Humboldt-Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät
Institut für Mathematik

## Forschungsseminar "Algebraische Geometrie"

Wintersemester 2019/20

Das Forschungsseminar findet mittwochs in der Zeit von 13:00 - 15:00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 3.007, statt.

 23.10.2019 Víctor González Alonso (Leibniz Universität Hannover) Title: Subbundles of the Hodge bundle, fibred surfaces and the Coleman-Oort conjecture. Abstract: The Hodge bundle of a semistable family of complex projective curves has two nested subbundles: the flat unitary subbundle (spanned by flat sections with respect to the Gauss-Manin connection), and the kernel of the Higgs field. The latter contains the flat subbundle, but it was not clear how strict the inclusion could be. In this talk I will show how to estimate the ranks of both subbundles and construct families where they are arbitrarily different. I will also discuss some implications of this result for the classification of fibred surfaces, as well as some connections with the Coleman-Oort conjecture on the (non-)existence of totally geodesic subvarieties in the Torelli locus of principally polarized abelian varieties. This is joint work with Sara Torelli. 30.10.2019 Andrey Soldatenkov (Humboldt-Universität zu Berlin) Title: André motives of projective hyperkähler manifolds. Abstract: A hyperkähler manifold is a compact simply-connected Kähler manifold, such that its space of holomorphic 2-forms is spanned by a symplectic form. Hyperkähler manifolds form an important class of varieties of Kodaira dimension zero, along with the complex tori and Calabi-Yau varieties. From the Hodge-theoretic point of view, hyperkähler manifolds are close to complex tori, the link being provided by the Kuga-Satake construction. We will discuss how to lift this construction to the category of André motives. This has several applications: we will see that for most of the known families of hyperkähler manifolds all Hodge cycles are absolute, and the Mumford-Tate conjecture holds for the even degree cohomology. 06.11.2019 This week the seminar is replaced by the Brill-Noether Workshop of the AG+ Semester. 13.11.2019 Ignacio Barros (Northeastern University Boston) Title: On product identities and the Chow ring of holomorphic symplectic manifolds. Abstract: We propose a series of basic conjectural identities in the Chow rings of hyperkähler varieties of K3 type that generalizes in higher dimensions a set of key properties of cycles on a K3 surface. We prove that they hold for the Hilbert scheme of n points on a K3 surface. The talk is based on joint work with Laure Flapan, Alina Marian, and Rob Silversmith. 20.11.2019 Ekaterina Amerik (Université de Paris-Sud) Title: On the characteristic foliation. Abstract: Let $$X$$ be a holomorphic symplectic manifold and $$D$$ a smooth hypersurface in $$X$$. Then the restriction of the symplectic form on $$D$$ has one-dimensional kernel at each point. This distribution is called the characteristic foliation. I shall survey a few results concerning the possible Zariski closure of a general leaf of this foliation by myself and Campana, myself and my former student L. Guseva, and more recently by my student R. Abugaliev. Alexandru Dimca (Université de Nice-Sophia Antipolis) Title: On two invariants of plane curves. Abstract: Let $$C$$ be a reduced curve in the complex projective plane. The first numerical invariant we consider is the global Tjurina number $$\tau(C)$$, which is the degree of the singular scheme of $$C$$. The second invariant is the minimal degree $$mdr(C)$$ of a non trivial logarithmic vector field along $$C$$. These invariants enter in a classical inequality due to A. du Plessis and C.T.C. Wall. In the talk we will survey a number of results and of open questions related to these invariants, in particular their relation with free curves and Terao's conjecture on line arrangements. 27.11.2019 Erwan Rousseau (Université d'Aix-Marseille and FRIAS) Title: Automorphisms of foliations. Abstract: We will discuss in various contexts the transverse finiteness of the group of automorphisms/birational transformations preserving a holomorphic foliation. This study provides interesting consequences for the distribution of entire curves on manifolds equipped with foliations and suggest some generalizations of Lang’s exceptional loci to non-special manifolds, in the analytic or arithmetic setting. This is a work in progress with F. Lo Bianco, J.V. Pereira and F. Touzet. 04.12.2019 Montag, 09.12., 13:15 Uhr Special Colloquium (room 1.013): Caucher Birkar (University of Cambridge) Title: Applications of algebraic geometry. Abstract: Algebraic geometry occupies a central place in modern mathematics. It has deep connections with various parts of mathematics. It is also deeply related to mathematical physics and has found applications in a wide range of topics. In this talk I will introduce some elements of algebraic geometry and then discuss some applications. About the speaker: Caucher Birkar is an iranian-kurdish mathematician who is Professor at the University of Cambridge. His fundamental contribution to algebraic geometry, in particular to the classification of algebraic varieties have been recognized with several important prizes, culminating with the Fields Medal in 2018. His talk will be designed for a general audience. Everbody is most welcome. 11.12.2019 Frank Gounelas (Technische Universität München) Title: Curves on K3 surfaces. Abstract: Bogomolov and Mumford proved that every projective K3 surface contains a rational curve. Since then, a lot of progress has been made by Bogomolov, Chen, Hassett, Li, Liedtke, Tschinkel and others, towards the stronger statement that any such surface in fact contains infinitely many rational curves. In this talk I will present joint work with Xi Chen and Christian Liedtke completing the remaining cases of this conjecture over the complex numbers, reproving some of the main previously known cases more conceptually and extending the result to arbitrary genus. 18.12.2019 Luca Cesarano (Universität Bayreuth) Title: On the canonical map of smooth ample divisors in abelian varieties. Abstract: The behavior and the degree of the Gauss map of the Theta divisor in a principal polarization have been widely studied. In the case of a general principally polarized abelian variety J of dimension g, the Theta Divisor is smooth, and the Gauss map is a dominant morphism of degree precisely g! which coincides with the canonical map. However, very little is known about the degree and the behavior of the canonical map of a general smooth ample divisor in the polarization class of a general non-principally polarized abelian variety. In this talk, we will give an overview of some open questions, conjectures, and recent progress in this research topic. In the case of general non-principally polarized abelian 3-folds, we will see that the canonical map turns out to be birational, and at least in some cases, a holomorphic embedding. 08.01.2020 Frank Sottile (Texas A&M University) Title: Galois groups in Enumerative Geometry and Applications. Abstract: In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem. I will describe this background and discuss some work in a long-term project to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry. A main focus is to understand Galois groups in the Schubert calculus, a well-understood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry. 15.01.2020 Margherita Lelli-Chiesa (Università degli studi Roma Tre) Title: Genus two curves on abelian surfaces. Abstract: Let $$(S,L)$$ be a general $$(d_1,d_2)$$-polarized abelian surfaces. The minimal geometric genus of any curve in the linear system $$|L|$$ is two and there are finitely many curves of such genus. In analogy with Chen's results concerning rational curves on K3 surfaces, it is natural to ask whether all such curves are nodal. In the seminar I will prove that this holds true if and only if $$d_2$$ is not divisible by $$4$$. In the cases where $$d_2$$ is a multiple of $$4$$, I will construct curves in $$|L|$$ having a triple, $$4$$-tuple or $$6$$-tuple point, and show that these are the only types of unnodal singularities a genus $$2$$ curve in $$|L|$$ may acquire. This is joint work with A. L. Knutsen. 17.01.2020 A meeting for Herbert Kurke's 80th birthday. 22.01.2020 Massimo Bagnarol (SISSA Trieste) Title: Betti numbers of stable map spaces to Grassmannians. Abstract: Being the basis of Gromov-Witten theory, Kontsevich's moduli spaces of stable maps are important in different areas of mathematics and theoretical physics. For this reason, it is interesting to study their cohomology. In fact, determining their Betti numbers is already a nontrivial problem. In this talk, I will present a method for computing the Betti/Hodge numbers of moduli spaces of genus $$0$$ stable maps to a Grassmann variety $$G(r,V)$$. First, by using the combinatorial properties of these spaces, I will show that the problem can be reduced to the computation of the Hodge numbers of the open locus parametrizing maps from nonsingular curves. Then, I will show how the latter can be explicitly calculated, by means of a suitable Quot scheme compactification of the space of morphism of fixed degree from the projective line to $$G(r,V)$$. 29.01.2020 05.02.2020 Kieran O'Grady (Università degli studi di Roma "La Sapienza") Title: Modular sheaves on HK varieties. Abstract: Since moduli of sheaves on K3 surfaces play a key role in Algebraic Geometry, and since K3's are the two dimensional hyperkähler (HK) manifolds, it is natural to investigate moduli of sheaves on higher dimensional HK's. We propose to focus attention on (coherent) torsion free sheaves on a HK variety $$X$$ whose discriminant in $$H^4(X)$$ satisfies a certain condition. These are the modular sheaves of the title. For example a sheaf whose discriminant is a multiple of $$c_2(X)$$ is modular. For HK's which are deformations of the Hilbert square of a K3 we prove an existence and uniqueness result for slope-stable vector bundles with certain ranks, $$c_1$$ and $$c_2$$. As a consequence we get that the period map from the moduli space of Debarre-Voisin varieties to the relevant period space is birational. 12.02.2020 Ulrike Riess (ETH Zürich) Title: Base loci of big and nef line bundles on irreducible symplectic varieties. Abstract: In the first part of this talk, I give a complete description of the divisorial part of the base locus of big and nef line bundles on irreducible symplectic varieties (under certain conditions). This is a generalization of well-known results of Mayer and Saint-Donat for K3 surfaces. In the second part, I will present what is currently known on the non-divisorial part, including the results of an ongoing cooperation with Daniele Agostini. Shengyuan Zhao (Université de Rennes I) Title: Birational Kleinian groups and birational structures. Abstract: A classical Kleinian group is by definition a subgroup of PGL(2,C) which preserves an open set and which acts properly discontinuously cocompactly on this open set. We consider the same problem for higher dimensional complex algebraic varieties, i.e. we replace the projective line with a projective variety and PGL(2,C) with the group of birational transformations. We give a classification in dimension two. The proof is an interplay between birational geometry of surfaces, representations of Kahler groups and holomorphic foliations. The main difficulty lies in the fact that the group of birational transformations is very large.

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