[Paper Review] Contraction algebra and singularity of three-dimensional flopping contraction

Zheng Hua|arXiv (Cornell University)|Oct 18, 2016

Algebraic structures and combinatorial models14 references4 citations

TL;DR

This paper proves that the contraction algebra of a 3-fold flopping contraction, equipped with its natural $\mathbb{Z}/2$-graded $A_\infty$-structure, determines the formal neighborhood of the singular point in the target threefold. By showing that the Hochschild cohomology of the $A_\infty$-enhanced contraction algebra recovers the Milnor algebra and the defining equation of the singularity, the authors establish that Morita equivalence of these $A_\infty$-algebras classifies the local geometry up to biholomorphic equivalence.

ABSTRACT

In [5], Donovan and Wemyss introduced the contraction algebra of flopping curves in 3-folds. They conjectured that the contraction algebra determines the formal neighborhood of the underlying singularity of the contraction. In this paper, we prove that the contraction algebra together with its natural $A_\infty$-structure constructed in [10], determine the formal neighborhood of the singularity.

Motivation & Objective

• To resolve a conjecture by Donovan and Wemyss on whether the contraction algebra determines the formal neighborhood of a 3-fold flopping singularity.
• To show that the $A_\infty$-structure on the contraction algebra is essential for recovering the local singularity geometry.
• To establish that Morita equivalence of $A_\infty$-algebras associated to flopping contractions implies isomorphism of the formal neighborhoods of the singularities.
• To connect the Hochschild cohomology of the contraction algebra to the Milnor algebra of the singularity, thereby recovering the defining equation of the hypersurface singularity.
• To demonstrate that the classical algebra structure alone is insufficient to recover the singularity, necessitating the $A_\infty$-enhancement.

Proposed method

• Use of Dyckerhoff's result that the Hochschild cohomology of the derived category of singularities $\mathrm{D}_{\mathrm{sg}}(Y)$ is isomorphic to the Milnor algebra of $Y$.
• Construction of a canonical $\mathbb{Z}/2$-graded $A_\infty$-structure on the contraction algebra via prior work with Toda.
• Establishing Morita equivalence between the $A_\infty$-contraction algebra and $\mathrm{D}_{\mathrm{sg}}(Y)$, so their Hochschild cohomologies are isomorphic.
• Application of Efimov's result on the $A_\infty$-structure of the minimal model of the endomorphism dg-algebra of the structure sheaf at the singular point.
• Use of dg-algebras of upper-triangular matrices (Happel, Buchweitz, Keller) to show that Hochschild cohomology classes are invariant under Morita equivalence.
• Identification of the class in Hochschild cohomology represented by the $A_\infty$-products with the defining equation $W$ of the singularity, via the isomorphism $HH^\bullet(A,A) \cong M_W$.

Experimental results

Research questions

• RQ1 Does the contraction algebra alone determine the formal neighborhood of a 3-fold flopping singularity, as conjectured by Donovan and Wemyss?
• RQ2 Is the $A_\infty$-structure on the contraction algebra necessary to recover the local geometry of the singularity?
• RQ3 Can the Hochschild cohomology of the $A_\infty$-enhanced contraction algebra reconstruct the Milnor algebra and the defining equation of the hypersurface singularity?
• RQ4 Is the class in Hochschild cohomology corresponding to the $A_\infty$-products isomorphic to the class of the defining equation $W$ in the Milnor algebra?
• RQ5 Does Morita equivalence of $A_\infty$-algebras imply isomorphism of the formal neighborhoods of the singularities?

Key findings

• The Hochschild cohomology of the $A_\infty$-enhanced contraction algebra is isomorphic to the Milnor algebra of the singularity.
• The class in Hochschild cohomology represented by the $A_\infty$-products corresponds precisely to the defining equation $W$ of the hypersurface singularity.
• The contraction algebra with its $A_\infty$-structure is Morita equivalent to the derived category of singularities $\mathrm{D}_{\mathrm{sg}}(Y)$, which controls the singularity.
• The Mather-Yau theorem implies that the Tjurina algebra determines the germ of the singularity, and since the Tjurina algebra is recovered from the Milnor algebra, the singularity is fully determined.
• The classical contraction algebra (without $A_\infty$-structure) has infinite-dimensional Hochschild cohomology in general, indicating that the $A_\infty$-enhancement is essential for singularity recovery.