[Paper Review] ON THE COHOMOLOGICAL CREPANT RESOLUTION CONJECTURE FOR WEIGHTED PROJECTIVE SPACES

Samuel Boissi Ere, Étienne Mann+1 more|arXiv (Cornell University)|Jan 1, 2007

Algebraic Geometry and Number Theory25 references7 citations

TL;DR

This paper proves a modified version of the Cohomological Crepant Resolution Conjecture for weighted projective spaces P(1;3;4;4) and P(1,…,1;n), establishing an isomorphism between the orbifold cohomology ring of the Gorenstein orbifold and the quantum-corrected cohomology ring of its crepant resolution, using Gromov-Witten invariants to define the deformation. The result supports the generalized McKay correspondence in the context of orbifolds and crepant resolutions.

ABSTRACT

We prove a modied version of the Cohomological Crepant Resolu- tion Conjecture for the weighted projective spaces P(1; 3; 4; 4) and P(1; : : : ; 1; n). The Cohomological Crepant Resolution Conjecture, as proposed by Y. Ruan (Rua06), predicts the existence of an isomorphism between the orbifold cohomology ring of a Gorenstein orbifold Y and the quantum corrected cohomology ring of any crepant resolution : Z ! jY j of the coarse moduli space jY j of Y , when it exists. The quantum corrected cohomology ring of Z is a deformation of the cohomology ring whose denition involves Gromov-Witten invariants associated to certain exceptional sets. The conjecture belongs to the so called generalized McKay correspondence which broadly speaking can be viewed as a duality between the algebra of nite groups and the geometry of crepant resolutions. The following examples was used to verify it: the Hilbert scheme of r points on a projective surface S (see (LQ02) for r = 2, (ELQ03) and (LL) for S = P 2 r = 3, and (FG03), (QW02), (Uri05) for r general and S with numerically trivial canonical class); the Hilbert scheme of r points on a quasi-projective surface S carrying a holomorphic symplectic form (see (LQW04), (LS01) and (Vas01));

Motivation & Objective

• To verify the Cohomological Crepant Resolution Conjecture in the context of weighted projective spaces.
• To establish an isomorphism between the orbifold cohomology ring of a Gorenstein orbifold and the quantum-corrected cohomology ring of its crepant resolution.
• To extend the generalized McKay correspondence to non-canonical orbifolds via cohomological invariants.
• To analyze the role of Gromov-Witten invariants in deforming the cohomology ring of crepant resolutions.

Proposed method

• Adapt the Cohomological Crepant Resolution Conjecture to weighted projective spaces with non-trivial orbifold structure.
• Compute the orbifold cohomology ring of the Gorenstein orbifold Y = P(1;3;4;4) and P(1,…,1;n).
• Construct the crepant resolution π: Z → |Y| of the coarse moduli space |Y|.
• Define the quantum-corrected cohomology ring of Z using Gromov-Witten invariants associated to exceptional divisors.
• Compare the orbifold cohomology ring of Y with the quantum-corrected cohomology ring of Z via generating functions.
• Verify the existence of an isomorphism between the two rings under the modified conjecture.

Experimental results

Research questions

• RQ1 Does the Cohomological Crepant Resolution Conjecture hold for weighted projective spaces P(1;3;4;4)?
• RQ2 Can the conjecture be modified to accommodate orbifolds with non-trivial stabilizers and non-canonical singularities?
• RQ3 Is there an isomorphism between the orbifold cohomology ring of P(1,…,1;n) and the quantum-corrected cohomology ring of its crepant resolution?
• RQ4 How do Gromov-Witten invariants contribute to the deformation of the cohomology ring in the resolution?
• RQ5 To what extent does the generalized McKay correspondence extend to weighted projective spaces?

Key findings

• The Cohomological Crepant Resolution Conjecture holds in a modified form for P(1;3;4;4) and P(1,…,1;n).
• An isomorphism exists between the orbifold cohomology ring of the Gorenstein orbifold Y and the quantum-corrected cohomology ring of its crepant resolution Z.
• The deformation of the cohomology ring of Z is fully determined by Gromov-Witten invariants associated to exceptional sets.
• The modified conjecture successfully accounts for the orbifold structure and singularities in weighted projective spaces.
• The results support the broader framework of the generalized McKay correspondence in algebraic geometry.
• The verification provides evidence for the conjecture beyond previously studied cases such as Hilbert schemes of points on surfaces.