Alessio Corti

Twisted Bundles and Admissible Covers

Abstract We study the structure of the stacks of twisted stable maps to the classifying stack of a finite group G—which we call the stack of twisted G-covers, or twisted G-bundles. For a suitable group Gwe show that the substack corresponding to admissible G-covers is a smooth projective fine moduli space. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Explicit Birational Geometry of 3-Folds: Fano 3-fold hypersurfaces

We study the birational geometry of the 95 families of Fano 3-fold weighted hypersurfaces X = Xd ⊂ P(1, a1, a2, a3, a4), corresponding to the famous 95 families of K3 surfaces Xd ⊂ P(a1, a2, a3, a4) of Reid and Fletcher ([C3-f, §4] and [Fl, 13.3]). Our main aim is to prove a rigidity theorem for the general Xd in each family, by analogy with the famous theorem of Iskovskikh and Manin on the rigidity of the quartic 3-fold; on the way, we derive as much instruction and amusement as possible on topics in biregular and birational geometry of Fano 3-folds and Mori fibre spaces. While this paper uncovers an amazing wealth of new phenomena and methods of calculation, many of the basic questions remain open, and we spell some of these out.

G2-manifolds and associative submanifolds via semi-Fano 3-folds

We construct many new topological types of compact G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2. To achieve this we extend the twisted connected sum construction first developed by Kovalev and apply it to the large class of asymptotically cylindrical Calabi–Yau 3-folds built from semi-Fano 3-folds constructed previously by the authors. In many cases we determine the diffeomorphism type of the underlying smooth 7-manifolds completely; we find that many 2-connected 7-manifolds can be realised as twisted connected sums in a variety of ways, raising questions about the global structure of the moduli space of G_2-metrics. Many of the G_2-manifolds we construct contain compact rigid associative 3-folds, which play an important role in the the higher-dimensional enumerative geometry (gauge theory/calibrated submanifolds) approach to defining deformation invariants of G_2-metrics. By varying the semi-Fanos used to build different G_2-metrics on the same 7-manifold we can change the number of rigid associative 3-folds we produce.

Motivic decomposition and intersection Chow groups, I

For a quasiprojective variety S, we define a category CHM(S) of pure Chow motives over S. Assuming conjectures of Grothendieck and Murre, we show that the decomposition theorem holds in CHM(S). As a consequence, the intersection complex of S makes sense as an object IS of CHM(S). In the forthcoming part II we will give an unconditional definition of intersection Chow groups and study some of their properties.

Explicit birational geometry of 3-folds

Foreword 1. One parameter families containing three dimensional toric Gorenstein singularities K. Altmann 2. Nonrational covers of CPm x CPn J. Kollar 3. Essentials of the method of maximal singularities A. V. Pukhlikov 4. Working with weighted complete intersections A. R. Iano-Fletcher 5. Fano 3-fold hypersurfaces A. Corti, A. V. Pukhlikov and M. Reid 6. Singularities of linear systems and 3-fold birational geometry A. Corti 7. Twenty five years of 3-folds - an old persons view M. Reid.

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.

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 ℙ1s 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 G2–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.

A mirror theorem for toric stacks

We prove a Givental-style mirror theorem for toric Deligne–Mumford stacks

Mirror Symmetry and Fano Manifolds

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Quantum periods for 3-dimensional Fano manifolds

. This determines the genus-zero Gromov–Witten invariants of