[Paper Review] Chern Classes Of Logarithmic Vector Fields For Locally-Homogenous Free Divisors

Xia Liao|arXiv (Cornell University)|May 17, 2012

Algebraic Geometry and Number Theory13 references3 citations

TL;DR

This paper establishes a numerical equivalence between the Chern-Schwartz-MacPherson class of the complement of a locally quasi-homogeneous free divisor and the total Chern class of its logarithmic derivations sheaf, after push-forward to projective space. Using Riemann-Roch and the logarithmic comparison theorem, the authors prove this relation holds for all locally quasi-homogeneous free divisors in projective space, generalizing prior results on hyperplane arrangements and confirming a conjecture in this setting.

ABSTRACT

Let $X$ be a nonsingular complex projective variety and $D$ a locally quasi-homogeneous free divisor in $X$. In this paper we study a numerical relation between the Chern class of the sheaf of logarithmic derivations on $X$ with respect to $D$, and the Chern-Schwartz-MacPherson class of the complement of $D$ in $X$. Our result confirms a conjectural formula for these classes, at least after push-forward to projective space; it proves the full form of the conjecture for locally quasi-homogeneous free divisors in $\mathbb P^n$. The result generalizes several previously known results. For example, it recovers a formula of M. Mustata and H. Schenck for Chern classes for free hyperplane arrangements. Our main tools are Riemann-Roch and the logarithmic comparison theorem of Calderon-Moreno, Castro-Jimenez, Narvaez-Macarro, and David Mond. As a subproduct of the main argument, we also obtain a schematic Bertini statement for locally quasi-homogeneous divisors.

Motivation & Objective

• To resolve a conjecture linking the Chern-Schwartz-MacPherson class of the complement of a divisor and the total Chern class of its logarithmic derivations sheaf.
• To identify precise conditions under which this class equality holds numerically after push-forward to projective space.
• To generalize known results on free hyperplane arrangements and isolated hypersurface singularities to a broader class of divisors.
• To establish a schematic Bertini-type theorem for locally quasi-homogeneous divisors as a byproduct of the main argument.

Proposed method

• Apply the logarithmic comparison theorem (LCT) to ensure that the Chern classes of the logarithmic derivations and the CSM class of the complement have the same degree.
• Use Riemann-Roch to compare degrees of the relevant classes in the Chow ring.
• Employ an exact sequence relating the logarithmic derivations on a hyperplane section to the normal bundle and the original sheaf, enabling inductive reduction.
• Leverage the projection formula and Chern class identities to relate push-forwards of classes on hyperplane sections to those on the original variety.
• Use induction on the dimension of the ambient projective variety, assuming the result holds for lower-dimensional cases.
• Utilize GAGA and analytic geometry to ensure the exactness of coherent sheaf sequences in both analytic and algebraic categories.

Experimental results

Research questions

• RQ1 Under what conditions does the Chern-Schwartz-MacPherson class of the complement of a divisor equal the total Chern class of its logarithmic derivations sheaf in the Chow ring?
• RQ2 Does the conjectural formula relating the CSM class and the Chern class of logarithmic derivations hold for locally quasi-homogeneous free divisors in projective space?
• RQ3 Can the logarithmic comparison theorem be used to establish degree equality between the CSM class and the Chern class of logarithmic derivations?
• RQ4 Is there a schematic Bertini-type theorem that preserves freeness and quasi-homogeneity under hyperplane section?
• RQ5 How do the Chern classes of logarithmic vector fields and the CSM class behave under restriction to hyperplane sections?

Key findings

• The formula $ c_{SM}(1_U) = c( ext{Der}_X(- ext{log } D)) imes [X] $ holds numerically after push-forward to $ b{P}^N $ for any locally quasi-homogeneous free divisor $ D $ in a nonsingular projective variety $ X o b{P}^N $.
• For $ X = b{P}^n $, the full formula in the Chow ring holds, confirming the conjecture in its entirety for this case.
• The degrees of the CSM class and the Chern class of the logarithmic derivations are equal, as established via the logarithmic comparison theorem and Riemann-Roch.
• A schematic Bertini theorem is proven: the freeness and quasi-homogeneity of $ (X,D) $ are preserved under intersection with a general hyperplane.
• The behavior of both the CSM class and the Chern class of logarithmic derivations under hyperplane restriction is governed by the same projection formula and Chern class identities.
• The inductive proof structure relies on the compatibility of the CSM class and the Chern class of logarithmic derivations with restriction to hyperplane sections, enabling dimension-by-dimension verification.