Fakultät für Mathematik: Seminar on complex geometry, RUB

Seminar on complex geometry

Wednesday 12.03.2025

10:00 am IA 1/109

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Elizaveta Vishnyakova (Belo Horizonte), About graded coverings

Abstract: In geometry, there is a well-known notion of a covering space. A classical example is the following universal covering: $p : R \to S^{1}$ , $t \mapsto \exp (i t)$ . Another example we can find in algebra: a flat covering or torsion-free covering in the theory of modules over rings. All of these covers satisfy some common universal properties.

In the paper ``Super Atiyah classes and obstructions to splitting of supermoduli space'', Donagi and Witten suggested a construction of a first obstruction class for splitting a supermanifold. It appeared that an infinite prolongation of the Donagi-Witten construction satisfies the universal properties of a covering. In other words, this is a covering of a supermanifold in the categorical category of graded manifolds corresponding to the non-trivial homomorphism $Z \to Z_{2}$ . In addition, the space of infinite jets can also be regarded as a covering of a (super)manifold in the category of graded manifolds corresponding to the homomorphism $Z \times Z_{2} \to Z_{2}$ , $(m, \bar{n}) \mapsto \bar{n}$ . (For usual manifolds this homomorphism is trivial $Z \to 0$ .) Another example is a multiplicity-free covering of a graded manifold. In fact, similar constructions have appeared before in Poisson geometry. In summary, covers in the category of graded manifolds we call graded. Our talk is devoted to the current state of the theory of graded coverings: general idea, particular examples, notion of a deck transformation group.

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20.11.2024:
Maxim Kukol (RUB), Symplectic Reduction for Actions of Selective Groups on Homogeneous Bounded Domains
Abstract: Let $X$ be a homogeneous bounded domain in $C^{d}$ . In this talk we will look at Hamiltonian actions of selective groups $H$ of the automorphism group of $X$ with momentum map $μ$ and show that the symplectic reduction $μ^{- 1} (0) / H$ is a Stein manifold and biholomorphic to $(H^{C} \cdot X) / H^{C}$ .
12.07.2024, Joint Seminar on Complex Algebraic Geometry and Complex Analysis:
Stéphanie Cupit-Foutou (Bochum), Toric quantum stacks: some review, a generalisation
Bingxiao Liu (Köln), Toeplitz-Fubini-Study forms and lowest eigenvalues of Toeplitz operators
Jörg Schürmann (Münster), Equivariant toric geometry and Euler-Maclaurin formulae
Further information can be found on the following webpage: (link).
23.05.2024:
Christian Zöller (RUB), Der lokale Struktursatz für Hamiltonsche Wirkungen auf kompakten Kähler-Mannigfaltigkeiten
17.04.2024:
Shang Li (Paris), Wonderful compactification over an arbitrary base scheme
Abstract: Wonderful compactifications of adjoint reductive groups over an algebraically closed field play an important role in algebraic geometry and representation theory. In this talk, we will construct an equivariant compactification for adjoint reductive groups over arbitrary base schemes, which parameterize classical wonderful compactifications of De Concini and Procesi as geometric fibers. Our construction is based on a variant of the Artin–Weil method of birational group laws. In particular, our construction gives a new intrinsic construction of wonderful compactifications. If time permits, we will also discuss several applications of our compactification in the study of torsors under reductive group schemes.
17.01.2024
Johanna Bimmermann (RUB), Capacities of disk tangent bundles
Abstract: In general rather little is known about symplectic capacities of open subsets of cotangent bundles. In this talk I will discuss methods to compute the Gromov width and the Hofer—Zehnder capacity of disk tangent bundles and demonstrate these methods for certain homogenous spaces (namely spheres and projective spaces). The key observation is that in these cases the disk bundle sits as complement of a divisor inside a coadjoint orbit. This makes it possible to explicitly construct symplectic embeddings of balls for the lower bound and find non-vanishing Gromov—Witten invariants for the upper bound.

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