1
vote
0answers
97 views

Why $D^b(S)\cong D^b_{\text{Car}}(\text{cosq}(X\rightarrow S))$?

Let $p: X\rightarrow S$ be a map between topological spaces and we can construct the simplicial space $\text{cosq}(X\rightarrow S)$ where $X_0=X$, $X_1=X\times_S X$ and $$ X_n=\underbrace{ X\times_S ...
3
votes
0answers
152 views

Does GKZ's reflexivity theorem imply the Plucker formula?

Let $S\subset\mathbb{P}^n$, Gelfand-Kapranov-Zelevinsky defined its dual variety $S^\vee\subset\mathbb{P}^{n^\ast}$. In this paper (http://arxiv.org/pdf/math/0111179v1.pdf), the author obtained the ...
3
votes
1answer
237 views

Reference for comparison of heart cohomology with standard cohomology

I'm looking for a reference for the following fact (which I believe to be true and should be easy for people who understand how spectral sequences arise from filtrations). Let A,B be two hearts of ...
1
vote
0answers
140 views

Derived category of product of complex manifolds

Let $X$ and $Y$ be a compact complex manifold. Is it possible to describe the derived category $D^b(X\times Y)$ of coherent sheaves in terms of $D^b(X)$ and $D^b(Y)$? I am particularly interested in ...
0
votes
0answers
112 views

Why “Fourier”-Mukai? [duplicate]

The Fourier-Mukai functor is one of the most important tools to work with in the derived category. While it is clear why the name of S.Mukai appears there, why does Joseph Fourier appear in the name ...
1
vote
1answer
191 views

Morphism between Fourier-Mukai functors implies the morphism between kernels?

Suppose $X,Y$ are smooth varieties over $\mathbb{C}$, and let $K_i \in D^b(X \times Y), i=1,2$ be objects in the derived category of bounded complex of coherent sheaves on $X \times Y$. Then there are ...
3
votes
2answers
281 views

Derived category of a hypersurface

Let $X$ be a smooth projective variety over $\mathbb{C}$, and $H \subset X$ be a smooth hypersurface. Many properties of an ambient variety $X$ could somehow inherit to the hypersurface $H$, I was ...
3
votes
1answer
168 views

flat descent for perverse sheaves

Let $E \in D^{b}_{c}(X,\overline{\mathbb{Q}}_{l})$ where $X$ is a $k$ scheme of finite type for a field $k$. Let $Y\rightarrow X$ a finite flat surjective morphism such that $f^{*}E$ is perverse and ...
0
votes
2answers
191 views

Injective resolution for right derived functor

This question is base on my previous question, and I repeat it here: Suppose $X$ is a projective variety and $D^{+}(X)$ is the derived cateogry of bounded below complexes of sheaf of ...
4
votes
2answers
476 views

Why is the derived tensor product only defined for bounded above derived categories?

In "Residues and Duality" by Hartshorne, the derived tensor $\otimes$ only defined for the bounded above categories (see Chapter I, section 4), that is one has $$\otimes: D^{-}(X) \times D^{-}(X) \to ...
8
votes
1answer
414 views

Are $D^b_{coh}(X)$ and $D^b(Coh(X))$ derived equivalent?

Let $X$ be a variety. Let $D^b(Coh(X))$ be the derived category of bounded complexes of coherent sheaves on $X$, and $D^b_{coh}(X)$ be the derived category of bounded complexes of sheaves of ...
5
votes
2answers
272 views

Equivariant derived category and invariant divisor

I'm looking for a reference of the following (folklore?) result. Let $X$ be a smooth projective variety equipped with a $G=\mathbb{Z}/2\mathbb{Z}$ action (we consider the simplest case, everything ...
4
votes
0answers
161 views

Grothendieck group of intersection of quadrics

Conjecturally, for a complete intersection $X$ the group $K_0(X)$ is finitely generated iff $H^{p,q}(X)=0$ for $p≠q$. This holds if and only if $X$ is a quadric, a cubic surface, or an ...
0
votes
0answers
165 views

Hodge structure of abelian surfaces

In my case, I have an abelian surface $A$ of (2,8)-polarization, and I have some finite group (may not be abelian group) $G$ acting on $A$ without fixed point. I want to understand when there is a ...
3
votes
0answers
421 views

unique enhancement for derived categories

I have two questions concerning the existence and uniqueness of enhancements in the following cases: i.) Let $A$ be a finite dimensional $k$ algebra of finite global dimension. Does the triangulated ...
6
votes
0answers
245 views

Reconstructing the Chow ring from the derived category

Let $X$ be a smooth projective variety and write $\mathbf{D}(X)$ for its triangulated category of perfect complexes of quasi-coherent sheaves. Recall that $\mathbf{D}(X)$ determines the Grothendieck ...
4
votes
1answer
261 views

Fourier-Mukai transform for abelian varieties

Let $A$ be an abelian variety over $\mathbb{C}$, $L$ be a very ample line bundle on $A$, then the dual abelian variety is $\hat{A} \cong A/K(L)$ with $K(L)$ the kernel of surjective morphism $A \to ...
-2
votes
1answer
295 views

Derived Category.

Question 1: Let $X$ be a scheme. Then generally for the complex $C^{\bullet}$ in $D^b(X)$, we define $R\Gamma(C^{\bullet})\colon$ = Complex obtained by applying $\Gamma$ to the injective resolution ...
2
votes
0answers
171 views

what is the zero locus of a morphism of the derived category?

For a morphism of quasi-coherent sheaves $v\colon E \to F$, on a scheme $X$, one can ask about the locus where $v =0$. When $F$ is a vector bundle, it's easy to see that this locus is closed.* Is ...
5
votes
1answer
306 views

Derived categories of singular varieties

Given my limited knowledge on derived categories, all the results on derived categories of complex of bounded sheaves are build upon smooth varieties, and people literally avoid singular case (as in ...
3
votes
1answer
159 views

Is there a blow-up formula for the derived category of a singular ambient variety?

For a nonsingular variety sitting inside a nonsingular ambient variety there is a semi-orthogonal decomposition of the derived category of the blow-up (with center that subvariety). What can be said ...
0
votes
1answer
181 views

Could we construct the Jacobian variety of a smooth curve $C$ with genus $>2$ from its derived category $D(C)$?

Let's consider a smooth curve $C$ over $\mathbb{C}$. We know that the Jacobian variety $Jac(C)$ of $C$ is the moduli space of the degree $0$ line bundles on $C$. $Jac(C)$ is an abelian variety of ...
1
vote
1answer
175 views

Could we extend the Atiyah class to the sheaf of poly-vector fields to get a Poisson bracket?

Let $X$ be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
3
votes
1answer
351 views

Can we define the tensor product in the derived category $D^b_{\text{coh}}(X)$ just from $D^b_{\text{coh}}(X)$ in certain cases?

This question arise from the comparision of the reconstruction theorems of Bondal-Orlov and Balmer and is inspired by Shizhuo Zhang's mathoverflow question: How to unify various reconstruction ...
9
votes
4answers
595 views

What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
3
votes
1answer
168 views

The Hochschild cohomology of a variety “with coefficient” in a vector bundle

This question is related to one of my previous question Do we have the following isomorphism for $\mathcal{Ext}$? Let $X$ be a smooth variety (over $\mathbb{C}$) and $\Delta: X \rightarrow X \times ...
9
votes
1answer
597 views

Why is proving fully-faithfulness of an integral functor locally analytically sufficient?

More than once I've come across a statement in a paper about derived categories in which it says something to the effect of "in order to prove that $\Phi:D^b(X)\rightarrow D^b(Y)$ is fully-faithful we ...
2
votes
1answer
223 views

why is the homological projective dual of this Lefschetz decomposition non-commutative?

I am reading these notes of an excellent course by Kuznetsov on Homological Projective Duality. On page 10 there is Example 1''. One starts with projective space, endowed with the identity embedding ...
6
votes
1answer
242 views

Derived categories of toric varieties and convex geometry

Toric varieties and convex polyhedra are intimately connected. Some of this can be found in standard text books (the connection between divisors and mixed volumes seems to be a popular example). One ...
4
votes
0answers
130 views

Derived category of toroidal varieties

This question comes from the first reduction step of Theorem 4.2 of Kawamata's paper on K-equivalent implies D-equivalent on toroidal varieties. But my question has little to do with this theorem. A ...
2
votes
1answer
150 views

admissible subcategories over non algebraically closed fields

Let $X$ be a smooth projective variety over a field $k$ and $D^b(X)$ its bounded derived category. Let $\bar{X}$ the base change to $\bar{k}$. Let $A$ be a triangulated subcategory of $D^b(X)$ that ...
7
votes
2answers
431 views

What is a good reference (preferably thorough) for the Derived Category of a scheme/orbifold/stack?

I've sort of circled around the idea of derived categories a few times, read a few introductory papers ("Derived Categories for the working mathematician", e.g.), and feel now that this is something ...
1
vote
1answer
176 views

Uniqueness of the canonical etale coverings

This is a construction [Definition 6.1] given in the paper D-equivalence and K-equivalence by Kawamata. Let $X$ be a normal quasiprojective variety such that the canonical divisor $K_X$ is a ...
1
vote
1answer
212 views

Derived equivalence of two varieties which are isomorphism over certain open subvarieties

Let $X,Y$ be varieties over $\mathbb{C}$, and $D(X),D(Y)$ be the derived categories of bounded complex of coherent sheaves. Let $U \subset X, V \subset Y$ be open subvarieties, and let $X-U, Y-V$ have ...
4
votes
1answer
254 views

Additive functors and Derived Categories

I have been learning about derived categories from Hartshorne's "Residues and Duality". One of the main theorems, as it is presented there seems to be the canonical isomorphism $RF \circ RG \cong R(F ...
1
vote
1answer
77 views

is the orthogonal complement of a saturated sequence saturated?

Suppose I have a smooth projective variety $X$, and a semi-orthogonal decomposition of its bounded derived category: $$D^b(X)= < A, E_1, E_2, ... , E_n >$$ where the $E_i$ are fully faithful, ...
2
votes
5answers
407 views

A statement for a triangulated category generated by a subset

Let $D$ be a triangulated category (the triangulated category in my mind is $D^{b}(X)$, that is the derived category of bounded complex of coherent sheaves on a smooth projective variety), $A \subset ...
2
votes
1answer
362 views

Calculate $Hom$ in derived category

Suppose $X$ is a smooth variety over $\mathbb{C}$. Let $K^{b}(X)$ be the homotopy category of bounded complex of coherent sheaves, and $D^{b}(X))$ be the derived category of bounded complex of ...
9
votes
0answers
309 views

McKay correspondence and tensor products

The theorem of Bridgeland-King-Reid says that if $M$ is a smooth quasi-projective complex variety of dimension at most $3$ on which a finite group $G$ acts such that the canonical sheaf $\omega_M$ is ...
1
vote
1answer
168 views

Compact generator of $D(\mathbb{P}^1)$

I suppose that Beilinson's compact generator (and, in fact, tilting object) $\mathcal{O} \oplus \mathcal{O}(1)$ in $D(\mathbb{P}^1)$ is the most well known example. I have the following simple ...
3
votes
1answer
186 views

what are mutations of sheaves all about?

Suppose I have a smooth projective variety $X$ and a semi-orthogonal decomposition of its bounded derived category of coherent sheaves $D^b(X)$. Then I can apply right or left mutations to the full ...
4
votes
0answers
196 views

What is the structure of the stack of complexes supported in dimension less than r?

Let $X$ be something. (smooth and projective variety over C are my assumptions) The stack $M$ parameterising coherent sheaves on $X$ splits as a disjoint union of open and closed substacks $M_\alpha$, ...
4
votes
2answers
385 views

Examples of tilting objects that don't come from exceptional sequences

This is a question on geometric tilting theory. On smooth projective variety it is possible to define in general tilting object as perfect complex that satisfy some properties, but are there examples ...
5
votes
0answers
194 views

Not isomorphic varieties with isomorphic tilting algebras

Let $X$ be a smooth projective variety over a field, than tilting object $T$ on $X$ is a perfect complex that is a compact generator of the derived category $\operatorname{D}(QCoh(X))$ and satisfies ...
6
votes
1answer
311 views

A question on the “natural metric” on the space of Bridgeland stability condition

I have a question on the "natural metric" on the space of Bridgeland stability condition. A stability condition $\sigma=(Z,\mathcal{P})$ on a triangulated category $\mathcal{D}$ consists of a linear ...
5
votes
0answers
381 views

On the derived category of constructible étale sheaves

The derived category $D^{\flat}_{c}(X,R)$ of constructible sheaves of $R$-modules on $X_{et}$ is defined as the full subcategory of $D^b(X,R)$ whose cohomology sheaves are all constructible. Clearly, ...
18
votes
3answers
2k views

Why are derived categories natural places to do deformation theory?

It seems to me that a lot of people do deformation theory (of schemes, sheaves, maps etc) in derived category (of an appropriate abelian category). For example, the cotangent complex of a morphism ...
5
votes
1answer
514 views

Kodaira-Spencer map as a “differential”

Using the laguage of derived category, the Kodaira-Spencer map $\kappa(x) : Ext^1_X(k(x), k(x)) \rightarrow Ext^1_X(\mathcal F_x, \mathcal F_x)$ can be described as a Fourier-Mukai transformation ...
4
votes
0answers
188 views

Leray degeneration for smooth projective morphisms and formality of families of compact Kähler manifolds

Let $\pi \colon X \to S$ be a smooth projective morphism of algebraic varieties, say over $\mathbf C$. By Deligne's argument ("Théorème de Lefschetz...", 1968), there is for each $i$ an injection $$ ...
3
votes
1answer
366 views

Semiorthogonal decompositions for Fano 3-folds and 4folds

Let $X$ be a projective Fano 3-fold or 4-fold and let $D^b(X)$ be the bounded derived category of coherent sheaves on $X$. For what $X$ is it known a semi orthogonal decomposition into indecomposable ...