Tangent Space to a Variety

9 Tangent Space to a Variety

Real directional derivatives and differentials

9.1 Let V be a finite dimensional real vector space, U ⊂ V be an open subset and f:U ∞ →WaC function. Let P ∈ U be a point. Then v

P (f ) ∈ R denotes the limit

f (P + tv) − f(P ) v P (f ) = lim

t→0

Show that the above limit indeed exists, and it depends only on the germ of f at P .

9.2 With the above notation, show that the following holds: v P (λ) = 0 for a constant function λ, v P (f + g) = v P (f ) + v P (f ), v P (f g) = f (P )v P (g) + g(P )v P (f ).

9.3 Let ∞ C V,P denote the local ring of germs of C functions at a point P ∈ V where

V is a finite dimensional real vector space, and let a P

⊂C ∞

V,P denote its maximal

ideal. Let a 2 P denote the square of the ideal. Show that if f ∈a P then v P (f ) = 0 for each v.

i , and if v = (v 1 ,...v n ), then show (using chain rule) that

9.4 n Exercise If V = R with linear coordinates x

9.5 Differential df P defined From the above exercise, conclude that the map

df P :V → R defined by v 7→ v P (f ) is a real linear functional on the vector space

V . This linear functional is denoted df P , and is called the differential of f at P . It depends only on the germ of f at P .

9.6 Exercise With notation as above, show that the map

U → Hom R (V, R) : P 7→ df P

is a ∞ C map from U to the dual vector space Hom R (V, R) of V .

9.7 C ∞ total derivative Let V and W be finite dimensional real vector spaces, and let v ∞ ∈ V . Let U ⊂ V be an open subset and let f : U → W be a C map. Let

P ∈ U be a point. Then v P (f ) ∈ W denotes the limit

f (P + tv) − f(P ) v P (f ) = lim

t→0

Show that the above limit indeed exists. If V = R n with linear coordinates x

i , if v = (v m

1 ,...v n ), and if W = R with linear coordinates y j so that f can be written as (f ∞

1 ,...,f m ) where each f i is in C (U ), then show that the column vector v P (f ) is given by applying the m × n Jacobian matrix

∂f k (P ) ∂x i

to the column vector v. Conclude that we thus get an R-linear homomorphism

Df P :V → W : v 7→ v P (f ) and as P varies we get a ∞ C map

Df : U → Hom R (V, W ) : P 7→ Df P

The linear map Df P ∈ Hom R (V, W ) is called the total derivative of f at P .

9.8 If P = (a ∞

1 ,...,a n ) ∈R is any point, show that any germ f ∈C R n ,P of a

C ∞ -function at P can be expressed as

1≤ i≤j≤n

where g ∞ i,j are germs of C -functions at P . In particular, if f (P ) = 0 and df P =0

then f ∞ ∈a P where a P is ideal formed by all germ of all C functions which take the value 0 at P .

Holomorphic directional derivatives and differentials

9.9 n Cauchy-Riemann equations Let V = C ,U ⊂ V an open subset, and f:U

→CaC n map (where U is regarded as an open subset of C =R 2 n and

C =R 2 where the identifications are made as usual by considering real and imaginary parts of coordinates. Then f is holomorphic if and only if for each P ∈ U we have

( −1v) P (f ) = −1 · v P (f )

for all vectors v n ∈C . This is equivalent to the condition that for each P ∈ U the R -linear map df P :V → C should actually be C-linear, that is,

df P ∈ Hom C (V, C) ⊂ Hom R (V, C)

√ If x n

i =s i + −1t i are the complex coordinates on C with real parts s i and imaginary parts t i , then show that the above condition is equivalent to the set of n equations

called the Cauchy-Riemann equations.

9.10 Holomorphic directional derivative Let V be a finite dimensional com- plex vector space, and let v ∈ V . Let U ⊂ V be an open subset and let f : U → C

be a holomorphic map. Let P ∈ U be a point. Then v P (f ) ∈ C denotes the limit

f (P + tv) − f(P ) v P (f ) = lim t→0 t

Show that the above limit indeed exists, and it depends only on the germ of f at P .

9.11 With the above notation, show that the following holds: v P (λ) = 0 for a constant function λ, v P (f + g) = v P (f ) + v P (f ), v P (f g) = f (P )v P (g) + g(P )v P (f ).

9.12 Let H V,P denote the local ring of germs of holomorphic functions at a point P ∈ V where V is a finite dimensional complex vector space, and let a P ⊂H V,P

denote its maximal ideal. Let a 2 P denote the square of the ideal. Show that if f ∈a P then v P (f ) = 0 for each v.

9.13 Holomorphic differentials With notation as above, we denote by df P the map V → C defined by v 7→ v P (f ). Show that this is a C-linear functional on V . The element df P ∈ Hom C (V, C) is called the holomorphic differential of f at P . It depends only on the germ of f at P .

9.14 Holomorphic total derivative Let V and W be finite dimensional complex vector spaces, and let v ∈ V . Let U ⊂ V be an open subset and let f : U → W be

a holomorphic map. Let P ∈ U be a point. Then v P (f ) ∈ W denotes the limit

f (P + tv) − f(P ) v P (f ) = lim t→0 t

Show that the above limit indeed exists. If V = C n with complex linear coordinates x m

i , if v = (v 1 ,...v n ), and if W = C with complex linear coordinates y j so that f can be written as (f 1 ,...,f m ) where each f i is in H(U), then show that the column vector v P (f ) is given by applying the m × n Jacobian matrix

∂f k (P ) ∂x i

to the column vector v. Conclude that we thus get a C-linear homomorphism

Df P :V → W : v 7→ v P (f )

and as P varies we get a holomorphic map

Df : U → Hom C (V, W ) : P 7→ Df P

The linear map Df P ∈ Hom C (V, W ) is called the holomorphic total derivative of f at P .

1 ,...,a n ) ∈C is any point, show that any germ f ∈H R n ,P of a holomorphic function at P can be expressed as

9.15 If P = (a n

1≤ i≤j≤n

where g i,j are germs of holomorphic functions at P . In particular, if f (P ) = 0 and df 2

P = 0 then f ∈a P where a P is ideal formed by all germ of all holomorphic functions which take the value 0 at P .

Tangent spaces to a manifold

9.16 ∞ C (respectively, holomorphic) Zariski tangent space Let V be a finite dimensional vector space over R (respectively, over C), and let X ⊂ V be an arbi-

trary subset. For any point P ∈ X, let T P (X) ⊂ V (respectively, let T P (X) ⊂V) denote the set of all vectors v ∞ ∈ V such that for each germ (U, f) of C -function

(respectively, of holomorphic function) on V at P such that f | X∩U = 0, we have v P (f ) = 0. Then show that T P

X is a complex) vec- tor subspace of V . This vector space T

X is a real (respectively, T P

X is called the C Zariski tangent space (respectively, T P

X is called the holomorphic Zariski tangent space) to X at P .

9.17 Let V be a finite dimensional vector space complex vector space of complex dimension d, and let X ⊂ V and let P ∈ V . Let the holomorphic Zariski tangent space T P X ⊂ V be the complex vector subspace defined above. We can regard

V as a real vector space of dimension 2d, so we have the ∞ C Zariski tangent space T P X ⊂ V as defined above, which is a real vector subspace of V . Then from the fact

that the real and imaginary components of a holomorphic function are ∞ C functions, deduce that T P X ⊂T P X.

9.18 With the above notation, if V is real (respectively, complex) and X ⊂V is a locally closed ∞ C -submanifold (respectively, holomorphic submanifold ) of V

X) is called the tangent space to the manifold X at P ∈ X. Show that if P is the centre of a

with P ∈ X, then the Zariski tangent space T P

X (respectively, T P

C ∞ (respectively, holomorphic) cubical coordinate chart (W ; u 1 ,...,u n ) in V (where n = dim(V ) such that X ∩ W is given by u d+1 =...=u n = 0 (where d = dim(X)),

then

T P (X) = {v ∈ V | v P (u i ) = 0 for all d + 1 ≤ i ≤ n}

(respectively, T P X= {v ∈ V | v P (u i ) = 0 for all d + 1 ≤ i ≤ n.) With respect to linear coordinates (x 1 ,...,x n ) on V , deduce that the tangent space T P (X) (respec- tively, T P

X) is described as follows. We have v = (v 1 ,...,v n ) ∈T P (X) (respectively, T P

X) if and only if

X ∂u i (P )v j = 0 for all d + 1 ≤i≤n

1≤ j≤n ∂x j

In particular, as the rank of the matrix (∂u i /∂x j ) d+1≤i≤n,1≤j≤n is n −d, it follows that dim R T P (X) = d = dim R (X) for any ∞ C (respectively, dim C T P (X) = d = dim C (X) for any holomorphic) manifold X and point P ∈ X.

9.19 With the above notation, if V is a complex vector space and X ⊂ V is

a locally closed holomorphic submanifold of V with P ∈ X, then as real vector subspaces of V we have an equality T P X=T P X ⊂ V . This equality follows from the description of these tangent spaces given above in terms of local coordinates.

9.20 Let V be a finite dimensional vector space over R, and let X ⊂ V be an arbitrary subset. For any point P ∈ X, let I X,P ⊂a P

be the ideal which consists of all germs (U, f ) which vanish on X ∗ ∩ U. Consider the linear subspace N

X,P of

V ∗ spanned by all differentials df P where f ∈I X,P . Then show that T P X ⊂ V is

the subspace annihilated by N ∗ X,P ⊂V under the dual pairing of V with V . In particular, dim(T ∗

X) = n − r where r = dim(N X,P ). State and prove an analogous statement for the holomorphic tangent space T P X.

9.21 Let V be a finite dimensional vector space over the field k = R (respectively, over k = C), and let X ∞ ⊂ V be an arbitrary subset. For any point P ∈ X, let C

V,P

(respectively, ∞ H V,P ) denote the local ring of germs of C -functions (respectively, holomorphic functions) on V at P , and let a P denote its maximal ideal. Let I X,P ⊂

be the ideal which consists of all germs (U, f ) which vanish on X ∩ U. We define

(where recall k denotes the field R or C as stated above). Note that v P (I X,P )=0 by definition of T

X and v P (a ) = 0 by 9.3 (respectively, by 9.12) so this map is well-defined. Show that that above map defines a non-degenerate bilinear pairing,

making T 2

X and a P /(I X,P +a P ) dual vector spaces of each other. Proof If v 6= 0 and x i are linear coordinates on V with P = (a 1 ,...,a n ), then

v(x i −a i ) 6= 0 for some i, showing the ‘left-kernel’ to be zero in the above pairing. To see the ‘right-kernel’ to be zero, suppose f ∈a P is such that v P (f ) = 0 for all v ∈T P

X (or T P X). Let f 1 ,...,f r ∈I X,P such that (df 1 ) P , . . . , d(f r ) P is a basis

for N ∗ X,P . As T P X ⊂ V is exactly the subspace annihilated by N X,P , and as df P

annihilates T P

c i d(f i ) P for constants c i . Hence P

X, df P is a linear combination

d(f −

c i f i ) P = 0. This shows using 9.8 or 9.15 that f − c i f i 2 ∈a P , so f = 0 in

a P 2 /(I X,P +a P ), as was to be shown.

Tangent spaces to a variety

9.22 Algebraic Zariski tangent space Let V be a finite dimensional vector space over C, and let X ⊂ V be an arbitrary subset. For any point P ∈ V , let T P (X) ⊂ V denote the set of all vectors v ∈ V such that for each germ (U, f) of regular function on V at P such that f | X∩U = 0, we have v P (f ) = 0. Then show that T P

X is called the algebraic Zariski tangent space to X at P .

X is a complex vector subspace of V . This vector space T P

X) denotes the tangent space in the C (respectively, in the holomorphic) category defined in 9.16, and if T P

9.23 If T ∞

X (respectively, T P

X denotes the tangent space in the algebraic category defined in 9.22, then as every regular germ is a holomorphic X denotes the tangent space in the algebraic category defined in 9.22, then as every regular germ is a holomorphic

T P X ⊂T P X ⊂T P X

If X is a locally closed algebraic subvariety with P ∈ X, then we get the equality T P X=T P

X as an immediate consequence of Proposition 10.12.

9.24 Let V is a finite dimensional complex vector space, and let X ⊂ V be an arbitrary subset. For any point P ∈ X, let O V,P denote the local ring of germs of regular functions on V at P , and let a P denote its maximal ideal. Let I X,P ⊂a P

be the ideal which consists of all germs (U, f ) which vanish on X ∩ U. We define a map

→ C : (v, f) 7→ v P (f )

I +a X,P 2 P

Note that v 2 P (I X,P ) = 0 by definition of T P

X and v P (a ) = 0 by 9.12, so this map

is well-defined. Let N ∗

be the vector space spanned by all differentials df P where f ∈I X,P . Show following the proof of 9.21 that above map defines a non- degenerate bilinear pairing, making T 2

X,P ⊂V

X and a P /(I X,P +a P ) dual vector spaces of each other.

9.25 Let V is a finite dimensional complex vector space, and let X ⊂ V be a locally closed subvariety. Let O X,P

be its maximal ideal. As above, let O V,P denote the local ring of germs of regular functions on V at P with maximal ideal a P , and let I X,P ⊂a P

be the local ring at P ∈ X and let m P ⊂O X,P

be the ideal which consists of all germs (U, f ) which vanish on X ∩ U. Then show that O X,P = O V,P /I X,P , m P =a P /I X,P , and consequently

I X,P +a P

Hence we have a non-degenerate pairing

P X × m 2 → C : (v, f) 7→ v P (f )