Canonical, tangent, and cotangent sheaf

Wanlong Zheng

January 7, 2020

This short article aims to explain some geometric intuitions behind these objects: canonical sheaf, tangent sheaf, and cotangent sheaf. Nothing appearing in this article is original.

Contents

1 Basics 1 1.1 Cotangent ...... 1 1.2 Tangent ...... 2 1.3 Euler sequence ...... 2

2 Cohomology of the tangent sheaf 3 2.1 Geometric understandings ...... 3 0 2.1.1 H of TX ...... 3 1 2.1.2 H of TX ...... 4 2.2 An example(s) with rings ...... 4

3 Canonical sheaves/divisors 5 3.1 Conormal, normal, adjunction ...... 5 3.2 Dualizing sheaf ...... 6 3.2.1 Relative dualizing sheaf ...... 7 3.3 Stability in terms of canonical sheaf ...... 7

1 Basics

1.1 Cotangent

Although it seems very unnatural, but cotangent (or conormal) is a more natural object than tangent (or normal) sheaf/bundle. We will start with cotangent. Cotangent is closely related with “differentials” of a function.

Definition. Let f : X Y be a morphism of schemes, and ∆ : X X ×Y X the diagonal morphism. Denote by W an open set of X ×Y X which contains ∆(X) as a closed subscheme. If I is the sheaf of ideals → →

1 of ∆(X) in W, then the sheaf of relative differentials of X over Y is the sheaf

∗ 2 ΩX/Y = ∆ (I/I ).

As a consequence (or another equivalent way to define this), we could cover X and Y by affine opens Y ⊃ U = Spec(A) and X ⊃ V = Spec(B) with f(V) ⊂ U, and the sheaf of relative differentials is obtained ∼ by glueing the sheaves (ΩB/A) associated to the modules of differentials ΩB/A.

We quickly recall here that the module ΩB/A is the free module generated by symbols {db | b ∈ B} dividing out by d(b + b0) = db − db0, d(bb0) = bdb0 + b0db and da = 0 for all a ∈ A.

Example.

2 • Take B = k[x, y] and A = k, then 3x dy + 4dx ∈ ΩB/A. In general, if B is generated by A by some xi’s, then taking dxi and the relations among them suffice to define Ω.

• If B = A/I, then db = 0 since any b is the image of some a. So Ω = 0. Similarly for localizations.

2 3 2 • Take B = C[x, y]/(y − x ), and A = C, then ΩB/A = (Bdx ⊕ Bdy)/(2ydy − 3x dx).

1.2 Tangent

Definition. The tangent sheaf of X is the dual TX = HomOX (ΩX/k, OX).

Generally we assume X is nonsingular, and in this case TX is locally free of rank n = dim X. Using the universal property of differentials:

Proposition. If B is an A-module, and Ω = ΩB/A the module of differentials and d : B Ω the natural map, then for any B-module M and for any A-derivation d0 : B M, there exists a unique B-module map f : Ω M with d0 = f ◦ d. → → → we see that elements in TX(Spec(A)) correspond to k-derivations A A, agreeing with our experi- ences in differential geometry. → At stalk level (cf bundle at a point) x ∈ X, tangent sheaf is simply the Zariski tangent space:

2 (TX)x = Homk(m/m , k) where m is the maximal ideal of the local ring OX,x.

1.3 Euler sequence

The version commonly considered in algebraic geometry is the short exact sequence:

⊕(n+1) 0 OPn OPn (1) TPn 0.

At the level of global sections (H0→also form→ a short exact sequence),→ → those of T (tangent sheaf/bundle) n n+1 are vector fields. Thinking of P as C \{0}, we have vector fields d/dxi, and for any set of n + 1 d n linear homogeneous functions v1, ... , vn+1, the vector field vi descends to a vector field on P . dxi And this map is surjective. P How about the kernel? Pretending Pn is a sphere, then we need a “normal” vector field in C3, which, d when we project down, would give 0. So the kernel is E = xi . dxi

2 In particular, this is really saying the Euler relation for a homogeneous polynomial of degree d (although symbolically I really should have used partials):

d xi f = d · f. dxi X The dual of this sequence gives:

⊕(n+1) 0 ΩPn/k OPn (−1) OPn 0.

→ → → → 2 Cohomology of the tangent sheaf

The standard reference is Hartshorne’s Deformation Theory. The goal is to explain (including the meaning of those words) the following results from deformation theory:

0 • H (X, TX) measures infinitesimal (first-order) automorphisms,

1 • H (X, TX) measures infinitesimal deformations,

2 2 • H (X, TX) measures obstructions to deformations (if H = 0 we call unobstructed).

To explain this, it is necessary to introduce the dual number D := Spec k[]/(2), which plays a very important role in deformation theory. The first observation is that, since dual number captures enough information (linear part of a function), tangent sheaf for an affine k-scheme X = Spec(A) is just:

TX = Homk(Spec(A ⊗ D), X).

2.1 Geometric understandings

0 2.1.1 H of TX

This section is devoted to explaining why the global sections of tangent sheaf for a curve X correspond to infinitesimal automorphisms of X. Here I quote from an answer in stackexchange that summarizes this result:

The basic idea that underlies their description is that given a manifold X, with a Lie group of automor- phisms G, every tangent vector at the origin of G (ie an element of the Lie algebra of G) induces a vector field V on X in the obvious way - namely given z ∈ Lie(G), at every point x ∈ X, the tangent vector Vx tz is represented by the curve {e (x)}t∈(−1,1) ⊂ X. In the cases described in the book, they’re claiming that this association yields an isomorphism between the appropriate spaces.

The rest is also taken from the same answer following the previous paragraph.

0 0 Definition. Let X be a scheme over k. A deformation of X0 over D is a pair (X , i), where X is a scheme flat over D, i : X , X0 is a closed immersion, and the induced map

0 → i ×D k : X X ×D k

0 0 is an isomorphism. Two deformations are equivalent→ if there is an isomorphism f : X1 X2 with i2 = f ◦ i1. →

3 0 0 Note some author simply requires X flat over D with X ×D k isomorphic to X. We could get this by dividing the action of the group of automorphisms of X over k. In any situation, it’s important to keep the intuitions in mind. Then the result we want it: Proposition. Let (X0, i) be a deformation of a k-scheme X over D. Then the automorphism group of (X0, i) is 0 naturally isomorphic to H (X, TX).

First notice that H0 is independent of the choice of deformation, so we could simply work with the 0 trivial deformation X = X ×k D.

We have a contravariant sheaf AX of automorphism groups associated to X: for any k-scheme S, AX(S) := AutS(X ×k S). Let’s assume this is really a sheaf (in a sufficiently fine Grothendieck topology that we care). In particular, Spec(k) , Spec(D) induces a restriction map AX(D) AX(k), and the automorphisms of the trivial deformation is the kernel of this group homomorphism (this will also be the version of definition given in the Stacks→ Project). →

In good cases, the sheaf AX is representable by a group scheme G. Then the above group homomor- phism becomes:

Homk(Spec(D), G) Homk(Spec(k), G),

G and the kernel of which you could convince yourself→ is the tangent space of at origin, or the Lie algebra of G. Example. Let X be a geometrically irreducible smooth proper genus g curve over C. Riemann-Roch says 0 1 dimk H (X, TX) = 3 if g = 0 (or X = P ). These 3 dimensional sections come from the 3 dimensional group scheme PGL(2, C) acting on P1. If g = 1, then elliptic curves have a group structure, so dim H0 = 1 comes from the action of X on itself. If g ≥ 2, there are only finitely many automorphism.

Note that we are talking about automorphisms of X and infinitesimal automorphisms of X at the same time. It should? be the case that the infinitesimal automorphisms form the Lie algebra of the Lie group of automorphisms, although we only care about the vector space structrue. Anyway, dimensions are the same.

1 2.1.2 H of TX

Suppose X is a smooth variety over k and X0 an infinitesimal deformation of X. There is a result called “Infinitesimal Lifting Property” which says over an affine piece, deformation must be trivial. 0 0 So we take an affine cover Ui of X, and correspondingly Ui of X . By the mentioned result, φi : 0 −1 Ui ×Spec(k) Spec(D) Ui is a trivialization, and φj φi gives a cocycle in a sheaf of automorphism group. But we just saw the this is in the tangent sheaf. → There is another argument in the context of complex manifold, which considers the deformation of the complex structures by perturbing the transition functions. See Kodaira’s Complex Manifolds and Deformation of Complex Structures chapter 4.

2.2 An example(s) with rings

This example is taken from Szendroi’s˝ notes on deformations. Suppose we have an associative ring A, and we want to perturb the multiplication, by

a ∗ b := ab + f(a, b)

4 for some f ∈ Hom(A ⊗ A, A) the space of all multiplications. Writing out the associative condition for ∗, and that 2 = 0, we need: af(b, c) − f(ab, c) + f(a, bc) − f(a, b)c = 0, with the equivalence of two new multiplication given by

f(a, b) − g(a)b − ag(b) + g(ab)

for some g ∈ Hom(A, A). The equivalence classes of deformations (over dual numbers, or infinitesimal), therefore, is the Hochschild cohomology (see algebra notes on webpage)

HH1(A, A),

and we will just write H1(A, A) here. Similarly, we could immediately work out that H0(A, A) is the derivations of A moduli inner deriva- tions, and H2(A, A) is the obstructions to deformations of A.

Example. Take A = k[Z/2Z]. Then for H1, all cocycles are coboundaries, and thus H1 = 0; there is no deformations at all. It is very “rigid”.

Example. Take A = k[x]/(x2). Then H1 is one dimensional.

Obstruction in the ring context is to say, given f in H1, there is an infinite order deformation in the direction of f: 2 fe= f +  f2 + ···

such that the new feis another deformation. This condition translates to d(∗) = 0 in H2. So if H2 = 0 then every thing is unobstructed; but this condition is only sufficient, but not necessary. k[x]/(x2) is such an example of nonvanishing H2 but everything is unobstructed.

3 Canonical sheaves/divisors

Definition. For a smooth variety X over k of dimension n, the canonical sheaf of X is defined to be

n ωX = ∧ ΩX/k.

Since ωX is an invertible sheaf, ωX = OX(KX) for some Cartier divisor KX, which we call the canonical divisor; or alternatively call any Weil divisor representing ωX the canonical divisor, also denoted by KX.

Definition. The geometric genus is

0 pg(X) = dimk H (X, ωX).

0 1 (Recall the arithmetic genus of a curve is pa(X) = 1 − dimk H (X, OX) + dimk H (X, OX))

The geometric genus is birationally invariant.

3.1 Conormal, normal, adjunction

We want to find an expression that calculates the canonical sheaf/divisor of a subvariety. This is done via the adjunction formula, which involves the normal sheaf.

5 Definition. If Y is a smooth subvariety of a smooth variety X with ideal sheaf I, then the conormal sheaf is the locally free sheaf I/I2, and the normal sheaf is its dual:

2 NY/X = HomOY (I/I , OY).

Proposition. In the setting as above, there is a short exact sequence

2 0 I/I ΩX/k ⊗ OY ΩY/k 0,

2 where ΩY/k is locally free, I is locally→ generated→ by codim(Y→, X) elements,→ and I/I is locally free of rank codim(Y, X).

Taking the dual of this sequence, we have:

0 TY TX ⊗ OY NY/X 0.

This roughly says the normal bundle→ is the→ tangent vectors→ of the→ ambient space modulo those of the subspace.

Theorem. (Adjunction formula) Let Y be a codim r smooth subvariety of a smooth variety X over k, then:

∼ r ωY = ωX ⊗ ∧ NY/X.

In the special case r = 1, then Y is a divisor with associated line bundle L = OX(Y), and

∼ ωY = ωX ⊗ L ⊗ OY.

In terms of divisors,

KY = (KX + Y)|Y.

To prove it, taking the determinant of the above ses and then the dual.

Example.

• On Pn, using ⊕(n+1) 0 ΩPn/k OPn (−1) OPn 0

ω n ∼ O(−n − 1) and taking top wedge, we have→ P =→ . → →

• On a smooth toric variety, if z1, ... , zn are coordinates of the big torus, then

dz dz dz 1 ∧ 2 ∧ ··· ∧ n z1 z2 zn

is invariant under the torus action, and is thus supported on the invariant divisors. 1 n−1 Locally on any invariant divisor X = Uσ isomorphic to A × Gm , we reduce to the case dz/z with a simple pole at 0. Thus KX + D ∼ 0 for D the sum of all invariant divisors. In particular, if X = Pn, then there are n + 1 invariant divisors all equivalent to H, and thus

KPn = −(n + 1)H.

3.2 Dualizing sheaf

A dualizing sheaf is a sheaf that makes duality works.

6 Definition. If X is a proper scheme of dimension n over a field k, a dualizing sheaf for X is a coherent sheaf ω on X together with a trace morphism t : Hn(X, ω) k such that for every coherent sheaves F on X, the natural pairing Hom(F, ω) × Hn(X, F) →Hn(X, ω)

t followed by gives an isomorphism →

∼ Hom(F, ω)∨ − Hn(X, F).

If it exists, it is unique up to a canonical isomorphism.→

n An explicit calculation shows P has a dualizing sheaf equivalent to its canonical sheaf ωPn . This is a general phenomenon. For a smooth (normal) projective scheme, the dualizing sheaf is the canonical sheaf. A form of Serre duality is the following: for a reasonable class of schemes of dimension n over k (Cohen-Macaulay + some other conditions), and for all locally free sheaves,

i ∼ n−i ∨ H (X, F) = H (X, Hom(F, ωX)) .

If Y is a local complete intersection of codimension r in X (locally complete intersections),

∼ r ωY = KX|Y ⊗ ∧ NY/X.

3.2.1 Relative dualizing sheaf

Relative version works better for families. Suppose π : X Y is a flat projective morphism of locally Noetherian schemes of relative dimension n, and all geometric fibers are Cohen-Macaulay (thus some → n reasonable class of families). Then there is a coherent sheaf ωX/Y along with a trace map R π∗ωX/Y OY such that for every finite rank locally free sheaves F on X, each of whose higher pushforwards are locally free on Y, we have a perfect pairing: →

i n−i ∗ ∨ n ∗ t R π∗F × R π (F ⊗ ωX/Y) R π ωX/Y − OY.

→ → 3.3 Stability in terms of canonical sheaf

Definition. A stable curve is a connected projective curve that

1. has only nodes as singularities;

2. has only finitely many automorphisms.

Currently we are not considering pointed curves. So finiteness condition is equivalent to saying that every smooth rational component of the curve has at least 3 nodes (intersect with other components in at least 3 points).

Lemma. Condition 2 is equivalent to saying the dualizing sheaf of the curve is ample.

Note that nodes are local complete intersections (xy = 0) and hence a stable curve is Gorenstein. In particular the dualizing sheaf exists and it is a line bundle. For a proof, see Liu’s book Algebraic geometry and arithmetic curves Corollary 10.3.13.

7 For some intuition (very very imprecise here), if ωX is (probably very) ample, usually we could think of X as negatively curved. ωX is ample, combined with Serre duality, is roughly equivalent to saying 0 ∨ 0 H (X, ωX ) = 0. The dual of ω is some (exterior power) of the tangent sheaf. But H of the tangent sheaf is related to the infinitesimal automorphisms. So they are indeed related. Now the pointed version is:

Lemma. Let (C, p1, ... , pn) be an n-pointed, connected, complete, at-worst-nodal curve of (arithmetic) genus g, with 2g − 2 + n > 0 (to make sure the curve exist). Then the automorphism group of the curve is finite iff

ωC( pi) is ample. P Of course, if we are considering a family f : X S, then the stable condition can also be written as

ωX/S( pi) is ample. → P

8