Let $f : Y \to X$ be a morphism of projective manifolds, with $(X,D)$ log smooth. Consider the exact sequence which defines the relative (logarithm) cotangent bundle (where $\Delta := f^{\ast} D$):

$$0 \to f^{\ast}\Omega_X^1(\log D) \to \Omega_Y^1(\log \Delta) \to \Omega_{Y/X}^1(\log \Delta) \to 0.$$

Let $\mathscr{M}$ be a line bundle on $Y$ with $H^0(Y, \mathscr{M}^m) \neq 0$, i.e., some multiple of $\mathscr{M}$ admits a section. Let $\psi : Z \to Y$ be a desingularization of the finite cyclic covering associated to taking roots of a section $\sigma \in H^0(Y, \mathscr{M}^m) \backslash \{ 0 \}$; so $H^0(Z, \psi^{\ast} \mathscr{M}) \neq 0$ also.

Set $h : = f \circ \psi$ and consider the pullback of the above exact sequence. I want to show that this induces a (decreasing) filtration $\mathscr{F}_k^{\bullet}$ ($k \geq 0$) of the bundle $\psi^{\ast} \Omega_Y^{\bullet}(\log \Delta))$ with

$$\mathscr{F}_k^{\bullet}/\mathscr{F}_{k+1}^{\bullet} \simeq \psi^{\ast} \left( \Omega_{Y/X}^{\bullet -k}(\log \Delta) \right) \otimes h^{\ast} \Omega_X^k(\log D).$$

I don't understand how such a filtration is constructed, is there some general result that I am missing? I apologise in advance if this result is well-known.

This problem arises from the study of hyperbolicity, specifically from the works of Eckhart Viehweg and Kang Zuo (2002, 2003). A more detailed treatment was given in section 2.2 of this article - https://arxiv.org/pdf/1809.05616.pdf

  • $\begingroup$ Welcome to Mathoverflow! Is the dot in the exponent just a variable, i.e. you don't mean to work with all the wedge powers at once? $\endgroup$
    – Will Sawin
    Jan 18, 2021 at 2:43
  • $\begingroup$ @WillSawin I assumed that it meant work with all the exterior powers at once. $\endgroup$
    – AshyK
    Jan 18, 2021 at 20:52

1 Answer 1


For $0 \to A \to B \to C \to 0$ an exact sequence of modules, we obtain a filtration on $\wedge^d B$ where $F^k / F^{k+1}$ is $\wedge^k A \otimes \wedge^{d-k} C$.

To see this, recall that $\wedge^d B$ is defined as the quotient of $B\otimes \dots \otimes B$ by relations of the form $x \otimes y = - y \otimes x$. We define $F^k $ to be the image in $\wedge^d A$ of the submodule generated by tensors $x_1 \otimes \dots \otimes x_d$ where at least $k$ of the factors $x_1,\dots, x_d$, come from $A$.

Using the relations, we can ensure that the first $k$ factors come from $A$, which gives a map to $\wedge^k A \otimes \wedge^{d-k} C$ by projecting the last $d-k$ factors from $C$ to $B$, and one can check this is an isomorphism.

Glancing at the paper, I think that they mean for the exact sequence to exist one $\cdot$ at a time, so this is a complete explanation, plugging in the terms of your exact sequence for $A,B, C$, but I didn't check 100%.


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