The cotangent bundle to T2 is (mathbbCast)2. But a theorem of Totaro (Internat. J. Math. 2 (1991), 563-566; MR1124283) implies that, if S is a closed orientable surface of negative Euler characteristic, there is no diffeomorphism, sending the symplectic orientation to the complex orientation, from T∗S to a smooth affine complex surface.
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