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§9. Numerical Invariants of Surfaces 375

(1) Let X be a curve. There is a differential d 1 : E p,q → E p+1,q and the possible
1
nonzero groups are:
1
1
1
H (X, O X ) −−−−→ H (X, ) = k
X
1 1 1
k = H (X, O X ) −−−−→ H (X, )
X
Here, g(X) = g = h 1,0 = h 0,1 is the genus of the curve X.
p,q p+1,q
(2) Let X be a surface. There is a differential d 1 : E → E and the possible
1 1
nonzero groups are:
1
2
2
2
2
H (X, O X ) −−−−→ H (X, ) −−−−→ H (X, ) = k
X X
1
2
1
1
1
H (X, O X ) −−−−→ H (X, ) −−−−→ H (X, )
X X
0 0 1 0 2
k = H (X, O X ) −−−−→ H (X, ) −−−−→ H (X, )
X X
1
2
Here q = dimH (X, O x ) is the irregularity of X,and p g = dimH (X, O X ) is the
geometric genus of the surface X.
p,q
As usual, the Betti numbers and Hodge numbers are related by b i = h
p+q=i
i
for the cohomology H (X, C) has a decreasing filtration
p
q
p
i
F H (X, C) with H (X, )
X
p
isomorphic to the quotient F H p+q (X, C)/F p+1 H p+q (X, C.
(9.2) Riemann–Roch. Riemann–Roch theorems have to do with the calculation of
the Euler–Poincar´ e characteristic
n
i i
χ(L) = (−1) dim k H (X, L).
i=0
There are two parts, firstly, a relation between χ(L) or χ(O(D)) for a divisor D and
χ(O X ) and, secondly, a relation between χ(O X ) and Chern classes of the tangent
bundle.
For curves: χ(L) = χ(O X ) + deg(L) for a line bundle L,and
χ(O(D)) = deg(D) + χ(O X ) = deg(D) + 1 − g(X)
1
for a divisor D on the curve X.Here g = g(X) = dimH (X, O X ) is the genus of
the curve X.
Moreover, χ(O X ) = (1/2)c 1 = 1 − g(X) where c 1 = c 1 (T X )[X]. This implies
the integrality assertion: c 1 is always an even integer.

For surfaces: χ(L) = χ(O X ) + (1/2)L · L ⊗ ω (−1)⊗ for a line bundle L,and
X