In this article, we give numerical restrictions on the Chern classes of Ulrich bundles on higher-dimensional manifolds, which are inspired by the results of Casnati in the case of surfaces. As a by-product, we prove that the only projective manifolds whose tangent bundle is Ulrich are the twisted cubic and the Veronese surface. Moreover, we prove that the cotangent bundle is never Ulrich.

Projective manifolds whose tangent bundle is Ulrich

Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II. Sheaves on Surfaces: 5. Construction methods 6. Moduli spaces on K3 surfaces 7. Restriction of sheaves to curves 8. Line bundles on the moduli space 9. Irreducibility and smoothness 10. Symplectic structures 11. Birational properties Glossary of notations References Index.

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The geometry of moduli spaces of sheaves

Homogeneous Vector Bundles

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

Proj ective manifolds with ample tangent bundles

Let S be a smooth complex algebraic surface and let L be a divisor on S. It is well-known that the linear system IL + KsI, where Ks is the canonical divisor, plays an important role in understanding the geometry of S. Several authors (see, for example, [8], [11]) studied various properties of this system. Our paper treats the same subject, but the point of view is different from the works mentioned above. The philosophy underlying our approach is almost classical: Points on S (more generally, effective 0-cycles) in special position with respect to IL + KsI (see definition below) contain information about the geometry of S. This point of view was recently revived in [5] (also [10]). In fact, this paper contains the technique of going from the effective 0-cycle Z in special position with respect to IL + KsI to the geometry of S. For the sake of completeness we give a definition and state a theorem which can be found in [5], [10].

Projective manifolds whose tangent bundle is Ulrich

References

The geometry of moduli spaces of sheaves

Homogeneous Vector Bundles

Higher-dimensional algebraic geometry

Proj ective manifolds with ample tangent bundles

Vector bundles of rank 2 and linear systems on algebraic surfaces

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