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Let $M$ be a closed spin$^c$ $4$-manifold with determinant line bundle $L$.

If $c_1^2(L)=2\chi(M)+3\tau(M)$, where $\chi$ and $\tau$ denote the Euler characteristic and signature of $M$ respectively, can we prove that the spin$^c$ structure is induced by some almost complex structure $J$ on $M$?

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    $\begingroup$ Is $M$ a 4-manifold, perhaps? $\endgroup$
    – Marco Golla
    Commented Feb 22 at 15:32
  • $\begingroup$ @MarcoGolla Yes. $\endgroup$
    – Zhiqiang
    Commented Feb 23 at 1:19

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Briefly, let $W^+$ denote the positive spin bundle of the given $Spin_c$- structure so that $det(W^+)=L$. Then the condition

$ c_1^2(L)=2\tau(M)+3\sigma(M) $

is equivalent, using a little characteristic class magic, to

$ c_2(W^+)=0. $

The latter means (on a 4-manifold) that $W^+$ admits a non-vanishing section $\Phi$ which WLOG has unit length. Clifford multiplication against this $\Phi$ induces a real isomorphism

$ \theta \in T^{*}M \mapsto c(\theta)\Phi \in W^- $

so the cotangent bundle inherits being a complex vector bundle from the complex vector bundle $W^-$ and hence $M$ an almost complex structure.

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    $\begingroup$ ^An almost complex structure on a Riemannian manifold is (by definition) a complex structure on its (co)tangent bundle, so $W^-$ complex $\Rightarrow$ $T^*M$ complex $\Rightarrow$ $M$ almost-complex. (Your last sentence parses as: $TT^*M$ complex.) Probably just semantics, since by "complex structure on bundle" I mean fiberwise (i.e. complex bundle) not the bundle as a manifold. $\endgroup$
    – Chris Gerig
    Commented Feb 24 at 2:06
  • $\begingroup$ Thanks Chris. Edited the answer a bit to take account of your comment. $\endgroup$
    – Tom Mrowka
    Commented Feb 25 at 12:10

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