Positivity in algebraic geometry

Preface 1. Polarity 2. Conics and quadrics 3. Plane cubics 4. Determinantal equations 5. Theta characteristics 6. Plane quartics 7. Cremona transformations 8. Del Pezzo surfaces 9. Cubic surfaces 10. Geometry of lines Bibliography Index.

Classical Algebraic Geometry: A Modern View

K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.

Lectures on K3 Surfaces

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 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 covering family, has non negative Kodaira dimension.

The pseudo-effective cone of a compact K\"ahler manifold and varieties of negative Kodaira dimension

A cubic fourfold is a smooth cubic hypersurface of dimension four; it is special if it contains a surface not homologous to a complete intersection. Special cubic fourfolds form a countably infinite union of irreducible families 
$$C_d $$

 , each a divisor in the moduli space 
$$C $$

 of cubic fourfolds. For an infinite number of these families, the Hodge structure on the nonspecial cohomology of the cubic fourfold is essentially the Hodge structure on the primitive cohomology of a K3 surface. We say that this K3 surface is associated to the special cubic fourfold. In these cases, 
$$C_d $$

 is related to the moduli space 
$$N_d $$

 of degree d K3 surfaces. In particular, 
$$C $$

 contains infinitely many moduli spaces of polarized K3 surfaces as closed subvarieties. We can often construct a correspondence of rational curves on the special cubic fourfold parametrized by the K3 surface which induces the isomorphism of Hodge structures. For infinitely many values of d, the Fano variety of lines on the generic cubic fourfold of 
$$C_d $$

 is isomorphic to the Hilbert scheme of length-two subschemes of an associated K3 surface.

/pdf/special-cubic-fourfolds-36ijp91sw8.pdf

Special Cubic Fourfolds

Higher-dimensional algebraic geometry studies the classification theory of algebraic varieties. This very active area of research is still developing, but an amazing quantity of knowledge has accumulated over the past twenty years. The author?s goal is to provide an easily accessible introduction to the subject. The book covers preparatory and standard definitions and results, moves on to discuss various aspects of the geometry of smooth projective varieties with many rational curves, and finishes in taking the first steps towards Mori?s minimal model program of classification of algebraic varieties by proving the cone and contraction theorems. The book is well organized and the author has kept the number of concepts that are used but not proved to a minimum to provide a mostly self-contained introduction to graduate students and researchers.

Higher-dimensional algebraic geometry

/pdf/classical-algebraic-geometry-3wzv0dreho.pdf

Classical Algebraic Geometry

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/pdf/addendum-inegalites-numeriques-pour-les-surfaces-de-type-2g6xhyt97o.pdf

Addendum : « Inégalités numériques pour les surfaces de type général »

Sur la variete des espaces lineaires contenus dans une intersection complete

We construct a new 20-dimensional family of projective hyper-Kahler fourfolds and prove that they are deformationequivalent to the second punctual Hilbert scheme of a K3 surface of genus 12.

/pdf/hyper-kahler-fourfolds-and-grassmann-geometry-299tfl10dv.pdf

Olivier Debarre

Papers

Higher-dimensional algebraic geometry

Classical Algebraic Geometry

Addendum : « Inégalités numériques pour les surfaces de type général »

Sur la variete des espaces lineaires contenus dans une intersection complete

Hyper-Kähler fourfolds and Grassmann geometry