Horseshoe lemma. ([208])

Horseshoe lemma. Let 𝑅 be a ring and let

0 //𝑀₁ ^𝛼 //𝑀₂ ^𝛽 //𝑀₃ //0 be an exact sequence of 𝑅-modules. Let

. . . //𝐹_𝑡 ^𝑓𝑡// . . . ^𝑓1 //𝐹₀ ^𝑓0 //𝑀₁ //0 and

. . . //𝐺_𝑡 ^𝑔𝑡// . . . ^𝑔1 //𝐺₀ ^𝑔0 //𝑀₃ //0

be two projective resolutions of 𝑀₁and 𝑀₃respectively (see“Injective and projective resolutions”). Then there is a projective resolution of 𝑀₂

. . . //𝐹_𝑡⊕ 𝐺_𝑡 ^𝑑𝑡 // . . . ^𝑑1 //𝐹₀⊕ 𝐺₀ ^𝑑0 //𝑀₂ //0

is commutative with exact rows and columns (where, for any 𝑡, the map 𝑖_𝑡is the obvi-ous injection and the map 𝑝_𝑡is the obvious projection).

The lemma takes its name from the shape of the given part of the diagram (the short exact sequence, and the given projective resolutions).

An analogous statement holds in case we have an injective resolution of 𝑀₁and an injective resolution of 𝑀₃.

Hurwitz’s theorem.

^See“Riemann surfaces (compact -) and algebraic curves”.

Hypercohomology of a complex of sheaves.

([79], [93]). We follow strict-ly [93].

Hyperplane bundles, twisting sheaves | 93 Let

⋅ ⋅ ⋅ 󳨀→^{F𝑝 𝑑}󳨀→^F𝑝+1󳨀→ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

be a complex of sheaves of Abelian groups on a topological space 𝑋 (see“Sheaves”).

We denote it by^F∗.

The cohomology sheaf^H𝑞(^F∗) is the sheaf on 𝑋 associated to the presheaf

𝑈 󳨃󳨀→ 𝐾𝑒𝑟 (^F𝑞(𝑈)󳨀→^{𝑑 F𝑞+1}(𝑈)) 𝐼𝑚 (^F𝑞−1(𝑈)󳨀→^{𝑑 F𝑞}(𝑈))

for any 𝑈 open subset of 𝑋. Let^U = {𝑈_𝛼}_𝛼∈𝐴be an open covering of 𝑋. Consider the bigraded complex 𝐶^𝑝(^U,^F𝑞) (see“Sheaves”and in particular the cohomology of the sheaves for the notation) with the differentials

𝛿 : 𝐶^𝑝(^U,^F𝑞) 󳨀→ 𝐶^𝑝+1(^U,^F𝑞), 𝑑 : 𝐶^𝑝(^U,^F𝑞) 󳨀→ 𝐶^𝑝(^U,^F𝑞+1),

where 𝑑 is induced by the differential of^F∗. Let 𝐶^∗(^U) be the associated single complex given by 𝐶^𝑛(^U) = ⊕_{𝑝+𝑞=𝑛}𝐶^𝑝(^U,^F𝑞) with differential 𝐷 = 𝑑 + 𝛿.

The hypercohomology of the complex of sheaves^F∗is defined to be the limit on^Uof the cohomology of the complex 𝐶^∗(^U), where the set of the open coverings is partially ordered by refinement (see“Limits, direct and inverse -”):

IHI^𝑞(𝑋,^F∗) = lim

→U

𝐻^𝑞(𝐶^∗(^U)) for any 𝑞.

By considering the two filtrations of 𝐶^∗Uand by passing to the limit for^U, we get two spectral sequences (see“Spectral sequences”) 𝐸_𝑟and 𝐸^󸀠_𝑟with 𝐸^𝑝,𝑞₂ ≅ 𝐻^𝑝(𝑋,^H𝑞(^F∗)) and 𝐸^󸀠₂^𝑝,𝑞≅ 𝐻_𝑑^𝑞(𝐻^𝑝(𝑋,^F∗)).

Proposition. If a map between complexes of sheaves induces an isomorphism on co-homology sheaves, then it also induces an isomorphism on hypercoco-homology.

Hyperelliptic Riemann surfaces.

^See“Riemann surfaces (compact -) and al-gebraic curves”.

Hyperelliptic or bielliptic surfaces.

^See“Surfaces, algebraic -”.

Hyperplane bundles, twisting sheaves.

([93], [107], [129]). Let 𝐾 be a field and 𝑠 ∈ ℤ. The line bundle 𝐻_𝑠on ℙ^𝑛_𝐾is defined in the following way: let the total space be

((𝐾^𝑛+1− {0}) × 𝐾)/ ∼, where ∼ is the following equivalence relation:

(𝑥₀, . . . , 𝑥_𝑛, 𝑡) ∼ (𝜆𝑥₀, . . . , 𝜆𝑥_𝑛, 𝜆^𝑠𝑡) ∀𝜆 ∈ 𝐾 − {0};

94 | Hyperplane bundles, twisting sheaves the projection

𝜋 : 𝐻_𝑠→ ℙ^𝑛_𝐾 is given by

𝜋([(𝑥₀, . . . , 𝑥_𝑛, 𝑡)]) = [𝑥₀:⋅ ⋅ ⋅:𝑥_𝑛].

We can take as trivializing subsets for 𝐻_𝑠the sets

𝑈_𝑖= {[𝑥₀:⋅ ⋅ ⋅:𝑥_𝑛] ∈ ℙ^𝑛_𝐾| 𝑥_𝑖 ̸= 0}

for 𝑖 = 0, . . . , 𝑛, and the trivializing function 𝜙_𝑖: 𝑈_𝑖× 𝐾 → 𝐻_𝑠|_𝑈𝑖is 𝜙_𝑖([𝑥₀:⋅ ⋅ ⋅:𝑥_𝑛], 𝑡) = [(𝑥₀, . . . , 𝑥_𝑛, 𝑥^𝑠_𝑖𝑡)].

Thus the transition functions are

𝑓_𝑖,𝑗([𝑥₀:⋅ ⋅ ⋅:𝑥_𝑛]) = 𝑥^−𝑠_𝑖 𝑥^𝑠_𝑗.

The bundle 𝐻₁is called a hyperplane bundle (since it is associated to the divisor given by a hyperplane) and is the dual of the universal bundle (see“Tautological (or univer-sal) bundle”). Obviously 𝐻_𝑠 = 𝐻^⊗𝑠. The sheaf^O(𝑠) (or, more precisely,^O_ℙ^𝑛_𝐾(𝑠)) is de-fined to be the sheaf of the sections of 𝐻_𝑠. Sometimes^O(𝑠) denotes also the line bundle itself. The sheaves^O(𝑠) are called Serre’s twisting sheaves.

For any sheaf^Eon ℙ^𝑛_𝐾, we denote^E⊗^O(𝑠) by^E(𝑠).

A more general way to define the hyperplane bundles and the twisting sheaves is the following one.

Let 𝑅 be a graded commutative ring with unity. For any graded 𝑅-module 𝑀, let^F𝑀

be the following sheaf on the scheme 𝑃𝑟𝑜𝑗(𝑅) (see“Schemes”): for every 𝑝 ∈ 𝑃𝑟𝑜𝑗(𝑅), let 𝑀(𝑝)be the group of the elements of degree 0 in the localization of 𝑀 with respect to the multiplicative system of the homogeneous elements in 𝑅 − 𝑝 (see“Localization, quotient ring, quotient field”); for every open subset 𝑈 of 𝑃𝑟𝑜𝑗(𝑅), let^F𝑀(𝑈) be the set of the functions

𝜎 : 𝑈 → ⊔_𝑝∈𝑈𝑀_(𝑝)

such that 𝜎(𝑝) ∈ 𝑀_(𝑝)for every 𝑝 ∈ 𝑈 and 𝜎 is locally a fraction, i.e., for all 𝑝 ∈ 𝑈, there exists a neighborhood 𝑉 of 𝑝 in 𝑈 and homogeneous elements 𝑚 ∈ 𝑀, 𝑟 ∈ 𝑅 of the same degree such that, for all 𝑝^󸀠∈ 𝑉, we have 𝑟 ̸∈ 𝑝^󸀠and 𝜎(𝑝^󸀠) = ^𝑚_𝑟.

For any graded 𝑅-module 𝑀 and 𝑠 ∈ ℤ, let 𝑀(𝑠) be the graded module whose part of degree 𝑑 is 𝑀_𝑑+𝑠(the part of degree 𝑑 + 𝑠 of 𝑀). For any 𝑠 ∈ ℤ, we define the twisting sheaf^O_𝑋(𝑠) on 𝑋 := 𝑃𝑟𝑜𝑗(𝑅) in the following way:

O_𝑋(𝑠) =^F_𝑅(𝑠),

i.e.,^O_𝑋(𝑠) is defined to be the sheaf associated to the 𝑅-module 𝑅(𝑠). We can prove that O_𝑋(𝑠) is locally free (see“Sheaves”) of rank 1 and that, for any 𝑠, 𝑡 ∈ ℤ,

O_𝑋(𝑠) ⊗^O_𝑋(𝑡) =^O_𝑋(𝑡 + 𝑠).

Injective and projective modules | 95 If we take 𝑋 = 𝑃𝑟𝑜𝑗(𝑅) where 𝑅 = 𝐾[𝑥0, . . . , 𝑥_𝑛] for some algebraically closed field 𝐾 (thus 𝑋 is the scheme corresponding to the projective space of dimension 𝑛 over 𝐾), we get the sheaves we have described before.

Theorem.

– Let 𝐴 be a Noetherian graded commutative ring with unity and let 𝑅 = 𝐴[𝑥₀, . . . , 𝑥_𝑛]. Let 𝑋 = 𝑃𝑟𝑜𝑗(𝑅). Denote^O_𝑋(𝑠) by^O(𝑠) for the sake of brevity.

– By Serre duality (see“Serre duality”)

ℎ⁰(𝑋,^O(𝑠)) = ℎ^𝑛(𝑋,^O(−𝑠 − 𝑛 − 1)) .

Theorem. Let 𝐾 be an algebraically closed field. Then any locally free sheaf of rank 1 on the scheme ℙ^𝑛_𝐾is isomorphic to^O(𝑠) for some 𝑠 ∈ ℤ.

I

Injective and projective modules.

([41], [62], [116], [159], [208]). Let 𝑅 be a ring. An module 𝐼 is said to be injective if for every injective morphism of 𝑅-modules 𝑓 : 𝑀 → 𝑁 and every 𝑅-morphism 𝑔 : 𝑀 → 𝐼 there exists an 𝑅-morphism

An 𝑅-module 𝑃 is said to be projective if for every surjective 𝑅-morphism of 𝑅-modules 𝑓 : 𝑀 → 𝑁 and every 𝑅-morphism 𝑔 : 𝑃 → 𝑁 there exists an 𝑅-morphism ℎ : 𝑃 → 𝑀

96 | Injective and projective resolutions

Proposition. Let 𝑅 be a ring with unity and consider only unital modules.

Free 𝑅-modules (i.e., direct sums of copies of 𝑅) are projective.

Any projective module is a direct summand of a free module.

Let 𝐾 be a field. Any finitely generated projective 𝐾[𝑥₀, . . . , 𝑥_𝑛]-module is free (see [159, Chapter IV, Theorem 3.15]).

Injective and projective resolutions.

([41], [62], [116], [208]). Let 𝑅 be a ring.

An injective resolution of an 𝑅-module 𝑀 is an exact sequence of 𝑅-modules 0 → 𝑀 → 𝐼₀→ ⋅ ⋅ ⋅ → 𝐼_𝑘→ ⋅ ⋅ ⋅

with 𝐼_𝑗injective 𝑅-modules.

A projective resolution of an 𝑅-module 𝑀 is an exact sequence of 𝑅-modules

⋅ ⋅ ⋅ → 𝑃_𝑘→ 𝑃_𝑘−1→ ⋅ ⋅ ⋅ → 𝑃₀→ 𝑀 → 0 with 𝑃_𝑗projective 𝑅-modules.

Integrally closed.

([12], [62], [164], [256]). Let 𝐵 be a commutative ring with unity.

Let 𝐴 be a subring of 𝐵 (so the unity belongs to 𝐴). We say that 𝑥 ∈ 𝐵 is integral over 𝐴 if there exists a monic polynomial 𝑝 with coefficients in 𝐴 such that 𝑝(𝑥) = 0.

The integral closure of 𝐴 in 𝐵 is defined to be the set 𝐶 := {𝑥 ∈ 𝐵 | 𝑥 integral over 𝐴}.

If 𝐴 = 𝐶, then we say that 𝐴 is integrally closed in 𝐵. If 𝐵 = 𝐶, then we say that 𝐵 is integral over 𝐴.

We say that a domain 𝐴 is integrally closed if it is integrally closed in its quotient field (see“Localization, quotient ring, quotient field”).

Proposition. ring, quotient field”for the notation 𝑆⁻¹𝐶).

Intersection of cycles | 97 (ii) Let 𝑅 be a commutative ring with unity and let 𝑅^󸀠, 𝑅^󸀠󸀠be two subrings of 𝑅 with 𝑅^󸀠󸀠 ⊂ 𝑅^󸀠⊂ 𝑅. If 𝑅 is integral over 𝑅^󸀠and 𝑅^󸀠is integral over 𝑅^󸀠󸀠, then 𝑅 is integral over 𝑅^󸀠󸀠.

(iii) Let 𝑅 be a domain. Then 𝑅 is integrally closed if and only if 𝑅_𝑝is integrally closed for every prime ideal 𝑝 of 𝑅 and this is true if and only if 𝑅𝑚is integrally closed for every maximal ideal 𝑚 of 𝑅 (see“Localization, quotient ring, quotient field”for the notation 𝑅𝑝and 𝑅𝑚).

(iv) A unique factorization domain is integrally closed, in particular, for any field 𝐾, the ring 𝐾[𝑥1, . . . , 𝑥_𝑛] is integrally closed.

Intersection of cycles.

([72], [74], [93], [107], [214], [227], [228]).

Intersection in topology. We follow mainly [93].

Let 𝑋 be an oriented 𝐶^∞manifold of (real) dimension 𝑛. Let 𝑐₁and 𝑐₂be two singular homology classes of cycles of complementary dimension, precisely,

𝑐₁∈ 𝐻_𝑘(𝑋, ℤ), 𝑐₂∈ 𝐻_𝑛−𝑘(𝑋, ℤ)

(see“Singular homology and cohomology” for the notation). Let 𝛾₁ and 𝛾₂ be two piecewise smooth cycles such that [𝛾₁] = 𝑐₁, [𝛾₂] = 𝑐₂, and 𝛾₁and 𝛾₂intersect trans-versely (i.e., every intersection point 𝑃 is smooth in 𝛾₁and 𝛾₂and 𝑇_𝑃𝑋 = 𝑇_𝑃𝛾₁⊕ 𝑇_𝑃𝛾₂, where 𝑇_𝑃denotes the tangent space at 𝑃). We define

𝑐₁⋅ 𝑐₂= ∑

𝑃∈𝛾1∩𝛾2

𝑖_𝑃(𝛾₁, 𝛾₂),

where 𝑖𝑃(𝛾₁, 𝛾₂) is defined as follows: take a positively oriented basis^B₁of 𝑇𝑃𝛾₁and a positively oriented basis^B₂of 𝑇_𝑃𝛾₂(observe that 𝛾₁and 𝛾₂have natural orientations given respectively by 𝑐1and 𝑐2); if^B₁,^B₂is a positively oriented basis of 𝑇𝑃𝑋, then we define 𝑖_𝑃(𝛾₁, 𝛾₂) to be 1, otherwise we set 𝑖_𝑃(𝛾₁, 𝛾₂) = −1. One can show that we can find 𝛾₁and 𝛾2as above and that 𝑐1⋅ 𝑐₂does not depend on the choice of 𝛾1and 𝛾2.

We can define also the intersection of two singular homology classes of cycles of non-complementary dimension. Precisely, let

𝑐₁∈ 𝐻_𝑛−𝑘1(𝑋, ℤ), 𝑐₂∈ 𝐻_𝑛−𝑘2(𝑋, ℤ),

with 𝑘₁+ 𝑘₂ < 𝑛. Let 𝛾₁and 𝛾₂be two piecewise smooth cycles such that [𝛾₁] = 𝑐₁, [𝛾₂] = 𝑐₂ and 𝛾₁and 𝛾₂intersect transversely almost everywhere (we say that they intersect transversely in a point 𝑃 if 𝑃 is smooth in 𝛾₁and 𝛾₂and 𝑇_𝑃𝛾₁+ 𝑇_𝑃𝛾₂= 𝑇_𝑃𝑋).

We define 𝑐₁⋅ 𝑐₂to be the element of 𝐻_{𝑛 −𝑘1−𝑘2}(𝑋, ℤ) given by 𝛾₁∩ 𝛾₂with the following orientation: if^Bis a positively oriented basis of 𝑇_𝑃(𝛾₁∩𝛾₂), where 𝑃 is a smooth point of 𝛾₁∩𝛾₂, and^B₁,^Bis a positively oriented basis of 𝑇_𝑃𝛾₁and^B,^B₂is a positively oriented basis of 𝑇_𝑃𝛾₂, then^B₁,^B,^B₂is a positively oriented basis of 𝑇_𝑃𝑋.

98 | Inverse image sheaf

One can prove that the intersection of cycles corresponds through the Poincaré duality to the cup product; see“Singular homology and cohomology”.

Intersection in algebraic geometry. Let 𝑋 be a smooth algebraic variety of dimension 𝑛 over an algebraic closed field 𝐾 and let 𝑉₁and 𝑉₂be two subvarieties of 𝑋 of co-dimension 𝑘1and 𝑘2, respectively. Suppose 𝑘1+ 𝑘₂≤ 𝑛. We say that 𝑉₁and 𝑉2intersect properly if the codimension of every irreducible component of 𝑉₁∩𝑉₂is equal to 𝑘₁+𝑘₂. Suppose that 𝑉1and 𝑉2intersect properly. We define the intersection cycle of 𝑉1and 𝑉₂, which we denote by 𝑉₁⋅𝑉₂or simply by 𝑉₁𝑉₂, to be the algebraic cycle (see“Cycles”)

∑

𝑊𝑖_𝑊(𝑉₁, 𝑉₂) 𝑊,

where the sum runs over all the irreducible components of 𝑉₁∩ 𝑉₂and the number 𝑖_𝑊(𝑉₁, 𝑉₂) is the so-called “intersection multiplicity” of 𝑉₁and 𝑉₂along 𝑊. There are several definitions of intersection multiplicity; we report Serre’s definition (see [227]):

if 𝑊 is an irreducible component of 𝑉₁∩ 𝑉₂, 𝑖_𝑊(𝑉₁, 𝑉₂) := ∑

𝑖 (−1)^𝑖𝑙𝑒𝑛𝑔𝑡ℎ (𝑇𝑜𝑟^𝐴_𝑖(𝐴/𝐼₁, 𝐴/𝐼₂)) ,

where 𝐴 is the local ring^O_𝑋,𝑥of 𝑋 at a generic point 𝑥 ∈ 𝑊 and 𝐼_𝑖is the ideal of 𝑉_𝑖in 𝐴 for 𝑖 = 1, 2 (see“Length of a module”,“Tor,^TOR”for the definitions of 𝑙𝑒𝑛𝑔𝑡ℎ and 𝑇𝑜𝑟). Serre showed that the numbers 𝑖_𝑊(𝑉₁, 𝑉₂) are nonnegative.

By linearity we can define the intersection cycle of any two cycles 𝑉1 and 𝑉2when they intersect properly, i.e., when the subvarieties of which 𝑉₁is a linear combination intersect properly the subvarieties of which 𝑉2is a linear combination.

Chow’s moving lemma states that if 𝑋 is a smooth quasi-projective algebraic variety and 𝑉1and 𝑉2are cycles of 𝑋 of codimension 𝑘1and 𝑘2, respectively, we can find a cycle 𝑉₁^󸀠rationally equivalent to 𝑉₁ and such that 𝑉₁^󸀠and 𝑉₂intersect properly (see

“Equivalence, algebraic, rational, linear -, Chow, Neron–Severi and Picard groups”

for the definition of rational equivalence). Moreover, one can show that if 𝑉₁and 𝑉₁^󸀠 are rationally equivalent and intersect properly 𝑉2, then 𝑉1⋅𝑉₂and 𝑉₁^󸀠⋅𝑉₂are rationally equivalent. This allows us to define the intersection of algebraic cycles (also intersect-ing not properly) up to rational equivalence.

Inverse image sheaf.

^See“Direct and inverse image sheaves”.

Irreducible topological space.

We say that a topological space is irreducible if it is not the union of two proper closed subsets.

Irregularity.

([93], [107]). Let 𝑋 be a complex manifold or an algebraic variety over a field 𝐾. The irregularity of 𝑋 is

ℎ¹(𝑋,^O_𝑋),

where^O_𝑋is the sheaf of the holomorphic functions, respectively of the regular func-tions, on 𝑋. The irregularity of 𝑋 is generally denoted by 𝑞(𝑋).

Jacobians of compact Riemann surfaces | 99 If 𝑋 is a compact Kähler manifold (see“Hermitian and Kählerian metrics”), then, by Dolbeault’s theorem and by Hodge’s theorem (see“Dolbeault’s theorem”,“Hodge the-ory”), we have that ℎ¹(𝑋,^O_𝑋) = ℎ^0,1(𝑋) = ℎ^1,0(𝑋) = ℎ⁰(𝑋, 𝛺¹_𝑋) (and somewhere the irregularity is defined to be ℎ^1,0(𝑋)).

If 𝑋 is a smooth projective algebraic surface over an algebraic closed field, then the irregularity is equal to 𝑝_𝑔(𝑋)−𝑝_𝑎(𝑋) (where 𝑝_𝑎(𝑋) and 𝑝_𝑔(𝑋) are respectively the arith-metic and the geometric genus of 𝑋; see“Genus, arithmetic, geometric, real, virtual -”

and“Dualizing sheaf”).

J

Jacobians of compact Riemann surfaces.

([93], [101], [102], [163], [165], [195]). To every compact Riemann surface we can associate a principally polarized Abelian variety, called its Jacobian. Jacobians were the first Abelian varieties to be studied.

Let 𝑋 be a compact Riemann surface of genus 𝑔 (see“Riemann surfaces (com-pact -) and algebraic curves”). By Riemann’s theorem, the complex vector space 𝐻⁰(𝑋,^O(𝐾_𝑋)), where 𝐾_𝑋is the canonical bundle of 𝑋 (see“Canonical bundle, canon-ical sheaf”), has dimension 𝑔. Consider the map

𝑝 : 𝐻₁(𝑋, ℤ) 󳨀→ 𝐻⁰(𝑋,^O(𝐾_𝑋))^∨ defined by

𝑝(𝛾) = ∫

𝛾

. It is injective, and the quotient

𝐻⁰(𝑋,^O(𝐾_𝑋))^∨ 𝑝(𝐻₁(𝑋, ℤ))

is a complex torus of dimension 𝑔 (see“Tori, complex - and Abelian varieties”). The Jacobian of 𝑋 is the complex torus above endowed with the following polarization:

let 𝐸 be the alternating form on 𝐻⁰(𝑋,^O(𝐾_𝑋))^∨obtained extending on ℝ the form on 𝐻₁(𝑋, ℤ) given by the intersection of 1-cycles; let

𝐻 : 𝐻⁰(𝑋,^O(𝐾_𝑋))^∨× 𝐻⁰(𝑋,^O(𝐾_𝑋))^∨󳨀→ ℂ, defined by

𝐻(𝑣, 𝑤) = 𝐸(𝑖𝑣, 𝑤) + 𝑖𝐸(𝑣, 𝑤),

for any 𝑣, 𝑤 ∈ 𝐻⁰(𝑋,^O(𝐾_𝑋))^∨. We endow the complex torus above with the polarization given by 𝐻. One can prove that it is a principal polarization. Thus the Jacobian of 𝑋 is a principally polarized Abelian variety.

The Jacobian of 𝑋 coincides with the Albanese variety of 𝑋 (see“Albanese varieties”).

100 | Jacobians of compact Riemann surfaces

Definition. Let 𝐷𝑖𝑣^𝑑(𝑋) be the set of divisors of degree 𝑑 on 𝑋. The map 𝐷𝑖𝑣⁰(𝑋) 󳨀→ 𝐽(𝑋)

defined by

∑

𝑎=1,...,𝑑

(𝑃_𝑎− 𝑄_𝑎) 󳨃󳨀→ ∑

𝑎=1,...,𝑑 𝑃𝑎

∫

𝑄_𝑎

,

for any 𝑑 ∈ ℕ, 𝑃_𝑎, 𝑄_𝑎 ∈ 𝑋, is called the Abel–Jacobi map of 𝑋 and denoted by 𝜇.

Obviously, if we fix a point 𝑃 on 𝑋, we can also define a map 𝐷𝑖𝑣^𝑑(𝑋) 󳨀→ 𝐽(𝑋)

(again called the Abel–Jacobi map and denoted by 𝜇) by composing the map 𝐷𝑖𝑣^𝑑(𝑋) 󳨀→ 𝐷𝑖𝑣⁰(𝑋),

𝐷 󳨃󳨀→ 𝐷 − 𝑑𝑃 with the Abel–Jacobi map 𝐷𝑖𝑣⁰(𝑋) 󳨀→ 𝐽(𝑋).

Abel–Jacobi theorem. The Abel–Jacobi map defines an isomorphism (again called the Abel–Jacobi map)

𝑃𝑖𝑐⁰(𝑋) 󳨀→ 𝐽(𝑋) .

(See “Equivalence, algebraic, rational, linear -, Chow, Neron–Severi and Picard groups”for the definition of the Picard group 𝑃𝑖𝑐⁰(𝑋); in case 𝑋 is a compact Riemann surface, 𝑃𝑖𝑐⁰(𝑋) is the set of the divisors of degree 0 up to linear equivalence.) Notation. Define 𝑉^𝑑 := 𝜇(𝑋^(𝑑)), where 𝑋^(𝑑)is the symmetric 𝑑-product of 𝑋, i.e., the set of effective divisors of degree 𝑑 on 𝑋.

Proposition. The dimension of 𝑉^𝑑is 𝑑 if 𝑑 ≤ 𝑔, while it is 𝑔 if 𝑑 ≥ 𝑔. If 𝐷 ∈ 𝑋^(𝑑), then 𝜇⁻¹(𝜇(𝐷)) = |𝐷| = ℙ(𝐻⁰(𝑋,^O(𝐷)),

where |𝐷| denotes the linear system of 𝐷 (see“Linear systems”) Let 𝑔 ≥ 1; then the map 𝜇 : 𝑋 󳨀→ 𝐽(𝑋) is injective and the map 𝜇 : 𝑋^(𝑑)󳨀→ 𝐽(𝑋) is generically injective for 𝑑 ≤ 𝑔.

Theorem. Let 𝛩 be the divisor in 𝐽(𝑋) associated to a section of a holomorphic line bundle on 𝐽(𝑋) defining the polarization. We have that the intersection number of 𝑉¹ and 𝛩 is 𝑔:

𝑉¹⋅ 𝛩 = 𝑔.

In addition, 𝛩 is a translate of the image through the Abel–Jacobi map of the set of the

Jacobians, Weil and Griffiths intermediate - | 101

effective divisors of degree 𝑔 − 1 on 𝑋, i.e., 𝛩 = 𝑉^𝑔−1+ 𝐾, for some 𝐾 ∈ 𝐽(𝑋).

Torelli’s theorem. The map from the set of compact Riemann surfaces up to isomor-phisms to the set of the principally polarized Abelian varieties up to isomorisomor-phisms, associating to every Riemann surface 𝑋 its Jacobian 𝐽(𝑋) is injective.

Poincaré’s formula. Let [𝑉^𝑑] be the class of 𝑉^𝑑 in the singular homology group 𝐻_{2𝑔−2𝑑}(𝐽(𝑋), ℤ). We have that

[𝑉^𝑑] = [𝛩](𝑔 − 𝑑)!^𝑔−𝑑,

where 𝛩 is the divisor associated to a section of a holomorphic line bundle defining the polarization on 𝐽(𝑋).

Jacobians, Weil and Griffiths intermediate -.

([92], [93], [166], [250]). We follow mainly [166].

Let 𝑋 be a compact Kähler manifold of dimension 𝑛 (see“Hermitian and Kählerian metrics”). Let 1 ≤ 𝑞 ≤ 𝑛. The Griffiths 𝑞-th intermediate Jacobian of 𝑋 is the complex torus (see“Tori, complex - and Abelian varieties”)

𝐽_𝑞(𝑋) = ⊕𝑖=𝑞,...,2𝑞−1𝐻^{2𝑞−1−𝑖,𝑖}(𝑋, ℂ) 𝑝(𝐻^2𝑞−1(𝑋, ℤ)) ,

where 𝑝 : 𝐻^2𝑞−1(𝑋, ℤ) → ⊕𝑖=𝑞,...,2𝑞−1𝐻^{2𝑞−1−𝑖,𝑖}(𝑋, ℂ) is given by the composition of the canonical map 𝑢 : 𝐻^2𝑞−1(𝑋, ℤ) → 𝐻^2𝑞−1(𝑋, ℂ) with the projection 𝐻^2𝑞−1(𝑋, ℂ) →

⊕𝑖=𝑞,...,2𝑞−1𝐻^{2𝑞−1−𝑖,𝑖}(𝑋, ℂ) induced by the Hodge decomposition (see“Hodge theory”and

“Singular homology and cohomology”, in particular the Universal Coefficient Theo-rem). If 𝑞 = 1 we have the Picard variety of 𝑋 (see“Equivalence, algebraic, rational, linear -, Chow, Neron–Severi and Picard groups”):

𝐽₁(𝑋) = 𝐻¹(𝑋,^O)

𝑝(𝐻¹(𝑋, ℤ)) = 𝑃𝑖𝑐⁰(𝑋).

If 𝑞 = 𝑛 we have the Albanese variety of 𝑋 (see“Albanese varieties”):

𝐽_𝑛(𝑋) = 𝐻^{𝑛−1,𝑛}(𝑋, ℂ)

𝑝(𝐻^2𝑛−1(𝑋, ℤ))≅ 𝐻^1,0(𝑋, ℂ)^∨

𝑗(𝐻₁(𝑋, ℤ)) ≅ 𝐻⁰(𝑋, 𝛺¹)^∨

𝑗(𝐻₁(𝑋, ℤ)) = 𝐴𝑙𝑏(𝑋),

where 𝑗 is the map 𝛾 󳨃→ ∫_𝛾(see“Dolbeault’s theorem”,“Hodge theory”,“Serre duality”

to understand the isomorphisms above).

We have

𝑃𝑖𝑐⁰(𝐴𝑙𝑏(𝑋)) = 𝐻^0,1(𝐴𝑙𝑏(𝑋), ℂ)

𝑝(𝐻¹(𝐴𝑙𝑏(𝑋), ℤ)) ≅ 𝐻𝑜𝑚_ℂ(𝐻^1,0(𝑋, ℂ), ℂ) 𝑝(𝐻¹(𝑋, ℤ))

≅ 𝐻¹(𝑋,^O)

𝑝(𝐻¹(𝑋, ℤ)) = 𝑃𝑖𝑐⁰(𝑋),

102 | Jacobians, Weil and Griffiths intermediate

-and analogously we also have an isomorphism 𝐴𝑙𝑏(𝑃𝑖𝑐⁰(𝑋)) ≅ 𝐴𝑙𝑏(𝑋). So 𝑃𝑖𝑐⁰(𝑋) and 𝐴𝑙𝑏(𝑋) are dual complex tori (see“Tori, complex - and Abelian varieties”). More gen-erally, we can prove that 𝐽𝑞(𝑋) and 𝐽_{𝑛−𝑞+1}(𝑋) are dual complex tori.

Now let 𝑋 be a smooth complex projective algebraic variety, and let 𝜔 be the Fubini-Study form restricted to 𝑋 (thus 𝜔 is a closed positive integer (1, 1)-form).

If 2𝑞 − 1 ≤ 𝑛, we can consider on 𝐽_𝑞(𝑋) the polarization with index defined by the following Hermitian form (if 2𝑞 − 1 > 𝑛, we define the polarization with index as the dual polarization with index of the one on 𝐽_{𝑛−𝑞+1}(𝑋)):

𝐻(𝜑, 𝜓) = 2𝑖(−1)^𝑞∫

𝑋

𝜔^{𝑛−2𝑞+1}∧ 𝜑 ∧ 𝜓;

the form 𝐻 is Hermitian; in fact it is ℂ-linear in the first variable and 𝐻(𝜓, 𝜑) = 2𝑖(−1)^𝑞∫

In general it is not positive definite, so in general the Griffiths intermediate Jacobian is not an Abelian variety; the Albanese variety and the Picard variety (of a smooth complex projective algebraic variety) are Abelian varieties.

Let 1 ≤ 𝑞 ≤ 𝑛. Let 𝐶 : 𝐻^𝑞(𝑋, ℂ) → 𝐻^𝑞(𝑋, ℂ) be the linear operator defined to be the multiplication by 𝑖^𝑎−𝑏 on 𝐻^𝑎,𝑏(𝑋); it takes 𝐻^𝑞(𝑋, ℝ) to 𝐻^𝑞(𝑋, ℝ) and, if 𝑞 is odd, we have 𝐶² = −1. The operator 𝐶 is called Weil’s operator. The 𝑞-th Weil intermediate Jacobian of 𝑋 is the following torus:

𝑊_𝑞(𝑋) = 𝐻^{2𝑞 −1}(𝑋, ℝ) 𝑢(𝐻^{2𝑞 −1}(𝑋, ℤ)),

with the complex structure given by −𝐶; here 𝑢 : 𝐻^{2𝑞 −1}(𝑋, ℤ) → 𝐻^{2𝑞 −1}(𝑋, ℝ) is the canonical map; its image is isomorphic to the free part of 𝐻^{2𝑞 −1}(𝑋, ℤ). We can prove that 𝑊_𝑞(𝑋) and 𝑊_{𝑛−𝑞+1}(𝑋) are dual complex tori.

Jumping lines and splitting type of a vector bundle on ℙ^𝑛 | 103 For 2𝑞 − 1 > 𝑛, we define the polarization on 𝑊𝑞(𝑋) to be the dual polarization of the one on 𝑊_{𝑛−𝑞+1}(𝑋).

One can prove that the Weil intermediate Jacobian is an Abelian variety (with the po-larization we have just defined).

As we have already said, Griffiths intermediate Jacobian is not an Abelian variety in general. The advantage of Griffiths Jacobian with respect to Weil Jacobian is that Grif-fiths Jacobian varies holomorphically in a family of smooth projective algebraic vari-eties (while Weil Jacobian does not).

See also“Jacobians of compact Riemann surfaces”.

In document Algebraic Geometry - A Concise Dictionary (gnv64).pdf (Page 102-113)