Extremal Rays Research Articles

Diffraction in the semiclassical approximation to Feynman's path integral representation of the Green function

We derive the semiclassical approximation to Feynman's path integral representation of the energy Green function of a massless particle in the shadow region of an ideal obstacle in a medium. The wavelength of the particle is assumed to be comparable to or smaller than any relevant length of the problem. Classical paths with extremal length partially creep along the obstacle and their fluctuations are subject to non-holonomic constraints. If the medium is a vacuum, the asymptotic contribution from a single classical path of overall length L to the energy Green function at energy E is that of a non-relativistic particle of mass E/ c 2 moving in the two-dimensional space orthogonal to the classical path for a time τ= L/ c. Dirichlet boundary conditions at the surface of the obstacle constrain the motion of the particle to the exterior half-space and result in an effective time-dependent but spatially constant force that is inversely proportional to the radius of curvature of the classical path. We relate the diffractive, classically forbidden motion in the “creeping” case to the classically allowed motion in the “whispering gallery” case by analytic continuation in the curvature of the classical path. The non-holonomic constraint implies that the surface of the obstacle becomes a zero-dimensional caustic of the particle's motion. We solve this problem for extremal rays with piecewise constant curvature and provide uniform asymptotic expressions that are approximately valid in the penumbra as well as in the deep shadow of a sphere.

Divisorial Zariski decompositions on compact complex manifolds

Using currents with minimal singularities, we introduce pointwise minimal multiplicities for a real pseudo-effective (1,1)-cohomology class α on a compact complex manifold X, which are the local obstructions to the numerical effectivity of α. The negative part of α is then defined as the real effective divisor N( α) whose multiplicity along a prime divisor D is just the generic multiplicity of α along D, and we get in that way a divisorial Zariski decomposition of α into the sum of a class Z( α) which is nef in codimension 1 and the class of its negative part N( α), which is an exceptional divisor in the sense that it is very rigidly embedded in X. The positive parts Z( α) generate a modified nef cone, and the pseudo-effective cone is shown to be locally polyhedral away from the modified nef cone, with extremal rays generated by exceptional divisors. We then treat the case of a surface and a hyper-Kähler manifold in some detail. Using the intersection form (respectively the Beauville–Bogomolov form), we characterize the modified nef cone and the exceptional divisors. The divisorial Zariski decomposition is orthogonal, and is thus a rational decomposition, which fact accounts for the usual existence statement of a Zariski decomposition on a projective surface, which is thus extended to the hyper-Kähler case. Finally, we explain how the divisorial Zariski decomposition of (the first Chern class of) a big line bundle on a projective manifold can be characterized in terms of the asymptotics of the linear series | kL| as k→∞.

Notes on toric varieties from Mori theoretic viewpoint

The main purpose of this notes is to supplement the paper by Reid: De- composition of toric morphisms, which treated Minimal Model Program (also called Mori's Program) on toric varieties. We compute lengths of negative extremal rays of toric varieties. As an application, a generalization of Fujita's c onjecture for singular toric varieties is obtained. We also prove that every toric variety has a small projective toric Q-factorialization. 0. Introduction. The main purpose of this notes is to supplement the paper by Reid (Re): Decomposition of toric morphisms, which treated Minimal Model Program (also called Mori's Program) on toric varieties. We compute lengths of negative extremal rays of toric varieties. This is an answer to (Ma, Remark-Question 10-3-6) for toric varieties, which is an easy exercise once we understand (Re). As a corollary, we obtain a strong version of Fujita's conjecture for singular toric varieties. Related topics are (Ft), (Ka), (La) and (Mu, Section 4). We will work, throughout this paper, over an algebraically closed field k of arbitrary characteristic.

Local Simple Connectedness of Resolutions of Log-Terminal Singularities

We study fundamental groups related with log-terminal singularities, and show that fundamental groups are preserved by a resolution of singularities. As corollaries, we show that fundamental groups are invariant under various fundamental operations in the Minimal Model Program, for example, by contractions of extremal rays, by flips, by pluricanonical morphisms of minimal varieties of general type.

Minimal Resolutions of Lattice Ideals and Integer Linear Programming

A combinatorial description of the minimal free resolution of a lattice ideal allows us to the connection of Integer Linear Programming and Algebra. The non null reduced homology spaces of some simplicial complexes are the key. The extremal rays of the associated cone reduce the number of variables.

Contractible classes in toric varieties

Let X be a smooth, complete toric variety. Let A1(X) be the group of algebraic 1-cycles on X modulo numerical equivalence and N1(X) = A1(X)⊗ZQ . Consider inN1(X) the coneNE(X) generated by classes of curves on X . It is a well-known result due to M. Reid [13] that NE(X) is closed, polyhedral and generated by classes of invariant curves on X . The varietyX is projective if and only if NE(X) is strictly convex; in this case, a 1-dimensional face ofNE(X) is called an extremal ray. It is shown in [13] that every extremal ray admits a contraction to a projective toric variety. We think of A1(X) as a lattice in the Q -vector space N1(X). Suppose that X is projective. For every extremal ray R ⊂ NE(X), we choose the primitive class in R ∩ A1(X); we call this class an extremal class. The set E of extremal classes is a generating set for the cone NE(X), namely NE(X) = ∑ γ∈E Q≥0 γ. For many purposes it would be useful to have a linear decomposition with integral coefficients: for instance, what can we say about curves having minimal degree with respect to some ample line bundle onX? It is an open question whether extremal classes generate NE(X) ∩ A1(X) as a semigroup. In this paper we introduce a set C ⊇ E of classes in NE(X) ∩ A1(X) which is a set of generators of NE(X) ∩ A1(X) as a semigroup. Classes in C are geometrically characterized by “contractibility”:

Towards Bounding Complexity of a Minimal Model

We give some effectivity results in birational geometry. We provide an upper bound on the rational constant in Rationality Theorem in terms of certain intersection numbers, under an additional condition on the variety that it admits a divisorial contraction. One consequence is an explicit bound on the number of certain extremal rays. Our main result tries to construct from a given set of ample divisors Hj on X with their intersection numbers bi, a certain set of ample divisors Lj on X' or X+ where X' or X+ arises from a contraction or a flip,such that the corresponding intersection numbers of Lj are uniformly bounded in terms of bi and the index of X. This gives a bound on the projective degree of a minimal model in special case.

The short resolution of a lattice ideal

The short resolution of a lattice ideal is a free resolution over a polynomial ring whose number of variables is the number of extremal rays in the associated cone. A combinatorial description of this resolution is given. In the homogeneous case. the regularity can be computed from this resolution.

On Buchsbaum simplicial affine semigroups

We give an arithmetic characterization which allow us to determine algorithmically when the semigroup ring associated to a simplicial affine semigroup is Buchsbaum. This characterization is based on a test performed on the Apery sets of the extremal rays of the semigroup. We use this method to obtain the cardinality of minimal presentations for semigroups with minimal Apery set.

Special rays in the Mori cone of a projective variety

AbstractLet X be a smooth n-dimensional projective variety over an algebraically closed field k such that KX is not nef. We give a characterization of non nef extremal rays of X of maximal length (i.e of length n – 1); in the case of Char(k) = 0 we also characterize non nef rays of length n – 2.

Automorphisms of Finite Order on Rational Surfaces: With an appendix by I. Dolgachev

We classify minimal pairs (X,G) for smooth rational projective surface X and finite group G of automorphisms on X. We also determine the fixed locus XG and the quotient surface Y=X/G as well as the fundamental group of the smooth part of Y. The realization of each pair is included. Mori's extremal ray theory and recent results of Alexeev and also Ambro on the existence of good anti-canonical divisors are used.

TOWARD A CLASSIFICATION OF ALMOST HOMOGENEOUS MANIFOLDS II — BAD MODELS

In this paper, we continue to deal with singular extremal rays with a Gm x SL (2, C) action which we started in [9] (see also [6]). We shall classify all Gm x SL(2, C) actions in [10] and state our results in Sec. 5 of this paper (see also [6]). Our idea is that once we have a singular extremal ray we can obtain enough information on the whole manifold which eventually leads to a classification. In this paper we concentrate on the most difficult case (5b) in the Proposition 4. For this reason we leave the proofs of Theorem 1 and Lemma 3 to [10]. But we give the proof of Theorem 3 for intuition of the whole process. Section 4 is the core of this paper where we construct the House Models and solve the (5b) case of Proposition 4 (Theorem 6). Once again (other than the proof of the Theorem 1 in [9]) we use non-algebraic argument for Theorem 3, a method we used very often and successfully. We also find in Sec. 4 of this paper that minimal projective 3-fold house models can have an arbitrary large second Betti number. Their effective curve cones are fully understood. The last section contains a criterion for projectivity which is very useful in our situations, and I cannot find it in the literature (Theorem 8). Most of this work was finished in Fall 1991. The original work was written in one paper as in [6 Chap. 1], but we now separate it into [9, 10] and this paper to make it more readable.

Blowing Ups of 3-dimensional Terminal Singularities

We study the blowing up π : X -> X of a 3-dimensional terminal singularity X of index m > 2 such that the exceptional locus of π consists of a prime divisor E with discrepancy 1/ m . A complete classification of such blowing ups is given and it is proved that these correspond to weighted blow ups by a certain kind of maximal weights except for the case where X is of type (cD/2). We shall treat the (cD/2) case later. These also give examples of contractions of extremal rays which contract a divisor to a point.

On the flip theorem for locally primitive extremal rays in dimension 3

The aim of this paper is to give an other proof of Mori's [Mo1] famous existence of flip for 3-folds in the case of extremal rays on 3-fold which are locally primitive. The proof starts with some original arguments from [Mo1] and reduces the proof to the case of index 1 terminal singularities. Nevertheless it does not use Kawamata's argument involving double canonical covers which is the method Mori used. Instead it uses the blowing up of the completion of the 3-fold along the curve defining an extremal ray.

Stochastic construction of a Feynman path integral representation for Green’s functions in radiative transfer

A construction of a stochastic, Lagrangian path integral representation is presented for classical Green’s functions or propagators for radiative transfer in random scattering medias. As Fermat rays comprising the initial collimated pulse (e.g., from a laser) enter the medium, they evolve into a bundle of random paths or trajectories due to scattering. Stochastically, this can be interpreted as a bundle of randomly evolving Markov trajectories traced out by a gas or ensemble of “Brownian-type” classical point photons undergoing multiple scatterings. The path integral is structurally of the same form as a Wick rotated Euclidean quantum Feynman integral with direct optical analogs of the Hamiltonian and Lagrangian emerging. However, the optical stochastic integral is real and is defined in real time becoming exponentially damped rather than oscillatory. The calculation also constitutes an alternative mathematical derivation of the time dependent diffusion equation from the radiative transfer theory. The limits of the path integral representation give a maximally diffused Gaussian distribution of the heat kernal form (the large scatter limit) and a Beer’s law exponential decay corresponding to the extremal Fermat rays obeying Euler–Lagrange equations (the zero scatter limit). The approach also highlights the direct structural analogs between classical radiative transfer and optical diffusion in real time (t) and ordinary quantum mechanics in Euclidean time (√(−1)t=it).

Cohomological invariants of complex manifolds coming from extremal rays

Cohomological invariants of complex manifolds coming from extremal rays

Quantum cohomology of projective bundles over ℙⁿ

In this paper we study the quantum cohomology ring of certain projective bundles over the complex projective space P n \mathbb {P}^{n} . Using excessive intersection theory, we compute the leading coefficients in the relations among the generators of the quantum cohomology ring structure. In particular, Batyrev’s conjectural formula for quantum cohomology of projective bundles associated to direct sum of line bundles over P n \mathbb {P}^{n} is partially verified. Moreover, relations between the quantum cohomology ring structure and Mori’s theory of extremal rays are observed. The results could shed some light on the quantum cohomology for general projective bundles.

Toward a Classification of Almost Homogeneous Manifolds I — Linearization of the Singular Extremal Rays

Toward a Classification of Almost Homogeneous Manifolds I — Linearization of the Singular Extremal Rays

Communication-Minimal Tiling of Uniform Dependence Loops

Tiling is a loop transformation that a compiler uses to create automatically blocked algorithms in order to improve the benefits of the memory hierarchy and reduce the communication overhead between processors. Motivated by existing results, this paper presents a conceptually simple approach to finding tilings with a minimal amount of communication between tiles. The development of almost all results is based primarily on the inequality of arithmetic and geometric means and the concept of extremal rays from convex cones. The key insight is that a tiling that is communication-minimal must induce the same amount of communication through all faces of a tile, which restricts the search space for optimal tilings to those tiling matrices whose rows are all extremal rays in a cone. For nested loops with several special forms of dependences, closed-form optimal tilings are derived. In the general case, a procedure is given that always returns optimal tilings. An efficient implementation of the procedure, along with experimental results, is presented. A detailed comparison of this work with some existing results is provided.

Ample vector bundles on singular varieties II

Theorem 0 Let X be a n-dimensional smooth projective variety over complex number C and let E be an ample vector bundle on X o f rank n + 1 such that c l (X) = cl(E). Then (X, E) ~ (~n, Qn+16~,.(1))" This also had been proved independently by Peternell [P1]. The theorem gives us a characterization of the projective spaces via ample vector bundles. The methods we used in our proof depend heavily on Mori ' s theory of the extremal rays. In particular, a theorem of Lazarsfeld [L] played an important role: I f X is smooth and f : ~ ~ X is surjective, then X ~ 1~. However, this is no longer true if X has singularities as showed by the following example: Example. Let Y = ~2 and let ~: [x, y, z ] --, [ x , y , z] be an involution on Y. Then X = Y / has a rational double point. The quotient map Y --, X is of degree two which ramified along I? 1. In this paper, our aim is to generalize Mukai ' s conjecture to projective varieties having at most log-terminal singularities. The precise statements of our results are as follows: