[Paper Review] Weight zero part of the first cohomology of complex algebraic varieties

Morihiko Saito|arXiv (Cornell University)|Apr 10, 2018

Algebraic Geometry and Number Theory17 references3 citations

TL;DR

This paper establishes that the weight zero part of the first rational cohomology of a complex algebraic variety is a topological invariant, providing explicit topological and sheaf-theoretic formulas for its dimension. It proves canonical isomorphisms between $ W_0H^1(X,bQ) $ and kernels of cohomology maps, and gives a dimension formula using the normalization and constructible sheaf $ c_X $.

ABSTRACT

We show that the weight 0 part of the first cohomology of a complex algebraic variety $X$ is a topological invariant, and give an explicit description of its dimension using a topological construction of the normalization of $X$, where $X$ can be reducible, but must be equidimensional. The first assertion is known in the $X$ compact case by A. Weber, where intersection cohomology is used. Note that the weight 1 or 2 part of the first cohomology is not a topological (or even analytic) invariant in the non-compact case by Serre's example.

Motivation & Objective

• To establish that the weight 0 part of the first rational cohomology of a complex algebraic variety is a topological invariant.
• To provide an explicit topological and sheaf-theoretic description of $ ext{dim } W_0H^1(X,bQ) $ for equidimensional, reduced complex algebraic varieties.
• To resolve a question posed by N. Budur on the topological invariance of $ W_0H^1(X,bQ) $ using mixed Hodge theory and normalization theory.
• To generalize known results in the compact case and extend them to non-compact, reducible varieties.

Proposed method

• Use the long exact sequence of mixed Hodge structures arising from the mapping cone construction of the normalization morphism $ ilde{X} o X $, leading to a canonical exact sequence involving $ H^0(X, c_X) $, $ H^0(X,bQ) $, $ H^0( ilde{X},bQ) $, and $ H^1 $ terms.
• Prove that $ H^1( ilde{X},bQ) $ has weight $ eq 0 $, using the injectivity of the pullback on cohomology from a resolution of singularities.
• Establish the injectivity of the canonical morphism $ H^1( ilde{X},bQ) o ext{IH}^1(X,bQ) $, linking the cohomology of the normalization to intersection cohomology.
• Show that the sheaf $ c_X = ext{Coker}(bQ_X o ilde{bQ}_{ ilde{X}}) $ has weight 0, using mixed Hodge module theory.
• Derive the dimension formula $ ext{dim } W_0H^1(X,bQ) = ext{dim } H^0(X,c_X) - b_0( ilde{X}) + b_0(X) $ via exact sequence and weight considerations.
• Construct examples using totally ramified cyclic coverings to illustrate nontrivial monodromy and non-rational strata in the normalization.

Experimental results

Research questions

• RQ1 Is the weight 0 part of the first cohomology of a complex algebraic variety a topological invariant, even when the variety is non-compact and reducible?
• RQ2 Can the dimension of $ W_0H^1(X,bQ) $ be expressed in terms of topological invariants of the normalization and the singular locus?
• RQ3 How does the constructible sheaf $ c_X $, defined as the cokernel of the normalization map, relate to the weight filtration on cohomology?
• RQ4 What is the role of intersection cohomology and mixed Hodge structures in characterizing $ W_0H^1(X,bQ) $?
• RQ5 Can explicit examples be constructed where $ ext{dim } H^0(X,c_X) $ increases under restriction, and what does this imply for the topology of the variety?

Key findings

• The weight 0 part $ W_0H^1(X,bQ) $ is canonically isomorphic to the kernel of the map $ H^1(X,bQ) o H^1( ilde{X},bQ) $, where $ ilde{X} $ is the normalization of $ X $.
• It is also isomorphic to the kernel of $ H^1(X,bQ) o ext{IH}^1(X,bQ) $, showing that $ W_0H^1(X,bQ) $ captures cohomological data not seen in intersection cohomology.
• The dimension of $ W_0H^1(X,bQ) $ is given by $ ext{dim } H^0(X,c_X) - b_0( ilde{X}) + b_0(X) $, a purely topological formula.
• The sheaf $ c_X $, supported on the non-unibranch locus, has weight 0, which is essential for the weight filtration decomposition.
• In the case of curves, $ ext{dim } c_{X,x} = r_{X,x} - 1 $, where $ r_{X,x} $ is the number of local irreducible components at $ x $, confirming the formula in the curve case.
• Examples show that $ ext{dim } H^0(X,c_X) $ can increase strictly under restriction to smaller open subsets, indicating non-local behavior of the weight 0 cohomology.