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Alessio Corti

Publications and source records attributed to Alessio Corti.

36 records · Page 2Linked to original sources
arXiv2013

G_2-manifolds and associative submanifolds via semi-Fano 3-folds

We provide a significant extension of the twisted connected sum construction of G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction. Some of the main contributions of the paper are: (i) We correct, clarify and extend several aspects of the K3 "matching problem" that occurs as a key step in the twisted connected sum construction. (ii) We show that the large class of asymptotically cylindrical Calabi-Yau 3-folds built from semi-Fano 3-folds (a subclass of weak Fano 3-folds) can be used as components in the twisted connected sum construction. (iii) We construct many new topological types of compact G_2-manifolds by applying the twisted connected sum to asymptotically Calabi-Yau 3-folds of semi-Fano type studied in arXiv:1206.2277. (iv) We obtain much more precise topological information about twisted connected sum G_2-manifolds; one application is the determination for the first time of the diffeomorphism type of many compact G_2-manifolds. (v) We describe "geometric transitions" between G_2-metrics on different 7-manifolds mimicking "flopping" behaviour among semi-Fano 3-folds and "conifold transitions" between Fano and semi-Fano 3-folds. (vi) We construct many G_2-manifolds that contain rigid compact associative 3-folds. (vii) We prove that many smooth 2-connected 7-manifolds can be realised as twisted connected sums in numerous ways; by varying the building blocks matched we can vary the number of rigid associative 3-folds constructed therein. This leads to speculation that the moduli space of G_2-metrics on a given 7-manifold may consist of many different connected components.

math.DG↗
arXiv2013

Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds

We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We pay particular attention to a subclass of weak Fano 3-folds that we call semi-Fano 3-folds. Semi-Fano 3-folds satisfy stronger cohomology vanishing theorems and enjoy certain topological properties not satisfied by general weak Fano 3-folds, but are far more numerous than genuine Fano 3-folds. Also, unlike Fanos they often contain P^1s with normal bundle O(-1) + O(-1), giving rise to compact rigid holomorphic curves in the associated ACyl Calabi-Yau 3-folds. We introduce some general methods to compute the basic topological invariants of ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds, and study a small number of representative examples in detail. Similar methods allow the computation of the topology in many other examples. All the features of the ACyl Calabi-Yau 3-folds studied here find application in arXiv:1207.4470 where we construct many new compact G_2-manifolds using Kovalev's twisted connected sum construction. ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds are particularly well-adapted for this purpose.

math.AG↗
arXiv2012

Mirror Symmetry and Fano Manifolds

We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas.

math.AG↗
arXiv2012

New outlook on the Minimal Model Program, II

We prove that the finite generation of adjoint rings proved in [Cascini and Lazić] implies all the foundational results of the Minimal Model Program: the Rationality, Cone and Contraction theorems, the existence of flips, and termination of flips with scaling in the presence of a big boundary.

math.AG↗
arXiv2008

Introduction to the Minimal Model Program and the existence of flips

The first aim of this note is to give a concise, but complete and self-contained, presentation of the fundamental theorems of Mori theory - the nonvanishing, base point free, rationality and cone theorems - using modern methods of multiplier ideals, Nadel vanishing, and the subadjunction theorem of Kawamata. The second aim is to write up a complete, detailed proof of existence of flips in dimension n assuming the minimal model program with scaling in dimension n-1.

math.AG↗
arXiv2008

The Crepant Resolution Conjecture for Type A Surface Singularities

Let X be an orbifold with crepant resolution Y. The Crepant Resolution Conjectures of Ruan and Bryan-Graber assert, roughly speaking, that the quantum cohomology of X becomes isomorphic to the quantum cohomology of Y after analytic continuation in certain parameters followed by the specialization of some of these parameters to roots of unity. We prove these conjectures in the case where X is a surface singularity of type A. The key ingredient is mirror symmetry for toric orbifolds.

math.AG↗
arXiv2008

Computing Genus-Zero Twisted Gromov-Witten Invariants

Twisted Gromov-Witten invariants are intersection numbers in moduli spaces of stable maps to a manifold or orbifold X which depend in addition on a vector bundle over X and an invertible multiplicative characteristic class. Special cases are closely related to local Gromov-Witten invariants of the bundle, and to genus-zero one-point invariants of complete intersections in X. We develop tools for computing genus-zero twisted Gromov-Witten invariants of orbifolds and apply them to several examples. We prove a "quantum Lefschetz theorem" which expresses genus-zero one-point Gromov-Witten invariants of a complete intersection in terms of those of the ambient orbifold X. We determine the genus-zero Gromov-Witten potential of the type A surface singularity C^2/Z_n. We also compute some genus-zero invariants of C^3/Z_3, verifying predictions of Aganagic-Bouchard-Klemm. In a self-contained Appendix, we determine the relationship between the quantum cohomology of the A_n surface singularity and that of its crepant resolution, thereby proving the Crepant Resolution Conjectures of Ruan and Bryan-Graber in this case.

math.AG↗
arXiv2007

The Quantum Orbifold Cohomology of Weighted Projective Spaces

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental's heuristic argument, which relates small quantum cohomology to S^1-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small J-function, a generating function for certain genus-zero Gromov-Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. We also obtain formulas for the small J-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.

math.AG↗
arXiv2007

Lectures on flips and minimal models

This document contains notes from the lectures of Corti, Kollár, Lazarsfeld, and Mustaţă at the workshop ``Minimal and canonical models in algebraic geometry" at MSRI, Berkeley, April 2007. The lectures give an overview of the recent advances on canonical and minimal models of algebraic varieties obtained by Hacon--McKernan and Birkar--Cascini--Hacon--McKernan.

math.AG↗
arXiv2006

Hypergeometric Equations and Weighted Projective Spaces

We compute the Hodge numbers of the variation of Hodge structure of the middle cohomology (with compact support) of the Landau-Ginzburg model dual to a weighted projective space. We state a conjectural formula for the Hodge numbers of general hypergeometric variations. We show that a general Landau-Ginzburg fibre is birational to a Calabi-Yau variety if and only if a general anticanonical section of the weighted projective space is Calabi-Yau. We analyse the 104 weighted 3-spaces with canonical singularities, and show that a general anticanonical section is not a K3 surface exactly in those 9 cases where a generic Landau-Ginzburg fibre is an elliptic surface of Kodaira dimension 1.

math.AG↗
arXiv2003

Birational Geometry of 3-fold Mori Fibre Spaces

We study the geography and birational geometry of 3-fold conic bundles over P^2 and cubic del Pezzo fibrations over P^1. We discuss many explicit examples and raise several open questions. This paper was submitted to the proceedings of the "Fano conference" held in Torino in October 2002.

math.AG↗
arXiv2002

Weighted Grassmannians

Many classes of projective algebraic varieties can be studied in terms of graded rings. Gorenstein graded rings in small codimension have been studied recently from an algebraic point of view, but the geometric meaning of the resulting structures is still relatively poorly understood. We discuss here the weighted projective analogs of homogeneous spaces such as the Grassmannian Gr(2,5) and orthogonal Grassmannian OGr(5,10) appearing in Mukai's linear section theorem for Fano 3-folds, and show how to use these as ambient spaces for weighted projective constructions. This is a first sketch of a subject that we expect to have many interesting future applications.

math.AG↗
arXiv2001

Twisted bundles and admissible covers

The cherry on top of this stacky paper is the following: for any g>1 we give a finite group G such that the moduli space of connected admissible G-covers of genus g is a smooth, fine moduli space, which is a Galois cover of the moduli space of stable curves. The proof relies on methods introduced by Looijenga and Pikaart-De Jong, and on the theory of twisted G-covers, a theory announced without proofs in math.AG/9811059, section 3, and developed in the first few sections of this paper. This includes a moduli description of the normalization of the Harris-Mumford space of admissible covers, and a study of moduli of stable curves with abelian and non-abelian level structure.

math.AG↗
arXiv2000

Explicit birational geometry of 3-folds, Foreword

This is the Foreword to the book ``Explicit birational geometry of 3-folds'', edited by A. Corti and M. Reid, CUP Jun 2000, ISBN: 0 521 63641 8, with papers by K. Altmann, A. Corti, A.R. Iano-Fletcher, J. Kollár, A.V. Pukhlikov and M. Reid. One of the main achievements of algebraic geometry over the last 20 years is the work of Mori and others extending minimal models and the Enriques--Kodaira classification to 3-folds. This book is an integrated suite of papers centred around applications of Mori theory to birational geometry. Four of the papers (those by Pukhlikov, Fletcher, Corti, and the long joint paper Corti, Pukhlikov and Reid) work out in detail the theory of birational rigidity of Fano 3-folds; these contributions work for the first time with a representative class of Fano varieties, 3-fold hypersurfaces in weighted projective space, and include an attractive introductory treatment and a wealth of detailed computation of special cases.

math.AG↗
arXiv1995

Semistable 3-fold flips

We piece together ingredients, which are well known and documented in the literature, into a new proof of the existence of semistable 3-fold flips

alg-geom↗
arXiv1995

Del Pezzo surfaces over Dedekind schemes

Let S be a Dedekind scheme with fraction field K. We study the following problem: given a Del Pezzo surface X, defined over K, construct a distinguished integral model of X, defined over all of S. We provide a satisfactory answer if S is a smooth complex curve, and a conjectural answer if X is a cubic Del Pezzo surface over (nearly) arbitrary S.

alg-geom↗