2,626 research outputs found
The numerical class of a surface on a toric manifold
In this paper, we give a method to describe the numerical class of a torus
invariant surface on a projective toric manifold. As applications, we can
classify toric 2-Fano manifolds of Picard number 2 or of dimension at most 4.Comment: 10 page
Diffeomorphism classes of Calabi-Yau varieties
In this article we investigate diffeomorphism classes of Calabi-Yau
threefolds. In particular, we focus on those embedded in toric Fano manifolds.
Along the way, we give various examples and conclude with a curious remark
regarding mirror symmetry.Comment: 10 pages; v2 minor changes: typos and exposition improved; to appear
in Rendiconti del Seminario Matematic
Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds
We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau
3-folds starting with (almost) any deformation family of smooth weak Fano
3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau
3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We
pay particular attention to a subclass of weak Fano 3-folds that we call
semi-Fano 3-folds. Semi-Fano 3-folds satisfy stronger cohomology vanishing
theorems and enjoy certain topological properties not satisfied by general weak
Fano 3-folds, but are far more numerous than genuine Fano 3-folds. Also, unlike
Fanos they often contain P^1s with normal bundle O(-1) + O(-1), giving rise to
compact rigid holomorphic curves in the associated ACyl Calabi-Yau 3-folds.
We introduce some general methods to compute the basic topological invariants
of ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds, and study a
small number of representative examples in detail. Similar methods allow the
computation of the topology in many other examples.
All the features of the ACyl Calabi-Yau 3-folds studied here find application
in arXiv:1207.4470 where we construct many new compact G_2-manifolds using
Kovalev's twisted connected sum construction. ACyl Calabi-Yau 3-folds
constructed from semi-Fano 3-folds are particularly well-adapted for this
purpose.Comment: 107 pages, 1 figure. v3: minor corrections, changed formattin
Calculation of Nonperturbative Terms in Open String Models
Nonperturbative corrections in type II string theory corresponding to Riemann
surfaces with one boundary are calculated in several noncompact geometries of
desingularized orbifolds. One of these models has a complicated phase structure
which is explored. A general condition for integrality of the numerical
invariants is discussed
Rationality properties of manifolds containing quasi-lines
Let X be a complex, rationally connected, projective manifold. We show that X
admits a modification X' that contains a quasi-line, ie a smooth rational curve
whose normal bundle is a direct sum of copies of O_{P^1}(1). For manifolds
containing quasi-lines, a sufficient condition of rationality is exploited:
There is a unique quasi-line from a given family passing through two general
points. We define a numerical birational invariant, e(X), and prove that X is
rational if and only if e(X)=1. If X is rational, there is a modification X'
which is strongly-rational We prove that strongly-rational varieties are stable
under smooth, small deformations. Finally, we relate the previous results and
formal geometry. This relies on \tilde{e}(X,Y), a numerical invariant of a
given quasi-line Y that depends only on the formal completion of X along Y. As
applications we show various instances in which X is determined by this formal
completion. We also formulate a basic question about the birational invariance
of \tilde{e}(X,Y).Comment: 25 page
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