Applications of noncommutative deformations

TL;DR

This paper explores noncommutative deformations in algebraic geometry, focusing on contractions of varieties and their applications to derived symmetries, providing new insights into the structure of these geometric transformations.

Contribution

It introduces a noncommutative enhancement of the contraction locus in algebraic varieties, advancing understanding of their derived symmetries and geometric properties.

Findings

Defined a noncommutative enhancement of the contraction locus

Connected noncommutative deformations to derived symmetries

Provided applications to the structure of contractions in algebraic geometry

Abstract

For a general class of contractions of a variety X to a base Y, I discuss recent joint work with M. Wemyss defining a noncommutative enhancement of the locus in Y over which the contraction is not an isomorphism, along with applications to the derived symmetries of X. This note is based on a talk given at the Kinosaki Symposium in 2016.

Equations

$$D(X)$$

$$\mathbb{R}{\mathrm{Hom}}_X(E,-)\underset{A}{\stackrel{\mathbb{L}}{\otimes }}E⟶{\mathrm{Id}}_{D(X)}⟶{\mathsf{T}}_E⟶;$$

$${\mathsf{T}}_E\cong {\left({\mathsf{F}}^{\prime }\circ \mathsf{F}\right)}^{-1}.$$

$$f^{-1}\mathbb{R}{f}_{\ast }\mathbb{R}\mathcal{H}o{m}_X(\mathcal{E},-)\underset{f^{-1}\mathcal{A}}{\stackrel{\mathbb{L}}{\otimes }}\mathcal{E}⟶{\mathrm{Id}}_{D(X)}⟶{\mathsf{T}}_{\mathcal{E}}⟶.$$

$$\mathcal{T}=f_{\ast }\mathcal{E}n{d}_X({\mathcal{O}}_X\oplus N)\cong \mathcal{E}n{d}_Y{f}_{\ast }({\mathcal{O}}_X\oplus N).$$

$$Y=\big{\{}M\in\operatorname{End}V\mathrel{\big{|}}\det M=0\,\big{\}},$$

$$X=\big{\{}(M,H)\in\operatorname{End}V\times\mathbb{P}V\mathrel{\big{|}}\operatorname{Im}M\subseteq H\,\big{\}}$$

Full text

The author is supported by World Premier International Research Center Initiative (WPI), MEXT, Japan, and JSPS KAKENHI Grant Number JP16K17561.

1. 1 Deformation algebras for $3$-folds
2. 2 General results
3. 3 Springer resolution example
4. 4 Conjectures

Derived symmetry groups of algebraic varieties extend classical symmetry groups to include contributions from symplectic geometry via homological mirror symmetry, and from birational geometry. In a recent joint paper [9], M. Wemyss and I construct, for a general class of birational contractions $f\colon X\to Y$​,  a sheaf of noncommutative algebras $\mathcal{A}$ on $Y$ which induces a derived symmetry of $X$ in an appropriate crepant setting. This short note explains key features of our results.

The sheaf $\mathcal{A}$ is supported on the locus of  $Y$ over which $f$ is not an isomorphism. In previous joint work [6, 8] we considered contractions of $3$-folds for which this locus is just a point. In this setting we studied an algebra of noncommutative deformations $A$ which allowed new constructions of derived symmetries, and extended and unified known invariants of such contractions. I begin by reviewing this, as $\mathcal{A}$ may be viewed as a sheafy version of the algebra $A$.

I also briefly discuss an example in which $f$ is a Springer resolution (§3), and indicate recent work in which deformation algebras are used to recover the geometry of contractions (§4).

Conventions

I work over the ground field $\mathbb{C}$, though this assumption can be weakened. Varieties $X$ are assumed quasi-projective, with bounded derived category of coherent sheaves denoted by $D(X)$. The variety of hyperplanes in a vector space $V$ is denoted by $\mathbb{P}V$.

1. Deformation algebras for $3$-folds

The theorem below applies noncommutative deformations to study derived symmetries of $3$-folds. Given smooth $3$-folds $X$ and $X^{\prime}$ related by a flop, Bridgeland [3] constructs certain canonical Fourier–Mukai equivalences

[TABLE]

These equivalences are not mutually inverse: the theorem explains this using deformations of curves on $X$.

Consider a $3$-fold $Y$ with an isolated rational singular point $p$, and a resolution $f\colon X\to Y$ of this singularity, with one-dimensional exceptional locus with components $C_{i}$ for $i=1,\dots,n$.

Theorem A**.**

[6, 7, 8]** Noting that the subvarieties $C_{i}$ of $X$​ are projective lines, we have that:

1. (1)

there exists a $\mathbb{C}^{n}$-algebra $A$ which represents the functor of noncommutative deformations of the sheaves $\mathcal{O}_{C_{i}}(-1)$ on $X$.

Write $E$ for the corresponding universal sheaf on $X$. If the contraction $f$ corresponds to a flop of $X$, then:

1. (2)

there is a Fourier–Mukai autoequivalence $\mathsf{T}_{\!E}$ of $D(X)$, fitting into a distinguished triangle of functors

[TABLE] 2. (3)

there is a natural isomorphism of functors

[TABLE]

In the simplest flopping situation, where $f$ contracts a $(-1,-1)$-curve, the autoequivalence $\mathsf{T}_{\!E}$ is a spherical twist in the sense of Seidel–Thomas [18]. For a contraction of a $(-2,0)$-curve, it is a generalized spherical twist as first constructed by Toda [19], who furthermore established the conclusion of Theorem A(3) in this case.

Remark*.*

The noncommutative deformation theory used here relies on work of Laudal [15], Eriksen [11], E. Segal [17], and Efimov–Lunts–Orlov [10].

Remark*.*

The algebra $A$ above, and similar noncommutative deformation algebras, have now been applied in settings including: enumerative geometry of curves on $3$-folds by Toda and Hua–Toda [20, 13]; flops of families of curves in higher dimensions by Bodzenta and Bondal [2]; construction of autoequivalences and exceptional objects by Kawamata [14]; and new braid-type groups of derived symmetries of $3$-folds by the author and Wemyss [8].

Remark*.*

The full statement of Theorem A does not require $X$ to be smooth: I leave details to the references.

2. General results

The following theorem from [9] gives a sheafy analogue of the deformation algebra $A$, applicable in higher dimensions. For a birational contraction $f\colon X\to Y$ satisfying the assumption below, we define a sheaf of algebras $\mathcal{A}$ on $Y$ which is supported on the locus over which $f$ is not an isomorphism. We furthermore construct an associated autoequivalence of $D(X)$.

Assumption**.**

Suppose that $f\colon X\to Y$ is a contraction with $\operatorname{dim}X\geq 2$, and that either:

1. (a)

*the variety $X$ has an *$f$*​-relative tilting generator with summand $\mathcal{O}_{X}$, where $f$ is crepant, and  *$Y$​ is Gorenstein;

or, alternatively,

1. (b)

the fibres of $f$ have dimension at most one.

Remark*.*

The tilting generator assumption from (a) is satisfied in a range of situations, including symplectic resolutions of quotient singularities as established by Bezrukavnikov and Kaledin [1], and contractions with fibres of dimension at most two under conditions of Toda and Uehara [21].

Write $Z$ for the locus in $Y$​ over which $f$ is not an isomorphism.

Theorem B**.**

[9]** Under the assumption above, there is a sheaf of algebras $\mathcal{A}$ on $Y$​, inducing an object $\mathcal{E}$ of $D(X)$, such that:

1. (1)

the support of $\mathcal{A}$ is $Z$.

For points $z$ of $Z$ such that $f^{-1}(z)$ is one-dimensional with components $C_{i}$, then:

1. (2)

the completion $\mathcal{A}_{z}$ is an algebra which prorepresents the functor of noncommutative deformations of the sheaves $\mathcal{O}_{C_{i}}(-1)$ on $X$, up to Morita equivalence; 2. (3)

*the restriction of  $\mathcal{E}$ to the formal fibre of *$f$​ over $z$ is a sheaf, namely the universal family corresponding to the prorepresenting object given in (2), up to summands of finite sums of sheaves.

If the following hold:

1. (i)

the contraction $f$ is crepant, 2. (ii)

the base $Y$ is complete locally a hypersurface at each point of $Z$,

and either $\operatorname{codim}Z\geq 3$ or, alternatively,

1. (iii)

the sheaf $\mathcal{A}$ is Cohen–Macaulay, and 2. (iv)

the object $\mathcal{E}$ is perfect,

then:

1. (4)

there is a Fourier–Mukai autoequivalence $\mathsf{T}_{\mathcal{E}}$ of $D(X)$, fitting into a distinguished triangle of functors

[TABLE]

I indicate the construction of the sheaf of algebras $\mathcal{A}$, and explain how it allows us to prove Theorem B. Under the assumption above, we have an $f$-relative tilting generator $\mathcal{O}_{X}\oplus N$, either by assertion in case (a), or by a theorem of Van den Bergh [22] in case (b). Let $\mathcal{T}$ denote the relative endomorphism algebra of $\mathcal{O}_{X}\oplus N$, a sheaf of algebras on $Y$. We establish that

[TABLE]

This allows us to make the following definition for $\mathcal{A}$. This is a sheafy version of a construction of the algebra $A$ from our previous work [6].

Definition**.**

[9] Let $\mathcal{A}=\mathcal{T}/\mathcal{I}$, a sheaf of algebras on $Y$​, where $\mathcal{I}$ is the ideal of sections of $\mathcal{T}$ which factor, at each stalk, through a sum of copies of $\mathcal{O}_{Y}$.

The prorepresenting property of $\mathcal{A}$ in Theorem B(2) is then proved as a sheafy version of the representing property of $A$ in Theorem A(1). The object $\mathcal{E}$ of $D(X)$ is defined as the image of $\mathcal{A}$ under an appropriate tilting equivalence: I refer to [9, Section 3] for a precise statement. Theorem B(3) show that this $\mathcal{E}$ has a universal property which is a sheafy version of the universal property of $E$ in Theorem A. We also have the following.

Proposition**.**

[9]** The support of $\mathcal{E}$ is contained in the exceptional locus of $f$.

The construction of a Fourier-Mukai autoequivalence $\mathsf{T}_{\mathcal{E}}$ in Theorem B(4) generalizes the construction of $\mathsf{T}_{\!E}$ from Theorem A(2). In particular, we have the following.

Remark*.*

When $Z$ is a point, the autoequivalence $\mathsf{T}_{\mathcal{E}}$ reduces to the autoequivalence $\mathsf{T}_{\!E}$ appearing in Theorem A(2).

Remark*.*

Although the tilting generator $\mathcal{O}_{X}\oplus N$, and thence the sheaf of algebras $\mathcal{A}$, is not canonically defined (see for instance the construction of Van den Bergh in [22]) it seems that the autoequivalence $\mathsf{T}_{\mathcal{E}}$ may be canonical, given a choice of contraction $f$. For instance, given two different tilting generators related by duplication of summands we obtain Morita equivalent sheaves of algebras $\mathcal{A}$, and thence isomorphic autoequivalences $\mathsf{T}_{\mathcal{E}}$.

Remark*.*

For $f$ a flopping contraction, it would be interesting to establish when $\mathsf{T}_{\mathcal{E}}$ is related to a flop-flop functor, as in Theorem A(3).

Remark*.*

It is tempting to speculate that the ‘tilting’ condition in the requirement for an $f$-relative tilting generator in (a) may be relaxed by upgrading $\mathcal{A}$ to an appropriate sheaf of differential graded algebras.

I record the following $3$-fold setting where the assumptions of Theorem B may be established.

Theorem C**.**

[9]** With $\operatorname{dim}X=3$, assume that

1. (i)

the contraction $f$ is crepant, 2. (ii)

the base $Y$ is complete locally a hypersurface at each point of $Z$, 3. (iii)

the exceptional fibres of $f$ are irreducible curves.

Then the assumptions of Theorem​ B hold, and there exists an associated autoequivalence $\mathsf{T}_{\mathcal{E}}$ of $D(X)$.

3. Springer resolution example

For an example in which the theory of the previous section applies to a contraction with higher-dimensional fibres, consider the Springer resolution of the variety of singular $d$-by-$d$ matrices. Namely, for a vector space $V$ of dimension $d$ with $d\geq 2$, take the singular cone

[TABLE]

which is a Gorenstein hypersurface. It has a resolution by

[TABLE]

whose natural projection​ $f$ to $\operatorname{End}V$ surjects onto $Y$. This resolution $f$ is crepant. Its exceptional fibres lie over points $M$ in $Y$ with $\operatorname{rk}M<d-1$, and are projective spaces of dimension $d-1-\operatorname{rk}M$.

A tilting generator for $X$ has been constructed by Buchweitz, Leuschke, and Van den Bergh [4], so that we are in the setting of Assumption (a). Conditions (i) and (ii) of Theorem B are noted above, and $\operatorname{codim}Z\geq 3$, so we can apply Theorem B(4) to obtain an autoequivalence $\mathsf{T}_{\mathcal{E}}$ of $D(X)$.

Remark*.*

For $d=2$, the variety $X$ is just a $3$-fold resolving a conifold $Y$ with exceptional fibre a $(-1,-1)$-curve, and we are in the setting of Theorem A.

Remark*.*

In joint work with E. Segal [5], I studied the resolution $f$, along with more general resolutions where $Y$ is the variety of $d$-by-$d$ matrices of rank at most $r$ for $0<r<d$. We constructed certain ‘Grassmannian twist’ autoequivalences of the corresponding $D(X)$ by quite different methods: it would be interesting to compare these with $\mathsf{T}_{\mathcal{E}}$.

Remark*.*

The sheaf of algebras $\mathcal{A}$ for this example may be computed from the presentation of the endomorphism algebra $\mathcal{T}$ in [4].

4. Conjectures

In the setting of a $3$-fold flopping contraction as in Theorem A, we made a conjecture [6, Conjecture 1.4] that the complete local neighbourhood of the $3$-fold $Y$ near the singularity $p$ is determined, up to isomorphism, by the deformation algebra $A$. This conjecture is clear in the following simple cases, namely the two kinds of flopping curve for which $A$ is commutative, but remains open more generally.

1. (1)

Contractions of $(-1,-1)$-curves. In this case $A=\mathbb{C}$, and the completion of $Y$ at $p$ is necessarily a conifold singularity. 2. (2)

Contractions of $(-2,0)$-curves. Here $A\cong\mathbb{C}[\varepsilon]/\varepsilon^{w}$ with $w\geq 2$, where $w$ is the width invariant of Reid [16], and the completion of $Y$ at $p$ is determined by $w$.

Hua and Toda subsequently proposed an $A_{\infty}$ version of the conjecture [13, Conjecture 5.3] in which $A$ is endowed with the structure of an $A_{\infty}$-algebra. They established their conjecture for contractions to weighted homogeneous hypersurface singularities [13, Theorem 5.5], and it has now been settled in general by Hua [12]. A key idea in these works is that the deformation algebra $A$ may be viewed as a noncommutative analogue of the Milnor algebra, and that the $A_{\infty}$ structure on it allows recovery of the Milnor algebra along with enough information to apply a version of the Mather–Yau theorem.

Remark*.*

It would be interesting to extend these results to higher dimensions, and to non-isolated singularities. For instance, it is natural to ask whether the the complete local neighbourhood of the variety $Y$​ near the non-isomorphism locus $Z$ is determined by $(Z,\mathcal{A})$, potentially along with some appropriate $A_{\infty}$ structure.