Derived noncommutative algebraic geometry

The proof starts out by analyzing two induced Serre functors on ${\displaystyle D^{b}(Y)}$ and finding an ismorphism between them. It particular, it shows there is an object ${\displaystyle \omega _{Y}=F(\omega _{X})}$ which acts like the dualizing sheaf on ${\displaystyle Y}$. The isomorphism between these two functors gives an isomorphism of the set of underlying points of the derived categories. Then, what needs to be check is an ismorphism ${\displaystyle F(\omega _{X}^{\otimes k})\cong \omega _{Y}^{\otimes k}}$, for any ${\displaystyle k\in \mathbb {N} }$, giving an isomorphism of canonical rings

${\displaystyle A(X)=\bigoplus _{k=0}^{\infty }H^{0}(X,\omega _{X}^{\otimes k})\cong \bigoplus _{k=0}^{\infty }H^{0}(Y,\omega _{Y}^{\otimes k})}$

If ${\displaystyle \omega _{Y}}$ can be shown to be (anti-)ample, then the proj of these rings will give an isomorphism ${\displaystyle X\to Y}$. All of the details are contained in Dolgachev's notes.

This theorem fails in the case ${\displaystyle X}$ is Calabi-Yau, since ${\displaystyle \omega _{X}\cong {\mathcal {O}}_{X}}$, or is the product of a variety which is Calabi-Yau. Abelian varieties are a class of examples where a reconstruction theorem could never hold. If ${\displaystyle X}$ is an abelian variety and ${\displaystyle {\hat {X}}}$ is it's dual, the Fourier–Mukai transform with kernel ${\displaystyle {\mathcal {P}}}$, the Poincare bundle,[4] gives an equivalence

${\displaystyle FM_{\mathcal {P}}:D^{b}(X)\to D^{b}({\hat {X}})}$

of derived categories. Since an abelian variety is generally not isomorphic to its dual, there are derived equivalent derived categories without isomorphic underlying varieties.[5] There is an alternative theory of tensor triangulated geometry where we consider not only a triangulated category, but also a monoidal structure, i.e. a tensor product. This geometry has a full reconstruction theorem using the spectrum of categories.[6]

K3 surfaces are another class of examples where reconstruction fails due to their Calabi-Yau property. There is a criterion for determining whether or not two K3 surfaces are derived equivalent: the derived category of the K3 surface ${\displaystyle D^{b}(X)}$ is derived equivalent to another K3 ${\displaystyle D^{b}(Y)}$ if and only if there is a Hodge isometry ${\displaystyle H^{2}(X,\mathbb {Z} )\to H^{2}(Y,\mathbb {Z} )}$, that is, an isomorphism of Hodge structure.[3] Moreover, this theorem is reflected in the motivic world as well, where the Chow motives are isomorphic if and only if there is an isometry of Hodge structures.[7]

One nice application of the proof of this theorem is the identification of autoequivalences of the derived category of a smooth projective variety with ample (anti-)canonical sheaf. This is given by

${\displaystyle \operatorname {Auteq} (D^{b}(X))\cong (\operatorname {Pic} (X)\rtimes \operatorname {Aut} (X))\times \mathbb {Z} }$

Where an autoequivalence ${\displaystyle F}$ is given by an automorphism ${\displaystyle f:X\to X}$, then tensored by a line bundle ${\displaystyle {\mathcal {L}}\in \operatorname {Pic} (X)}$ and finally composed with a shift. Note that ${\displaystyle \operatorname {Aut} (X)}$ acts on ${\displaystyle \operatorname {Pic} (X)}$ via the polarization map, ${\displaystyle g\mapsto g^{*}(L)\otimes L^{-1}}$.[8]

Although localization of categories is defined for a class of morphisms ${\displaystyle \Sigma }$ in the category closed under composition, we can construct such a class from a triangulated subcategory. Given a full triangulated subcategory ${\displaystyle {\mathcal {N}}\subset {\mathcal {T}}}$ the class of morphisms ${\displaystyle \Sigma ({\mathcal {N}})}$, ${\displaystyle s}$ in ${\displaystyle {\mathcal {T}}}$ where ${\displaystyle s}$ fits into a distinguished triangle

${\displaystyle X{\xrightarrow {s}}Y\to N\to X[+1]}$

with ${\displaystyle X,Y\in {\mathcal {T}}}$ and ${\displaystyle N\in {\mathcal {N}}}$. It can be checked this forms a multiplicative system using the octahedral axiom for distinguished triangles. Given

${\displaystyle X{\xrightarrow {s}}Y{\xrightarrow {s'}}Z}$

with distinguished triangles

${\displaystyle X{\xrightarrow {s}}Y\to N\to X[+1]}$

${\displaystyle Y{\xrightarrow {s'}}Z\to N'\to Y[+1]}$

where ${\displaystyle N,N'\in {\mathcal {N}}}$, then there are distinguished triangles

${\displaystyle X\to Z\to M\to X[+1]}$

${\displaystyle N\to M\to N'\to N[+1]}$ where ${\displaystyle M\in {\mathcal {N}}}$ since ${\displaystyle {\mathcal {N}}}$ is closed under extensions. This new category has the following properties

• It is canonically triangulated where a triangle in ${\displaystyle {\mathcal {T}}/{\mathcal {N}}}$ is distinguished if it is isomorphic to the image of a triangle in ${\displaystyle {\mathcal {T}}}$
• The category ${\displaystyle {\mathcal {T}}/{\mathcal {N}}}$ has the following universal property: any exact functor ${\displaystyle F:{\mathcal {T}}\to {\mathcal {T}}'}$ where ${\displaystyle F(N)\cong 0}$ where ${\displaystyle N\in {\mathcal {N}}}$, then it factors uniquely through the quotient functor ${\displaystyle Q:{\mathcal {T}}\to {\mathcal {T}}/{\mathcal {N}}}$, so there exists a morphism ${\displaystyle {\tilde {F}}:{\mathcal {T}}/{\mathcal {N}}\to {\mathcal {T}}'}$ such that ${\displaystyle {\tilde {F}}\circ Q\simeq F}$.

• If ${\displaystyle X}$ is a regular scheme, then every bounded complex of coherent sheaves is perfect. Hence the singularity category is trivial
• Any coherent sheaf ${\displaystyle {\mathcal {F}}}$ which has support away from ${\displaystyle \operatorname {Sing} (X)}$ is perfect. Hence nontrivial coherent sheaves in ${\displaystyle D_{sg}(X)}$ have support on ${\displaystyle \operatorname {Sing} (X)}$.
• In particular, objects in ${\displaystyle D_{sg}(X)}$ are isomorphic to ${\displaystyle {\mathcal {F}}[+k]}$ for some coherent sheaf ${\displaystyle {\mathcal {F}}}$.

Kontsevich proposed a model for Landau–Ginzburg models which was worked out to the following definition:[14] a Landau–Ginzburg model is a smooth variety ${\displaystyle X}$ together with a morphism ${\displaystyle W:X\to \mathbb {A} ^{1}}$ which is flat. There are three associated categories which can be used to analyze the D-branes in a Landau–Ginzburg model using matrix factorizations from commutative algebra.

With this definition, there are three categories which can be associated to any point ${\displaystyle w_{0}\in \mathbb {A} ^{1}}$, a ${\displaystyle \mathbb {Z} /2}$-graded category ${\displaystyle DG_{w_{0}}(W)}$, an exact category ${\displaystyle \operatorname {Pair} _{w_{0}}(W)}$, and a triangulated category ${\displaystyle DB_{w_{0}}(W)}$, each of which has objects

${\displaystyle {\overline {P}}=(p_{1}:P_{1}\to P_{0},p_{0}:P_{0}\to P_{1})}$ where ${\displaystyle p_{0}\circ p_{1},p_{1}\circ p_{0}}$ are multiplication by ${\displaystyle W-w_{0}}$.

There is also a shift functor ${\displaystyle [+1]}$ send ${\displaystyle {\overline {P}}}$ to

${\displaystyle {\overline {P}}[+1]=(-p_{0}:P_{0}\to P_{1},-p_{1}:P_{1}\to P_{0})}$.

The difference between these categories are their definition of morphisms. The most general of which is ${\displaystyle DG_{w_{0}}(W)}$ whose morphisms are the ${\displaystyle \mathbb {Z} /2}$-graded complex

${\displaystyle \operatorname {Hom} ({\overline {P}},{\overline {Q}})=\bigoplus _{i,j}\operatorname {Hom} (P_{i},Q_{j})}$

where the grading is given by ${\displaystyle (i-j){\bmod {2}}}$ and differential acting on degree ${\displaystyle d}$ homogeneous elements by

${\displaystyle Df=q\circ f-(-1)^{d}f\circ p}$

In ${\displaystyle \operatorname {Pair} _{w_{0}}(W)}$ the morphisms are the degree ${\displaystyle 0}$ morphisms in ${\displaystyle DG_{w_{0}}(W)}$. Finally, ${\displaystyle DB_{w_{0}}(W)}$ has the morphisms in ${\displaystyle \operatorname {Pair} _{w_{0}}(W)}$ modulo the null-homotopies. Furthermore, ${\displaystyle DB_{w_{0}}(W)}$ can be endowed with a triangulated structure through a graded cone-construction in ${\displaystyle \operatorname {Pair} _{w_{0}}(W)}$. Given ${\displaystyle {\overline {f}}:{\overline {P}}\to {\overline {Q}}}$ there is a mapping code ${\displaystyle C(f)}$ with maps

${\displaystyle c_{1}:Q_{1}\oplus P_{0}\to Q_{0}\oplus P_{1}}$ where ${\displaystyle c_{1}={\begin{bmatrix}q_{0}&f_{1}\\0&-p_{1}\end{bmatrix}}}$

and

${\displaystyle c_{0}:Q_{0}\oplus P_{1}\to Q_{1}\oplus P_{0}}$ where ${\displaystyle {\displaystyle c_{0}={\begin{bmatrix}q_{1}&f_{0}\\0&-p_{0}\end{bmatrix}}}}$

Then, a diagram ${\displaystyle {\overline {P}}\to {\overline {Q}}\to {\overline {R}}\to {\overline {P}}[+1]}$ in ${\displaystyle DB_{w_{0}}(W)}$ is a distinguished triangle if it is isomorphic to a cone from ${\displaystyle \operatorname {Pair} _{w_{0}}(W)}$.

Using the construction of ${\displaystyle DB_{w_{0}}(W)}$ we can define the category of D-branes of type B on ${\displaystyle X}$ with superpotential ${\displaystyle W}$ as the product category

${\displaystyle DB(W)=\prod _{w\in \mathbb {A} _{1}}DB_{w_{0}}(W).}$

This is related to the singularity category as follows: Given a superpotential ${\displaystyle W}$ with isolated singularities only at ${\displaystyle 0}$, denote ${\displaystyle X_{0}=W^{-1}(0)}$. Then, there is an exact equivalence of categories

${\displaystyle DB_{w_{0}}(W)\cong D_{sg}(X_{0})}$

given by a functor induced from cokernel functor ${\displaystyle \operatorname {Cok} }$ sending a pair ${\displaystyle {\overline {P}}\mapsto \operatorname {Coker} (p_{1})}$. In particular, since ${\displaystyle X}$ is regular, Bertini's theorem shows ${\displaystyle DB(W)}$ is only a finite product of categories.

There is a Fourier-Mukai transform ${\displaystyle \Phi _{Z}}$ on the derived categories of two related varieties giving an equivalence of their singularity categories. This equivalence is called Knörrer periodicity. This can be constructed as follows: given a flat morphism ${\displaystyle f:X\to \mathbb {A} ^{1}}$ from a separated regular Noetherian scheme of finite Krull dimension, there is an associated scheme ${\displaystyle Y=X\times \mathbb {A} ^{2}}$ and morphism ${\displaystyle g:Y\to \mathbb {A} ^{1}}$ such that ${\displaystyle g=f+xy}$ where ${\displaystyle xy}$ are the coordinates of the ${\displaystyle \mathbb {A} ^{2}}$-factor. Consider the fibers ${\displaystyle X_{0}=f^{-1}(0)}$, ${\displaystyle Y_{0}=g^{-1}(0)}$, and the induced morphism ${\displaystyle x:Y_{0}\to \mathbb {A} ^{1}}$. And the fiber ${\displaystyle Z=x^{-1}(0)}$. Then, there is an injection ${\displaystyle i:Z\to Y_{0}}$ and a projection ${\displaystyle q:Z\to X_{0}}$ forming an ${\displaystyle \mathbb {A} ^{1}}$-bundle. The Fourier-Mukai transform

${\displaystyle \Phi _{Z}(\cdot )=\mathbf {R} i_{*}q^{*}(\cdot )}$

induces an equivalence of categories

${\displaystyle D_{sg}(X_{0})\to D_{sg}(Y_{0})}$

called Knörrer periodicity. There is another form of this periodicity where ${\displaystyle xy}$ is replaced by the polynomial ${\displaystyle x^{2}+y^{2}}$.[15][16] These periodicity theorems are the main computational techniques because it allows for a reduction in the analysis of the singularity categories.

If we take the Landau–Ginzburg model ${\displaystyle (\mathbb {C} ^{2k+1},W)}$ where ${\displaystyle W=z_{0}^{n}+z_{1}^{2}+\cdots +z_{2k}^{2}}$, then the only fiber singular fiber of ${\displaystyle W}$ is the origin. Then, the D-brane category of the Landau–Ginzburg model is equivalent to the singularity category ${\displaystyle D_{\text{sing}}(\operatorname {Spec} (\mathbb {C} [z]/(z^{n})))}$. Over the algebra ${\displaystyle A=\mathbb {C} [z]/(z^{n})}$ there are indecomposible objects

${\displaystyle V_{i}=\operatorname {Coker} (A{\xrightarrow {z^{i}}}A)=A/z^{i}}$

whose morphisms can be completely understood. For any pair ${\displaystyle i,j}$ there are morphisms ${\displaystyle \alpha _{j}^{i}:V_{i}\to V_{j}}$ where

• for ${\displaystyle i\geq j}$ these are the natural projections
• for ${\displaystyle i<j}$ these are multiplication by ${\displaystyle z^{j-i}}$

where every other morphism is a composition and linear combination of these morphisms. There are many other cases which can be explicitly computed, using the table of singularities found in Knörrer's original paper.[16]