There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

인공고관절의 세라믹 볼 헤드의 기계적 안정성 평가를 위한 3 차원 유한요소 해석

Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.

/pdf/existence-of-minimal-models-for-varieties-of-log-general-43vm28b5my.pdf

Existence of minimal models for varieties of log general type

Dans son travail sur la log-concavite des multiplicites, Okounkov montre au passage que l'on peut associer un corps convexe a un systeme lineaire sur une variete projective, puis utiliser la geometrie convexe pour etudier ces systemes lineaires. Bien qu'Okounkov travaille essentiellement dans le cadre classique des fibres en droites amples, il se trouve que sa construction s'etend au cas d'un grand diviseur arbitraire. De plus, ce point de vue permet de rendre transparentes de nombreuses proprietes de base des invariants asymptotiques des systemes lineaires, et ouvre la porte a de nombreuses extensions. Le but de cet article est d'initier un developpement systematique de la theorie et de donner quelques applications et exemples.

/pdf/convex-bodies-associated-to-linear-series-2k3c3rzc0k.pdf

Convex bodies associated to linear series

We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of " movable curves " , which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive (1, 1)-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a good covering family, has non-negative Kodaira dimension.

/pdf/the-pseudo-effective-cone-of-a-compact-kahler-manifold-and-53afhv29x2.pdf

The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension

Using the techniques of [20] and [10], we prove that certain log forms may be lifted from a divisor to the ambient variety. As a consequence of this result, following [22], we show that: For any positive integer n there exists an integer r
 n
 such that if X is a smooth projective variety of general type and dimension n, then $\phi_{rK_X}\colon X\dasharrow\mathbb{P}(H^0(\mathcal{O}_{X}(rK_X)))$
 is birational for all r≥r
 n
 .

/pdf/boundedness-of-pluricanonical-maps-of-varieties-of-general-1xjxbmc2so.pdf

Boundedness of pluricanonical maps of varieties of general type

We prove the rational connectedness conjecture of V. V. Shokurov in [20] which, in particular, implies that the fibres of a resolution of a variety with divisorial log terminal singularities are rationally chain connected

/pdf/on-shokurov-s-rational-connectedness-conjecture-2o2a6u9zqs.pdf

On Shokurov's rational connectedness conjecture

We show that log canonical thresholds satisfy the ACC.

/pdf/acc-for-log-canonical-thresholds-y3ippbgqhn.pdf

Christopher D. Hacon

Papers

Existence of minimal models for varieties of log general type

Boundedness of pluricanonical maps of varieties of general type

On Shokurov's rational connectedness conjecture

ACC for log canonical thresholds

ACC for log canonical thresholds