New exotic symplectic 4-manifolds with nonnegative signatures and exotic smooth structures on small 4-manifolds
Başlık çevirisi mevcut değil.
- Tez No: 601921
- Danışmanlar: DR. ANAR AKHMEDOV
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 2018
- Dil: İngilizce
- Üniversite: Unıversıty Of Mınnesota
- Enstitü: Yurtdışı Enstitü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
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Özet (Çeviri)
The focus of this thesis is twofold. First one is the geography problem of symplectic and smooth 4-manifolds with nonnegative signatures. We construct new non-spin, irreducible, symplectic and smooth 4-manifolds with nonnegative signatures, with more than one smooth structures and small topology. These manifolds are interesting with respect to the symplectic and smooth geography problems. More speci cally, we construct in nite families of smooth, closed, simply-connected, minimal, symplectic and non-symplectic 4-manifolds with nonnegative signatures that have the smallest Euler characteristics among the all known such manifolds, and with more than one smooth structures. The second focus of this thesis is the study of brations of complex curves of genus two and constructing exotic 4-manifolds with small Euler characteristics. In [93, 94] Namikawa and Ueno gave complete classi cation of all singular bers in pencils of genus two curves, where each pencil is a family of complex curves of genus two over the 2-disc with one singular curve over the origin. They gave the list of all singular bers arising in such families. In the constructions of singularities they used algebrogeometric techniques. In this thesis, we topologically construct certain singularity types in the Namikawa-Ueno's list. More precisely, we nd pencils of genus two curves in Hirzebruch surfaces and from which we obtain speci c types of Namikawa-Ueno's genus two singular bers and sections, precisely. In addition to constructing these singularities topologically, we also introduce a deformation technique of the singular bers of certain types Lefschetz brations over the 2-sphere. Then by using them and via symplectic surgeries, we build new exotic minimal symplectic 4-manifolds with small topology.
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