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In this paper we address the problem of generic vanishing for (negative) harmonic spinors of Dirac operators coupled with variable metric connections. 0. Introduction Let M be a compact spin manifold with spinor bundle S and Dirac operator 6∂. In [L] Lichnerowicz proved a very strong vanishing theorem: 6∂ has no harmonic spinors (dim ker 6∂ = 0), if the scalar curvature of M is non-negative and does not vanish identically (see [LM] for complete details). When M is even-dimensional, S splits into a direct sum of positive and negative spinors, S = S+⊕S−. In this case it would be nice to have a less restrictive Lichnerowicz-type theorem, ensuring the vanishing of, say, only the negative harmonic spinors. To our knowledge, such a theorem does not exist for arbitrary M . There is evidence, though, that it should (cf. [CM], where M is a homogeneous space). On a related note, in [S] Singer stressed the importance of vanishing of negative harmonic spinors in particle physics. He claimed, without proof, that for twisted Dirac operators 6∂A, obtained by coupling (S, 6∂) with a fixed Hermitian vector bundle E equipped with a variable metric connection A, this vanishing is generic in A. Here, of course, index(6∂+A), which is independent of A, is assumed to be non-negative. In this paper we address the problem of generic vanishing for (negative) harmonic spinors for Dirac operators coupled with variable metric connections. Our treatment is elementary and direct, avoiding the customary use of the Sard-Smale theorem. In fact, we feel that, for the problem at hand, this direct way is better suited and definitely more revealing. First we develop a general tool for analyzing minA∈C dim ker(D + A), where {D +A}A∈C is a certain family of unbounded Fredholm operators on an arbitrary Hilbert space (Theorems 1.1, 1.2, and 1.3). Next we use this tool to address the generic vanishing problem stated above, when either dimM ≤ 4 (Theorem 1.5) or Received by the editors May 9, 1995. 1991 Mathematics Subject Classification. Primary 58G25; Secondary 47A53, 53C05.

Generic vanishing for harmonic spinors of twisted Dirac operators