Hilbert Basis theorem. ([12], [164] [256])

Theorem “Hilbert basis theorem”. If 𝑅 is a Noetherian ring, then 𝑅[𝑥] is Noethe-rian.

Corollary. If 𝑅 is a Noetherian ring, then 𝑅[𝑥₁, . . . , 𝑥_𝑛] is Noetherian.

See“Noetherian, Artinian”for the definition of Noetherian ring. Since every field is Noetherian, from the Hilbert basis theorem we get the following corollary:

84 | Hilbert’s Nullstellensatz

Corollary. If 𝐾 is a field, then 𝐾[𝑥1, . . . , 𝑥_𝑛] is Noetherian.

Hilbert’s Nullstellensatz.

([12], [104], [256]).

Notation. For any ideal 𝐽 in a ring 𝑅, let √𝐽 denote the radical ideal of 𝐽, i.e.,

√𝐽 := {𝑓 ∈ 𝑅| ∃ 𝑟 ∈ ℕ − {0} such that 𝑓^𝑟∈ 𝐽} . Notation. Let 𝐾 be a field.

– For any 𝑆 ⊂ 𝐾[𝑥1, . . . , 𝑥_𝑛], let 𝑍(𝑆) denote the zero locus of 𝑆 in the affine space 𝔸^𝑛_𝐾, i.e.,

𝑍(𝑆) := {𝑃 ∈ 𝔸^𝑛_𝐾| 𝑓(𝑃) = 0 ∀𝑓 ∈ 𝑆}.

– For any 𝑋 ⊂ 𝔸^𝑛_𝐾, let 𝐼(𝑋) denote the ideal of 𝑋, i.e.,

𝐼(𝑋) := {𝑓 ∈ 𝐾[𝑥₁, . . . , 𝑥_𝑛]| 𝑓(𝑃) = 0 ∀𝑃 ∈ 𝑋} .

Theorem “Hilbert’s Nullstellensatz”. Let 𝐾 be an algebraically closed field. We have that

𝐼(𝑍(𝐽)) = √𝐽 for any ideal 𝐽 of 𝐾[𝑥₁, . . . , 𝑥_𝑛].

In particular, if 𝑍(𝐽) = 0, then √𝐽 = 𝐾[𝑥₁, . . . , 𝑥_𝑛] and then 𝐽 = 𝐾[𝑥₁, . . . , 𝑥_𝑛].

Corollary. Let 𝐾 be an algebraically closed field. There is a bijection between the set of the algebraic subsets of 𝔸^𝑛_𝐾and the set of the radical ideals of 𝐾[𝑥1, . . . , 𝑥_𝑛].

Notation. Let 𝐾 be a field.

– For any subset 𝑆 of homogeneous elements of 𝐾[𝑥0, . . . , 𝑥_𝑛], we denote 𝑍(𝑆) = {𝑃 ∈ ℙ^𝑛_𝐾| 𝑓(𝑃) = 0 ∀𝑓 ∈ 𝑆}.

For any homogeneous ideal 𝐼 (see“Homogeneous ideals”), we define 𝑍(𝐼) to be the zero locus of the set of the homogeneous elements of 𝐼.

– For any 𝑋 ⊂ ℙ^𝑛_𝐾, we denote by 𝐼(𝑋) the ideal generated by

{𝑓 ∈ 𝐾[𝑥₀, . . . , 𝑥_𝑛]| 𝑓 homogeneous and 𝑓(𝑃) = 0 ∀𝑃 ∈ 𝑋};

obviously it is a homogeneous ideal.

Theorem “Projective Hilbert’s Nullstellensatz”. Let 𝐾 be an algebraically closed field.

Let 𝐽 be a homogeneous ideal of 𝐾[𝑥₀, . . . , 𝑥_𝑛].

We have that 𝑍(𝐽) = 0 if and only if there exists 𝑑 ∈ ℕ such that the part of degree 𝑑 of 𝐾[𝑥₀, . . . , 𝑥_𝑛] is contained in 𝐽 for any 𝑑 ≥ 𝑑.

If 𝑍(𝐽) ̸= 0, then 𝐼(𝑍(𝐽)) = √𝐽.

See also“Weierstrass preparation theorem and Weierstrass division theorem”.

Hilbert function and Hilbert polynomial | 85

Hilbert function and Hilbert polynomial.

([12], [104], [107], [164]).

Definition. We say that a polynomial 𝑃(𝑥) ∈ ℚ[𝑥] is numerical if 𝑃(𝑠) ∈ ℤ for any 𝑠 ∈ ℤ with 𝑠 >> 0.

Proposition. If 𝑃(𝑥) ∈ ℚ[𝑥] is a numerical polynomial of degree 𝑑, then it is a linear combination with integer coefficients of the polynomials (^𝑥_𝑖) := 𝑥(𝑥−1)⋅⋅⋅(𝑥−𝑖+1)

𝑖! for 𝑖 = 0, . . . , 𝑑 (where (^𝑥₀) = 1). In particular 𝑃(𝑠) ∈ ℤ for every 𝑠 ∈ ℤ.

Let 𝐾 be an algebraically closed field.

Definition. Let 𝑀 = ⊕_𝑠∈ℤ𝑀_𝑠be a graded module over 𝐾[𝑥₀, . . . , 𝑥_𝑟], where 𝑀_𝑠is its part of degree 𝑠. The Hilbert function of 𝑀, we denote 𝑓_𝑀, is defined in the following way:

𝑓_𝑀(𝑠) := 𝑑𝑖𝑚_𝐾𝑀_𝑠 ∀𝑠 ∈ ℤ .

Theorem (Hilbert–Serre). Let 𝑀 be a graded finitely generated module on 𝐾[𝑥₀, . . . , 𝑥_𝑟]. There exists a unique polynomial, called Hilbert polynomial of 𝑀, 𝑃_𝑀(𝑥) ∈ ℚ[𝑥] such that

𝑃_𝑀(𝑠) = 𝑓_𝑀(𝑠) for 𝑠 >> 0, 𝑠 ∈ ℕ.

Let 𝑋 be a projective algebraic variety in ℙ^𝑁:= ℙ^𝑁_𝐾. Its Hilbert function and its Hilbert polynomial are defined to be the Hilbert function and the Hilbert polynomial of its homogeneous coordinates ring. Thus the Hilbert function is

𝑓_𝑋(𝑠) = 𝑑𝑖𝑚_𝐾𝐻⁰(^O_ℙ^𝑁(𝑠)) 𝐻⁰(^I_𝑋(𝑠)) and the Hilbert polynomial is

𝑃_𝑋(𝑠) = 𝜒(^O_𝑋(𝑠)),

where^O_𝑋is the sheaf of the regular functions on 𝑋,^I_𝑋is the ideal sheaf of 𝑋 and 𝜒 is the Euler–Poincaré characteristic. In fact, by Serre’s theorem (see“Cartan–Serre theorems”, for 𝑠 >> 0, we have ℎ^𝑖(^I_𝑋(𝑠)) = ℎ^𝑖(^O_𝑋(𝑠)) = 0 for all 𝑖 > 0, thus, for 𝑠 >> 0, we have

𝑑𝑖𝑚_𝐾𝐻⁰(^O_ℙ^𝑁(𝑠))

𝐻⁰(^I_𝑋(𝑠)) = ℎ⁰(^O_𝑋(𝑠)) = 𝜒(^O_𝑋(𝑠)) by the sequence

0 →^I_𝑋→^O_ℙ^𝑁 →^O_𝑋→ 0.

We can prove that the degree of 𝑃_𝑋is the dimension 𝑛 of 𝑋, and we may define the degree (see“Degree of an algebraic subset”) of 𝑋 to be 𝑛! times the leading coefficient of 𝑃_𝑋.

86 | Hilbert syzygy theorem Examples.

– Let 𝜈_𝑛,𝑑 : ℙ^𝑛 → ℙ^𝑁, where 𝑁 = (^𝑛+𝑑_𝑑 ) − 1, be the Veronese embedding (see

“Veronese embedding”). Let 𝑉𝑛,𝑑be the Veronese variety 𝜈𝑛,𝑑(ℙ^𝑛). Its Hilbert func-tion is

𝑓_{𝑉𝑛,𝑑}(𝑠) = ℎ⁰(^O_ℙ^𝑁(𝑠))/ℎ⁰(^I_{𝑉𝑛,𝑑}(𝑠)) = ℎ⁰(^O_{𝑉𝑛,𝑑}(𝑠)) = ℎ⁰(^O_ℙ^𝑛)(𝑑𝑠)) = (𝑠𝑑 + 𝑛𝑛 ), where the second equality holds since the Veronese variety is projectively normal (see“Normal, projectively -, 𝑘-normal, linearly normal”). Since 𝑓_{𝑉𝑛,𝑑}(𝑠) is a poly-nomial we have that

𝑝_{𝑉𝑛,𝑑}(𝑠) = 𝑓_{𝑉𝑛,𝑑}(𝑠).

In particular for 𝑛 = 1, i.e., for the rational normal curve, we get 𝑝_𝑉1,𝑑(𝑠) = 𝑠𝑑 + 1.

– Let 𝐶 be a smooth projective curve of degree 𝑑 and genus 𝑔. For 𝑠 >> 0 ℎ⁰(^O_𝐶(𝑠)) = 𝑠𝑑 − 𝑔 + 1,

by the Riemann–Roch theorem (see“Riemann surfaces (compact -) and algebraic curves”), since^O_𝐶(𝑠) is a line bundle of degree 𝑠𝑑 on 𝐶 (the restriction of 𝑠 times the hyperplane bundle to 𝐶, i.e., 𝑠 times the line bundle given the embedding of 𝐶 in the projective space). Thus

𝑝_𝐶(𝑠) = 𝑠𝑑 − 𝑔 + 1.

Hilbert schemes.

^See“Moduli spaces”

Hilbert syzygy theorem.

([62], [159], [164]).

Hilbert syzygy theorem. Let 𝐾 be a field and 𝑅 = 𝐾[𝑥₀, . . . , 𝑥_𝑛]. Every finitely gener-ated graded 𝑅-module 𝑀 has a finite graded free resolution of length ≤ 𝑛 by finitely generated free 𝑅-modules, that is, there exists a graded exact sequence (i.e., an exact sequence where the maps preserve the degree)

0 → 𝐹_𝑛→ ⋅ ⋅ ⋅ → 𝐹₁→ 𝐹₀→ 𝑀 → 0, with 𝐹_𝑖finitely generated free 𝑅-modules.

See also“Horrocks’ theorem”.

Hironaka’s decomposition of birational maps.

([107], [117], [118]). Let 𝐾 be a field of characteristic zero and let 𝑉 and 𝑉^󸀠be two smooth projective algebraic varieties over 𝐾. Let 𝑓 : 𝑉 → 𝑉^󸀠be a birational map. Then there exists a morphism 𝑔 : ̃𝑉 → 𝑉 such that 𝑔 is the composition of a finite sequence of blow-ups along smooth subvarieties and the birational map 𝑓 ∘ 𝑔 is a morphism.

Hirzebruch-Riemann-Roch theorem | 87 In the case where 𝑉 and 𝑉^󸀠are surfaces, we have a stronger statement; see “Structure of birational maps on surfaces” in“Surfaces, algebraic -”.

Finally, we want to mention that it is known that in dimension ≥ 3 not every birational morphism is the composition of blow-ups along smooth subvarieties (see [229]).

See also [2].

Hirzebruch surfaces.

^See“Surfaces, algebraic -”.

Hirzebruch–Riemann–Roch theorem.

([13], [29], [107], [119], [146]). Let 𝑋 be a compact complex manifold and let 𝐸 be a vector bundle on 𝑋 of rank 𝑟. Write the Chern polynomial (see“Chern classes”)

𝑐(𝐸)(𝑡) = 𝑐₀(𝐸) + 𝑐₁(𝐸)𝑡 + ⋅ ⋅ ⋅ + 𝑐_𝑟(𝐸)𝑡^𝑟 as

𝑐(𝐸)(𝑡) = 𝛱_{𝑖=1,...,𝑟}(1 + 𝑎_𝑖𝑡).

We define the Chern character to be

𝑐ℎ(𝐸) := ∑

𝑖=1,...,𝑟𝑒^𝑎𝑖

(where 𝑒^𝑥= 1 + 𝑥 +¹₂𝑥²+ ⋅ ⋅ ⋅ ). Furthermore, we define the Todd class by 𝑡𝑑(𝐸) = 𝛱_{𝑖=1,...,𝑟} 𝑎_𝑖

1 − 𝑒^−𝑎𝑖, where_1−𝑒^𝑥−𝑥is the series 1 +¹₂𝑥 +₁₂¹𝑥²−₇₂₀¹ 𝑥⁴. . . .

We recall also that the symbol ∪ denotes the cup product; see“Singular homology and cohomology”for the definition.

Hirzebruch–Riemann–Roch theorem. Let 𝑋 be a compact complex manifold of dimen-sion 𝑛 and let 𝐸 be a holomorphic vector bundle on 𝑋 of rank 𝑟. We have that

𝜒(^O(𝐸)) = ∫

𝑋

𝑐ℎ(𝐸) ∪ 𝑡𝑑(𝑇_𝑋^1,0),

that is, 𝜒(^O(𝐸)) (the Euler–Poincaré characteristic of^O(𝐸)) is equal to the component in 𝐻^2𝑛(𝑋, ℝ) of 𝑐ℎ(𝐸) ∪ 𝑡𝑑(𝑇_𝑋^1,0) evaluated in the fundamental class of 𝑋, i.e., in the class of the 2𝑛-cycle determined by the natural orientation of 𝑋.

In the case where 𝑋 is a compact Riemann surface of genus 𝑔 (see“Riemann surfaces (compact -) and algebraic curves”) and 𝐸 is a line bundle, we get the usual Riemann–

Roch theorem:

ℎ⁰(𝑋,^O(𝐸)) − ℎ¹(𝑋,^O(𝐸)) = 𝑑𝑒𝑔(𝐸) − 𝑔 + 1;

in fact, the Chern polynomial is 𝑐(𝐸)(𝑡) = 1 + 𝑐1(𝐸)𝑡 and the Chern character is 𝑐ℎ(𝐸) = 𝑒^𝑐1(𝐸) = 1 + 𝑐₁(𝐸); in addition, 𝑇_𝑋^1,0 = −𝐾_𝑋, and thus 𝑡𝑑(𝑇_𝑋^1,0) = 1 − ¹₂𝑐₁(𝐾_𝑋).

88 | Hodge theory

The component of (1 + 𝑐1(𝐸)) ∪ (1 −¹₂𝑐₁(𝐾_𝑋)) in 𝐻²(𝑋) is 𝑐₁(𝐸) − ¹₂𝑐₁(𝐾_𝑋); thus we get 𝜒(^O(𝐸)) = deg(𝐸) − 12𝑑𝑒𝑔(𝐾_𝑋).

Taking 𝐸 to be trivial, we get that the degree of 𝐾_𝑋is 2𝑔 − 2; hence, by substituting, we get the Riemann–Roch formula for any holomorphic line bundle 𝐸 on a compact Riemann surface.

Analogously, in the case where 𝑋 is a surface and 𝐸 is a holomorphic line bundle, we get the usual Riemann–Roch theorem for surfaces (see“Surfaces, algebraic -”).

The theorem also holds for nonsingular projective algebraic varieties over alge-braically closed fields; we will not state it in this context, since it requires the gen-eralization of some concepts, such as Chern classes and intersection theory, for such varieties. The statement for complex nonsingular projective varieties is due to Hirze-bruch, the one for compact complex manifolds to Atiyah–Singer, the one for nonsin-gular projective varieties over algebraically closed fields to Grothendieck (see [29]).

Hodge theory.

([44], [90], [93], [245]). Let 𝑋 be a compact complex manifold of (complex) dimension 𝑛. Let ℎ be a Hermitian metric on 𝑋 and let 𝜔 be the associated (1, 1)-form (see“Hermitian and Kählerian metrics”).

Let us denote by 𝑇^{∨ (𝑝,𝑞)}𝑋 the bundle ∧^𝑝𝑇^1,0𝑋^∨ ⊗ ∧^𝑞𝑇^0,1𝑋^∨, where 𝑇^1,0𝑋 and 𝑇^0,1𝑋 are the holomorphic and the antiholomorphic tangent bundles (see“Almost complex manifolds, holomorphic maps, holomorphic tangent bundles”).

For any 𝑥 ∈ 𝑋, the metric ℎ induces a Hermitian metric on 𝑇_𝑥^{∨ (𝑝,𝑞)}𝑋, we call again ℎ, defined in the following way: let 𝜔 = ₂^𝑖∑_{𝑗=1,...,𝑛}𝑧_𝑗∧ 𝑧_𝑗in local coordinates 𝑧₁, . . . , 𝑧_𝑛 around 𝑥; let ℎ be such that the 𝑧_𝐼 ∧ 𝑧_𝐽for |𝐼| = 𝑝 and |𝐽| = 𝑞 form an orthogonal basis in 𝑇_𝑥^{∨ (𝑝,𝑞)}𝑋 and the norm of every 𝑧_𝐼∧ 𝑧_𝐽is 2^𝑝+𝑞. For any 𝑝, 𝑞 ∈ ℕ, let ( , ) be the following positive definite product on the set 𝐴^𝑝,𝑞(𝑋) of the 𝐶^∞(𝑝, 𝑞)-forms on 𝑋:

(𝜂, 𝛾) = ∫

𝑋

ℎ(𝜂(𝑧), 𝛾(𝑧)) 𝜔^𝑛(𝑧) 𝑛! .

Let 𝜕^∗be the adjoint operator of 𝜕 : 𝐴^𝑝,𝑞(𝑋) → 𝐴^𝑝,𝑞+1(𝑋) with respect to ( , ), i.e., let

𝜕^∗: 𝐴^𝑝,𝑞(𝑋) → 𝐴^{𝑝,𝑞−1}(𝑋) be the operator such that (𝜕^∗𝜂, 𝜃) = (𝜂, 𝜕𝜃) for all 𝜂 ∈ 𝐴^𝑝,𝑞(𝑋), 𝜃 ∈ 𝐴^{𝑝,𝑞−1}(𝑋).

Let 𝛥_𝜕: 𝐴^𝑝,𝑞(𝑋) → 𝐴^𝑝,𝑞(𝑋) be the so-called Laplacian operator:

𝛥_𝜕:= 𝜕^∗𝜕 + 𝜕 𝜕^∗.

The forms 𝜂 s.t 𝛥_𝜕𝜂 = 0 are called 𝜕-harmonic. Let^H𝑝,𝑞_𝜕 (𝑋) be the space of the 𝜕-harmonic (𝑝, 𝑞)-forms. Observe that 𝛥_𝜕𝜂 = 0 if and only if 𝜕𝜂 = 0 and 𝜕^∗𝜂 = 0, in fact (⋅, ⋅) is positive definite and

(𝛥_𝜕𝜂, 𝜂) = ((𝜕 𝜕^∗+ 𝜕^∗𝜕)𝜂, 𝜂) = (𝜕^∗𝜂, 𝜕^∗𝜂) + (𝜕𝜂, 𝜕𝜂).

Hodge theory | 89 Theorem.

(i) The space^H𝑝,𝑞_𝜕 (𝑋) is finite dimensional. Thus, we can define an orthogonal pro-jection 𝐻 : 𝐴^𝑝,𝑞(𝑋) →^H_𝜕^𝑝,𝑞(𝑋).

(ii) There exists an operator (called Green’s operator) 𝐺 : 𝐴^𝑝,𝑞(𝑋) → 𝐴^𝑝,𝑞(𝑋) such that 𝐺(^H𝑝,𝑞_𝜕 (𝑋)) = 0, 𝐺 commutes with 𝜕 and 𝜕^∗and, for all 𝜂 ∈ 𝐴^𝑝,𝑞(𝑋),

𝜂 = 𝐻𝜂 + 𝛥_𝜕𝐺𝜂.

The above decomposition is called Hodge decomposition.

(iii) Let 𝜂 ∈ 𝐴^𝑝,𝑞(𝑋); there exists 𝛾 ∈ 𝐴^𝑝,𝑞(𝑋) such that 𝛥_𝜕𝛾 = 𝜂 if and only if 𝜂 is orthogonal to^H𝑝,𝑞_𝜕 (𝑋).

Observe that the implication ⇒ of (iii) is obvious and the other implication follows from (ii); in fact, let 𝜂 ⊥^H𝑝,𝑞_𝜕 (𝑋); by (ii) we have 𝜂 = 𝐻𝜂 + 𝛥_𝜕𝐺𝜂 = 𝛥_𝜕𝐺𝜂.

Corollary. Every 𝜕-closed form 𝜂 is 𝜕-homologous to a 𝜕-harmonic form. Thus H^𝑝,𝑞

𝜕 (𝑋) ≅ 𝐻_𝜕^𝑝,𝑞(𝑋) .

In fact, let 𝜂 be such that 𝜕𝜂 = 0; by Hodge decomposition, we have 𝜂 = 𝐻𝜂 + 𝜕 𝜕^∗𝐺𝜂 + 𝜕^∗𝜕𝐺𝜂 = 𝐻𝜂 + 𝜕 𝜕^∗𝐺𝜂 + 𝐺𝜕^∗𝜕𝜂 = 𝐻𝜂 + 𝜕 𝜕^∗𝐺𝜂, so we have found a harmonic form, 𝐻𝜂, that is 𝜕-homologous to 𝜂.

Thus, by Dolbeault’s theorem,^H𝑝,𝑞_𝜕 (𝑋) ≅ 𝐻^𝑞(𝑋, 𝛺^𝑝) (see“Dolbeault’s theorem”).

An analogous theory can be developed for a Riemannian manifold and the operator 𝑑 instead of the operator 𝜕 (and 𝛥_𝑑 := 𝑑 𝑑^∗+ 𝑑^∗𝑑 instead of 𝛥_𝜕, where 𝑑^∗is the adjoint operator of 𝑑 with respect to ∫ ⋅ ∧ ∗⋅, where ∗ is the star operator; see“Star operator”).

If 𝑋 is a Kähler manifold, we have that

2𝛥_𝜕= 𝛥_𝑑= 2𝛥_𝜕.

In particular,^H𝑝,𝑞_𝜕 (𝑋) =^H𝑝,𝑞_𝜕 (𝑋) =^H𝑝,𝑞_𝑑 (𝑋) (where^H𝑝,𝑞_𝑑 (𝑋) is the set of the (𝑝, 𝑞)-forms 𝜂 such that 𝛥_𝑑𝜂 = 0).

Since 𝛥_𝑑= 2𝛥_𝜕, the operator 𝛥_𝑑preserves the bidegree; hence H^𝑟_𝑑(𝑋) = ⊕_{𝑝+𝑞=𝑟}^H_𝑑^𝑝,𝑞(𝑋);

moreover 𝛥_𝑑is “real”, thus

H^𝑝,𝑞_𝑑 (𝑋) =^H𝑞,𝑝_𝑑 (𝑋).

By Hodge decomposition for 𝑑, we have 𝐻_𝐷𝑅^𝑟 (𝑋) = ^H𝑟(𝑋) and 𝐻^𝑝,𝑞(𝑋) = ^H𝑝,𝑞_𝑑 (𝑋), where 𝐻_𝐷𝑅^𝑟 (𝑋) is the set the 𝑑-closed 𝑟-forms (over ℂ) modulo the set of the 𝑑-exact 𝑟-forms (over ℂ) and 𝐻_𝑑^𝑝,𝑞(𝑋) is the set of the 𝑑-closed (𝑝, 𝑞)-forms modulo the 𝑑-exact (𝑝, 𝑞)-forms. Thus we get the following theorem:

90 | Holomorphic

Theorem. If 𝑋 is a compact Kähler manifold, then 𝐻^𝑟(𝑋, ℂ) = ⊕_{𝑝+𝑞=𝑟}𝐻^𝑝,𝑞(𝑋),

𝐻^𝑝,𝑞(𝑋) = 𝐻^𝑞,𝑝(𝑋) .

Holomorphic.

^See“Almost complex manifolds, holomorphic maps, holomorphic tangent bundles”.

Homogeneous bundles.

([30], [210]). Let 𝑋 be a 𝐺-homogeneous complex alge-braic variety (see“Homogeneous varieties”) and 𝐸 be a vector bundle on 𝑋. We say that 𝐸 is homogeneous if there is an action of 𝐺 on 𝐸 (that is, a homomorphism from 𝐺 to the group of automorphisms of 𝐸) such that

𝑔𝐸_𝑥⊂ 𝐸_𝑔𝑥

for all 𝑔 ∈ 𝐺 and for all 𝑥 ∈ 𝑋 (where 𝐸_𝑥denotes the fibre on 𝑥).

If we write 𝑋 as 𝐺/𝑃 where 𝑃 is the isotropy subgroup of a point of 𝑋, we can easily prove that 𝐸 is homogeneous if and only if it comes from the principal bundle 𝐺 → 𝐺/𝑃 and a representation 𝜌 : 𝑃 → 𝐺𝐿(𝑟, ℂ), where 𝑟 is the rank of 𝐸 (see“Bundles, fibre -”

and precisely principal bundles), i.e., 𝐸 is homogeneous if and only if 𝐸 ≅ 𝐺 ×_𝜌ℂ^𝑟:= 𝐺 × ℂ^𝑟/ ∼,

where ∼ is the equivalence relation such that (𝑔, 𝑣) ∼ (𝑔𝑝, 𝜌(𝑝⁻¹)𝑔) for any 𝑝 ∈ 𝑃.

For a vector bundle 𝐸 on a homogeneous rational variety 𝐺/𝑃, with 𝐺 simply con-nected semisimple group, 𝑃 parabolic subgroup (see“Lie groups”), we have that 𝐸 is homogeneous if and only if 𝑙^∗_𝑔𝐸 ≅ 𝐸 for every 𝑔 ∈ 𝐺, where 𝑙_𝑔 : 𝐺 → 𝐺 is the left multiplication by 𝑔.

Homogeneous ideals.

([73], [185], [256]). Let 𝐾 be a field and 𝐼 be an ideal of 𝐾[𝑥₀, . . . , 𝑥_𝑛]. We say that 𝐼 is a homogeneous ideal if and only if the following property holds: if we write an element 𝐹 of 𝐼 as sum of homogeneous polynomials, 𝐹 = ∑_{𝑖=1,...,𝑘}𝐹_𝑖, we have that 𝐹_𝑖∈ 𝐼 for 𝑖 = 1, . . . , 𝑘.

Proposition.

– An ideal in 𝐾[𝑥0, . . . , 𝑥_𝑛] is homogeneous if and only if it is generated by homoge-neous polynomials.

– The sum, product, intersection of homogeneous ideals are homogeneous, the rad-ical of a homogeneous ideal is homogeneous.

– A homogeneous ideal 𝐼 is prime if and only if, for any 𝑓, 𝑔 ∈ 𝐾[𝑥0, . . . , 𝑥_𝑛] with 𝑓, 𝑔 homogeneous and such that 𝑓𝑔 ∈ 𝐼, we have either 𝑓 ∈ 𝐼 or 𝑔 ∈ 𝐼.

Horrocks’ theorem | 91

Homogeneous varieties.

([28], [210]). Let 𝐺 be an algebraic group, respectively a topological group. Let 𝑋 be an algebraic variety, respectively a manifold. We say that 𝐺 acts on 𝑋 if there is a morphism 𝐺 × 𝑋 → 𝑋, (𝑔, 𝑥) 󳨃→ 𝑔𝑥 such that 1𝑥 = 𝑥 for any 𝑥 ∈ 𝑋, 𝑔₁(𝑔₂𝑥) = (𝑔₁𝑔₂)𝑥 for any 𝑥 ∈ 𝑋 and for any 𝑔₁, 𝑔₂∈ 𝐺; we say that the action is transitive if, for any 𝑥, 𝑥^󸀠∈ 𝑋, there exists 𝑔 ∈ 𝐺 such that 𝑔𝑥 = 𝑥^󸀠. We say that 𝑋 is 𝐺-homogeneous if 𝐺 acts transitively on it.

Remark. Every homogeneous variety is smooth.

Theorem (Borel–Remmert). A homogeneous compact Kähler manifold is isomorphic to the product of a complex torus and a rational homogeneous projective algebraic variety (see“Tori, complex - and Abelian varieties”,“Rational varieties”).

Furthermore, a rational homogeneous projective algebraic variety is isomorphic to 𝐺₁/𝑃₁× ⋅ ⋅ ⋅ × 𝐺_𝑘/𝑃_𝑘

for some simple Lie groups 𝐺_𝑖and 𝑃_𝑖parabolic subgroups (see“Lie groups”).

Homology, Singular -.

^See“Singular homology and cohomology”.

In document Algebraic Geometry - A Concise Dictionary (gnv64).pdf (Page 93-101)