Mori Theory Research Articles

On the geometry of Sasakian-Einstein 5-manifolds

On simply connected five manifolds Sasakian-Einstein metrics coincide with Riemannian metrics admitting real Killing spinors which are of great interest as models of near horizon geometry for three-brane solutions in superstring theory [24]. We expand on the recent work of Demailly and Kollar [14] and Johnson and Kollar [20] who give methods for constructing Kahler-Einstein metrics on log del Pezzo surfaces. By a previous result of the first two authors [9], circle V-bundles over log del Pezzo surfaces with Kahler-Einstein metrics have Sasakian-Einstein metrics on the total space of the bundle. Here these simply connected 5-manifolds arise as links of isolated hypersurface singularities which by the well known work of Smale [36] together with [11] must be diffeomorphic to S5#l(S2×S3). More precisely, using methods from Mori theory in algebraic geometry we prove the existence of 14 inequivalent Sasakian-Einstein structures on S2×S3 and infinite families of such structures on #l(S2×S3) with 2≤l≤7. We also discuss the moduli problem for these Sasakian-Einstein structures.

Toric Fano varieties and birational morphisms

In this paper we study smooth toric Fano varieties using primitive relations and toric Mori theory. We show that for any irreducible invariant divisor D in a toric Fano variety X, we have $0\leq\rho_X-\rho_D\leq 3$, for the difference of the Picard numbers of X and D. Moreover, if $\rho_X-\rho_D>0$ (with some additional hypotheses if $\rho_X-\rho_D=1$), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have $\rho_X\leq 9$. In the second part of the paper, we study equivariant birational morphisms f whose source is Fano. We give some general results, and in dimension 4 we show that f is always a composite of smooth equivariant blow-ups. Finally, we study under which hypotheses a non-projective toric variety can become Fano after a smooth equivariant blow-up.

Projectively Dual Varieties

This text is a draft of the review paper on projectively dual varieties. Topics include dual varieties, Pyasetskii pairing, discriminant complexes, resultants and schemes of zeros, secant and tangential varieties, Ein theorems, applications of projective differential geometry and Mori theory to dual varieties, degree and multiplicities of discriminants, self-dual varieties, etc.

Toric varieties whose blow-up at a point is Fano

We classify smooth toric Fano varieties of dimension $n\geq 3$ containing a toric divisor isomorphic to the $(n-1)$-dimensional projective space. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\geq 3$ with a fixed point $x\in X$ such that the blow-up $B_x(X)$ of $X$ at $x$ is Fano is isomorphic either to the $n$-dimensional projective space or to the blow-up of the $n$-dimensional projective space along an invariant linear codimension two subspace. As expected, such results are proved using toric Mori theory due to Reid.

Open Algebraic Surfaces with Finite Group Actions

Suppose that a finite group G acts algebraically on a log projective surface (V, A) in such a way that Δ is G-stable. Suppose κ (V - Δ U SingV) = -∞. We first look for the conditions under which the Mori theory can be applied equivariantly and observe what we obtain from the Mori theory. We then consider the P 1 -fibrations which are preserved by the actions of G.

On the simulation of inhomogeneous ESR lineshape of a spin probe in a liquid crystalline polymer

A procedure for the analysis of the electron spin resonance (ESR) lineshape of a cholestane paramagnetic tracer dissolved in a liquid crystalline polymer is described. Due to its rodlike geometry, the cholestane probe is subjected to an anisotropic reorientational motion in the polymer matrix. On the basis of the generalized Mori theory, simulation approaches to the lineshape have been developed. The ESR lineshape has been calculated for a molecule reorienting in a homogeneous molecular site according to diffusion or jump motion. Moreover, the case of inhomogeneous sites with continuous or discrete distribution functions was also accounted for. Systematic comparison with experimental spectra has been carried out, and a two-δ distribution function of the molecular sites has been singled out as best description of the experiment. The temperature dependence of the two species has been fully investigated and the spatial character of the heterogeneity has been proved.

Effective Potential and Quantum Dynamical Correlators

A new approach to the study of the properties of quantum dynamical systems by calculations of classical type is suggested. This approach is based on Mori theory and projection techniques. Bibliography: 14 titles.

Towards a Mori theory on compact Kähler threefolds III

Utilisant les resultats de la premiere et de la deuxieme partie de ce travail, nous considerons des varieties kahleriennes minimales X de dimension 3, i.e. dont le fibre canonique K X est nef. Alors K X est un fibre « good », i.e. dont la dimension de Kodaira est egale a la dimension de Kodaira numerique, sous l'exception possible que X est simple, (i.e. il n'existe pas une sous-variete compacte contenant un points tres general) et X non Kummer. Le deuxieme theoreme dit que les varietes kahleriennes X de dimension 3 avec des singularites terminales de sorte que K X n'est pas nef, ont des contractions de Mori.

Mori dream spaces and GIT.

The main goal of this paper is to study varieties with the best possible Mori theoretic properties (measured by the existence of a certain decomposition of the cone of effective divisors). We call such a variety a Mori Dream Space. There turn out to be many examples, including quasi-smooth projective toric (or more generally, spherical) varieties, many GIT quotients, and log Fano 3-folds. We characterize Mori dream spaces as GIT quotients of affine varieties by a torus in a manner generalizing Cox's construction of toric varieties as quotients of affine space. Via the quotient description, the chamber decomposition of the cone of divisors in Mori theory is naturally identified with the decomposition of the G-ample cone from geometric invariant theory. In particular every rational contraction of a Mori dream space comes from GIT, and all possible factorizations of a rational contraction can be read off from the chamber decomposition.

Application of free rotational models of molecular reorientation to the explanation of high frequency effects in dielectric relaxation

Exact solutions obtained by Gross [J. Chem. Phys. 23, 1415 (1955)] and Sack [Proc. Phys. Soc. London, Sect. B 70, 402 (1957)] for the complex polarizability of assemblies of nonelectrically interacting rotators subjected to a variety of collisions and various approximations to that quantity, specifically the Rocard equation are reappraised in view of recent attempts to use a variety of forms of that equation for the interpretation of far infrared resonance absorption spectra. It is shown that for small values of the inertial parameter (heavy damping) the Rocard equation yields a really good approximation for the complex polarizability only for the small collision model considered by Gross and Sack. In the case of large inertial parameter values it is emphasized by means of plots of the complex polarizability that such an approximation always exhibits behavior characteristic of a sharply resonant system, i.e., a pronounced absorption peak well in excess of the Debye peak and a strongly negative real part, while the exact complex polarizability spectrum for the same parameter values merely displays inertia corrected Debye relaxation. Therefore, an explanation of the resonant term other than that based on a Rocard equation with a large inertial parameter must be sought as that equation strictly applies to inertia corrected Debye (heavily damped) relaxation only. The application of the itinerant oscillator model and the three variable Mori theory to the problem is discussed in view of this conclusion.

Effective potential and quantum dynamical correlators

The approach to the calculation of quantum dynamical correlation functions is presented in the framework of the Mori theory. An unified treatment of classic and quantum dynamics is given in terms of Weyl representation of operators and holomorphic variables. The range of validity of an approximate quantum molecular dynamics is discussed.

Towards a Mori theory on compact Kähler threefolds, II

Utilisant les resultats de la premiere et de la deuxieme partie de ce travail, nous considerons des varieties kahleriennes minimales X de dimension 3, i.e. dont le fibre canonique K X est nef. Alors K X est un fibre « good », i.e. dont la dimension de Kodaira est egale a la dimension de Kodaira numerique, sous l'exception possible que X est simple, (i.e. il n'existe pas une sous-variete compacte contenant un points tres general) et X non Kummer. Le deuxieme theoreme dit que les varietes kahleriennes X de dimension 3 avec des singularites terminales de sorte que K X n'est pas nef, ont des contractions de Mori.

Fano 4-Folds with scroll structure

Abstract.We classify smooth Fano 4-folds with second betti number b2 = 2, possesing adjunction theoretic scroll structure. These manifolds occur to be projectivisations of coherent sheaves over Fano manifolds. The paper deals mainly with projectivisations of non-locally free rank 2 Fano sheaves over Fano 3-folds with b2 = 1, using Mori theory. By the way, we classify nef and big rank 2 bundles over P2 with the first Chern class (3).

Motives of Uniruled 3-Folds

J. Murre has conjectured that every smooth projective variety X of dimension d admits a decomposition of the diagonal δ=p are orthogonal projectors which lift the Künneth components of the identity map in étale cohomology. If this decomposition induces an intrinsic filtration on the Chow groups of X, we call it a Murre decomposition. In this paper we propose candidates for such projectors on 3-folds by using fiber structures. Using Mori theory, we prove that every smooth uniruled complex 3-fold admits a Murre decomposition.

Memory-function approach to the Hall constant in strongly correlated electron systems

The anomalous properties of the Hall constant in the normal state of high-$T_c$ superconductors are investigated within the single-band Hubbard model. We argue that the Mori theory is the appropriate formalism to address the Hall constant, since it aims directly at resistivities rather than conductivities. More specifically, the frequency dependent Hall constant decomposes into its infinite frequency limit and a memory function contribution. As a first step, both terms are calculated perturbatively in $U$ and on an infinite dimensional lattice, where $U$ is the correlation strength. If we allow $U$ to be of the order of twice the bare band width, the memory function contribution causes the Hall constant to change sign as a function of doping and to decrease as a function of temperature.

Towards a Mori Theory on Compact Kähler Threefolds, I

AbstractIn this paper we construct from non‐splitting families of rational curves special contractions on compact Kähler threefolds, i.e., morphisms which are analogous to those in the Mori theory in algebraic geometry.

Theory of the hall effect in strongly correlated Fermi systems

The anomalous properties of the Hall constant in the normal state of high-Tc superconductors are investigated within the Hubbard model. In Mori theory, the frequency dependent Hall constant is given as the sum of its infinite frequency limit and a memory function contribution. The first term (R H ∞ ) was already considered by Shastry et al.1 In perturbation theory and in the limit of infinite spatial dimensions, the memory function contribution causes the Hall constant to change sign as a function of doping (δ) and to decrease as a function of temperature, if we allow U to be ∼ 2W (U: Coulomb repulsion; W: bandwidth). In the limit U → ∞, the memory function is calculated via its moments. For frequencies W ≪ ω ≪ U, this leads to a decrease of R H ∞ by a factor of (1 + δ)/2.

Selfconsistent approximations in Mori's theory

The constitutive quantities in Mori's theory, the residual forces, are expanded in terms of time-dependent correlation functions and products of operators at t = 0, where it is assumed that the time derivatives of the observables are given by products of them. As a first consequence the Heisenberg dynamics of the observables are obtained as an expansion of the same type. The dynamic equations for correlation functions result to be selfconsistent nonlinear equations of the type known from mode-mode coupling approximations. The approach yields a necessary condition for the validity of the presented equations. As a third consequence the static correlations can be calculated from fluctuation-dissipation theorems, if the observables obey a Lie algebra. For a simple spin model the convergence of the expansion is studied. As a further test, dynamic and static correlations are calculated for a Heisenberg ferromagnet at low temperatures, where the results are compared to those of a Holstein-Primakoff treatment.

Introduction to the Theory of the Integer Quantum Hall Effect

Part 1 Introduction - Basic Facts: The Integer Quantum Hall Effect Classical Dynamics Quantizing Magnetic Fields The Eigenvalue Problem The Landau Model Models of Confinement Bloch Representation Disorder Broadening. Part 2 Quantum Hall Effect for Pedestrians - High Field Model: Confined Cylinder Model Spectral Conditions for the Quantum Hall Effect Systems with Contacts Robustness of the Quantum Hall Effect. Part 3 Linear Response Isothermal Susceptibilities: Dynamic Susceptibilities Conductivity Tensor Spectral Decomposition Conductivitie in Spectral Decomposition Hall Conductivity in Terms of the Center Coordinates - Multi-Probe Systems Conductance and Geometry Problems in Magnetotransport Calculations. Part 3 Phenomenology of Global Conductivities - Quantum Langevin Equation: Mori Theory Localization Criteria Hall Plateaus and Mobility Edges Finite Temperatures. Part 4 Localization in High Landau Bands - Quantum Corrections to the Conductivity: Quantum Wires Weak Localization Regime in the Quantum Hall Effect Localization in the Tails of High Landau Bands. Part 5 Averaging Green Functions - Gaussian Path Integrals: Supersymmetry Method Replica Tyick The Instanton Free Energy. Part 6 Localization in the Lowest Landau Band Density of States: Improved Perturbation Methods Inverse Participation Number Instanton Method - Density of States Localization in Band Tails.

A remark on elementary contractions

LetXbe a smooth projective variety of dimensionnover the field of complex numbers. We denote byKxthe canonical bundle ofX. By Mori's theory, ifKxis not numerically effective (i.e. if there exists a curve onXwhich has negative intersection number withKx), then there exists an extremal ray ℝ+[C] onXand an elementary contractionfR: X → Y associated with ℝ+[C].fRis called a small contraction if it is bi-rational and an isomorphism in co-dimension one.