I’ll explain two opposing pieces of work: (1) Markman’s proof of theHodge conjecture for general Weil type abelian fourfolds of discriminant 1, and (2) Kontsevich’s tropical approach to finding a counterexample to the Hodgeconjecture for Weil type…
Categoria evento: Algebraic and Arithmetic Geometry Seminar
Hodge-to-singular correspondence – Mirko Mauri (IST Austria)
We show that the cohomology of moduli spaces of Higgs bundles decomposes in elementary summands depending on the topology of the symplectic singularities on a (fixed!) master object and/or the combinatorics of certain posets and lattice polytopes.…
(Non-archimedean) SYZ fibrations for Calabi-Yau hypersurfaces – Enrica Mazzon (University of Regensburg)
The SYZ conjecture is a conjectural geometric explanation of mirror symmetry. Based on this, Kontsevich and Soibelman proposed a non-archimedean approach, which led to the construction of non-archimedean SYZ fibrations by Nicaise-Xu-Yu. A recent…
Diophantine methods and S-unit equations – Samuel Le Fourn (Institut Fourier, Grenoble)
Baker’s method (based on linear forms in logarithms) and Runge’s method (based on the pigeonhole principle) both allow to bound heights of integral points on curves (or even varieties) in certain situations which turn out to be rather different. In…
More efficient algorithms using stack-theoretic weighted blow-ups – David Rydh (Stockholm)
Abramovich, Temkin and Wlodarczyk has recently given an easier and more efficient algorithm for resolution of singularities using stack-theoretic weighted blow-ups. Weighted blow-ups generalize root stacks and ordinary blow-ups but are far more…
Moduli of Higgs bundles and Hecke operators on surfaces – Olivier Schiffmann (CNRS and Université de Paris-Saclay)
We will introduce and describe an algebra $H(S)$ acting on the cohomology of various moduli spaces of sheaves on a smooth complex surface $S$. We will provide some application to a generalization of Markman’s theorem in the semistable (as opposed to…
Categorified Beauville-Laszlo theorem (and related problems) – Mauro Porta (Université de Strasbourg)
Sheaves of Azumaya algebras were introduced by Grothendieck to represent classes in the cohomological Brauer group of schemes, i.e. $Br(X) := H^2_{ét}(X;G_m)$, along the same lines every class in $H^1_{ét}(X;G_m)$ is representable by a line bundle…
Degenerations of Hilbert schemes and relative VGIT – Lars Halvard Halle (Università di Bologna)
Let G be a reductive group acting on a projective variety X. In Mumford’s Geometric Invariant Theory (GIT), the formation of a quotient in this situation depends on the choice of a G-linearized ample line bundle on X. In “Variation of GIT” (VGIT),…
Fano 4-folds with $b_2>12$ are products of surfaces – Cinzia Casagrande (Università di Torino)
Let $X$ be a smooth, complex Fano 4-fold, and $b_2$ its second Betti number. We will discuss the following result: if $b_2>12$, then X is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions $f:…
Bounded height problems and applications – Francesco Amoroso (Université de Caen)
We shall report on some recent joint works with D. Masser and U. Zannier.Let $C$ be a curve defined over $\mathbb{Q}$. In 1999, Bombieri, Masser, and Zannier proved a result which may be rephrased as a toric analogue of Silverman’s Specialization…