[Paper Review] On the definition of mixed Hodge modules

Morihiko Saito|arXiv (Cornell University)|Jul 8, 2013

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper simplifies the definition of mixed Hodge modules by employing Beilinson’s maximal extension and systematic stability under subquotients, replacing earlier recursive constructions. The key contribution is a streamlined, intrinsic definition that ensures compatibility and stability of cohomological functors, validated through induction on dimension and well-definedness of nearby/vanishing cycle functors.

ABSTRACT

We give some details of a simpler definition of mixed Hodge modules which has been announced in some papers. Compared with earlier arguments, this new definition is simplified by using Beilinson's maximal extension together with stability by subquotients systematically.

Motivation & Objective

• To provide a simplified, intrinsic definition of mixed Hodge modules that avoids recursive constructions.
• To establish the independence of the definition from choices of local functions and neighborhoods.
• To prove that the resulting category is stable under essential cohomological functors such as direct images, nearby and vanishing cycles, and external products.
• To justify the use of derived functors in the bounded derived category of mixed Hodge modules.
• To clarify the role of admissible variations and the necessity of the new definition in the analytic and algebraic settings.

Proposed method

• Defining mixed Hodge modules via a local condition involving a function $ g $ on a Zariski-open neighborhood, where the restriction to $ U' = U \setminus g^{-1}(0) $ is an admissible variation of mixed Hodge structure.
• Using Beilinson’s maximal extension to simplify the construction of mixed Hodge modules from pure Hodge modules.
• Applying stability under subquotients to reduce the problem to verifying conditions on the support's dimension via induction.
• Ensuring well-definedness of nearby and vanishing cycle functors $ \psi_g $ and $ \varphi_{g,1} $ for the local definition.
• Constructing derived functors $ f_* $, $ f_! $, $ f^* $, $ f^! $, $ \psi_g $, $ \varphi_{g,1} $, $ \boxtimes $, $ \otimes $, $ \mathcal{H}om $ on the bounded derived category $ D^b\mathrm{MHM}(X,A) $.
• Reducing verification of functoriality to the underlying perverse sheaves with $ A $-coefficients, where standard results apply.

Experimental results

Research questions

• RQ1 Can the definition of mixed Hodge modules be simplified by replacing recursive constructions with maximal extension and subquotient stability?
• RQ2 Is the condition for being a mixed Hodge module independent of the choice of local function $ g $ and neighborhood $ U_x $?
• RQ3 Does the category of mixed Hodge modules remain stable under cohomological functors such as $ \mathcal{H}^j f_* $, $ \psi_g $, and $ \varphi_{g,1} $?
• RQ4 Can derived functors on $ D^b\mathrm{MHM}(X,A) $ be canonically defined and compatible with the underlying perverse sheaves?
• RQ5 How does the new definition handle non-normal varieties, particularly in verifying admissibility of variations at singular points?

Key findings

• The new definition of mixed Hodge modules is independent of the choice of local function $ g $ and neighborhood $ U_x $, as proven in Theorem 1.
• The category $ \mathrm{MHM}(X,A) $ is stable under all standard cohomological functors $ \mathcal{H}^j f_* $, $ \mathcal{H}^j f_! $, $ \mathcal{H}^j f^* $, $ \mathcal{H}^j f^! $, $ \psi_g $, $ \varphi_{g,1} $, and $ \boxtimes $, as stated in Theorem 2.
• The derived category $ D^b\mathrm{MHM}(X,A) $ admits canonical functors $ f_* $, $ f_! $, $ f^* $, $ f^! $, $ \psi_g $, $ \varphi_{g,1} $, $ \boxtimes $, $ \otimes $, $ \mathcal{H}om $, with cohomology functors matching the $ \mathcal{H}^j $ versions, as shown in Corollary 1.
• The use of Beilinson’s maximal extension and subquotient stability eliminates the need for repeated restrictions via smooth projections, simplifying earlier constructions in [Sa3].
• The definition ensures that a weakly mixed Hodge module becomes a mixed Hodge module if its restriction to a dense open subset is an admissible variation and the nearby/vanishing cycle functors are well-defined and preserve the category.
• The proof of stability relies on induction on the dimension of the support and reduction to the underlying perverse sheaves, where standard results on admissible variations and cycle functors apply.