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

## Forschungsseminar "Arithmetische Geometrie"

Wintersemester 2018/19

Das Forschungsseminar findet dienstags in der Zeit von 13:15 - 15:00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 3.006 (Haus 3, Erdgeschoss), statt.

 23.10.2018 Giulio Codogni (U Roma Tre) Title: Gauss map, singularities of the theta divisor and trisecants Abstract: The Gauss map is a finite rational dominant map naturally defined on the theta divisor of an irreducible principally polarised abelian varieties. In the first part of this talk, we study the degree of the Gauss map of the theta divisor of principally polarised complex abelian varieties. Thanks to this analysis, we obtain a bound on the multiplicity of the theta divisor along irreducible components of its singular locus. We spell out this bound in several examples, and we use it to understand the local structure of isolated singular points. We further define a stratification of the moduli space of ppavs by the degree of the Gauss map. In dimension four, we show that this stratification gives a weak solution of the Schottky problem, and we conjecture that this is true in any dimension. This is a joint work with S. Grushevsky and E. Sernesi. In the second part of this talk, we will study the relation between the Gauss map and trisecant of the Kummer variety. Fay's trisecant formula shows that the Kummer variety of the Jacobian of a smooth projective curve has a four dimensional family of trisecant lines. We study when these lines intersect the theta divisor of the Jacobian, and prove that the Gauss map of the theta divisor is constant on these points of intersection, when defined. We investigate the relation between the Gauss map and multisecant planes of the Kummer variety as well. This is a joint work with R. Auffarth and R. Salvati Manni. 30.10.2018 Nero Budur (KU Leuven) Title: Absolute sets and the Decomposition Theorem Abstract: The celebrated Monodromy Theorem states that the eigenvalues of the monodromy of a polynomial are roots of unity. In this talk we give an overview of recent results on local systems giving a generalization of the Monodromy Theorem. We end up with a conjecture of André-Oort type for special loci of local systems. If true in general, it would provide a simple conceptual proof for all semi-simple perverse sheaves of the Decomposition Theorem, assuming only the geometric case of perverse sheaves constructed from the constant sheaf (Beilinson-Bernstein-Deligne-Gabber). We prove the conjecture in rank one. Thus we have a new proof of the Decomposition Theorem for perverse sheaves constructed from rank one local systems. Joint work with Botong Wang. 06.11.2018 Peter Jossen (ETH Zürich) Title: Exponential Periods and Exponential Motives Abstract: I begin by explaining Nori's formalism and how to use it to construct abelian categories of motives. Then, following ideas of Katz and Kontsevich, I show how to construct a tannakian category of "exponential motives" by applying Nori's formalism to rapid decay cohomology, which one thinks of as the Betti realisation. This category of exponential motives contains the classical mixed motives à la Nori. We then introduce the de Rham realisation, as well as a comparison isomorphism with the Betti realisation. When k = IQ, this comparison isomorphism yields a class of complex numbers, "exponential periods", which includes special values of the gamma and the Bessel functions, the Euler-Mascheroni constant, and other interesting numbers which are not expected to be periods of classical motives. In particular, we attach to exponential motives a Galois group which conjecturally governs all algebraic relations among their periods. 13.11.2018 Thorsten Herrig (HU Berlin) Title: Fixed points and entropy of endomorphisms on complex tori Abstract: We investigate the fixed-point numbers and entropies of endomorphisms on complex tori. Motivated by an asymptotic perspective that has turned out in recent years to be so fruitful in Algebraic Geometry, we study how the number of fixed points behaves when the endomorphism is iterated. In this talk I show that the fixed-points function can have only three kinds of behaviour, and I characterize them in terms of the analytic eigenvalues. An interesting follow-up question is to determine criteria to decide of which type an endomorphism is. I will provide such criteria for simple abelian varieties in terms of the possible types of endomorphism algebras. The gained insight into the occurring eigenvalues can be applied to questions about the entropy of an endomorphism. I will give criteria for an endomorphism to be of zero or positive entropy and answer the important question whether the entropy can be the logarithm of a Salem number. 20.11.2018 kein Seminar 27.11.2018 Philipp Reichenbach (TU Berlin) Title: Vector bundles on elliptic curves and an associated Tannakian category Abstract: In 1957 Atiyah classified all vector bundles on elliptic curves for an algebraically closed ground field. Moreover, for the characteristic 0 case he completely described the multiplicative structure, i.e. the behavior of the tensor product. In this talk we review the essential results due to Atiyah and will interpret them in the light of Tannakian categories. Namely, allowing only morphisms of vector bundles on elliptic curves that respect the Harder-Narasimhan filtration leads to a neutral Tannakian category. For characteristic 0 we discuss some properties of the corresponding affine group scheme and give complete classifications of certain Tannakian sub-categories. Finally, some known results for the characteristic $p$ case are stated and questions for future research will be formulated. 04.12.2018 Jürg Kramer (HU Berlin) Title: Sup-norm bounds of automorphic forms Abstract: In our talk we will talk about new approaches for establishing optimal sup-norm bounds for Maass forms. 11.12.2018 Marco D'Addezio (FU Berlin) Title: Finiteness of perfect torsion points of an abelian variety and $F$-isocrystals Abstract: I will report on a joint work with Emiliano Ambrosi. Let $k$ be a field which is finitely generated over the algebraic closure of a finite field. As a consequence of the theorem of Lang-Néron, for every abelian variety over $k$ which does not admit any isotrivial abelian subvariety, the group of $k$-rational torsion points is finite. We show that the same is true for the group of torsion points defined on a perfect closure of $k$. This gives a positive answer to a question posed by Hélène Esnault in 2011. To prove the theorem we translate the problem into a certain question on morphisms of $F$-isocrystals. Then we handle it by studying the monodromy groups of the $F$-isocrystals involved. 18.12.2018 Christian Liedtke (TU München) Title: Rigid rational curves in positive characteristic Abstract: Rational curves are central to higher-dimensional algebraic geometry. If a rational curve “moves” on a variety, then the variety is uniruled and in characteristic zero, this implies that the variety has negative Kodaira dimension. Over fields of positive characteristic, varieties can be inseparably uniruled without having negative Kodaira dimension. However, I will show in my talk that in the case that a rational curve moves on a surface of non-negative Kodaira dimension, then this rational curve must be “very singular”. In higher dimensions, there is a similar result that is more complicated to state. I will also give examples that show the results are optimal. This is joint work with Kazuhiro Ito and Tetsushi Ito. 08.01.2019 Renjie Lyu (U Amsterdam) Title: A result of algebraic cycles on cubic hypersurfaces Abstract: The Chow group of algebraic cycles of a smooth projective variety is an important subject in algebraic geometry, which in general, is too massive to grasp. In this talk, we show that the Chow group of a smooth cubic hypersurface $X$ can be recovered by the algebraic cycles of its Fano variety of lines $F(X)$. It generalizes $M$. Shen’s previous work of 1-cycles on cubics. The proof relies on some birational geometry concerning the Hilbert square of cubic hypersurfaces recently studied by E. Shinder, S. Galkin and C. Voisin. As applications, when $X$ is a complex smooth 4-fourfold, the result we obtained could prove the integral Hodge conjecture for 1-cycles on the polarised hyper-Kähler variety $F(X)$. In the arithmetic aspect, C. Schoen addressed the integral analog of the Tate conjecture, which is predicted to be true for 1-cycles of any smooth projective variety defined over finite fields. We will show how to use our result to prove this conjecture for 1-cycles on the Fano variety $F(X)$ if $X$ is a smooth cubic 4-fold over a finitely generated field. 15.01.2019 Paola Frediani (U Pavia) Title: On the geometry of some special subvarieties contained in the Torelli locus Abstract: Almost all the examples of special subvarieties of $A_g$ contained in the Torelli locus are given by families of Jacobians of Galois covers of the projective line or of elliptic curves. They satisfy a sufficient condition that I will explain. I will first show that this condition is also necessary in the case of double covers of elliptic curves. In fact I will prove that the bielliptic locus is not totally geodesic for $g>3$. Finally I will discuss the geometry of some examples of Galois covers of elliptic curves yielding special subvarieties of $A_3$, which are fibered in totally geodesic curves and hence contain countably many Shimura curves. 22.01.2019 kein Seminar 29.01.2019 Jan Hendrik Bruinier (TU Darmstadt) Title: Arithmetic degrees of special cycles and derivatives of Siegel Eisenstein series Abstract: Let $V$ be a rational quadratic space of signature $(m,2)$. A conjecture of Kudla relates the arithmetic degrees of top degree special cycles on an integral model of a Shimura variety associated with $SO(V)$ to the coefficients of the central derivative of a Siegel Eisenstein series of genus $m+1$. We report on joint work with Tonghai Yang proving this conjecture for the coefficients of non-singular index $T$ under certain conditions on $T$. To this end we establish some new local arithmetic Siegel-Weil formulas at the archimedean and non-archimedean places. 05.02.2019 Hsueh-Yung Lin (U Bonn) Title: Algebraic approximations of compact Kähler manifolds of algebraic codimension 1 Abstract: Let $X$ be a compact Kähler manifold. The so-called Kodaira problem asks whether $X$ has arbitrarily small deformations to some projective varieties. In this talk, we will explain our solution to the Kodaira problem for compact Kähler manifolds of algebraic dimension $a(X) = dim(X) − 1$. This is partly joint work with B. Claudon and A. Höring. 12.02.2019 kein Seminar

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Wintersemester 2007/08
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