Assuming a basic knowledge of algebraic geometry, this post serves as a complilation of my study on how derived geometry has been recently used in the classification of Fano varieties. This is a very natural question to consider early in one's study of algebraic geometry, since the vast majority of examples one sees turn out to be Fano — the prime example is of course \( \P^n \) since \( \omega_{\P^n} \cong \OO_{\P^n}(-n - 1) \), and furthermore any hypersurface of degree \( d \leq n \) via the adjunction formula (this may also be generalized to smooth complete intersections). Exercise 11.3.J in Vakil shows that such hypersurfaces are always uniruled, which implies that any two points \( p, q \in Y \) can be connected by a rational curve. Unfortunately this statement is not true in its full glory for non-smooth Fano variety (take the cone over a smooth plane cubic); however, Kollar et al. show in Chapter V of Rational Curves on Algebraic Varieties that any two points \( p, q\) on an arbitrary Fano variety may be connected by a (finite) chain of rational curves.
It becomes apparent that the more arbitrary Fano varieties enjoy many of the nice properties that their hypersurface families do. Similar to how a smooth hypersurface can be classified by its degree, there are ways to generalize this idea to Fano varieties under the right assumptions. The most important assumption that we will use throughout this set of notes is motivated by the Noether Lefschetz Theorem:
The requirement that \( d \geq 4 \) is obviously strict since a cubic surface \(Y\) is isomorphic to \( \P^2 \) blown up at 6 points, giving \( \pic(Y) \cong \Z^7 \). However, if our degree \(d\) hypersurface \(X_d\) is of higher dimension — say \( X_d \subset \P^n \) for \( n \geq 4 \) — it is easy to show that \( \pic X_d \cong \Z \) by using the \(d \)-uple embedding together with the Lefschetz Hyperplane theorem. Hence, we call a Fano variety \(X\) prime if \( \pic X \cong \Z \). Under this assumption, we have that \( \pic X \) is generated by a hyperplane section \( H \) (corresponding to \( \OO(1) \)) so that \( -K_X = i_X H \) for some positive integer \( i_X \) — this integer will be known as the index of the prime Fano variety \(X\). Additionally, define the degree of \( X \) to be \( d := (H)^{\dim(X)} \).
If we further assume that our Fano variety is smooth over an algebraically closed field \( k \), then Kodaira vanishing implies that
$$ H^j(X, mH) = H^j(X, K_X + (i_X + m)H) = 0 $$for \(j > 0\) and \( m > -i_X \), so that the Hilbert polynomial \( P(m) = \chi(X, mH) \) is precisely \( h^0(X, mH) \). Since \( h^0(X, mH) \) vanishes for \( -i_X < m < 0 \), this implies that \( i_X \leq n + 1 \), as described by Oliver Debarre in his notes How to Classify Fano Varieties?. The classification of prime Fano varieties by their index \( i_X \) saw a substantial amount of progress in the late 80's and early 90's due to a select few mathematicians making great strides; briefly reiterating the listing given by Debarre in his paper Gushel-Mukai Varieties, it was shown
Additionally, Shokurov was able to show by manipulating the Hilbert polynomial that \( h^0(X, H) = g + n - 1 \); as the linear series \( |H| \) is ample and thus base-point free, it defines a morphism \( X \to \P^{g + n - 2} \) which was shown to be an embedding for \( g \geq 4 \).
It is not my goal to recount the entire history of Fano varieties since there are many rich topics left out of the above discussion; however, the methods used in more recent years to further scrutinize the above cases rely a great deal on the derived geometry. Thus, it is important to explain the ideas at play very briefly.
The main idea of derived geometry is to effectively reduce an object to its (co)homological information — instead of observing the object by itself, we should consider any resolution (projective, injective, flat, etc.) as the same object from a different perspective. For example, the celebrated Hilbert Syzygy Theorem states that when \( X \) is a smooth projective variety of dimension \(n\) and \( \mathcal{E} \) is a coherent sheaf on \( X \), then there exists a resolution by (finite rank) locally free sheaves:
As a projective resolution, this complex should have vanishing homology in degrees other than \( 0\); thus the morphism of complexes
gives matching homological information between \( \mathcal{E} \) and the resolution \( (\mathcal{F}_i) \), and should be treated as the same 'sheaf' from different perspectives. This (along with other motivations) led Grothendieck and his student Verdier to develop the subject of derived categories in the 1960's and 1970's using a number of accessory tools such as triangulated categories.
I will not give a rigorous intro to triangulated categories, as this subject is already vividly captured in Fourier-Mukai Transforms in Algebraic Geometry by Daniel Huybrechts; however, the basic idea is that without an abelian category structure, one must use something more general than exact sequences to relate subobjects and quotients. In our case of derived categories of coherent sheaves, the objects one works with are equivalence classes of complexes (of coherent sheaves) which loose their abelian category structure after quotienting by homotopy equivalence — thus, given a short exact sequence of complexes
$$ \begin{CD} 0 @>>> A_\bullet @>>> B_\bullet @>>> C_\bullet @>>> 0 \end{CD} $$we cannot reasonably say whether the sequence remains exact after quotienting by homotopy equivalence since there is not any notion of quotients and cokernels. Nonetheless, we know that at the level of (co)homology the short exact sequence above extends to a long exact sequence:
$$ \begin{CD} \dots @>>> H^i(A_\bullet) @>>> H^i(B_\bullet) @>>> H^i(C_\bullet) @>>> H^{i+1}(A_\bullet) @>>> \dots \end{CD} $$As (co)homological information is really all that we're interested in, this allows us to instead consider the original short exact sequence as a triangle
$$ \begin{CD} A @>>> B @>>> C @>>> A[1] \end{CD} $$(the name triangle comes from an alternative way to write the above quadrupel, where \( C \) sits below the middle of the map \( A \to B \) and one draws a dotted arrow from \( C \) back to \(A\), thus making a triangle). It is also important to mention that we no longer consider \( C \) as the cokernel of the map \( A \to B \) (for one because \( A[1] \) need not be zero), but instead consider \( C \) to be the cone of our map \( A \to B \); when \( A \) and \(B\) are complexes concentrated in degree 0 (so we can consider them as classical sheaves) and \( f : A \to B \) is a morphism of complexes, the cone \( C = \textrm{Cone}(f) \) is simply the complex with \( A \) in index \(-1\) and \(B\) in index \( 0 \) connected by the map \(f\) with \( 0 \) elsewhere. It is easy to check that in this instance, the homology groups give both the cokernel and the kernel.
Similar to how short exact sequences define the Grothendieck group of an abelian category, triangles give a natural way to define the Grothendieck group of the derived category:
This concept of the Grothendieck group will play an important role in our study of Fano varieties, and has been used to motivate several conjectures along the topic — the most notable one that comes to mind is the mysterious duality between Fano threefolds of index 1 and index 2 discovered by Kuznetsov in Derived Categories of Fano Threefolds. However, another important property of the numerical Grothendieck groups that we will see is it respects (semiorthogonal) decompositions of our derived category. More precisely, consider the following definitions:
It is now easy to show that the numerical Grothendieck group splits semiorthogonal decompositions. The inclusion functor \( j_k : \mathcal{A}_k \hookrightarrow D^b(X) \) clearly induces an inclusion \( K_0(\mathcal{A}_k) \subset K_0(D^\flat(X)) \) for each \( 1 \leq k \leq n \) — the groups \( K_0(\mathcal{A}_i) \cap K_0(\mathcal{A}_j) \) must intersect trivially, as otherwise we could take any nonzero object \(T\) in their intersection, and the identity map \( \textrm{id}_T : T \to T \) would provide a nonzero morphism from higher index to lower index (contradicting semiorthogonality). Thus, we get the internal direct sum \( K_0(\mathcal{A}_1) \oplus \dots \oplus K_0(\mathcal{A}_n) \) is contained in \( K_0(D^\flat(X)) \). For reverse containment, for any \( [\mathcal{E}_\bullet] \in K_0(D^\flat(X)) \) we can find a filtration of a representative \( \mathcal{E}_\bullet \)
$$ \begin{CD} 0 = T_n @>>{f_n}> T_{n-1} @>>{f_{n-1}}> \dots @>>{f_2}> T_1 @>>{f_1}> T_0 = \mathcal{E}_\bullet \end{CD} $$by our second axiom in the definition above, where \( A_i = \textrm{Cone}(f_i) \in \mathcal{A}_i \). In the Grothendieck group, we have \( [\mathcal{E}_\bullet] = [T_1] + [A_1] \) and \( [T_i] = [T_{i+1}] + [A_{i+1}] \); since \( T_n = 0 \), this allows us to write \( [\mathcal{E}_\bullet] = [A_1] + \dots + [A_n] \) in \( K_0(D^\flat(X)) \). Thus, we obtain
Using significantly more advanced methods involving dg-enhancements, as laid out in Theorem 5.1 of On Differentially Graded Categories by B. Keller, this decomposition extends to the higher algebraic K-groups thus making semiorthogonal decompositions relevant to K-theory.
A natural question to ask after defining a concept is how to construct examples of what we just defined; in other words, how do we obtain semiorthogonal decompositions of \( D^\flat(X) \) for some smooth projective variety \(X\)? As you may expect, any non-trivial answer (i.e. \( D^\flat(X) = \langle D^\flat(X) \rangle\)) deeply depends on the geometry of the variety \(X\). Even the seemingly trivial answer of \( D^\flat(X) = \langle \mathcal{A}^\perp, \mathcal{A} \rangle \) for any admissible full subcategory \( \mathcal{A} \subset D^\flat(X) \) need not exist: Calabi-Yau manifolds never admit non-trivial semiorthogonal decompositions (this can be easily shown after we define the Serre functor). However, when we are not in the Calabi-Yau setting, considering decompositions that look like \( D^\flat(X) = \langle \mathcal{A}^\perp, \mathcal{A} \rangle \) actually isnt a terrible start — to strengthen the idea, we want to refine our \( \mathcal{A} \) to be something generated by 'simple objects' (from a homological perspective).
In this case, every exceptional collection \( \mathcal{E}_1, \dots, \mathcal{E}_n \) generates an admissible subcategory \( \mathcal{A} = \langle \mathcal{E}_1, \dots, \mathcal{E}_n \rangle \) (where the brackets mean the smallest full triangulated subcategory containing all \(\mathcal{E}_i\) closed under shifts, extensions and cones) and thus gives a semiorthogonal decomposition \( D^\flat(X) = \langle \mathcal{A}^\perp, \mathcal{A} \rangle \).
Over the past few decades, the subcategory \( \mathcal{A}^\perp \) associated to an exceptional collection in the bounded derived category of a projective variety has gained a significant amount of attention. This is, for the most part, due to Alexander Kuznetsov's significant contributions to how this 'residue' determined the geometry of the underlying scheme \(X\). Thus, it is not uncommon to see the orthogonal complement \( \mathcal{A}^\perp \) referred to as the Kuznetsov component of \( D^\flat(X) \) in most recent mathematical literature, sometimes written as \( \mathcal{Ku}(X) := \mathcal{A}^\perp \).
One important thing to point out is that the notation \( \mathcal{Ku}(X) \) seems to imply that the admissible subcategory somehow does not depend on the choice of exceptional collection; this would of course be completely false. For example, consider a cubic threefold \( X = \{ f = 0 \} \subset \mathbb{P}^4 \) (for example, take the Kleinian cubic \( f(x,y,z,w, v) = v^2 w + w^2x + x^2 y + y^2 z + z^2v \))
By the Lefschetz hyperplane theorem, the Picard group \( \text{Pic}(X) \cong \mathbb{Z} \). Taking \( H \) to be a hyperplane section which generates \( \text{Pic}(X) \) (up to linear equivalence, we may simply say \( H = X \cap \{ x = 0 \} \)), the long exact sequence applied to the ideal exact sequence
$$ \begin{CD} 0 @>>> \mathcal{O}_{\mathbb{P}^4}(-3) @>>{f}> \mathcal{O}_{\mathbb{P}^4} @>>> \mathcal{O}_X @>>> 0 \end{CD} $$we obtain
$$ \begin{align*} \text{Hom}^i(\mathcal{O}_X, \mathcal{O}_X) = \text{Hom}^i(\mathcal{O}_X(H), \mathcal{O}_X(H)) = H^i(X, \mathcal{O}_X) = \begin{cases} k & i = 0 \\ 0 & i \neq 0 \end{cases} \end{align*} $$Applying a similar argument to the ideal exact sequence of our divisor \( H \) yields \( \textrm{Hom}^i(\mathcal{O}_X, \mathcal{O}_X(H)) = k\) for \( i = 0 \) and \( 0\) otherwise. However, there are no maps \( \mathcal{O}_X(H) \to \mathcal{O}_X \) for the same reason that \( \mathcal{O}_X(-1) \) does not have any global sections. Therefore, we see that \( (\mathcal{O}_X, \mathcal{O}_X(H)) \) is an exceptional collection in the derived category of our cubic threefold; however, we trivially also have that \( (\mathcal{O}_X(H)) \) is an exceptional collection! This leads to some ambiguity: we could either have \( \mathcal{Ku}_1(X) := \langle \mathcal{O}_X, \mathcal{O}_X(H) \rangle^\perp \) or \( \mathcal{Ku}_2(X) := \langle \mathcal{O}_X(H) \rangle^\perp \) depending on which exceptional collection we were considering. In this case though, \( \mathcal{Ku}_2(X) = \langle \mathcal{Ku}_1(X), \mathcal{O}_X \rangle \) so that the two are certainly not equal.
The above example was a bit trivial and can be easily remedied by requiring that the Kuznetsov component be defined for maximal exceptional collections \( \mathcal{E}_1, \dots, \mathcal{E}_n \) (in the sense that adding any other exceptional object \( \mathcal{F} \) will not change the subcategory \( \langle \mathcal{E}_1, \dots, \mathcal{E}_n \rangle \)). Alas, this also does not give us a definition of the Kuznetsov component that is independent of choice of exceptional collection! A 2014 paper by Böhning, von Bothmer, and Sosna gave an example of a projective variety (namely the Godeaux surface) which admits two distinct maximal exceptional collections — one generated by 9 objects and the other by 11. By taking the ranks of the Grothendieck groups of the associated right orthogonals, it can be shown that the two 'Kuznetsov components' obtained from these two collections cannot be the same! As a consequence of the above discussion, the Kuznetsov component is not something generally defined for all projective varieties, but is often defined in literature for specific classes of projective varieties (e.g. cubic fourfolds, Gushel-Mukai varieties, etc).
Another thing that should be mentioned at some point in the discourse of exceptional collections is that, when they exist, the existence of a exceptional collection almost always implies the existence of others — this phenomenon is a simple result of a funcorial, called mutation, which effectively acts as a permutation.
To explain how this mutation will be used, whenever we have an exceptional collection
$$ (\mathcal{E}_1, \dots, \mathcal{E}_{i-1}, \mathcal{E}_i, \mathcal{E}_{i+1}, \dots, \mathcal{E}_n) $$then both
$$ (\mathcal{E}_1, \dots, \mathcal{E}_{i-1}, \mathbb{L}_{\mathcal{E}_i} \mathcal{E}_{i+1}, \mathcal{E}_i, \dots, \mathcal{E}_n ) $$and
$$ (\mathcal{E}_1, \dots, \mathcal{E}_{i}, \mathbb{R}_{\mathcal{E}_i} \mathcal{E}_{i-1}, \mathcal{E}_{i+1}, \dots, \mathcal{E}_n ) $$are exceptional collections! Proving that the above is true requires an additional fact that mutation through exceptional objects takes the convenient form as a cone of evaluation:
$$ \begin{CD} \mathcal{E} \otimes \textrm{RHom}^\bullet(\mathcal{E}, \mathcal{F}) @>{\textrm{ev}}>> \mathcal{F} @>>> \mathbb{L}_\mathcal{E} \mathcal{F} \end{CD} $$ $$ \begin{CD} \mathbb{R}_\mathcal{E} \mathcal{F} @>>> \mathcal{E} \otimes \textrm{RHom}^\bullet(\mathcal{F}, \mathcal{E})^\vee \end{CD} $$from here, it is straightforward to show that if \( \mathcal{E}_i, \mathcal{E}_{i+1} \) exceptional with \( \mathcal{E}_i \subset \mathcal{E}_{i+1}^\perp \) then \( \mathbb{L}_{\mathcal{E}_i} E_{i+1} \) is exceptional (since tensoring by \( \textrm{RHom}^\bullet(\mathcal{E}_i, \mathcal{E}_{i+1}) \) is equivalent to tensoring by \( 0 \)).
There is also a secondary application of mutation functors to semiorthogonal decompositions: suppose \( \langle \mathcal{A}_1, \dots, \mathcal{A}_n \rangle \) is a semiorthogonal decomposition of \( D^\flat(X) \). For any \( 1 \leq i \leq n \), we may also take \( D^\flat(X) = \langle \mathcal{A}_i, \,^\perp\mathcal{A}_i \rangle\) to be a semiorthogonal decomposition. From the second axiom in our above definition, for any \(\mathcal{E}_\bullet \in D^\flat(X)\) there exists a filtration
$$ \begin{CD} 0 = T_2 @>{f_2}>> T_1 @>{f_1}>> T_0 = \mathcal{E}_\bullet \end{CD} $$such that the cone of \( f_2 \) is in \( \mathcal{A}_i \) and the cone of \( f_1 \) is in \( \,^\perp\mathcal{A}_i \). Denote \( R_{i,1} : D^\flat(X) \to \,^\perp\mathcal{A}_i \) to be the functor sending \( \mathcal{E}_\bullet \mapsto \textrm{Cone}(f_1) \) and \( R_{i,2} : D^\flat(X) \to \mathcal{A}_i \) the functor sending \( \mathcal{E}_\bullet \) to \( \textrm{Cone}(f_2) \) (which by the triangle axioms must be \( T_1 \)). It was shown by Bondal & Kapranov in one of their seminal papers to the study of semiorthogonal decompositions that right mutation of \( \mathcal{A}_{i-1} \) through \( \mathcal{A}_i \) is equivalent to the projection \( R_{i,1} \); in a similar vain, by considering the semiorthogonal decomposition \( \langle \mathcal{A}_i^\perp, \mathcal{A}_i \rangle \), left mutation of \( \mathcal{A}_{i+1}\) through \( \mathcal{A}_{i} \) is equivalent to the projection on to \( \mathcal{A}_i^\perp \).
The last concept that should be explained in our discussion of derived categories is that of a Serre functor. Recall that when \( X \) is a smooth projective variety of dimension \( n \) over some field \( k \), there is a natural isomorphism \( \textrm{Ext}^i(\mathcal{E}, \omega_X) \cong H^{n-i}(X, \mathcal{E})^\vee \). In particular, when \( \mathcal{E} \) is locally free this says that \( H^i(X, \mathcal{E}) \cong H^{n-i}(X, \mathcal{E}^\vee \otimes \omega_X )^\vee \). Grothendieck generalized this duality drastically to the case where \( X \) is a proper scheme of finite type over \(k\), in which case there exists a dualizing complex \( \omega_X^\bullet \) (c.f. Hartshorne III.7) such that
$$ \textrm{Hom}_X(\mathcal{E}, \omega_X^\bullet) \cong \textrm{Hom}_X(\OO_X, \mathcal{E})^\vee $$in the derived category. When \( X \) is Cohen-Macaulay, \( \omega_X^\bullet \) is precisely \( \omega_X[n] \) (i.e. the canonical bundle shifted by the dimension). Therefore, for smooth projective varieties, we define the functor \( S_X : D^\flat(X) \to D^\flat(X) \) given by \( \mathcal{E} \mapsto \mathcal{E} \otimes \omega_X[n] \) as the Serre functor of \(X\). Generalizing this property drastically to more arbitrary triangulated categories:
It should now hopefully be easy to see why exceptional objects do not exist in the derived category of a Calabi-Yau variety \(X\): since the canonical bundle \( \omega_X \) is trivial, the Serre functor \( S_{D^\flat(X)} \) is simply homological shift by the dimension. Thus if \( \mathcal{E} \) were an exceptional object, we would have
$$ \textrm{Hom}^i(\mathcal{E}, \mathcal{E}) = \textrm{Hom}^i(\mathcal{E}, S_{D^\flat(X)}\,\mathcal{E} ) = \textrm{Hom}^{i + n}(\mathcal{E}, \mathcal{E}) $$thus forcing \( X \) to be a point — a contradiction.
In fact, a stronger result using the Hochschild-Konstant-Rosenberg theorem shows that the derived category of a Calabi-Yau cant even have any non-trivial semiorthogonal decompositions! Thus, the use of derived categories to study Calabi-Yau varieties takes a significantly different approach than what is typically used for Fano varieties.