Se encontraron 5 investigaciones
Understand a moduli of Sasakian manifolds that allows the variation of the associated CR-structures. The natural object of study is a set that gives a configuration of different CR-structures compatible with a contact structure. We think that the notion of the bouquet of Sasaki cones given in Boyer [B] encodes important information that can be useful to understand the connectedness of a moduli of Sasakian structures on a manifold. Please see ANEXO for more details.
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The purpose of this project is to investigate and describe the equivariant intersection cohomology of possibly singular projective equivariant embeddings of reductive groups. Such a description is only available in the case of equivariant compactifications of tori, i.e. toric varieties, by work of Barthel, Brasselet, Fieseler and Kaup. Our goal is to extend their results to more general group embeddings, and to write down, explicitly and combinatorially, the equivariant intersection cohomology of possibly singular projective group embeddings, without placing any assumptions on the singular locus. Crucial to our description is the fact that all of these embeddings can be realized as projectivizations of affine monoids. In this setting, we anticipate that a suitable extension of GKM theory for the equivariant intersection cohomology of singular varieties with torus actions could lead to our main results. This is a further step in the program started in [G2]. The outcome should reveal certain aspects of the geometry of projective group embeddings which are not captured by equivariant bivariant theories.
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The purpose of this project is to investigate the cone of projective group embeddings with the same Cox ring R(X). The ring R(X) is a unique factorization domain. Q-factorial projective varieties with a finitely generated Cox ring are characterized using Mori theory. If X is a spherical variety, then R(X) is a finitely generated algebra, and it carries a natural grading by the classes of effective divisors. The Cox ring of spherical varieties has been described by Brion and Gagliardi. In the case when X is a G-embedding, we can associate a certain cone H, in the class group Cl(X), so that any element y of H defines a group embedding X_y satisfying R(X_y) = R(X). It seems that H corresponds to the movable cone of X. The purpose of this project is to calculate H explicitely, using the knowledge of NE(X), the cone of effective curves on X, and nef(X), the cone of numerically effective divisors on X, and certain monoid data. Moreover, we would like to understand, and quantify, how the cones H and H', associated to different G-embeddings, are related.
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Please see file Anexo for a detailed explanation of certain definitions. It is known that many contact manifolds admit symplectic fillings, that is, they can be realized as boundary of compact symplectic manifolds. It is also known that one can extend this definition at the level of complex geometry. In this situation the analogue is given holomorphic fillings:a compact complex manifold (N,J) is a holomorphic filling of the contact manifold (M, D) if N (M,J) is strictly pseudo-convex and $D=TMcap JTM$. Due to a result of Marinescu, G., Yeganefer, N (see [MY]) Sasakian manifolds, which are of contact type, admit holomorphic fillings and thus Kählerian fillings. There is classification of Sasakian manifolds in terms of the basic Chern class (one obtains positive, negative and null structures). We are interested in simply connected 5 manifolds admitting both positive and null structures. Positive Sasakian manifolds usually can be realized as S^1-Seifert bundles over a log Del Pezzo surfaces. In the null case, they are S^1-Seifert bundles over K3 orbifolds. We think, for certain cases: it is possible to determine the (strong) symplectic filling of associated to the null case and the holomorphic filling of the positive case. The reason to choose these two structures is trying to understand an interesting connection discovered by Alexeev and Nikulin in the classification of certain Del Pezzo surfaces using K3 lattice theory. We expect to reinterpret this connection at the level of fillings.
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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.
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