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One of the main results is a precise vanishing criterion for $H^1(X,\\Theta_X (-K_X))$.\n  The proof is based on the geometric interpretation of non-zero cohomology classes of $H^1(X,\\Theta_X (-K_X))$. This interpretation in turn uses higher rank vector bundles on $X$.\n  We apply our methods to the long standing conjecture saying that the irregularity of surfaces in $\\PP^4$ is at most 2. 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