[Paper Review] Direct image of logarithmic complexes and infinitesimal invariants of cycles

Morihiko Saito|ArXiv.org|Jun 1, 2005

Algebraic Geometry and Number Theory14 references5 citations

TL;DR

This paper establishes a filtered decomposition theorem for the direct image of logarithmic de Rham complexes on projective morphisms with normal crossing divisors, showing it splits as a direct sum of filtered logarithmic complexes associated to variations of Hodge structures. The key contribution is a natural definition of total infinitesimal invariants of algebraic cycles via cohomology of these filtered complexes, which are equivalent to the cycle's cohomology class and generalize earlier invariants in Hodge theory.

ABSTRACT

We show that the direct image of the filtered logarithmic de Rham complex is a direct sum of filtered logarithmic complexes with coefficients in variations of Hodge structures, using a generalization of the decomposition theorem of Beilinson, Bernstein and Deligne to the case of filtered $D$-modules. The advantage of using the logarithmic complexes is that we have the strictness of the Hodge filtration by Deligne after taking the cohomology group in the projective case. As a corollary, we get the total infinitesimal invariant of a (higher) cycle in a direct sum of the cohomology of filtered logarithmic complexes with coefficients, and this is essentially equivalent to the cohomology class of the cycle.

Motivation & Objective

• To generalize the decomposition theorem of Beilinson, Bernstein, and Deligne to filtered $D$-modules in the context of logarithmic de Rham complexes.
• To establish a canonical isomorphism between the direct image of the filtered logarithmic de Rham complex and a direct sum of filtered logarithmic complexes with coefficients in variations of Hodge structures.
• To define total infinitesimal invariants of higher cycles in algebraic geometry using cohomology of filtered logarithmic complexes.
• To show that these invariants are equivalent to the cohomology class of the cycle, extending previous results in Hodge theory and cycle theory.

Proposed method

• Use of the filtered derived category to analyze the direct image $\mathbf{R}f_*(\Omega_X^\bullet(\log Y), F)$ of the logarithmic de Rham complex on a projective morphism $f: X \to S$ with $Y = f^*D$ a normal crossing divisor.
• Application of a generalized decomposition theorem for filtered $D$-modules to decompose $\mathbf{R}f_*(\Omega_X^\bullet(\log Y), F)$ into a direct sum of $\mathrm{DR}(M^i), F)[-i]$ for regular holonomic $D$-modules $M^i$.
• Construction of a split increasing filtration $L$ on the direct image complex, with associated graded pieces isomorphic to $\mathrm{DR}(M^i), F)[-i]$, ensuring compatibility with Hodge filtrations.
• Use of Deligne's extension $\widetilde{V}^i$ of local systems to define $\mathrm{DR}_{\log}(\widetilde{V}^i)$, which carries a natural Hodge filtration $F^p$.
• Definition of total infinitesimal invariants $\delta_{S,D}(\xi)$ and $\overline{\delta}_{S,D}(\xi)$ via cohomology groups $\mathbf{H}^i(S, F^p\,\mathrm{DR}(M^{2p-n-i}))$ and $\mathbf{H}^i(S, \mathrm{Gr}_F^p\,\mathrm{DR}(M^{2p-n-i}))$, respectively.
• Leveraging the strictness of the Hodge filtration after cohomology in the projective case to ensure well-definedness and independence of choices under vanishing conditions.

Experimental results

Research questions

• RQ1 How can the direct image of the filtered logarithmic de Rham complex be decomposed in the context of projective morphisms with normal crossing divisors?
• RQ2 What is the relationship between the cohomology of filtered logarithmic complexes and the infinitesimal invariants of algebraic cycles?
• RQ3 Can the total infinitesimal invariant of a higher cycle be naturally defined using filtered logarithmic complexes with coefficients in variations of Hodge structures?
• RQ4 Under what conditions is the total infinitesimal invariant independent of the choice of splitting in the decomposition theorem?
• RQ5 How does the cohomology class of a cycle relate to its infinitesimal invariants in the filtered logarithmic complex framework?

Key findings

• The direct image $\mathbf{R}f_*(\Omega_X^\bullet(\log Y), F)$ admits a noncanonical isomorphism in the filtered derived category to a direct sum of $\mathrm{DR}(M^i), F)[-i]$, with canonical graded pieces isomorphic to $\mathrm{DR}(M^i), F)[-i]$.
• When $D$ is a divisor with normal crossings, the decomposition holds with $\mathrm{DR}(M^i)$ replaced by $\mathrm{DR}_{\log}(\widetilde{V}^i)$, linking the result to Deligne's canonical extension of Hodge structures.
• The total infinitesimal invariant $\delta_{S,D}(\xi)$ of a cycle $\xi \in \mathrm{CH}^p(X\setminus Y, n)$ lies in $\bigoplus_{i \geq 0} \mathbf{H}^i(S, F^p\,\mathrm{DR}(M^{2p-n-i}))$, and is independent of splitting if lower-order invariants vanish.
• The invariant $\overline{\delta}_{S,D}(\xi)$, defined in the graded Hodge filtration, is similarly well-defined and equivalent to the cohomology class of the cycle.
• In the case $S$ is Stein or affine, the cohomology groups $\mathbf{H}^i(S, F^p\,\mathrm{DR}_{\log}(\widetilde{V}^q))$ are computed by the complex $\Gamma(S, \Omega_S^\bullet(\log D) \otimes F^{p-\bullet}\widetilde{V}^q)$.
• For $s=1$, the paper shows $\delta_S^s(\xi_{a,\mathcal{X}}) \neq 0$ for one of $a=1,2$ under genericity assumptions, implying non-torsionness of the cycle class in the geometric generic fiber.