Journal of Algebraic Geometry 

About: Journal of Algebraic Geometry is an academic journal published by American Mathematical Society. The journal publishes majorly in the area(s): Moduli space & Cohomology. It has an ISSN identifier of 1056-3911. Over the lifetime, 520 publications have been published receiving 19649 citations. The journal is also known as: Algebraic geometry.

Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties

Abstract: We consider families ${\cal F}(\Delta)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $\Delta$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed $n$-dimensional Newton polyhedron $\Delta$ in $n$-dimensional algebraic torus ${\bf T} =({\bf C}^*)^n$. If the family ${\cal F}(\Delta)$ defined by a Newton polyhedron $\Delta$ consists of $(n-1)$-dimensional Calabi-Yau varieties, then the dual, or polar, polyhedron $\Delta^*$ in the dual space defines another family ${\cal F}(\Delta^*)$ of Calabi-Yau varieties, so that we obtain the remarkable duality between two {\em different families} of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau $3$-folds. Our method allows to construct many new examples of Calabi-Yau $3$-folds and new candidats for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families ${\cal F}(\Delta)$ and ${\cal F}(\Delta^*)$.

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

Abstract: 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.

Erratum to “The homogeneous coordinate ring of a toric variety”

Abstract: This paper will introduce the homogeneous coordinate ring S of a toric variety X. The ring S is a polynomial ring with one variable for each one-dimensional cone in the fan ∆ determining X, and S has a natural grading determined by the monoid of effective divisor classes in the Chow group A n−1 (X) of X (where n = dim X). Using this graded ring, we will show that X behaves like projective space in many ways. The paper is organized into four sections as follows. In §1, we define the homogeneous coordinate ring S of X and compute its graded pieces in terms of global sections of certain coherent sheaves on X. We also define a monomial ideal B ⊂ S that describes the combinatorial structure of the fan ∆. In the case of projective space, the ring S is just the usual homogeneous coordinate ring C[x 0 ,. .. , x n ], and the ideal B is the \" irrelevant \" ideal x 0 ,. .. , x n. Projective space P n can be constructed as the quotient (C n+1 −{0})/C *. In §2, we will see that there is a similar construction for any toric variety X. In this case, the algebraic group G = Hom Z (A n−1 (X), C *) acts on an affine space C ∆(1) such that the categorical quotient (C ∆(1) − Z)/G exists and is isomorphic to X. The exceptional set Z is the zero set of the ideal B defined in §1. If X is simplicial (meaning that the fan ∆ is simplicial), then X ≃ (C ∆(1) − Z)/G is a geometric quotient, so that elements of C ∆(1) − Z can be regarded as \" homogeneous coordinates \" for points of X. For any toric variety, we will see in §3 that finitely generated graded S modules give rise to a coherent sheaves on X, and when X is simplicial, every coherent sheaf arises in this way. In particular, every closed subscheme of X is determined by a graded ideal of S. We will also study the extent to which this correspondence fails to be is one-to-one. Another feature of P n is that the action of P GL(n + 1, C) on P n lifts to an action of GL(n + 1, C) on C n+1 − {0}. In §4, we will see that …

Motivic Igusa zeta functions

Abstract: We define motivic analogues of Igusa's local zeta functions. These functions take their values in a Grothendieck group of Chow motives. They specialize to p-adic Igusa local zeta functions and to the topological zeta functions we introduced several years ago. We study their basic properties, such as functional equations, and their relation with motivic nearby cycles. In particular the Hodge spectrum of a singular point of a function may be recovered from the Hodge realization of these zeta functions.

A new six-dimensional irreducible symplectic variety

Abstract: There are three types of “building blocks” in the Bogomolov decomposition [B, Th.2] of compact Kahlerian manifolds with torsion c1, namely complex tori, CalabiYau varieties, and irreducible symplectic manifolds. We are interested in the last type, i.e. simply-connected compact Kahlerian manifolds carrying a holomorphic symplectic form which spans H. (The holonomy of a Ricci-flat Kahler metric is equal to Sp(r), hence these manifolds are hyperkahler [B].) The stock of available irreducible symplectic manifolds appears to be quite scarce, expecially if we think of the many examples of CalabiYau’s. Every known irreducible symplectic manifold is a deformation of one of the following varieties: the Hilbert scheme parametrizing zero-dimensional subschemes of a K3 of fixed length [B], the generalized Kummer variety parametrizing zero-dimensional subschemes of a complex torus of fixed length and whose associated 0-cycle sums up to 0 [B], the (10-dimensional) desingularization of the moduli space of rank-two semistable torsion-free sheaves on a K3 with c1 = 0, c2 = 4 constructed by the author [O1]. Briefly: all known examples are deformations of an irreducible factor in the Bogomolov decomposition of a moduli space of semistable sheaves on a surface with trivial canonical bundle or, as in the last case, of a symplectic desingularization of such a moduli space. This paper provides a new example in dimension 6: the manifold in question is an irreducible factor in the Bogomolov decomposition of a symplectic desingularization of a moduli space of sheaves on an abelian surface. To put our result in perspective we recall some results on moduli spaces of sheaves on a surface with trivial canonical bundle. Let X be such a surface and D an ample divisor on it: given a vector w ∈ H∗(X;Z), we let Mw(X,D) be the moduli space of D-semistable torsion-free sheaves F on X with Mukai vector