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376 19. Higher Dimensional Analogs of Elliptic Curves: Calabi–Yau Varieties

1 1
χ(O(D)) = D · (D − K X ) + χ(O X ) = D · (D − K X ) + 1 − q + p g
2 2
1
for a divisor D on the surface X.Here q = dimH (x, O X ) is the irregularity and
2
p g = dimH (X, O X ) is the geometric genus of the surface X.
2
Moreover, χ(O X ) = (1/12)(c + c 2 ) = 1 − q + p g where c i = c i (T X )[X] for
1
2
i = 1, 2. This implies the integrality assertion: c + c 2 is always an integer divisible
1
by 12.
(9.3) Role of Serre Duality. Recall that Serre duality is a nondegenerate pairing

i n−i
H (X, L) × H X,ω X ⊗ L (−1)⊗ → k
In particular, the two cohomology vector spaces have the same dimension. For divi-
sors, this takes the form
i
dimH (X, O(D)) = dimH n−i (X, O(K − D)).
In the special case of curves, this leads directly to
1
0
deg(K) = 2g − 2and g = dimH (O X ) = dimH (ω X ).
As an application of the Riemann–Roch formula, Grothendieck gave an algebraic
proof of the following theorem.
(9.4) Theorem (Algebraic Index Theorem). On R ⊗ NS(X), the intersection form
is of signature (1,ρ(X) − 1).
−
+
(9.5) Index Theorem for Complex Surfaces. Let σ = b −b be the signature or
2
index of the cup product quadratic form on H (X, R).
1 1
+ − 2 2
σ = b − b = (c − 2c 2 ) = (K − 2e).
1
3 3
For nonalgebraic complex surfaces this and the Riemann–Roch formula are
proved by applying the Atiyah–Singer index formula.
(9.6) Genus Formula. Let D > 0 be a divisor, and let O D be the structure sheaf on
the scheme D defined by the ideal sheaf J D = O(−D). From the exact sequence
0 → J D = O(−D) → O X → O D → 0.
we have the Euler–Poincar´ e characteristic relation
χ(O D ) = χ(O X ) − χ(O(−D))

which by Riemann–Roch is the genus formula
1
χ(O D ) =− D · (D + K).
2
For an irreducible curve C we recover the usual genus formula

2
2g(C) − 2 = C · K + C .