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Unformatted text preview: ume. The root vector corresponding to the
extreme right vertex is equal to the vector r + e, where r generates the kernel of
the lattice E2,3,6 embedded naturally in E2,3,8 and the vector e is the root vector
corresponding to the extreme right vector in the subdiagram deﬁning the sublattice
isomorphic to E2,3,7 . 24 IGOR V. DOLGACHEV 5. Automorphisms of algebraic surfaces
5.1. Quadratic lattices associated to an algebraic surface. Let X be a complex projective algebraic surface. It has the underlying structure of a compact
smooth oriented 4-manifold. Thus the cohomology group H 2 (X, Z) is a ﬁnitely
generated abelian group equipped with a symmetric bilinear form
H 2 (X, Z) × H 2 (X, Z) → Z
deﬁned by the cup-product.
When we divide H 2 (X, Z) by the torsion subgroup we obtain a unimodular
quadratic lattice HX . We will denote the value of the bilinear form on HX induced
by the cup product by x · y and write x2 if x = y .
To compute its signature one uses the Hodge decomposition (depending on the
complex structure of X )
H 2 (X, C) = H 2,0 (X ) ⊕ H 1,1 (X, C) ⊕ H 0,2 (X ),
where dim H 2,0 = dim H 0,2 and is equal to the dimension pg (X ) of the space
of holomorphic diﬀerential 2-forms on X . It is known that under the complex
conjugation on H 2 (X, C) the space H 1,1 (X ) is invariant and the space H 2,0 (X )
is mapped to H 0,2 (X ), and vice versa. One can also compute the restriction of
the cup-product on H 2 (X, C) to each H p,q to conclude that the signature of the
cup-product on H 2 (X, R) is equal to (b+ , b− ), where b+ = 2pg + 1.
The parity of the lattice HX depends on the property of its ﬁrst Chern class
c1 (X ) ∈ H 2 (X, Z). If is divisible by 2 in H 2 (X, Z), then HX is an even lattice.
The lattice HX contains an important primitive sublattice which depends on the
complex structure of X . For any complex irreducible curve C on X its fundamental
class [C ] deﬁnes a cohomology class in H 2 (X, Z). The Z-span of these classes deﬁnes
a subgroup of H 2 (X, Z), and its image SX in HX is called the Neron-Severi lattice
(or the Picard lattice ) of X . It is a sublattice of HX of signature (1, ρ − 1), where
ρ = rank SX . An example of an element of positive norm is the class of a hyperplane
section of X in any projective embedding of X . Thus the lattice SX is hyperbolic
in the sense of previous sections, and we can apply the theory of reﬂection groups
to SX .
It is known that the image in HX of the ﬁrst Chern class c1 (X ) ∈ H 2 (X, Z)
belongs to the Picard lattice. The negative KX = −c1 (X ) is the canonical class
of X . It is equal to the image in H 2 (X, Z) of a divisor of zeros and poles of a
holomorphic diﬀerential 2-form on X . We denote the image of KX in HX by kX .
For any x ∈ SX we deﬁne
pa (x) = x2 + x · kX .
It is always an even integer. If x is the image in HX of the fundamental class of a
nonsingular complex curve C on X , then pa (x) is equal to 2g −2, where g is the genus
of the Riemann surface C (the adjunction formula ). If C is an irreducible complex
curve with ﬁn...
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