Searcharxiv⌕ Search

arXiv subjects

Simon Felten

Publications and source records attributed to Simon Felten.

7 recordsLinked to original sources
arXiv2026

Generically log smooth families via generators and relations

Let $f\colon X \to \mathbb{A}^1_t$ be an affine flat morphism of finite type, and let $V = f^{-1}(0)$. Then, we obtain a morphism of log schemes $f\colon (X|V) \to (\mathbb{A}^1_t|0)$. In this article, we develop algorithmic tools to study the log-geometric properties of $f$ by means of a presentation \[\Gamma(X,\mathcal{O}_X) = \Bbbk[t,x_1,\ldots,x_n]/(f_1,\ldots,f_r).\] We obtain similar tools for projective flat morphisms when the homogeneous coordinate ring is given by generators and relations. We provide an implementation of our algorithms in Macaulay2. In a slightly different direction, we give some results on the sheaf $\mathcal{LS}_V$ of log smooth structures on a toroidal crossing scheme $(V,\mathcal{P},\bar\rho)$.

math.AG↗
arXiv2023

Global logarithmic deformation theory

A classical problem in algebraic geometry is to construct smooth algebraic varieties with prescribed properties. In the approach via smoothings, one first constructs a degenerate scheme with the prescribed properties, and then shows the existence of a smooth variety degenerating to this scheme. Logarithmic geometry has given important new impulses to the second step of this approach, which we explore in this book. Degenerations, in particular in the context of mirror symmetry, often enjoy similar formal properties as smooth morphisms once considered from the logarithmic perspective. Logarithmic deformation theory has therefore become an effective tool for the construction of smoothings and the transfer of properties between smooth nearby fibers and the singular special fiber. The strongest existence result for deformations in classical algebraic geometry is the Bogomolov--Tian--Todorov theorem for Calabi--Yau varieties. A logarithmic variant, once established, constructs log smooth deformations. However, the logarithmic Bogomolov--Tian--Todorov theorem has resisted efforts to its proof for a while. Finally, a method to prove it was discovered in 2019 by Chan, Leung, and Ma. In this book, we explore this new approach to the logarithmic Bogomolov--Tian--Todorov theorem. We prove several variants of the abstract unobstructedness theorem, some of which are new and stronger than previously known results. We investigate its application to the global deformation theory of log smooth and mildly log singular spaces, obtaining unobstructedness results for log Calabi--Yau spaces, some log Fano spaces, and line bundles. Special care is taken to allow sufficiently mild log singularities, including all log singularities that appear in the Gross--Siebert construction of toric log Calabi--Yau mirror pairs.

math.AG↗
arXiv2021

Deformations of log Calabi-Yau pairs can be obstructed

We exhibit examples of pairs $(X,D)$ where $X$ is a smooth projective variety and $D$ is an anticanonical reduced simple normal crossing divisor such that the deformations of $(X,D)$ are obstructed. These examples are constructed via toric geometry.

math.AG↗
arXiv2020

Log Smooth Deformation Theory via Gerstenhaber Algebras

We construct a $k[[Q]]$-linear predifferential graded Lie algebra $L^*_{X/S}$ associated to a log smooth and saturated morphism $f: X \rightarrow S$ and prove that it controls the log smooth deformation functor. This provides a geometric interpretation of a construction by Chan-Leung-Ma whereof $L^*_{X/S}$ is a purely algebraic version. Our proof crucially relies on studying deformations of the Gerstenhaber algebra of polyvector fields and interestingly does not need to keep track of the log structure. The method of using Gerstenhaber algebras is closely related to recent developments in mirror symmetry.

math.AG↗
arXiv2019

Smoothing toroidal crossing spaces

We prove the existence of a smoothing for a toroidal crossing space under mild assumptions. By linking log structures with infinitesimal deformations, the result receives a very compact form for normal crossing spaces. The main approach is to study log structures that are incoherent on a subspace of codimension two and prove a Hodge-de Rham degeneration theorem for such log spaces which also settles a conjecture by Danilov. We show that the homotopy equivalence between Maurer-Cartan solutions and deformations combined with Batalin-Vilkovisky theory can be used to obtain smoothings. The construction of new Calabi-Yau and Fano manifolds as well as Frobenius manifold structures on moduli spaces are potential applications.

math.AG↗