Abstract: Abstract: Moduli spaces of polarized Enriques surfaces have several components, even if one fixes the degree of the polarization. In this talk I will discuss results concerning detecting irreducibile components and proving their unirationality. This is work in progress in collaboration with Th. Dedieu, C. Galati and A. Title: Volume-preserving birational maps of 3-fold Mori fibred Calabi-Yau pairs. Title: A simplicity criterion for normal isolated singularities. Abstract: The link of an isolated singular point of a complex variety is an analytic invariant of the singularity.

It is natural to ask how much information the link carries about the singularity; for instance, the link of a smooth point is a sphere, and one can ask whether the converse is true. Work of Mumford and Brieskorn has shown that this is the case for normal surface singularities but not in higher dimensions. Recently, McLean asked whether more structure on the link may provide a way to characterize smooth points. In this talk, I will discuss how CR geometry can be used to define a link-theoretic invariant of singularities that distinguishes smooth points. The proof relies on a partial solution to the complex Plateau problem.

Title: Polar Geometry, generalizations and applications. Abstract: this talk will be an attempt at conveying the importance of projective geometry in more applied settings. Polar classes are very classical objects in projective geometry. They have recently been rediscovered in connection with interesting applications of algebraic methods in imaging and variety sampling.

I will try to explain these connections and give some new generalizations. Paltin Ionescu , University of Ferrara, Italy. Title: A remark on boundedness of manifolds embedded with small codimension. Abstract: For embedded projective manifolds of small codimension we give an explicit bound for their degree, depending on the Castelnuovo--Mumford regularity of their structure sheaf.

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As an application, we obtain bounds for the degree of such manifolds whose structure sheaf is arithmetically Cohen--Macaulay in a weak sense and whose canonical map is not birational. Title: Positivity of cotangent sheaves for projective varieties with trivial canonical class. Abstract: Let X be a complex projective manifold. If X is not covered by rational curves, a theorem of Miyaoka tells us that the restriction of the cotangent sheaf to a sufficiently positive general complete intersection curve is nef. In this talk I will explain why this theorem can not be improved for projective manifolds with trivial canonical class.

This is based on joint work with Thomas Peternell. Title: Moduli of curves on Enriques surfaces. Abstract: Curves on K3 surfaces have been studied a lot and play a relevant role in the study of the moduli space of curves. By contrast, not so much is known about curves on Enriques surfaces, except that the rich geometry of the surface in particular the existence of many elliptic pencils make curves on Enriques surfaces quite special from the point of view of Brill-Noether theory.

Moreover, the existence of a non-trivial 2-torsion bundle on the surface the canonical bundle endows every curve on an Enriques surface with a natural structure of a Prym curve.

I will report on joint work in collaboration with C. Ciliberto, Th.

- Constructions and classifications of projective Poisson varieties?
- Ten!
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Dedieu and C. Galati, where we in particular prove that, with a few exceptions, a general Prym curve lying on an Enriques surface lies on a unique such surface. Conic bundles with big discriminant loci , Izv. Nauk Ser.

A remark on the topology of cyclic coverings of algebraic varieties , Boll. A 5 , Tome 18 no. Algebra , Tome 31 no. Some special Cremona transformations , Amer. Notes Tome 6 , pp. On the degrees of Fano four-folds of Picard number 1 , J. Reine Angew. Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1 , Ann. Fourier Grenoble , Tome 57 no. Birationality of the tangent map for minimal rational curves , Asian J.

## Constructions and classifications of projective Poisson varieties

Severi varieties and their varieties of reductions , J. A Barth-type theorem for branched coverings of projective space , Math. We then describe several constructions of Poisson threefolds, outlining the classification in the regular case, and the case of rank-one Fano threefolds such as projective space. We close with a discussion of log symplectic manifolds with simple normal crossings degeneracy divisor, including a new proof of the classification in the case of rank-one Fano manifolds.

This paper is an introduction to the geometry of holomorphic Poisson structures, i. Poisson brackets on the ring of holomorphic or algebraic functions on a complex manifold or algebraic variety and sometimes on more singular objects, such as schemes and analytic spaces. The theme for the course was how the methods of algebraic geometry can be used to construct and classify Poisson brackets. Hence, this paper also serves a second purpose: it is an overview of results on the classification of projective Poisson manifolds that have been obtained by several authors over the past couple of decades, with some added context for the results and the occasional new proof.

However, there are many situations in which one naturally encounters Poisson brackets that are actually holomorphic:.

- 4-folds with numerically effective tangent bundles and second Betti numbers greater than one.
- 4-folds with numerically effective tangent bundles and second Betti numbers greater than one.
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- A classification theorem on Fano bundles.
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While the two settings have much in common, there are also many important differences. These differences are a consequence of the rigidity of holomorphic and algebraic functions, and they will play a central role in our discussion. The problem is that a complex manifold X may have very few global holomorphic functions; for instance, if X is compact and connected, then the maximum principle implies that every holomorphic function on X is constant.

Thus, in order to define the bracket on X , we must define it in local patches which can be glued together in a globally consistent way. In other words, we should replace the ring of global functions with the corresponding sheaf:. The Jacobi identity for the bracket is equivalent to the condition. The bivector can then be restricted to the leaf, and inverted to obtain a holomorphic symplectic form.

Thus, X has a natural foliation by holomorphic symplectic leaves.

## A classification theorem on Fano bundles

This foliation will typically be singular, in the sense that there will be leaves of many different dimensions. Indeed, the powerful tools of algebraic geometry give much tighter control over the local and global behaviour of holomorphic Poisson structures. Thus, our aim is to give some introduction to how an algebraic geometer might think about Poisson brackets. We will focus on the related problems of construction and classification : how do we produce holomorphic Poisson structures on compact complex manifolds, and how do we know when we have found them all? The paper is organized as a sort of induction on dimension.

We begin in Sect.

In Sect. We highlight the intriguing phenomenon of excess dimension that is commonplace for these loci, as formulated in a conjecture of Bondal. We close in Sect. Although the material in this section was not covered in the lecture series, the author felt that it should be included here, given the focus on classification.

itlauto.com/wp-includes/require/4662-override-application-root.php The lecture series on which this article is based was intended for an audience that already has some familiarity with Poisson geometry, but has potentially had less exposure to algebraic or complex geometry. We have tried to keep this article similarly accessible. On the other hand, we hope that because of the focus on examples, experts in algebraic geometry will also find this article useful, both as an introduction to the geometry of Poisson brackets, and as a guide to the growing literature on the subject.

There are essentially two types of classifications: local and global. In the local case, one is looking for nice local normal forms for Poisson brackets—essentially, coordinate systems in which the Poisson bracket takes on a simple standard form. While these issues will come up from time to time, they will not be the main focus of this paper.