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商品描述
This monograph provides the first systematic treatment of the logarithmic Bogomolov-Tian-Todorov theorem. Providing a new perspective on classical results, this theorem guarantees that logarithmic Calabi-Yau spaces have unobstructed deformations. Part I develops the deformation theory of curved Batalin-Vilkovisky calculi and the abstract unobstructedness theorems which hold in quasi-perfect curved Batalin-Vilkovisky calculi. Part II presents background material on logarithmic geometry, families of singular log schemes, and toroidal crossing spaces. Part III establishes the connection between the geometric deformation theory of log schemes and the purely algebraic deformation theory of curved Batalin-Vilkovisky calculi. The last Part IV explores applications to the Gross-Siebert program, to deformation problems of log smooth and log toroidal log Calabi-Yau spaces, as well as to deformations of line bundles and deformations of log Fano spaces. Along the way, a comprehensive introduction to the logarithmic geometry used in the Gross-Siebert program is given. This monograph will be useful for graduate students and researchers working in algebraic and complex geometry, in particular in the study of deformation theory, degenerations, moduli spaces, and mirror symmetry.
商品描述(中文翻譯)
這本專著首次系統性地探討了對數的 Bogomolov-Tian-Todorov 定理。該定理提供了對經典結果的新視角，保證了對數 Calabi-Yau 空間具有無障礙的變形。
第一部分發展了彎曲 Batalin-Vilkovisky 計算的變形理論以及在準完美彎曲 Batalin-Vilkovisky 計算中成立的抽象無障礙性定理。第二部分介紹了對數幾何、奇異對數方案的族以及環形交叉空間的背景材料。第三部分建立了對數方案的幾何變形理論與彎曲 Batalin-Vilkovisky 計算的純代數變形理論之間的聯繫。最後的第四部分探討了對 Gross-Siebert 計畫的應用、對對數光滑和對數環形 Calabi-Yau 空間的變形問題，以及對線束和對數 Fano 空間的變形。在此過程中，還提供了對 Gross-Siebert 計畫中使用的對數幾何的全面介紹。
這本專著將對研究代數幾何和複幾何的研究生和研究人員特別有用，尤其是在變形理論、退化、模空間和鏡像對稱的研究中。
作者簡介
Simon Felten received his PhD in 2021 from the University of Mainz. He has since been a visitor at the Institut des Hautes Études Scientifiques and a postdoctoral researcher at Columbia University and the University of Oxford. His main research interests are logarithmic algebraic geometry and its applications to singularities, infinitesimal deformation theory, and the construction of algebraic varieties
作者簡介(中文翻譯)
西門·費爾滕於2021年在美因茨大學獲得博士學位。此後，他曾在高等科學研究所（Institut des Hautes Études Scientifiques）擔任訪問學者，並在哥倫比亞大學和牛津大學擔任博士後研究員。他的主要研究興趣包括對數代數幾何及其在奇異性、無窮小變形理論和代數簇構造中的應用。
