[Paper Review] The McKay correspondence, tilting equivalences, and rationality

Morgan V. Brown, Ian Shipman|arXiv (Cornell University)|Dec 13, 2013

Algebraic Geometry and Number Theory25 references3 citations

TL;DR

This paper establishes a connection between the derived McKay correspondence and tilting equivalences in dimensions two and three by introducing a two-step tilting procedure using weak central charges and torsion pairs. It proves that any smooth projective surface admitting a full, strong, exceptional collection of line bundles is rational, providing a new criterion for rationality via tilting bundles.

ABSTRACT

We consider the problem of comparing t-structures under the derived McKay correspondence and for tilting equivalences. We relate the t-structures using certain natural torsion theories. As an application, we give a criterion for rationality for surfaces with a tilting bundle. In particular we show that every smooth projective surface which admits a full, strong, exceptional collection of line bundles is rational.

Motivation & Objective

• To understand the relationship between t-structures in derived categories under the derived McKay correspondence and tilting equivalences.
• To develop a two-step tilting framework using weak central charges and torsion pairs to relate coherent sheaves on a variety to modules over a finite-dimensional algebra.
• To provide a criterion for rationality of smooth projective surfaces using the existence of a tilting bundle composed of line bundles.
• To show that the derived McKay correspondence in dim 2 and 3 can be interpreted as a tilting equivalence via universal G-constellations.

Proposed method

• The authors define a weak central charge on the Grothendieck group of a Noetherian abelian category, which induces a torsion pair via the vanishing of the imaginary part of the central charge.
• They introduce upper and lower tilts with respect to such torsion pairs, generalizing classical tilting to handle objects with cohomology in degrees 0, 1, and 2.
• The derived McKay correspondence is analyzed via the equivalence between equivariant sheaves on C^n and coherent sheaves on a crepant resolution Y, identifying the image of coh(Y) in D(G-Coh(C^n)) via cohomological support.
• A key construction uses the universal θ-stable G-constellation over the moduli space M_θ to define a tilting bundle on M_θ, which induces a derived equivalence.
• The method applies Reimann–Roch techniques and cohomological vanishing to analyze line bundle differences and prove rationality.
• The framework is applied to show that if a surface has a tilting bundle decomposed into line bundles, then it must be rational.

Experimental results

Research questions

• RQ1 How do t-structures on derived categories relate under the derived McKay correspondence in dimensions two and three?
• RQ2 Can the derived McKay correspondence be interpreted as a tilting equivalence via a two-step tilting process?
• RQ3 What conditions on a smooth projective surface ensure its rationality via the existence of a tilting bundle of line bundles?
• RQ4 How do weak central charges and torsion pairs facilitate the reconstruction of coherent sheaf categories from derived equivalences?
• RQ5 What is the role of the universal G-constellation in realizing tilting bundles on moduli spaces of G-constellations?

Key findings

• Every smooth projective surface admitting a full, strong, exceptional collection of line bundles is rational.
• If a surface has a tilting bundle that is a direct sum of line bundles, then the global dimension constraint on the endomorphism algebra forces the existence of at least one pair of line bundles with h^0(D_j - D_i) > 1, implying rationality.
• The derived McKay correspondence in dimensions two and three can be interpreted as a tilting equivalence, with the universal G-constellation on M_θ giving rise to a tilting bundle.
• The category of coherent sheaves on a crepant resolution Y is recovered as a full subcategory of the derived category of G-equivariant sheaves on C^n via two successive tilts.
• The existence of a tilting bundle of line bundles implies that the surface is a blowup of a rational ruled surface, hence rational.
• The method using weak central charges and torsion pairs provides a uniform framework to relate t-structures across derived equivalences in low-dimensional McKay correspondences.