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c © 2005 International Press Vol. 9, No. 3, pp. 407–448, September 2005 007 GEOGRAPHY AND THE NUMBER OF MODULI OF SURFACES OF GENERAL TYPE∗ IGOR REIDER To the memory of Andrei Nikolaevich Tyurin Abstract. The paper considers a relationship between the Chern numbers K2 X , c2(X) of a smooth minimal surface X of general type and the dimension of the space of infinitesimal deformations of X, i.e. h(ΘX), where ΘX is the holomorphic tangent bundle of X. We prove that if the ratio of the Chern numbers α(X) = c2(X) K2 X ≤ 3 8 and KX is ample then h(ΘX) ≤ 9(3c2 − K ). On the geometric side it is shown that a smooth surface of general type X with α(X) ≤ 3 8 and h(ΘX) ≥ 3 has two distinguished effective divisors F and E such that H (ΘX) admits a direct sum decomposition H(ΘX) = V1 ⊕ V0, where V1 is identified with a subspace of H (OX(F )) while V0 is identified with a subspace of H(ΘX ⊗ OE(E)). This gives a geometric interpretation of the cohomology classes in H(ΘX) and allows to bound the dimension of V0 (resp. V1) in terms of geometry of the divisor E (resp. F ). The main idea of the paper is to use the natural identification H(ΘX ) = Ext (ΩX ,OX) where ΩX is the holomorphic cotangent bundle of X. Then the ”universal” extension gives rise to a certain vector bundle whose study constitutes the essential part of the paper.

Geography and the Number of Moduli of Surfaces of General Type