1,720,998 research outputs found
Log contractions and equidimensional models of elliptic threefolds
No full textWe use Minimal model theory to link
birational maps of log surfaces (log contractions) to equidimensional fibrations of
elliptic threefolds. In particular we give a necessary and sufficient condition for an
elliptic threefold X → S to be birationally equivalent to an equidimensional elliptic fibratio
Birational geometry old and new
Full text linkA classical problem in algebraic geometry is to describe quantities that are invariants under birational equivalence as well as to determine some convenient birational model for each given variety, a minimal model. One such quantity is the ring of objects which transform like a tensor power of a differential of top degree, known as the canonical ring. The histories of the existence of minimal models and the finite generation of the canonical ring are intertwined; minimal models and canonical rings constitute the major building blocks for the birational classification of algebraic varieties. In this paper we will discuss some of the ideas involved, recent advances on the existence of minimal models, some applications, and the (algebraic-geometric proof of the) finite generation of the canonical ring. These results have been long standing conjectures in algebraic geometry
Divisors on elliptic Calabi-Yau 4-folds and the superpotential in F-theory — I
No full textEach smooth elliptic Calabi-Yau 4-fold determines both a three-dimensional physical theory (a compactification of “M-theory”) and a four-dimensional physical theory (using the “F-theory” construction). A key issue in both theories is the calculation of the “superpotential” of the theory, which by a result of Witten is determined by the divisors D on the 4-fold satisfying X(D = 1. We propose a systematic approach to identify these divisors, and derive some criteria to determine whether a given divisor indeed contributes. We then apply our techniques in explicit examples, in particular, when the base B of the elliptic fibration is a toric variety or a Fano 3-fold.
When B is Fano, we show how divisors contributing to the superpotential are always “exceptional” (in some sense) for the Calabi-Yau 4-fold X. This naturally leads to certain transitions of X, i.e., birational tranformations to a singular model (where the image of D no longer contributes) as well as certain smoothings of the singular model. The singularities which occur are “canonical”, the same type of singularities of a (singular) Weierstrass model. We work out the transitions. If a smoothing exists, then the Hodge numbers change.
We speculate that divisors contributing to the superpotential are always “exceptional” (in some sense) for X, also in M-theory. In fact we show that this is a consequence of the (log)-minimal model algorithm in dimension 4, which is still conjectural in its generality, but it has been worked out in various cases, among which are toric varieties
Elliptic threefolds with high Mordell-Weil rank
Full text linkWe present the first examples of smooth elliptic Calabi-Yau threefolds with
Mordell-Weil rank 10, the highest currently known value. They are given by the
Schoen threefolds introduced by Namikawa; there are six isolated fibers of
Kodaira Type IV. We explicitly compute the Shioda homomorphism for the
generators of the Mordell-Weil group and their induced height pairing.
Compactification of F-theory on these threefolds gives an effective theory in
six dimensions which contains ten abelian gauge group factors. We compute the
massless matter spectrum. In particular, we show that the charged singlet
matter need not reside at enhancement loci of Type , as previously
believed. We relate the multiplicities of the massless spectrum to genus-zero
Gopakumar-Vafa invariants and other geometric quantities of the Calabi-Yau. We
show that the gravitational and abelian anomaly cancellation conditions are
satisfied. We prove a Geometric Anomaly Cancellation equation and we deduce
birational equivalence for the quantities in the spectrum. We explicitly
describe a Weierstrass model over of the Calabi-Yau threefolds as
a log canonical model and compare it to a construction by Elkies and classical
results of Burkhardt.Comment: Final versio
Higher Dimensional Elliptic Fibrations and Zariski Decompositions
Full text linkWe study the existence and properties of birationally equivalent models for elliptically fibered
varieties. In particular these have either the structure of Mori fiber spaces or, assuming some standard
conjectures, minimal models with a Zariski decomposition compatible with the elliptic fibration. We prove
relations between the birational invariants of the elliptically fibered variety, the base of the fibration and of
its Jacobian
Spectrum bounds in geometry
Full text linkFilipazzi, Hacon, and Svaldi proved that there are only finitely many topological types of elliptically fibered Calabi–Yau threefolds. We explore the implications of their results on the boundedness of the geometric quantities in the massless spectrum of the F-theory Calabi–Yau compactifications. A key ingredient is what we call the geometric anomaly equation, and the extension of the gravitational anomaly cancellation in physics, also to singular spaces. We review and extend the dictionary between geometry and physics. We conclude with explicit bounds
Anomalies and the euler characteristic of elliptic Calabi-Yau threefolds
Full text linkWe investigate the delicate interplay between the types of singular fibers in elliptic fibrations of Calabi-Yau threefolds (used to formulate F-theory) and the "matter" representation of the associated Lie algebra. The main tool is the analysis and the appropriate interpretation of the anomaly formula for six-dimensional supersymmetric theories. We find that this anomaly formula is geometrically captured by a relation among codimension two cycles on the base of the elliptic fibration, and that this relation holds for elliptic fibrations of any dimension. We introduce a "Tate cycle" that efficiently describes this relationship, and which is remarkably easy to calculate explicitly from the Weierstrass equation of the fibration. We check the anomaly cancellation formula in a number of situations and show how this formula constrains the geometry (and in particular the Euler characteristic) of the Calabi-Yau threefold
ON THE HODGE CONJECTURE FOR HYPERSURFACES IN TORIC VARIETIES
Full text linkWe show that for very general hypersurfaces in odd-dimensional simplicial projective toric varieties satisfying an effective combina- torial property the Hodge conjecture holds. This gives a connection between the Oda conjecture and Hodge conjecture. We also give an explicit criterion which depends on the degree for very general hypersurfaces for the combinatorial condition to be verified
Matter from geometry without resolution
Full text linkWe utilize the deformation theory of algebraic singularities to study charged matter in compactifications of M-theory, F-theory, and type IIa string theory on elliptically fibered Calabi-Yau manifolds. In F-theory, this description is more physical than that of resolution. We describe how two-cycles can be identified and systematically studied after deformation. For ADE singularities, we realize non-trivial ADE representations as sublattices of ZN, where N is the multiplicity of the codimension one singularity before deformation. We give a method for the determination of Picard-Lefschetz vanishing cycles in this context and utilize this method for one-parameter smooth deformations of ADE singularities. We give a general map from junctions to weights and demonstrate that Freudenthal's recursion formula applied to junctions correctly reproduces the structure of high-dimensional ADE representations, including the 126 of SO(10) and the 43,758 of E6. We identify the Weyl group action in some examples, and verify its order in others. We describe the codimension two localization of matter in F-theory in the case of heterotic duality or simple normal crossing and demonstrate the branching of adjoint representations. Finally, we demonstrate geometrically that deformations correctly reproduce the appearance of non-simply-laced algebras induced by monodromy around codimension two singularities, showing the reduction of D4to G2in an example. A companion mathematical paper will follow. © SISSA 2013
Geometric transitions, del Pezzo surfaces and open string instantons
No full textWe continue the study of a class of geometric transitions proposed by Aganagic and Vafa which exhibit open string instanton corrections to Chern-Simons theory. In this paper we consider an extremal transition for a local del Pezzo model which predicts a highly nontrivial relation between topological open and closed string amplitudes. We show that the open string amplitudes can be computed exactly using a combination of enumerative techniques and Chern-Simons theory proposed by Witten some time ago. This yields a striking conjecture relating all genus topological amplitudes of the local del Pezzo model to a system of coupled Chern- Simons theories
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