One of the major discoveries of the last two decades of the twentieth century in algebraic geometry is the realization that the theory of minimal models of surfaces can be generalized to higher dimensional varieties. This generalization, called the minimal model program or Mori's program, has developed into a powerful tool with applications to diverse questions in algebraic geometry and beyond. This book provides the a comprehensive introduction to the circle of ideas developed around the program, the prerequisites being only a basic knowledge of algebraic geometry. It will be of great interest to graduate students and researchers working in algebraic geometry and related fields.

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Ferroelectric materials are widely used in modern electric devices such as memory elements, filtering devices and high-performance insulators. Ferroelectric crystals have a spontaneous electric polarization arising from the coherent arrangement of electric dipoles (specifically, a polar displacement of anions and cations). First-principles calculations and electron density analysis of ferroelectric materials have revealed that the covalent bond between the anions and cations, or the orbital hybridization of electrons on both ions, plays a key role in establishing the dipolar arrangement. However, an alternative model-electronic ferroelectricity-has been proposed in which the electric dipole depends on electron correlations, rather than the covalency. This would offer the attractive possibility of ferroelectric materials that could be controlled by the charge, spin and orbital degrees of freedom of the electron. Here we report experimental evidence for ferroelectricity arising from electron correlations in the triangular mixed valence oxide, LuFe(2)O(4). Using resonant X-ray scattering measurements, we determine the ordering of the Fe(2+) and Fe(3+) ions. They form a superstructure that supports an electric polarization consisting of distributed electrons of polar symmetry. The polar ordering arises from the repulsive property of electrons-electron correlations-acting on a frustrated geometry.

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Using Lorenz microscopy and small-angle electron diffraction, we directly present that the chiral magnetic soliton lattice (CSL) continuously evolves from a chiral helimagnetic structure in small magnetic fields in Cr(1/3)NbS2. An incommensurate CSL undergoes a phase transition to a commensurate ferromagnetic state at the critical field strength. The period of a CSL, which exerts an effective potential for itinerant spins, is tuned by simply changing the field strength. Chiral magnetic orders observed do not exhibit any structural dislocation, indicating their high stability and robustness in Cr(1/3)NbS2.

The oxygen evolution reaction that occurs during water oxidation is of considerable importance as an essential energy conversion reaction for rechargeable metal–air batteries and direct solar water splitting. Cost-efficient ABO3 perovskites have been studied extensively because of their high activity for the oxygen evolution reaction; however, they lack stability, and an effective solution to this problem has not yet been demonstrated. Here we report that the Fe4+-based quadruple perovskite CaCu3Fe4O12 has high activity, which is comparable to or exceeding those of state-of-the-art catalysts such as Ba0.5Sr0.5Co0.8Fe0.2O3−δ and the gold standard RuO2. The covalent bonding network incorporating multiple Cu2+ and Fe4+ transition metal ions significantly enhances the structural stability of CaCu3Fe4O12, which is key to achieving highly active long-life catalysts.

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