Given a smooth, projective variety X and an effective divisor \(D\,\subseteq \, X\), it is well-known that the (topological) obstruction to the deformation of the fundamental class of D as a Hodge class, lies in \(H^2({{\,\mathrm{{\mathcal {O}}}\,}}_X)\). In this article, we replace \(H^2({{\,\mathrm{{\mathcal {O}}}\,}}_X)\) by \(H^2_D({{\,\mathrm{{\mathcal {O}}}\,}}_X)\) and give an analogous topological obstruction theory. We compare the resulting local topological obstruction theory with the geometric obstruction theory (i.e., the obstruction to the deformation of D as an effective Cartier divisor of a first order infinitesimal deformations of X). We apply this to study the jumping locus of families of linear systems and the Noether–Lefschetz locus. Finally, we give examples of first order deformations \(X_t\) of X for which the cohomology class [D] deforms as a Hodge class but D does not lift as an effective Cartier divisor of \(X_t\).

给定一个光滑的射影变数X和一个有效的除数\（D \，\ subseteq \，X \），众所周知，D的基本类（作为Hodge类）的变形的（拓扑）障碍在于在\（H ^ 2（{{\，\ mathrm {{\ mathcal {O}}} \，}} _ X）\）中。在本文中，我们将\（H ^ 2（{{\，\ mathrm {{\ mathcal {O}}} \，}} _ X）\）替换为\（H ^ 2_D（{{\，\ mathrm {{ \ mathcal {O}}} \，}} _ X）\）并给出类似的拓扑障碍理论。我们将产生的局部拓扑阻塞理论与几何阻塞理论进行比较（即，D的变形作为D阶一阶无穷小变形的有效Cartier除数的形式）X）。我们将其用于研究线性系统族的跳跃轨迹和Noether–Lefschetz轨迹。最后，我们给出X的一阶变形\（X_t \）的示例，其同调类[ D ]变形为Hodge类，但D不能提升为\（X_t \）的有效Cartier除数。