Minimal Models on 5-dimensional Seifert bundles

Acronym

P_MM-SB5

Consortium Coordinator

Cuadros Valle, Jaime

Start Date

March 14, 2015

End Date

March 14, 2016

Status

https://purl.org/pe-repo/concytec/estadoProyecto#concluido

Tipo de proyecto

https://purl.org/pe-repo/ocde/tipoProyecto#investigacionBasica

Description

It is well-known that the singularities of a $K3$ surface can be resolved in two (seemingly different) ways: one can resolve algebraically the singularities using the classical techniques of complex algebraic geometry (to obtain a smooth $K3$ surface) or one can consider the total space of the associated $S^1$-Seifert bundle as a (real) resolution of the underlying complex $K3$ orbifold. The purpose of this work is to build a connection between these two kinds of resolution. A motivation for this is the classical work of Orlik and Wagreich on isolated surface singularities with $C^*$-action (see [OW1]). Observe, that the classification of surfaces admitting $C^*$-action, due to Orlik and Wagreich, do not include K3 surfaces. Nevertheteless we can exploit the natural $C^*$-action on the affine cone over the K3 surface to deal with this problem. In fact, we anticipate an extension of the plumbing techniques of Orlik and Wagreich to the associated (complex) Kähler cones. Furthermore, it is expected to obtain an effective algorithm for constructing the Milnor fiber $F$ (up to diffeomorphism) as a handlebody from the polynomials $f_1,ldots, f_r$ that define a complex subvariety $Wsubset C^{n+1}.$ This requires an understanding of the antisymmetric intersection form on $H_3(F)$ (see [S]). Note that in the surface case the corresponding intersection form on the homology of the Milnor fiber is symmetric, and many invariants (e.g. signature) determine the link $K$ completely. So these techniques do not necessarily generalize to our case. Even in the case of surfaces, it does not seem to be known whether the Milnor fiber can be constructed as a handlebody from information of the resolution graph. This is conjectured in [M] page 58.

Keywords

Modelos mínimos

;

Fibrados de Seifert

;

Geometría diferencial

;

Dimensión cinco

Área de conocimiento

Natural sciences

Campo OCDE

https://purl.org/pe-repo/ocde/ford#1.01.01