arXiv subjects
Publications and source records attributed to Alexis Aumonier.
We propose an orthogonal approach to the stable homotopy type of spaces of holomorphic maps to projective space. We study the Weiss towers of the unitary functors of holomorphic and continuous maps to $\mathbb{P}(V)$, and show that the former is polynomial and completely compute the latter. As an application we give a new proof of a stable splitting of Cohen--Cohen--Mann--Milgram.
We explain how results comparing the homology of spaces of algebraic and continuous maps to projective spaces can be leveraged to compare moduli stacks of families of algebraic and continuous maps.
We show that the spaces of holomorphic and continuous maps from a smooth complex projective variety to a projective space have the same homology in a range depending on the degree of the maps.
We study the space of smooth marked hypersurfaces in a given linear system. Specifically, we prove a homology h-principle to compare it with a space of sections of an appropriate jet bundle. Using rational models, we compute its rational cohomology in a range of degrees, and deduce a homological stability result for hypersurfaces of increasing degree. We also describe the Hodge weights on the stable cohomology, and thereby connect our topological result to motivic results of Howe.
We study the locus of smooth hypersurfaces inside the Hilbert scheme of a smooth projective complex variety. In the spirit of scanning, we construct a map to a continuous section space of a projective bundle, and show that it induces an isomorphism in integral homology in a range of degrees growing with the ampleness of the hypersurfaces. When the ambient variety is a curve, this recovers a result of McDuff about configuration spaces. We compute the rational cohomology of the section space and exhibit a phenomenon of homological stability for hypersurfaces with first Chern class going to infinity. For simply connected varieties, the rational cohomology is shown to agree with the stable cohomology of a moduli space of hypersurfaces, with a peculiar tangential structure, as studied by Galatius and Randal-Williams.
We compare spaces of non-singular algebraic sections of ample vector bundles to spaces of continuous sections of jet bundles. Under some conditions, we provide an isomorphism in homology in a range of degrees growing with the jet ampleness. As an application, when $\mathcal{L}$ is a very ample line bundle on a smooth projective complex variety, we prove that the rational cohomology of the space of non-singular algebraic sections of $\mathcal{L}^{\otimes d}$ stabilises as $d \to \infty$ and compute the stable cohomology. We also prove that the integral homology does not stabilise using tools from stable homotopy theory.