Logarithmic Cotangent And Tangent Sheaf And Bundle

Definition

Let \(X\) be a smooth complex algebraic variety with normal crossings divisor \(D\). Let \(U = X \setminus D\) and let \(j:U\hookrightarrow X\) be the open inclusion.

• The logarithmic cotangent sheaf on \(X\) is \(\Omega_X^1(\operatorname{log}D)\), the sheaf of algebraic one-forms with at worst logarithmic poles along \(D\).
• The logarithmic tangent sheaf on \(X\) is \(\Theta_X = T_X(-\operatorname{log}D) = \Hom(\Omega_X^1(\operatorname{log}D), \mathcal O_X)\). It consists of vector fields which vanish along \(D\); if \(D\) is given locally by \(z_1\cdot...\cdot z_k = 0\) then \(\Theta_X\) is the locally free \(\mathcal O_X\)-module generated by \[z_1 \frac{\partial}{\partial z_1},...,z_k\frac{\partial}{\partial z_k}, \frac{\partial}{\partial z_{k+1}},...,\frac{\partial}{\partial z_n}.\]