COCOMPACT-ARITHMETIC-SUBGROUPS-OF-PU(n-1,-1)-WITH-EULER-POINCAR′E-CHARACTERISTIC-n-AND-A-FAKE-P-4.pdfVIP

COCOMPACT ARITHMETIC SUBGROUPS OF PU(n − 1, 1) WITH ´ 4 EULER-POINCARE CHARACTERISTIC n AND A FAKE P Gopal Prasad and Sai Kee Yeung 1. Introduction The purpose of this paper is to determine cocompact arithmetic subgroups of the real Lie group PU(n − 1, 1), for n 3, whose orbifold Euler-Poincar´e characteristic (i. e., the Euler-Poincar´e characteristic in the sense of C.T.C. Wall, see [Se]) is n (= χ(Pn−1)). We C will prove that such an arithmetic subgroup exists only if n = 5. We will say that a smooth complex projective algebraic variety V (of dimension n − 1) is a fake Pn−1 if V is the quotient of the unit ball Bn−1 in Cn−1by a torsion-free cocompact discrete subgroup of PU(n − 1, 1), and all the Betti numbers of V are equal to those of Pn−1. C If the fundamental group of a fake Pn−1 is an arithmetic subgroup of PU(n − 1, 1), then we will say that it is an arithmetic fake Pn−1 . It follows from the results of this paper that arithmetic fake Pn−1 can exist only if n = 3 or 5. We have shown in [PY] that there are at least twelve distinct classes of fake P2 . We will show here that there exists an arithmetic fake P4 . It is an immediate consequence of the Hirzebruch proportionality principle, see [Se], Proposition 23, that the Euler-Poincar´e characteristic of any arithmetic subgroup of PU(n − 1, 1), for n even, is negative. Therefore, we may (and we will) assume in the sequel that n is an odd integer 5. Let us a