Publicación

Sasaki-Einstein 7-Manifolds and Orlik’s Conjecture

Jaime Cuadros Valle · Joe Lope Vicente
2023 Annals of Global Analysis and Geometry DOI: 10.1007/s10455-023-09930-z

Resumen

We study the homology groups of certain 2-connected 7-manifolds admitting quasiregular Sasaki-Einstein metrics, among them, we found 52 new examples of Sasaki-Einstein rational homology 7-spheres, extending the list given by Boyer, Galicki and Nakamaye in [6]. As a consequence, we exhibit new families of positive Sasakian homotopy 9-spheres given as cyclic branched covers, determine their diffeomorphism types and find out which elements do not admit extremal Sasaki metrics. We also improve previous results given by Boyer [12] showing new examples of Sasaki-Einstein 2-connected 7-manifolds homeomorphic to connected sums of S3 × S4. Actually we show that manifolds of the form #k(S3 × S4) admit Sasaki-Einstein metrics for 22 different values of k. All these links arise as Thom-Sebastiani sums of chain type singularities and cycle type singularities where Orlik's conjecture holds due to a recent result by Hertling and Mase [19]. Mathematics Subject Classification 53C25; 57R60.

Autores y colaboradores

Authors

Palabras clave

Links of weighted hypersurfaces Orlik’s conjecture Rational homology 7-spheres Sasaki–Einstein metrics

Proyectos

Research Projects

Loading...
Project Logo
Item type:Research Project,
Minimal Models on 5-dimensional Seifert bundles
P_MM-SB5
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.
P_MM-SB5
Full item page