularity and the zeta functions. This will open the possibility to establish for them various new bounds/relations."

Wissenschaftliches Gebiet

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BEN-GURION UNIVERSITY OF THE NEGEV

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€ 100 000,00

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84105 Beer Sheva
Israel

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Higher or Secondary Education Establishments

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Daphna Tripto (Ms.)

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