04.10.15 · algebraic-geometry / moduli

Derived GIT and magic windows

shipped3 tiersLean: none

Anchor (Master): Halpern-Leistner 2015 *The derived category of a GIT quotient* (J. Amer. Math. Soc. 28, 871-912 — the magic window theorem and the categorical Kempf-Ness theta-stratification); Van den Bergh 2004 *Non-commutative crepant resolutions* (in *The Legacy of Niels Henrik Abel*, Springer, 749-770 — the NCCR definition); Špenko-Van den Bergh 2017 *Non-commutative resolutions of quotient singularities for reductive groups* (Invent. Math. 210, 3-67 — the systematic construction of magic windows for linear quotient singularities); Donovan-Segal 2014 *Window shifts, flop equivalences and Grassmannian twists* (Adv. Math. 261, 56-93 — the window-shift formula and the Grassmannian-flop derived equivalence); Bondal-Orlov 2002 *Derived categories of coherent sheaves* (Proc. ICM Beijing II, 47-56 — the flop derived equivalence conjecture); Halpern-Leistner-Sam 2020 *Combinatorial constructions of derived equivalences* (J. Amer. Math. Soc. 33, 735-773); Mumford-Fogarty-Kirwan *Geometric Invariant Theory* 3rd ed. (Springer Ergebnisse 34, 1994); Ballard-Favero-Katzarkov 2019 *Variation of geometric invariant theory quotients and derived categories* (J. reine angew. Math. 746, 235-303)

Intuition Beginner

Classical geometric invariant theory builds a projective quotient $X / /_{L} G$ from a variety $X$ acted on by a reductive group $G$ and a linearisation $L$ . The construction discards the unstable locus, identifies semistable orbits modulo the closure relation, and outputs an ordinary projective scheme. Information is lost: orbits with nontrivial stabilisers get folded in, and the boundary of the quotient hides the geometry of the orbits that got discarded.

Derived geometric invariant theory keeps everything. Instead of throwing away the unstable locus, one studies the entire quotient stack $[X / G]$ — a categorical object whose sheaves remember the full $G$ -action, including the stabilisers and the unstable orbits. The bounded derived category of coherent sheaves on this stack is too big to be a single GIT quotient, but it contains a controlled subcategory — the magic window — that recovers the derived category of any single quotient at a given linearisation. Different linearisations give different windows. The same big stack-level category accommodates them all.

The payoff is dramatic. Crossing a wall in the GIT fan no longer means rebuilding the quotient from scratch: instead, one rotates the window inside the fixed stack-level derived category, and the two resulting quotients turn out to share their derived category up to an equivalence. The Bondal-Orlov flop conjecture — that birational varieties related by a flop have equivalent derived categories — becomes a clean construction in the global-quotient setting, and the entire Mumford GIT picture lifts from points to sheaves.

Visual Beginner

A schematic of the bounded derived category of the quotient stack $[X / G]$ drawn as a large box, with a shifted polytope-like window $W$ drawn inside as a smaller rectangle. Arrows mark the linearisations $ℓ_{-}$ and $ℓ_{+}$ on either side of a wall; the same window position recovers $D^{b} (X_{ℓ_{-}}^{ss} / G)$ on the left and $D^{b} (X_{ℓ_{+}}^{ss} / G)$ on the right after a shift.

The picture compresses the central idea of Halpern-Leistner 2015. The magic window is a tiny piece of the stack-level derived category, carved out by a numerical constraint on the characters of line bundles. Each linearisation positions one window inside the same big category, and the window itself is equivalent to the derived category of the GIT quotient at that linearisation. Walls in the GIT fan correspond to shifts of the window, and the resulting derived equivalence is the categorical refinement of the variation-of-GIT flip.

Worked example Beginner

The cleanest example is $G = GL_{n}$ acting on $X = Mat_{n \times m}$ by left multiplication, with $n \leq m$ . A point of $X$ is an $n \times m$ matrix; the action sends a matrix $A$ to $g A$ for $g \in GL_{n}$ . The standard determinantal linearisation gives the GIT quotient the structure of the Grassmannian $Gr (n, m)$ .

Step 1. Identify the stable locus. A matrix $A$ is stable when its rows are linearly independent — equivalently, when $A$ has full rank $n$ . The unstable locus is the closed subvariety of matrices of rank less than $n$ .

Step 2. Identify the quotient as a Grassmannian. The full-rank $n \times m$ matrices modulo left multiplication by $GL_{n}$ are precisely the $n$ -dimensional subspaces of $C^{m}$ — the row spans. So the GIT quotient is $X^{ss} / GL_{n} = Gr (n, m)$ , the Grassmannian of $n$ -planes in $m$ -space.

Step 3. Describe the magic window. The characters of $GL_{n}$ are integer tuples $χ = (a_{1}, \dots, a_{n})$ . The Halpern-Leistner magic window for this linearisation consists of the line bundles $O (χ)$ on the stack $[X / GL_{n}]$ whose character $χ$ lies in the box $$ -m + n \le a_n \le a_{n-1} \le \cdots \le a_1 \le 0. $$ This is a shifted version of the Hilbert-Mumford polytope adapted to the determinantal linearisation. The window contains finitely many line bundles, and they form a generating set of the magic-window subcategory of $D^{b} ([X / GL_{n}])$ .

Step 4. Recognise the Grassmannian's derived category. Kapranov's exceptional collection on $Gr (n, m)$ is a list of finitely many line bundles indexed by partitions inside the same $n \times (m - n)$ rectangle. Halpern-Leistner's magic window theorem applied to this $(G, X, L)$ data identifies the magic window with the derived category of the Grassmannian, restricting each magic-window line bundle to the GIT quotient and producing exactly Kapranov's collection.

What this tells us: the abstract magic window — a subcategory carved by a numerical condition on the quotient stack — recovers, in this small example, a concrete object known classically since the 1980s: Kapranov's exceptional collection on the Grassmannian. The Halpern-Leistner construction is the universal reason every such resolution exists. For $n = 1, m = 2$ , the magic window has two line bundles $O (0)$ and $O (- 1)$ , and they restrict to the exceptional collection $⟨ O, O (- 1)⟩$ on $P^{1}$ — Beilinson's resolution of the diagonal in the simplest case.

Check your understanding Beginner

Exercise (easy, multiple choice).

The magic window subcategory of $D^{b} ([X / G])$ in Halpern-Leistner 2015 is best described as:

The subcategory of all coherent sheaves on the quotient stack
The subcategory generated by line bundles $\mathcal{O}(\chi)$ whose character $\chi$ lies in a shifted Hilbert-Mumford box determined by the linearisation, equivalent to the derived category of the GIT quotient at generic stability
The kernel of the restriction functor to the unstable locus
The subcategory of objects with vanishing higher Ext groups

Hint

The magic window is defined by a numerical condition on the characters of line bundles relative to the Kempf-Ness weights of the linearisation, and the content of the theorem is that this small subcategory is equivalent to the derived category of the quotient at a generic stability.

Answer

B. The magic window is generated by line bundles $O (χ)$ whose character lies in a specific box, shifted by the Kempf-Ness weights of the linearisation $ℓ$ . Halpern-Leistner's theorem identifies this subcategory with the bounded derived category of the GIT quotient at the generic stability $ℓ$ . The other options miss the key feature: the window is a small, numerically defined piece of the big stack-level category, not the whole category or a vanishing locus.

Exercise (easy, true-false).

True or false: as one crosses a wall in the GIT fan, the two magic windows on either side give two distinct bounded derived categories of GIT quotients, and these two categories are related by a derived equivalence.

Hint

This is exactly the content of the Donovan-Segal window-shift formula and the construction of the Bondal-Orlov flop derived equivalence in the global-quotient case.

Answer

True. Crossing a wall in the GIT fan from one chamber to an adjacent chamber takes the chosen linearisation $ℓ_{-}$ to $ℓ_{+}$ , and the corresponding GIT quotients $Q_{-} = X_{ℓ_{-}}^{ss} / G$ and $Q_{+} = X_{ℓ_{+}}^{ss} / G$ are related by a flip. The two magic windows are two distinct positions of a window inside the same fixed stack-level derived category $D^{b} ([X / G])$ . They are related by the window-shift autoequivalence of Donovan-Segal 2014, and the consequence is a derived equivalence $D^{b} (Q_{-}) ≃ D^{b} (Q_{+})$ refining the Bondal-Orlov flop conjecture in the global-quotient case.

Exercise (easy, numeric).

For $G = GL_{2}$ acting on $X = Mat_{2 \times 3}$ with the determinantal linearisation, how many line bundles generate the magic window subcategory of $D^{b} ([X / G])$ ?

Hint

For $n = 2$ and $m = 3$ , the magic-window box consists of integer pairs $(a_{1}, a_{2})$ with $- 1 \leq a_{2} \leq a_{1} \leq 0$ . Count the integer points in this box.

Answer

$3$ . The box is $- 1 \leq a_{2} \leq a_{1} \leq 0$ , so the integer points are $(0, 0)$ , $(0, - 1)$ , and $(- 1, - 1)$ . There are exactly three line bundles in the magic window. The resulting subcategory of $D^{b} ([X / GL_{2}])$ restricts to the bounded derived category of $Gr (2, 3)$ , which carries Kapranov's exceptional collection $⟨ O, S^{*}, det S^{*} ⟩$ of length $3 = (2 3)$ , where $S$ is the tautological rank- $2$ subbundle. The count matches the number of partitions inside the $2 \times 1$ rectangle.

Formal definition Intermediate+

Let $k$ be an algebraically closed field of characteristic zero, let $G$ be a connected reductive linear algebraic group over $k$ , and let $X$ be a smooth quasi-projective $k$ -variety carrying a $G$ -action together with a $G$ -linearised ample line bundle $L$ . Write $T \subset G$ for a maximal torus and $X^{*} (T)$ for the character lattice of $T$ . The equivariant Picard group $Pic^{G} (X)$ contains $X^{*} (T)$ as the lattice of line bundles plain on $X$ but twisted by a character of $T$ .

Definition (quotient stack). The quotient stack of the $G$ -action on $X$ is the stack $[X / G]$ whose category of points over a test scheme $S$ is the groupoid of pairs $(P \to S, ϕ : P \to X)$ where $P \to S$ is a principal $G$ -bundle and $ϕ$ is a $G$ -equivariant morphism. The bounded derived category $D^{b} ([X / G])$ is, by descent, equivalent to the bounded derived category $D^{b} (Coh^{G} (X))$ of $G$ -equivariant coherent sheaves on $X$ .

Definition (semistable locus). For a stability $ℓ \in Pic^{G} (X)_{Q}$ generic with respect to the GIT chamber decomposition of $Amp^{G} (X)$ (see 04.10.09), the semistable locus $X_{ℓ}^{ss} \subseteq X$ is the open $G$ -invariant subvariety where every $G$ -invariant section of some positive power of $ℓ$ does not vanish. The GIT quotient is $X / /_{ℓ} G := X_{ℓ}^{ss} / G$ .

Definition (Kempf-Ness weights / theta-stratification). Let $ℓ$ be a generic stability. Hilbert-Mumford produces a finite collection of pairs $(β, Z_{β})$ where $β$ is a one-parameter subgroup of $G$ and $Z_{β} \subset X$ is a component of the $β$ -fixed locus; the Kempf-Ness weight $η_{β} = η_{β} (ℓ)$ measures the asymptotic $ℓ$ -weight of $β$ on $Z_{β}$ . The theta-stratification of the unstable locus $X ∖ X_{ℓ}^{ss}$ is the locally closed decomposition into strata $S_{β} = G \cdot Y_{β}^{KN}$ refining the Kirwan stratification (see 04.10.08).

Definition (magic window). Let $ℓ$ be a generic stability and let $η = (η_{β})_{β}$ be the Kempf-Ness weights. Choose a real parameter $δ \in R$ , called the window-shift. The magic window of width $δ$ centred at $ℓ$ is the full triangulated subcategory $W (ℓ, δ) \subset D^{b} ([X / G])$ generated by the line bundles $O (χ)$ for characters $χ \in X^{*} (T)$ in the Hilbert-Mumford box $$ -\tfrac{1}{2}\eta_\beta - \delta < \langle \chi, \beta \rangle \le \tfrac{1}{2}\eta_\beta - \delta \quad \text{for every } \beta \in \mathrm{KN}(\ell), $$ where $KN (ℓ)$ is the finite set of Kempf-Ness one-parameter subgroups associated with $ℓ$ . For $δ = 0$ this is the centred magic window $W (ℓ)$ .

Definition (noncommutative crepant resolution). Following Van den Bergh 2004 ^{[Van den Bergh 2004]}, for a normal Gorenstein integral domain $S$ , a noncommutative crepant resolution (NCCR) is a reflexive finitely generated $S$ -module $M$ such that $A := End_{S} (M)$ is a Cohen-Macaulay $S$ -algebra of finite global dimension equal to $dim S$ , with the crepant condition that $A$ is symmetric: $Hom_{S} (A, S) ≅ A$ as $A$ -bimodules. When $Spec S$ admits a crepant resolution $Y \to Spec S$ as a scheme, the NCCR conjecturally satisfies $D^{b} (A) ≃ D^{b} (Y)$ , and the magic window of Špenko-Van den Bergh 2017 ^{[Špenko-Van den Bergh 2017]} realises this equivalence for linear quotient singularities $Spec S = V / / G$ .

Counterexamples to common slips

The magic window depends on the choice of stability $ℓ$ and on the window-shift parameter $δ$ . Different $δ$ 's give different windows inside the same $D^{b} ([X / G])$ , and they are not, in general, the same subcategory; they are related by the window-shift autoequivalence.
The Hilbert-Mumford box must be defined with strict inequality on one side and weak inequality on the other to avoid double-counting characters on the boundary. The Halpern-Leistner convention is $- \frac{1}{2} η - δ < ⟨ χ, β ⟩ \leq \frac{1}{2} η - δ$ ; flipping the strict-versus-weak side produces a non-canonically different window.
The magic window theorem requires $ℓ$ to be generic — strictly inside a GIT chamber, not on a wall. At a wall the semistability-equals-stability property fails, the Kempf-Ness weights collapse, and the window construction degenerates.
A noncommutative crepant resolution always exists for a linear quotient singularity by a reductive group, by Špenko-Van den Bergh 2017. The deeper claim — that the NCCR algebra has the same derived category as a commutative crepant resolution when the latter exists — is conjectural in general and is established in the linear-quotient case by Halpern-Leistner's magic window theorem.

Key theorem with proof Intermediate+

Theorem (magic window theorem, Halpern-Leistner 2015 Theorem 2.10). Let $G$ be a connected reductive group acting on a smooth quasi-projective variety $X$ with a $G$ -linearised ample line bundle $L$ , and let $ℓ$ be a generic stability inside a GIT chamber. The restriction functor $$ \mathrm{res}\ell : \mathcal{W}(\ell) ;\longrightarrow; D^b(X^{ss}\ell / G), \qquad \mathcal{F} \mapsto \mathcal{F}|{X^{ss}\ell} $$ is an equivalence of triangulated categories between the centred magic window $W (ℓ) \subset D^{b} ([X / G])$ and the bounded derived category of the GIT quotient. ^{[Halpern-Leistner 2015]}

Proof. The argument has four steps. First, set up the theta-stratification of the unstable locus and the associated semiorthogonal decomposition of $D^{b} ([X / G])$ . Second, identify the magic window as a particular semiorthogonal block in this decomposition. Third, show that the restriction to $X_{ℓ}^{ss}$ is fully faithful on the magic window. Fourth, verify essential surjectivity using the Beilinson-style resolution of the diagonal on the GIT quotient.

Step 1: Theta-stratification and semiorthogonal decomposition. The Hilbert-Mumford theta-stratification associates to the generic stability $ℓ$ a finite collection $KN (ℓ) = {β_{1}, \dots, β_{N}}$ of one-parameter subgroups, ordered by decreasing instability, together with locally closed Kirwan strata $S_{β_{i}} = G \cdot Y_{β_{i}}^{KN}$ exhausting the unstable locus (see 04.10.08). Halpern-Leistner 2015 §3 proves that the bounded derived category of the stack decomposes as a semiorthogonal sum $$ D^b([X / G]) = \langle D^b([S_{\beta_N} / G]),, \ldots,, D^b([S_{\beta_1} / G]),, D^b([X^{ss}\ell / G]) \rangle, $$ with each unstable piece $D^b([S{\beta_i} / G]) $f u r t h er d eco m p ose d in t o t h e * c a t e g or i c a l K e m p f - N ess b l oc k s * in d e x e d b y in t e g er c ha r a c t er so f$ \beta_i $. T h ese mi or t h o g o na l i t y f o l l o w s f r o ma c a t e g or i c a l v er s i o n o f t h eK e m p f - N ess l e mma : t h e u n s t ab l es t r a t a co n t r a c t u n d er$ \beta_i $t o t h e f i x e d - co m p o n e n t$ Z_{\beta_i} $, an d t h e d er i v e d c a t e g or y o f t h eco n t r a c t e d p i ece i s g e n er a t e d b y l in e b u n d l es w h ose$ \beta_i$-weight lies outside a controlled window.

Step 2: Magic window as semiorthogonal block. Inside the semiorthogonal decomposition, choose the single block corresponding to all $β_{i}$ -weights lying in the half-open interval $(- \frac{1}{2} η_{β_{i}}, \frac{1}{2} η_{β_{i}}]$ simultaneously. This block is generated by the line bundles $O (χ)$ for $χ$ in the centred Hilbert-Mumford box, and is by construction the centred magic window $W (ℓ)$ . The choice of strict-versus-weak inequality on opposite sides of the interval guarantees that the block is closed under shifts and that no character is double-counted across strata.

Step 3: Full faithfulness of $res_{ℓ}$ . Restrict any object $F \in W (ℓ)$ along the open immersion $j : X_{ℓ}^{ss} ↪ X$ and compute the higher direct images $R j_{*} F ∣_{X_{ℓ}^{ss}}$ . Halpern-Leistner 2015 Lemma 3.13 shows that for $F, G \in W (ℓ)$ , the higher Ext groups computed on $[X_{ℓ}^{ss} / G]$ agree with the Ext groups computed on $[X / G]$ , because the differences are concentrated on the unstable strata where every object of $W (ℓ)$ has vanishing relevant cohomology — a consequence of the Hilbert-Mumford box constraint. Concretely, for $F = O (χ_{1})$ and $G = O (χ_{2})$ with $χ_{1}, χ_{2}$ in the box, $$ \mathrm{Ext}^{[X^{ss}\ell / G]}(\mathcal{O}(\chi_1)|{ss}, \mathcal{O}(\chi_2)|{ss}) \cong \mathrm{Ext}^_{[X / G]}(\mathcal{O}(\chi_1), \mathcal{O}(\chi_2)), $$ and the Hom spaces between objects of $W (ℓ)$ are preserved by restriction.

Step 4: Essential surjectivity. The GIT quotient $Q = X / /_{ℓ} G$ is smooth (since $ℓ$ is generic and $X$ is smooth, the stable locus equals the semistable locus and the stabilisers are finite), and its bounded derived category $D^{b} (Q)$ is generated by the restrictions of equivariant line bundles from $[X_{ℓ}^{ss} / G]$ . Among the equivariant line bundles, the ones whose character lies in the Hilbert-Mumford box already generate the entire restricted derived category — a finite-dimensional generation argument using the Kempf-Ness deformation of any character into the box modulo the relations imposed by the theta-stratification. Hence $res_{ℓ}$ is essentially surjective on $D^{b} (Q)$ . Combined with Step 3, the functor is an equivalence. $□$

Bridge. The magic window theorem builds toward the categorical wall-crossing theory and the construction of derived equivalences between birationally distinct moduli spaces, and the central insight is that the bounded derived category of every GIT quotient appears as a numerically defined subcategory of one fixed stack-level derived category. The bridge is that the magic window approach identifies the derived category of a quotient with a tiny piece of the derived category of the quotient stack, and putting these together with the variation-of-GIT chamber picture (see 04.10.09) the foundational reason wall-crossing produces derived equivalences is that two adjacent chambers index two adjacent windows inside the same big category, related by a window-shift autoequivalence rather than by an external comparison. This appears again in 04.10.13 (K-stability), where the infinite-dimensional analogue produces derived autoequivalences associated with one-parameter degenerations of Fano varieties, and the foundational reason the construction generalises is exactly that the magic window is defined by a numerical condition on characters — the same kind of condition that defines test configurations in K-stability. The pattern recurs across moduli theory: the bridge is that GIT operates on points but its derived enhancement operates on sheaves, identifying every chamber-dependent quotient with one chamber-dependent window inside a chamber-independent big category, and the central insight is that the dependence on the chamber is then a window shift rather than a category change.

Exercises Intermediate+

Exercise 1 (easy, numeric).

For $G = GL_{3}$ acting on $X = Mat_{3 \times 5}$ with the determinantal linearisation, count the integer points in the magic-window box $- 2 \leq a_{3} \leq a_{2} \leq a_{1} \leq 0$ and identify the number with a partition count.

Hint

The integer points $(a_{1}, a_{2}, a_{3})$ with $- 2 \leq a_{3} \leq a_{2} \leq a_{1} \leq 0$ are in bijection with partitions fitting inside the $3 \times 2$ rectangle, i.e., partitions with at most $3$ parts each $\leq 2$ .

Answer

$10$ . The integer points are in bijection with partitions $λ = (λ_{1}, λ_{2}, λ_{3})$ with $2 \geq λ_{1} \geq λ_{2} \geq λ_{3} \geq 0$ , where $λ_{i} = - a_{i}$ . The number of such partitions is $(3 3 + 2) = 10$ , which is exactly $(n m)$ for $m = 5, n = 3$ — the dimension of $H^{0} (Gr (3, 5), O (1))$ via the Plücker embedding. The magic window for this action recovers Kapranov's $10$ -term exceptional collection on $Gr (3, 5)$ , with each line bundle indexed by a partition inside the $3 \times 2$ rectangle. Rubric: full credit for the count $10$ and the identification with $(n m)$ .

Exercise 2 (easy, multiple choice).

Which of the following best describes the role of the window-shift parameter $δ$ in the magic window construction of Halpern-Leistner 2015?

It parametrises the choice of stability inside a fixed GIT chamber
It shifts the Hilbert-Mumford box by a real number, producing a family of magic windows inside $D^b([X / G])$ related by autoequivalences
It determines the codimension of the unstable locus
It controls the dimension of the GIT quotient

Hint

For fixed $ℓ$ , different choices of $δ$ give different magic windows; the comparison between them is the Donovan-Segal window-shift autoequivalence.

Answer

B. The shift $δ$ moves the Hilbert-Mumford box rigidly inside the character lattice, giving a one-real-parameter family of magic windows $W (ℓ, δ) \subset D^{b} ([X / G])$ . Each window is equivalent to $D^{b} (X_{ℓ}^{ss} / G)$ via restriction, and any two windows in this family are related by an autoequivalence of $D^{b} ([X / G])$ — the window-shift autoequivalence of Donovan-Segal 2014. Across a wall, the two relevant windows correspond to the two adjacent chambers, and the shift implements the flop derived equivalence.

Exercise 3 (medium, symbolic).

Consider $G = C^{\times}$ acting on $X = C^{4}$ with weights $(+ 1, + 1, - 1, - 1)$ (the Atiyah flop setup). Describe the two GIT chambers, the two GIT quotients $Q_{\pm}$ , the two magic windows inside $D^{b} ([X / G])$ , and explain how the window-shift autoequivalence realises the Bondal-Orlov flop derived equivalence $D^{b} (Q_{-}) ≃ D^{b} (Q_{+})$ .

Hint

The two chambers correspond to the signs of the equivariant character. Each gives a small resolution of the conifold ${X Z - Y W = 0}$ . The two magic windows each consist of one $Z$ -graded interval of characters of $C^{\times}$ .

Answer

Setup. Write coordinates $(x_{1}, x_{2}, y_{1}, y_{2})$ for $C^{4}$ with $C^{\times}$ -weights $(+ 1, + 1, - 1, - 1)$ . The GIT chambers are the two half-lines of the equivariant $Q$ -Picard group $Q$ : $ℓ > 0$ and $ℓ < 0$ .

Chamber $ℓ > 0$ . The semistable locus is ${(x_{1}, x_{2}) \neq = (0, 0)}$ ; the unstable locus is the linear subspace ${x_{1} = x_{2} = 0}$ . The GIT quotient is the total space of the line bundle $O (- 1) \oplus O (- 1) \to P^{1}$ , with $[x_{1} : x_{2}]$ on the base and $(y_{1}, y_{2})$ in the fibre. This is one of the two small resolutions of the conifold singularity ${x_{1} y_{1} + x_{2} y_{2} = 0}$ .

Chamber $\ell < 0$ . By symmetry the semistable locus is ${(y_{1}, y_{2}) \neq = (0, 0)}$ and the quotient is the other small resolution, $O (- 1) \oplus O (- 1) \to P^{1}$ with the roles of $(x_{1}, x_{2})$ and $(y_{1}, y_{2})$ swapped. The two quotients $Q_{\pm}$ are birational but non-isomorphic — they are related by the Atiyah flop.

Magic windows. The character lattice of $C^{\times}$ is $Z$ . The Kempf-Ness weights for the two destabilising one-parameter subgroups are $η_{+} = 2$ (from $β_{+} = + 1$ ) and $η_{-} = 2$ (from $β_{-} = - 1$ ). The centred magic window for $ℓ > 0$ consists of characters $a \in Z$ with $- 1 < a \leq 1$ , i.e., $a \in {0, 1}$ — two line bundles, $O (0)$ and $O (1)$ . The magic window for $\ell < 0$ consists of $a \in {0, - 1}$ — two line bundles, $O (0)$ and $O (- 1)$ .

Window shift and the flop equivalence. The two magic windows share the line bundle $O (0)$ but differ in their second generator; the window-shift autoequivalence of $D^{b} ([X / C^{\times}])$ taking $W (ℓ_{-})$ to $W (ℓ_{+})$ is realised by tensoring with $O (1)$ followed by a left mutation. Composing the equivalences $D^{b} (Q_{-}) ≃ W (ℓ_{-}) \sim W (ℓ_{+}) ≃ D^{b} (Q_{+})$ yields the Bondal-Orlov flop derived equivalence $Φ : D^{b} (Q_{-}) \sim D^{b} (Q_{+})$ . The kernel of $Φ$ is the structure sheaf of the fibre product $Q_{-} \times_{Spec (conifold)} Q_{+}$ — exactly Bondal-Orlov's original Fourier-Mukai construction. Rubric: full credit for identifying the two chambers, the two quotients as small resolutions of the conifold, the two magic windows by their character constraint, and the window-shift autoequivalence as the implementing equivalence.

Exercise 4 (medium, short-answer).

State the noncommutative crepant resolution definition of Van den Bergh 2004, and explain why an NCCR $A = End_{S} (M)$ for a linear quotient singularity $S = k [V]^{G}$ ( $G$ reductive acting linearly on $V$ ) gives rise to a derived equivalence $D^{b} (A) ≃ D^{b} (V / / G)$ for any commutative crepant resolution $V / / G$ , when such a commutative resolution exists.

Hint

Use the Špenko-Van den Bergh 2017 construction: $M = ⨁_{χ} k [V] \otimes V_{χ}$ , the sum over characters $χ$ in a magic-window box.

Answer

Definition (Van den Bergh 2004). For a Gorenstein normal integral domain $S$ over $k$ , a noncommutative crepant resolution is a reflexive finitely generated $S$ -module $M$ such that $A = End_{S} (M)$ is a Cohen-Macaulay $S$ -algebra of finite global dimension $dim S$ and $Hom_{S} (A, S) ≅ A$ as $A$ -bimodules (the symmetric / Calabi-Yau condition). Equivalently: $A$ is homologically homogeneous of dimension $dim S$ , and Auslander's higher Calabi-Yau axiom holds.

NCCR for $S = k [V]^{G}$ . When $G$ is a connected reductive group acting linearly on a finite-dimensional $k$ -vector space $V$ , Špenko-Van den Bergh 2017 construct an NCCR by taking the magic-window box of Halpern-Leistner — characters $χ$ inside the shifted Hilbert-Mumford rectangle — and forming the reflexive module $M = ⨁_{χ \in box} k [V] \otimes V_{χ}$ , with $V_{χ}$ the irreducible $G$ -representation of highest weight $χ$ . The endomorphism algebra $A = End_{S} (M)$ is an NCCR.

Derived equivalence to a commutative crepant resolution. Suppose $Y \to V / / G$ is a commutative crepant resolution. Halpern-Leistner's magic window theorem identifies the bounded derived category $D^{b} (Y)$ with the magic window $W \subset D^{b} ([V / G])$ . The NCCR algebra $A = End_{S} (M)$ acts on the module $M$ , and the categorical equivalence $D^{b} (A - Mod_{fg}) ≃ W$ follows from the fact that $M$ is a tilting object whose endomorphisms are exactly $A$ and whose tilting equivalence is $RHom (M, -)$ . Composing, $$ D^b(A) \simeq \mathcal{W} \simeq D^b(\widetilde{V /!!/ G}). $$ The two derived categories are the commutative and noncommutative incarnations of the same magic window inside $D^{b} ([V / G])$ . Rubric: full credit for the NCCR definition, the explicit module $M$ , the tilting argument, and the chain of equivalences through the magic window.

Exercise 5 (medium, short-answer).

State the window-shift formula of Donovan-Segal 2014 and explain how it produces the Bondal-Orlov flop derived equivalence in the global-quotient case.

Hint

The window-shift autoequivalence is generated by line-bundle twists composed with appropriate left/right mutations of the semiorthogonal decomposition.

Answer

Window-shift formula (Donovan-Segal 2014). For a generic stability $ℓ$ and a real shift $δ$ , let $T_{δ} : W (ℓ, 0) \to W (ℓ, δ)$ be the comparison functor between the two windows centred at the same $ℓ$ . Donovan-Segal 2014 prove that $T_{δ}$ is an autoequivalence of $D^{b} ([X / G])$ given as a composition of two operations: (i) tensoring with the line bundle $O (χ_{δ})$ for a specific character $χ_{δ}$ determined by the shift, and (ii) left mutation through the unstable semiorthogonal blocks corresponding to those characters that exit the window $W (ℓ, 0)$ as one shifts to $W (ℓ, δ)$ .

Wall-crossing as flop equivalence. Consider two adjacent GIT chambers separated by a wall, with linearisations $ℓ_{-}, ℓ_{+}$ on either side. By the magic window theorem, $$ D^b(Q_-) \simeq \mathcal{W}(\ell_-) \subset D^b([X / G]) \supset \mathcal{W}(\ell_+) \simeq D^b(Q_+). $$ Halpern-Leistner 2015 Theorem 3.16 shows that the two windows $W (ℓ_{-})$ and $W (ℓ_{+})$ are related by a window-shift autoequivalence — the wall is a transition where the Kempf-Ness weights change sign on a specific subset of one-parameter subgroups, and the window must shift to accommodate this. The induced equivalence $Φ_{-+} : D^{b} (Q_{-}) \to D^{b} (Q_{+})$ is the derived flop equivalence of Bondal-Orlov 2002, with the explicit Fourier-Mukai kernel given by the structure sheaf of the fibre product $Q_{-} \times_{X / /_{wall} G} Q_{+}$ — exactly Bondal-Orlov's construction. Rubric: full credit for the window-shift formula statement, the realisation as tensor-plus-mutation, and the identification of the wall-crossing equivalence with the Bondal-Orlov flop equivalence.

Exercise 6 (medium, short-answer).

Verify the magic window theorem in the case $G = C^{\times}$ acting on $C^{n + 1} ∖ {0}$ by scaling, with the standard linearisation, and identify the resulting magic window with the Beilinson exceptional collection on projective space.

Hint

The GIT quotient is $P^{n}$ . The Kempf-Ness weight for the single destabilising direction (towards the origin) is $η = n + 1$ . The centred magic window box is $- (n + 1) /2 < a \leq (n + 1) /2$ , giving $n + 1$ characters.

Answer

Setup. $G = C^{\times}$ acts on $C^{n + 1}$ by scaling with weight $1$ . The standard linearisation gives the semistable locus $C^{n + 1} ∖ {0}$ , with GIT quotient $Q = P^{n}$ . The unstable locus is the single point ${0}$ , with destabilising one-parameter subgroup $β = 1$ and fixed component $Z_{β} = {0}$ .

Kempf-Ness weight. The weight of $β = 1$ acting on the tangent space $T_{0} C^{n + 1} = C^{n + 1}$ is $1$ in each of the $n + 1$ coordinate directions, so the total $β$ -weight is $η_{β} = n + 1$ .

Magic window. The centred Hilbert-Mumford box is $- \frac{n + 1}{2} < a \leq \frac{n + 1}{2}$ for $a \in Z$ . The integer points are $a \in {- ⌊ n /2 ⌋, \dots, ⌈ n /2 ⌉}$ — a contiguous interval of $n + 1$ integers. By shifting, one can take the integer points ${0, - 1, - 2, \dots, - n}$ , giving the line bundles $O, O (- 1), O (- 2), \dots, O (- n)$ on $[C^{n + 1} / C^{\times}]$ .

Comparison with Beilinson. Restricting these $n + 1$ line bundles along the open immersion $C^{n + 1} ∖ {0} ↪ C^{n + 1}$ and descending to $P^{n}$ gives the line bundles $O, O (- 1), \dots, O (- n)$ on $P^{n}$ . This is precisely Beilinson's exceptional collection $⟨ O, O (- 1), \dots, O (- n)⟩$ on $P^{n}$ , established in Beilinson 1978 via the resolution of the diagonal in $P^{n} \times P^{n}$ . The magic window theorem identifies $W ≃ D^{b} (P^{n})$ with the magic window generated by these $n + 1$ line bundles, recovering Beilinson's result as the simplest application beyond the empty case. Rubric: full credit for the Kempf-Ness weight $η = n + 1$ , the magic-window line bundles, and the identification with Beilinson's collection.

Exercise 7 (hard, short-answer).

Explain how Halpern-Leistner-Sam 2020 use the magic window construction together with combinatorial data on the GIT fan to produce derived equivalences for moduli of representations of quivers, and indicate the application to Reineke-style wall-crossing in Donaldson-Thomas theory.

Hint

For a quiver $Q$ with dimension vector $d$ , the representation space $Rep (Q, d)$ carries a $GL_{d} = \prod_{i} GL_{d_{i}}$ -action. The stability is a tuple of integers (the King stability). The combinatorial magic window gives derived equivalences between two moduli of King-semistable representations across a stability wall.

Answer

Setup. A finite quiver $Q$ with vertex set $I$ and edge set $A$ has representation space $Rep (Q, d) = ⨁_{a : i \to j} Hom (k^{d_{i}}, k^{d_{j}})$ on which the group $GL_{d} = \prod_{i} GL_{d_{i}}$ acts by change of basis at each vertex. A King stability $θ \in Z^{I}$ with $\sum_{i} θ_{i} d_{i} = 0$ gives a $GL_{d}$ -linearised line bundle on $Rep (Q, d)$ whose semistable locus is the locus of $θ$ -semistable representations, and the GIT quotient $M_{θ} (Q, d)$ is King's moduli of $θ$ -stable representations.

Combinatorial magic window. Halpern-Leistner-Sam 2020 identify the Hilbert-Mumford theta-stratification for the action of $GL_{d}$ on $Rep (Q, d)$ with a combinatorial structure on the lattice of characters $Z^{\sum_{i} d_{i}}$ . The Kempf-Ness weights are computed from the dimensions of subquiver representations, and the magic window for stability $θ$ is the subcategory of $D^{b} ([Rep (Q, d) / GL_{d}])$ generated by line bundles $O (χ)$ for $χ$ in an explicit polytope determined by $θ$ .

Wall-crossing and derived equivalences. The stability $θ$ lives in a chamber of the GIT fan of the $GL_{d}$ -action; crossing a wall to a stability $θ^{'}$ produces an adjacent magic window, and the resulting derived equivalence $D^{b} (M_{θ} (Q, d)) ≃ D^{b} (M_{θ^{'}} (Q, d))$ is the categorical refinement of Reineke's wall-crossing formula for the Euler characteristics of King moduli (Reineke 2003 Invent. Math. 152).

Application to DT theory. Donaldson-Thomas invariants of the quiver $Q$ are Euler characteristics of the King moduli weighted by Behrend's constructible function. Wall-crossing for DT invariants (Kontsevich-Soibelman 2008, Joyce-Song 2012) lifts Reineke's identity to motivic invariants. Halpern-Leistner-Sam 2020 lift it one further step to the derived category: the wall-crossing autoequivalence is the categorical generator of the entire motivic wall-crossing identity, and the DT wall-crossing formula is the trace of this autoequivalence on the Grothendieck group. Rubric: full credit for the King stability setup, the combinatorial magic window, the wall-crossing derived equivalence, and the link to DT wall-crossing.

Exercise 8 (hard, short-answer).

Prove or sketch a proof of full faithfulness of the restriction functor $res_{ℓ} : W (ℓ) \to D^{b} (X_{ℓ}^{ss} / G)$ in the simplest case beyond the identity action: $G = C^{\times}$ acting on $X = C^{2}$ with weights $(+ 1, + 1)$ and the standard linearisation. Compute the Ext groups between $O$ and $O (- 1)$ on the stack and on the quotient $P^{1}$ and verify they agree.

Hint

The unstable locus is the origin ${0} \subset C^{2}$ , a single point. Compute the local cohomology supported at the origin for both $O$ and $O (- 1)$ , and use the localisation triangle.

Answer

Setup. $G = C^{\times}$ acts on $X = C^{2}$ with weights $(+ 1, + 1)$ , semistable locus $X^{ss} = C^{2} ∖ {0}$ , GIT quotient $Q = P^{1}$ , Kempf-Ness weight $η = 2$ , magic-window characters $a \in {- 1, 0}$ .

Ext on the stack. The stack $[C^{2} / C^{\times}]$ has coherent-sheaf category equivalent to $Z$ -graded modules over the polynomial ring $C [x, y]$ with $∣ x ∣ = ∣ y ∣ = 1$ . The line bundle $O (a)$ corresponds to the shifted free module $C [x, y] ⟨ a ⟩$ . The graded Ext is $$ \mathrm{Ext}^*_{[\mathbb{C}^2 / \mathbb{C}^\times]}(\mathcal{O}, \mathcal{O}(-1)) = \mathrm{Hom}{\mathbb{Z}\text{-gr}}(\mathbb{C}[x, y], \mathbb{C}[x, y]\langle -1 \rangle) = \mathbb{C}[x, y]{-1}. $$ The graded piece of degree $- 1$ in $C [x, y]$ vanishes (the ring is non-negatively graded), so $Ext^{0} = 0$ . Higher Ext groups vanish because $O$ is free on the affine stack.

Ext on the quotient. On $P^{1}$ , the line bundles $O$ and $O (- 1)$ have $$ \mathrm{Ext}^_{\mathbb{P}^1}(\mathcal{O}, \mathcal{O}(-1)) = H^(\mathbb{P}^1, \mathcal{O}(-1)) = 0, $$ the standard cohomology vanishing on $P^{1}$ ( $O (- 1)$ has no global sections and is acyclic in higher cohomology because $dim P^{1} = 1$ and $- 1 > - 2 = - dim - 1$ ).

Comparison. Both Ext groups vanish in every degree. The restriction $res$ sends the zero space to the zero space, an isomorphism. Repeating the calculation in the reverse direction with $O (- 1)$ and $O$ gives $$ \mathrm{Ext}^_{[\mathbb{C}^2 / \mathbb{C}^\times]}(\mathcal{O}(-1), \mathcal{O}) = \mathbb{C}[x, y]_1 = \mathbb{C} \cdot x \oplus \mathbb{C} \cdot y, $$ a two-dimensional space concentrated in degree $0$ . On $P^{1}$ , $\mathrm{Ext}^(\mathcal{O}(-1), \mathcal{O}) = H^*(\mathbb{P}^1, \mathcal{O}(1)) = \mathbb{C}^2 $, t h e tw o - d im e n s i o na l s p a ceo f l in e a r f or m so n$ \mathbb{P}^1 $— s am e an s w er . F u l l f ai t h f u l n ess i s v er i f i e d o nb o t h p ai r so f g e n er a t or so f t h e ma g i c w in d o w . R u b r i c : f u l l cr e d i t f or t h e tw o g r a d e d - m o d u l e E x t co m p u t a t i o n s, t h e tw o$ \mathbb{P}^1$-cohomology computations, and the agreement.

Advanced results Master

Theorem (categorical Kempf-Ness, Halpern-Leistner 2015 §3). Let $G$ act on a smooth quasi-projective variety $X$ with a generic stability $ℓ$ , and let $β \in KN (ℓ)$ be a Hilbert-Mumford one-parameter subgroup with fixed component $Z_{β}$ and Kempf-Ness weight $η_{β}$ . The bounded derived category of the corresponding Kirwan stratum $S_{β} = G \cdot Y_{β}^{KN}$ admits a $Z$ -indexed semiorthogonal decomposition $$ D^b([S_\beta / G]) = \bigoplus_{w \in \mathbb{Z}} D^b([Z_\beta / L_\beta])w, $$ *where $L\beta $i s t h e L e v i ce n t r a l i ser o f$ \beta $in s i d e$ G $an d$ D^b([Z_\beta / L_\beta])_w $i s t h e f u l l s u b c a t e g or y o f o bj ec t so n w hi c h$ \beta $a c t s w i t h w e i g h t$ w $. T h ese mi or t h o g o na l i t y i s g e n er a t e d b y t h e$ \beta$-weight ordering.* ^{[Halpern-Leistner 2015]}

The categorical Kempf-Ness theorem is the structural input to the magic window theorem. It refines the Kirwan stratification (04.10.08) to a derived-categorical decomposition, identifies the unstable strata with Levi quotients on the fixed components, and assembles the magic window as a specific block in the resulting semiorthogonal decomposition.

Theorem (Špenko-Van den Bergh NCCR for linear quotient singularities). Let $G$ be a connected reductive group acting linearly on a finite-dimensional $k$ -vector space $V$ , and let $S = k [V]^{G}$ be the invariant ring. The shifted-rectangle module $$ M = \bigoplus_{\chi \in \mathrm{box}\delta} k[V] \otimes V\chi, $$ where $V_{χ}$ is the irreducible $G$ -representation of highest weight $χ$ and the sum is over characters $χ$ in the centred Hilbert-Mumford box, is a reflexive $S$ -module whose endomorphism algebra $A = End_{S} (M)$ is a noncommutative crepant resolution of $S$ in the sense of Van den Bergh 2004. Moreover, $D^{b} (A) ≃ W (ℓ, δ)$ as triangulated categories. ^{[Špenko-Van den Bergh 2017]}

The Špenko-Van den Bergh theorem constructs an NCCR uniformly for every linear quotient singularity by a connected reductive group. This is the noncommutative side of the magic window programme: the NCCR algebra acts on the module $M$ , and the tilting equivalence between $A$ -modules and the magic window is the noncommutative incarnation of the derived equivalence with the commutative crepant resolution (when the latter exists). For groups without a commutative resolution — for example, $SL_{n}$ acting on $V = Mat_{n \times n}$ for $n \geq 4$ — the NCCR is the only crepant resolution available, and the magic window furnishes it.

Theorem (Donovan-Segal window-shift, 2014). For a generic stability $ℓ$ and a real shift $δ$ , the comparison functor $T_{δ} : W (ℓ, 0) \to W (ℓ, δ)$ is an autoequivalence of $D^{b} ([X / G])$ given as a composition of a line-bundle twist by an explicit character $χ_{δ}$ and a sequence of left mutations through the unstable semiorthogonal blocks indexed by characters exiting the centred box. The kernel of $T_{δ}$ in the Fourier-Mukai presentation is the structure sheaf of the diagonal twisted by $χ_{δ}$ , restricted along the wall-crossing locus. ^{[Donovan-Segal 2014]}

The Donovan-Segal formula computes the window-shift autoequivalence explicitly. The line-bundle-twist part is straightforward; the mutation part records the categorical analogue of the change in semistability across the wall, and the composition is what realises the Bondal-Orlov flop equivalence when $T_{δ}$ is interpreted as the window-shift between two adjacent chambers. The formula has been verified in many cases — Grassmannian flops (Donovan-Segal 2014), spinor flops (Segal 2011), determinantal flops (Buchweitz-Leuschke-Van den Bergh) — and is the categorical organising principle for derived wall-crossing.

Theorem (Bondal-Orlov flop derived equivalence, global-quotient case, Halpern-Leistner 2015 Corollary 5.5). Let $G$ act on a smooth quasi-projective variety $X$ and let $ℓ_{-}, ℓ_{+}$ be two generic stabilities in adjacent GIT chambers. The two GIT quotients $Q_{\pm} = X_{ℓ_{\pm}}^{ss} / G$ are smooth projective varieties related by a flip. The composition of the magic window equivalences, $$ \Phi_{-+} : D^b(Q_-) \xrightarrow{\sim} \mathcal{W}(\ell_-) \xrightarrow{T_\delta} \mathcal{W}(\ell_+) \xrightarrow{\sim} D^b(Q_+), $$ is a triangulated equivalence whose Fourier-Mukai kernel is the structure sheaf of the fibre product $Q_{-} \times_{Q_{wall}} Q_{+}$ , where $Q_{wall} = X_{ℓ_{wall}}^{ss} / / G$ is the GIT quotient at the wall. ^{[Halpern-Leistner 2015]}

This is the global-quotient realisation of the Bondal-Orlov conjecture: birational varieties related by a flop have equivalent bounded derived categories. The conjecture is open in general; the magic window construction proves it whenever the flop can be realised by a VGIT wall-crossing on a smooth quotient stack. Halpern-Leistner-Sam 2020 ^{[Halpern-Leistner-Sam 2020]} extend the case list to all toric flops and to a large class of quiver-moduli flops, completing the conjecture's verification within the global-quotient regime.

Theorem (derived GIT for stacks with non-reductive stabilisers, Halpern-Leistner 2018 onwards). Let $X$ be a smooth quasi-projective variety carrying an action of a not-necessarily-reductive linear algebraic group $G$ whose unipotent radical $R_{u} (G)$ is graded in the sense of Bérczi-Kirwan 2016. The magic window construction extends to this setting via the reductive envelope $G^{+}$ of Doran-Kirwan (04.10.14), and the resulting subcategory of $D^{b} ([X / G])$ is equivalent to $D^{b} (X_{ℓ}^{ss} / G)$ for the non-reductive GIT quotient at a generic stability $ℓ$ .

The non-reductive extension is the natural generalisation of the magic window theorem to the framework of 04.10.14. The same Hilbert-Mumford box defines the window, with the Kempf-Ness weights computed via the reductive envelope. Applications include moduli of parabolic Higgs bundles, jet-differential moduli, and the curvilinear Hilbert scheme, where the non-reductive ambient quotient stack is the natural home of the derived category and the magic window furnishes the bounded derived category of the corresponding moduli space.

Theorem (Kempf-Ness via the moment map and Morse theory on the moduli of objects, Halpern-Leistner-Sjamaar-Tolman). The Hilbert-Mumford theta-stratification of $X$ corresponds, on the symplectic side via Kempf-Ness (04.10.04), to a stratification of the moduli of objects of $D^{b} ([X / G])$ by the gradient flow of $∥ μ ∥^{2}$ , where $μ$ is the moment map of a compatible Hamiltonian $K$ -action ( $K$ a maximal compact of $G$ ). The Morse-Bott critical sets of $∥ μ ∥^{2}$ are the categorical analogues of the Kempf-Ness fixed-component sets $Z_{β}$ , and the resulting equivariant Poincaré polynomial of $D^{b} ([X / G])$ matches the GIT-side Kirwan formula.

This is the symplectic-categorical dictionary that underpins the entire programme. The moment map gives a Morse function on the moduli of objects in $D^{b} ([X / G])$ , the critical sets index the unstable strata, and the Morse-theoretic decomposition realises the semiorthogonal decomposition of the derived category at the categorical level. Halpern-Leistner-Sam 2020 ^{[Halpern-Leistner-Sam 2020]} formalise this dictionary for stacks with controlled instability, with Donaldson-Thomas wall-crossing of 04.10.13 entering as the moduli-of-objects analogue of the Hilbert-Mumford weight inequality.

Theorem (Donaldson-Thomas wall-crossing via derived GIT, Toda 2009 and Halpern-Leistner-Sam 2020). For a smooth projective Calabi-Yau threefold $X$ , the moduli stack of $σ$ -semistable objects in $D^{b} (X)$ for a Bridgeland stability $σ$ carries a theta-stratification compatible with the moment-map gradient flow of 04.10.04. Across a Bridgeland wall in the space of stability conditions, the magic window framework produces a derived equivalence between the moduli of $σ_{-}$ -semistable and $σ_{+}$ -semistable objects, refining the Joyce-Song / Kontsevich-Soibelman wall-crossing formula for Donaldson-Thomas invariants from the motivic level to the derived-categorical level. ^{[Toda 2009]}

The Donaldson-Thomas application is the highest-stakes use case for derived GIT. DT invariants on a Calabi-Yau threefold count Bridgeland-stable objects in the derived category, and they jump in a controlled way as the stability condition crosses a wall. The motivic wall-crossing formula of Kontsevich-Soibelman lifts this jump to an identity in the motivic Hall algebra; the derived GIT framework lifts it one further step to a derived equivalence between the two moduli stacks, with the magic window playing the role of the universal stability-independent receptacle. Toda 2009 ^{[Toda 2009]} and Halpern-Leistner-Sam 2020 ^{[Halpern-Leistner-Sam 2020]} provide the rigorous categorical wall-crossing in the cases where the magic window construction extends to the moduli of objects.

Synthesis. Derived geometric invariant theory replaces the point-set GIT quotient $X / /_{ℓ} G$ by the bounded derived category of the quotient stack $[X / G]$ together with a finite-dimensional shifted-Hilbert-Mumford window inside it, and the central insight is that the derived category of every GIT quotient appears as a numerically defined subcategory of one fixed stack-level derived category, with stability dependence reduced to a window shift rather than a category change. Three apparently distinct constructions — the categorical Kempf-Ness theta-stratification, the noncommutative crepant resolution of Van den Bergh, and the window-shift autoequivalence of Donovan-Segal — fit together as one identity: each stability $ℓ$ produces a window, each window is equivalent to the derived category of the quotient at $ℓ$ , and crossing a wall in the GIT fan identifies adjacent windows via an autoequivalence whose explicit form is the line-bundle twist plus mutation of Donovan-Segal. Putting these together, the foundational reason wall-crossing yields derived equivalences in the global-quotient setting is that two adjacent chambers index two adjacent windows inside the same big stack-level category, and the bridge is that the Bondal-Orlov flop conjecture in this setting is a one-line consequence of magic window comparison rather than a separate construction. This is exactly the categorical refinement of variation of GIT, and the central insight is that GIT operates on points but its derived enhancement operates on sheaves, identifying every chamber-dependent quotient with one chamber-dependent window inside a chamber-independent big category.

The pattern recurs across moduli theory. For K-stability and the Yau-Tian-Donaldson conjecture (04.10.13) the infinite-dimensional analogue of the magic window produces derived autoequivalences associated with one-parameter degenerations of Fano varieties, with the Mabuchi functional playing the role of the Hilbert-Mumford weight. For Donaldson-Thomas wall-crossing of 04.10.13 the magic window applied to Bridgeland-stable objects on a Calabi-Yau threefold lifts the motivic Kontsevich-Soibelman wall-crossing formula to a derived equivalence between the two stability moduli. For the Bondal-Orlov flop conjecture the global-quotient case becomes a theorem rather than a conjecture, with the explicit Fourier-Mukai kernel given by the structure sheaf of the fibre product over the wall quotient. The bridge is that all of these wall-crossings — GIT, Bridgeland, K-stability — are governed by the same shifted-Hilbert-Mumford-box mechanism, and the central insight is that the noncommutative crepant resolution of Špenko-Van den Bergh provides the algebra side of the magic window even when no commutative crepant resolution exists. The synthesis is structural: every classical flop derived equivalence in the literature — Atiyah, Grassmannian, spinor, determinantal — is a corollary of the magic window theorem with the appropriate $(G, X, L)$ -data, and the magic window is the universal organising principle.

Full proof set Master

Proposition (magic window theorem for $G = C^{\times}$ acting on $C^{n + 1}$ with the standard linearisation). The bounded derived category of the quotient stack $[C^{n + 1} / C^{\times}]$ contains a magic window of cardinality $n + 1$ generated by the line bundles $O, O (- 1), \dots, O (- n)$ , and the restriction functor to $D^{b} (P^{n})$ is an equivalence of triangulated categories.

Proof. Write $X = C^{n + 1}$ with $C^{\times}$ acting by scaling. The coherent-sheaf category $Coh^{C^{\times}} (X)$ is equivalent to the category of $Z$ -graded finitely generated modules over the polynomial ring $R = C [x_{0}, \dots, x_{n}]$ with grading $∣ x_{i} ∣ = 1$ . Line bundles correspond to shifted copies of $R$ : $O (a) \leftrightarrow R ⟨ a ⟩$ .

The unstable locus is the origin ${0} \subset X$ , the destabilising one-parameter subgroup is $β = 1$ , and the Kempf-Ness weight is $η = n + 1$ (the sum of the $β$ -weights on $T_{0} X = C^{n + 1}$ ). The centred Hilbert-Mumford box is $- \frac{n + 1}{2} < a \leq \frac{n + 1}{2}$ . After a uniform shift, take the box to be $- n \leq a \leq 0$ , giving $n + 1$ integer characters: $a \in {- n, - n + 1, \dots, 0}$ .

The magic window $W$ is the full triangulated subcategory of $D^{b} ([X / C^{\times}])$ generated by $O, O (- 1), \dots, O (- n)$ . By Beilinson 1978, $D^{b} (P^{n})$ is generated as a triangulated category by $O, O (- 1), \dots, O (- n)$ with a known $Ext$ -pattern: $Ext^{k} (O (a), O (b)) = H^{k} (P^{n}, O (b - a))$ is non-zero only for $k = 0$ with $b - a \geq 0$ or $k = n$ with $b - a \leq - n - 1$ . For $a, b \in {- n, \dots, 0}$ the second condition never holds (since $b - a \geq - n$ , not $\leq - n - 1$ ), and the first gives $Hom (O (a), O (b)) = H^{0} (P^{n}, O (b - a))$ , the space of homogeneous polynomials of degree $b - a$ in $C [x_{0}, \dots, x_{n}]$ .

On the stack $[X / C^{\times}]$ , the analogous calculation reads $Ext^{k} (O (a), O (b)) = R_{b - a}$ for $k = 0$ and $0$ for $k > 0$ (since $R$ is free over itself). For $a, b \in {- n, \dots, 0}$ with $b - a \geq 0$ , this is the space of polynomials of degree $b - a$ , matching the $P^{n}$ calculation. For $b - a < 0$ , both sides vanish. The restriction functor is therefore fully faithful on the chosen generators.

Essential surjectivity follows from Beilinson's generation: the line bundles $O, O (- 1), \dots, O (- n)$ generate $D^{b} (P^{n})$ via the Beilinson resolution of the diagonal in $P^{n} \times P^{n}$ . Since restriction preserves the generation, the magic window restricts to a triangulated subcategory of $D^{b} (P^{n})$ containing the Beilinson generators, hence is all of $D^{b} (P^{n})$ . The restriction functor is an equivalence. $□$

Proposition (Atiyah flop derived equivalence via window shift). For the $C^{\times}$ -action on $C^{4}$ with weights $(+ 1, + 1, - 1, - 1)$ , the two small resolutions $Q_{\pm}$ of the conifold ${x_{1} y_{1} + x_{2} y_{2} = 0}$ have equivalent bounded derived categories, and the equivalence is realised as the window-shift autoequivalence between the two magic windows.

Proof. Write $X = C^{4}$ with coordinates $(x_{1}, x_{2}, y_{1}, y_{2})$ and $C^{\times}$ acting with weights $(+ 1, + 1, - 1, - 1)$ . The two GIT chambers are $ℓ > 0$ and $ℓ < 0$ , with semistable loci ${(x_{1}, x_{2}) \neq = 0}$ and ${(y_{1}, y_{2}) \neq = 0}$ respectively. Each chamber's GIT quotient is a small resolution of the conifold $V = {x_{1} y_{1} + x_{2} y_{2} = 0} \subset C^{4} / / C^{\times}$ (the affine quotient at the wall).

The Kempf-Ness weights are $η_{+} = 2$ and $η_{-} = 2$ for the two destabilising one-parameter subgroups $β_{\pm} = \pm 1$ . For chamber $ℓ > 0$ , the centred magic window box is $- 1 < a \leq 1$ , giving characters $a \in {0, 1}$ . For chamber $ℓ < 0$ , the box is $- 1 \leq a < 1$ , giving $a \in {0, - 1}$ .

By the proposition above (adapted to this $G$ -action), restriction to $Q_{+}$ realises $W (ℓ_{+}) ≃ D^{b} (Q_{+})$ via the line bundles $O, O (1)$ , and restriction to $Q_{-}$ realises $W (ℓ_{-}) ≃ D^{b} (Q_{-})$ via $O, O (- 1)$ .

The window-shift autoequivalence $T : W (ℓ_{-}) \to W (ℓ_{+})$ is computed by Donovan-Segal: tensor with $O (1)$ , then left-mutate through the unstable block at the origin. Explicitly, $T (O) = O (1)$ and $T (O (- 1)) = O$ (after mutation). The composition $$ \Phi : D^b(Q_-) \xrightarrow{\sim} \mathcal{W}(\ell_-) \xrightarrow{T} \mathcal{W}(\ell_+) \xrightarrow{\sim} D^b(Q_+) $$ is a triangulated equivalence, and unwinding the definitions identifies $Φ$ with the Bondal-Orlov flop Fourier-Mukai equivalence whose kernel is $O_{Q_{-} \times_{V} Q_{+}}$ — exactly the structure sheaf of the fibre product over the conifold. $□$

Proposition (NCCR for the conifold via Špenko-Van den Bergh). For the conifold singularity $V = {x_{1} y_{1} + x_{2} y_{2} = 0}$ realised as the linear quotient $C^{4} / / C^{\times}$ with the wall linearisation, the module $M = R \oplus R (1)$ over $S = R^{C^{\times}} = k [V]$ is reflexive and its endomorphism algebra $A = End_{S} (M)$ is a noncommutative crepant resolution.

Proof sketch. The Špenko-Van den Bergh box for $G = C^{\times}$ on $C^{4}$ at the wall gives characters $a \in {0, 1}$ , so the module $M = (k [V] \otimes V_{0}) \oplus (k [V] \otimes V_{1}) = R^{C^{\times}} \oplus R^{C^{\times}} (1)$ in the natural $Z$ -grading. Reflexivity follows from $R$ being a polynomial ring (hence Cohen-Macaulay) and $S$ being a hypersurface singularity (the conifold). The endomorphism algebra $A = End_{S} (M)$ is the conifold quiver algebra of Klebanov-Witten (1998) — the path algebra of the conifold quiver modulo the Klebanov-Witten relations $A_{i} B_{j} A_{k} = A_{k} B_{j} A_{i}$ and $B_{i} A_{j} B_{k} = B_{k} A_{j} B_{i}$ .

Global dimension equals $3 = dim V$ by an explicit projective resolution of the two simple $A$ -modules using the four arrows of the conifold quiver. Cohen-Macaulayness of $A$ over $S$ follows from $M$ being a maximal Cohen-Macaulay $S$ -module. The symmetric condition $Hom_{S} (A, S) ≅ A$ as bimodules is the Calabi-Yau- $3$ property of the conifold, verified by computing the trace pairing on $A$ via the canonical module of $S$ (which is the identity bimodule because $V$ is Gorenstein).

By the Špenko-Van den Bergh theorem, $D^{b} (A) ≃ W (ℓ_{wall}, 0) \subset D^{b} ([C^{4} / C^{\times}])$ , and by the magic window theorem applied to either chamber, $W (ℓ_{wall}, 0)$ is equivalent to $D^{b} (Q_{\pm})$ for either of the two small resolutions. Hence $D^{b} (A) ≃ D^{b} (Q_{+}) ≃ D^{b} (Q_{-})$ , with the two equivalences related by the window-shift autoequivalence. $□$

Theorem (Halpern-Leistner magic window, full statement, restated). Under the hypotheses of the magic window theorem, the centred magic window $W (ℓ) \subset D^{b} ([X / G])$ is equivalent via restriction to $D^{b} (X_{ℓ}^{ss} / G)$ . ^{[Halpern-Leistner 2015]} The proof of the theorem in the Intermediate section assembles the categorical Kempf-Ness theta-stratification, identifies the magic window as a semiorthogonal block, checks full faithfulness via the box constraint, and verifies essential surjectivity using a Beilinson-style resolution of the diagonal on the quotient. The detailed proof occupies Halpern-Leistner 2015 §3-§5. $□$

Theorem (Donovan-Segal window-shift, stated without full proof here — full proof in Donovan-Segal 2014 §3 and Halpern-Leistner 2015 §3.5 ^{[Donovan-Segal 2014]}). The window-shift autoequivalence is a composition of a line-bundle tensor twist and a finite sequence of left mutations through unstable semiorthogonal blocks. The explicit formula is computed in terms of the Kempf-Ness fixed-component data $(Z_{β}, η_{β})$ .

Theorem (Špenko-Van den Bergh NCCR for linear quotient singularities, stated without full proof here — full proof in Špenko-Van den Bergh 2017 ^{[Špenko-Van den Bergh 2017]}). The construction takes the shifted-rectangle module $M = ⨁_{χ} k [V] \otimes V_{χ}$ over the invariant ring $S$ , proves reflexivity via the linear-quotient symmetry, computes the global dimension of $A = End_{S} (M)$ as $dim S$ via an explicit Koszul-style resolution, and verifies the Calabi-Yau / symmetric condition through the trace pairing.

Connections Master

Geometric invariant theory 04.10.02. Derived GIT lifts the classical Mumford construction from the point-set quotient $X / /_{ℓ} G$ to the bounded derived category $D^{b} ([X / G])$ together with a numerically defined magic window. Every concrete GIT quotient enters the derived picture as the image of a magic window under restriction, and putting these together with the variation-of-GIT framework the foundational reason wall-crossing produces derived equivalences is that two adjacent chambers index two adjacent windows inside the same big stack-level category.
Hilbert-Mumford numerical criterion 04.10.03. The magic window box is defined by Kempf-Ness weights, which are exactly the weights of one-parameter subgroups in the Hilbert-Mumford criterion. The shift from point-set stability to derived-categorical stability rotates the same Mumford weights into the role of bounds on character lattices, identifying the categorical theta-stratification with the classical Hilbert-Mumford theta-stratification on the quotient stack.
Kirwan stratification 04.10.08. The categorical Kempf-Ness theta-stratification refines the Kirwan stratification of the unstable locus to a derived-categorical semiorthogonal decomposition. Each Kirwan stratum $S_{β}$ contributes a $Z$ -indexed block $D^{b} ([S_{β} / G]) = ⨁_{w} D^{b} ([Z_{β} / L_{β}])_{w}$ , and the magic window is the single block where every $β$ -weight lies in the Hilbert-Mumford box.
Variation of GIT 04.10.09. Classical VGIT produces a chamber-wall decomposition of the equivariant ample cone with flips between adjacent quotients. Derived GIT lifts this to derived equivalences between adjacent magic windows, with the flip on the point-set side becoming a window-shift autoequivalence on the derived side. The Bondal-Orlov flop conjecture in the global-quotient case becomes a one-line consequence of magic window comparison.
Non-reductive GIT 04.10.14. Halpern-Leistner's magic window construction extends to non-reductive group actions via the reductive envelope of Doran-Kirwan, with the Kempf-Ness weights computed inside the enveloping reductive group. Applications include parabolic Higgs bundles, jet differentials in the hyperbolicity programme, and curvilinear Hilbert schemes — moduli problems whose underlying group is non-reductive but whose derived categories are accessible via the non-reductive magic window.
K-stability and the Yau-Tian-Donaldson conjecture 04.10.13. The infinite-dimensional analogue of derived GIT on the moduli of Fano varieties produces categorical analogues of the Donaldson-Futaki invariant and derived autoequivalences associated with one-parameter degenerations. The Mabuchi functional plays the role of the Hilbert-Mumford weight, and the magic window provides the bounded derived category of moduli of K-stable Fano varieties as a subcategory of the stack-level derived category.
Derived functors and Ext 04.03.06. The magic window theorem is a statement about the bounded derived category of coherent sheaves, with full faithfulness verified by computing Ext groups on the stack and on the quotient and showing they agree. The Ext computations use the localisation triangle for the open immersion $X^{ss} ↪ X$ together with the vanishing of unstable Ext contributions implied by the Hilbert-Mumford box constraint.
Canonical sheaf 04.08.02. The crepant condition in the noncommutative crepant resolution definition is the symmetric / Calabi-Yau condition $Hom_{S} (A, S) ≅ A$ as $A$ -bimodules, which is the noncommutative analogue of the triviality of the canonical bundle on a commutative crepant resolution. For Calabi-Yau three-fold flops the canonical bundles of $Q_{\pm}$ vanish, and the magic window equivalence preserves the Calabi-Yau structure on both sides.

Historical & philosophical context Master

The derived-categorical refinement of geometric invariant theory has a long prehistory. Beilinson's 1978 resolution of the diagonal on $P^{n}$ identified the bounded derived category of projective space with the category generated by the line bundles $O, O (- 1), \dots, O (- n)$ , foreshadowing the magic window construction in the simplest case. Kapranov's 1988 work on Grassmannians and flag varieties extended this to a rich combinatorial theory of exceptional collections indexed by Young diagrams. Mukai's 1981 Fourier-Mukai transform between abelian varieties and their duals introduced the kernel-on-the-product viewpoint that later became the standard tool for constructing derived equivalences.

The flop derived equivalence conjecture of Alexei Bondal and Dmitri Orlov was formulated in Derived categories of coherent sheaves (Proc. ICM Beijing 2002, Vol. II, 47-56) ^{[Bondal-Orlov 2002]}: birational varieties related by a flop should have equivalent bounded derived categories. The conjecture was verified in low-dimensional cases (Bridgeland 2002 for threefold flops) but remained open in general. Michel Van den Bergh's Non-commutative crepant resolutions (in The Legacy of Niels Henrik Abel, Springer 2004, 749-770) ^{[Van den Bergh 2004]} introduced the NCCR framework as a noncommutative substitute for commutative crepant resolutions, conjecturally producing derived equivalences whenever both kinds of resolution exist.

The breakthrough came from Daniel Halpern-Leistner's 2014 thesis and his 2015 Journal of the American Mathematical Society paper The derived category of a GIT quotient (J. Amer. Math. Soc. 28, 871-912) ^{[Halpern-Leistner 2015]}. Halpern-Leistner introduced the magic window — the precise shifted-Hilbert-Mumford rectangle of characters — and proved that the bounded derived category of every GIT quotient is recovered as a magic window inside the bounded derived category of the quotient stack. Will Donovan and Ed Segal in Window shifts, flop equivalences and Grassmannian twists (Adv. Math. 261, 56-93, 2014) ^{[Donovan-Segal 2014]} independently identified the window-shift autoequivalence and used it to construct derived equivalences for Grassmannian flops, with the Bondal-Orlov flop equivalence emerging as a one-line consequence of magic window comparison. Špela Špenko and Michel Van den Bergh in Non-commutative resolutions of quotient singularities for reductive groups (Invent. Math. 210, 3-67, 2017) ^{[Špenko-Van den Bergh 2017]} systematically constructed NCCRs for all linear quotient singularities by connected reductive groups, identifying the shifted-rectangle module as the universal construction.

The programme has since been extended in several directions. Ballard-Favero-Katzarkov (Variation of GIT quotients and derived categories, J. reine angew. Math. 746, 235-303, 2019) extended the magic window framework to VGIT chains with multiple walls. Yukinobu Toda's Curve counting theories via stable objects (Adv. Math. 222 / Geom. Topol. 13, 2009) ^{[Toda 2009]} applied derived GIT to Donaldson-Thomas invariants on Calabi-Yau threefolds, lifting Kontsevich-Soibelman wall-crossing from motivic invariants to derived equivalences. Halpern-Leistner and Steven Sam in Combinatorial constructions of derived equivalences (J. Amer. Math. Soc. 33, 735-773, 2020) ^{[Halpern-Leistner-Sam 2020]} gave combinatorial recipes producing the window-shift autoequivalence for toric and quiver moduli, completing the Bondal-Orlov conjecture in these cases. Špenko-Van den Bergh 2020s work extended the NCCR construction to non-reductive linear quotient singularities, connecting derived GIT to the non-reductive framework of Bérczi-Kirwan (04.10.14).

Bibliography Master

@article{HalpernLeistner2015,
  author  = {Halpern-Leistner, Daniel},
  title   = {The derived category of a {GIT} quotient},
  journal = {Journal of the American Mathematical Society},
  volume  = {28},
  year    = {2015},
  pages   = {871--912}
}

@incollection{VandenBergh2004,
  author    = {Van den Bergh, Michel},
  title     = {Non-commutative crepant resolutions},
  booktitle = {The Legacy of Niels Henrik Abel},
  publisher = {Springer-Verlag},
  year      = {2004},
  pages     = {749--770}
}

@article{SpenkoVdB2017,
  author  = {{\v S}penko, {\v S}pela and Van den Bergh, Michel},
  title   = {Non-commutative resolutions of quotient singularities for reductive groups},
  journal = {Inventiones Mathematicae},
  volume  = {210},
  year    = {2017},
  pages   = {3--67}
}

@article{DonovanSegal2014,
  author  = {Donovan, Will and Segal, Ed},
  title   = {Window shifts, flop equivalences and {G}rassmannian twists},
  journal = {Advances in Mathematics},
  volume  = {261},
  year    = {2014},
  pages   = {56--93}
}

@inproceedings{BondalOrlov2002,
  author    = {Bondal, Alexei and Orlov, Dmitri},
  title     = {Derived categories of coherent sheaves},
  booktitle = {Proceedings of the {ICM} {B}eijing 2002},
  volume    = {II},
  year      = {2002},
  pages     = {47--56}
}

@book{MumfordFogartyKirwan1994,
  author    = {Mumford, David and Fogarty, John and Kirwan, Frances},
  title     = {Geometric Invariant Theory},
  edition   = {3rd},
  publisher = {Springer-Verlag},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  volume    = {34},
  year      = {1994}
}

@article{BallardFaveroKatzarkov2019,
  author  = {Ballard, Matthew and Favero, David and Katzarkov, Ludmil},
  title   = {Variation of geometric invariant theory quotients and derived categories},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume  = {746},
  year    = {2019},
  pages   = {235--303}
}

@article{HalpernLeistnerSam2020,
  author  = {Halpern-Leistner, Daniel and Sam, Steven V.},
  title   = {Combinatorial constructions of derived equivalences},
  journal = {Journal of the American Mathematical Society},
  volume  = {33},
  year    = {2020},
  pages   = {735--773}
}

@article{Toda2009,
  author  = {Toda, Yukinobu},
  title   = {Curve counting theories via stable objects},
  journal = {Advances in Mathematics},
  volume  = {222},
  year    = {2009},
  pages   = {1014--1071}
}

@article{Beilinson1978,
  author  = {Beilinson, Alexander A.},
  title   = {Coherent sheaves on {$\mathbb{P}^n$} and problems in linear algebra},
  journal = {Functional Analysis and its Applications},
  volume  = {12},
  year    = {1978},
  pages   = {214--216}
}

@article{Kapranov1988,
  author  = {Kapranov, Mikhail M.},
  title   = {On the derived categories of coherent sheaves on some homogeneous spaces},
  journal = {Inventiones Mathematicae},
  volume  = {92},
  year    = {1988},
  pages   = {479--508}
}

@article{Bridgeland2002,
  author  = {Bridgeland, Tom},
  title   = {Flops and derived categories},
  journal = {Inventiones Mathematicae},
  volume  = {147},
  year    = {2002},
  pages   = {613--632}
}

@article{KontsevichSoibelman2008,
  author  = {Kontsevich, Maxim and Soibelman, Yan},
  title   = {Stability structures, motivic {D}onaldson-{T}homas invariants and cluster transformations},
  year    = {2008},
  note    = {arXiv:0811.2435}
}

Prerequisites

04.10.02
04.10.03
04.10.08
04.10.09
04.03.06
04.08.02

Tier anchors

beginner: Strogatz-style: the derived category of a GIT quotient is recovered inside the derived category of the quotient stack as a controlled window of line bundles, and varying the window across a wall produces a derived equivalence between two birationally distinct quotients
intermediate: Halpern-Leistner 2015 *J. Amer. Math. Soc.* 28; Špenko-Van den Bergh 2017 *Invent. Math.* 210; Donovan-Segal 2014 *Adv. Math.* 261; Van den Bergh 2004 *The Legacy of Niels Henrik Abel*
master: Halpern-Leistner 2015 *The derived category of a GIT quotient* (J. Amer. Math. Soc. 28, 871-912 — the magic window theorem and the categorical Kempf-Ness theta-stratification); Van den Bergh 2004 *Non-commutative crepant resolutions* (in *The Legacy of Niels Henrik Abel*, Springer, 749-770 — the NCCR definition); Špenko-Van den Bergh 2017 *Non-commutative resolutions of quotient singularities for reductive groups* (Invent. Math. 210, 3-67 — the systematic construction of magic windows for linear quotient singularities); Donovan-Segal 2014 *Window shifts, flop equivalences and Grassmannian twists* (Adv. Math. 261, 56-93 — the window-shift formula and the Grassmannian-flop derived equivalence); Bondal-Orlov 2002 *Derived categories of coherent sheaves* (Proc. ICM Beijing II, 47-56 — the flop derived equivalence conjecture); Halpern-Leistner-Sam 2020 *Combinatorial constructions of derived equivalences* (J. Amer. Math. Soc. 33, 735-773); Mumford-Fogarty-Kirwan *Geometric Invariant Theory* 3rd ed. (Springer Ergebnisse 34, 1994); Ballard-Favero-Katzarkov 2019 *Variation of geometric invariant theory quotients and derived categories* (J. reine angew. Math. 746, 235-303)

References

Halpern-Leistner — The derived category of a GIT quotient · J. Amer. Math. Soc. 28 (2015), 871–912
Van den Bergh — Non-commutative crepant resolutions · in *The Legacy of Niels Henrik Abel* (Springer, 2004), 749–770
Špenko, Van den Bergh — Non-commutative resolutions of quotient singularities for reductive groups · Inventiones Math. 210 (2017), 3–67
Donovan, Segal — Window shifts, flop equivalences and Grassmannian twists · Advances in Mathematics 261 (2014), 56–93
Bondal, Orlov — Derived categories of coherent sheaves · Proceedings of the ICM Beijing 2002, Vol. II, 47–56
Mumford, Fogarty, Kirwan — Geometric Invariant Theory · 3rd ed., Springer Ergebnisse 34, 1994
Ballard, Favero, Katzarkov — Variation of GIT quotients and derived categories · J. reine angew. Math. 746 (2019), 235–303
Halpern-Leistner, Sam — Combinatorial constructions of derived equivalences · J. Amer. Math. Soc. 33 (2020), 735–773
Toda — Curve counting theories via stable objects · Adv. Math. 222 (2009), 1014–1071; Geom. Topol. 13 (2009), 2349–2378

Estimated time

beginner: 15m
intermediate: 40m
master: 80m