Non-Hermitian Yang-Mills bundles

This couple of months, we were working with Dima Kaledin on extension of Simpson's theory to the bundles with arbitrary Chern class. This work is based on the analogy between flat unitary bundles and holomorphic bundles with Hermitian Yang-Mills connection. Namely, when first Chern classes vanishes, every connection which is Hermitian Yang-Mills is also flat (and vice versa, of course). Hence it seems plausible that a big part of Simpson's theory is extended to Hermitian Yang-Mills bundles of arbitrary Chern class.

The natural analogue of (not necessary Hermitian) flat bundles is then the non-Hermitian Yang-Mills bundles (NHYM), that is, not necessary Hermitian bundles with (1,1)-curvature $\Theta$ such that $\Lambda \Theta=0$ (or constant times identity, but we are mostly working with $deg B =0$).

We found that a big part of Simpson's theory is indeed extended to this case. Namely, there exist a hyperkaehler structure on the moduli of stable NHYM, preferred metric (which is analogue of the Corlette-Simpson's harmonic metric). The projection to the (1,0)-part of the connection gives a Lagrangean fibration, with the the base moduli of semistable holomorphic bundles. We are working further in this direction, because our picture seems to be much more general than Simpson's, but also beautiful. Unfortunately, almost none of Simpson's arguments is translated directly.

This is mostly interesting when the base manifold $M$ is hyperkaehler. Then NHYM is in many cases equivalent (and always implied by) the following property:

(*): The curvature $\Theta$ is invariant under the natural action of $SU(2)$ on 2-forms on a hyperkaehler manifold.

Also, set of NHYM satisfying (*) is open and closed in the space of all NHYM. Connection satisfying (*) is called autodual (in fact, in 4 real dimensions, (*) is equivalent to anti-autoduality and to NHYM). The results for NHYM are carried over to autodual connections.

The deformation space $X$ of a stable holomorphic bundle $B$ on $M$ is naturally hyperkaehler when the first two Chern classes of $B$ are $SU(2)$-invariant (my article of 1991, Hyperholomorphic Bundles over a Hyperkaehler Manifold). We prove that moduli of autodual connections on $B$ are identified with the space $Sec(X)$ of sections of the twistor projection $Tw(X) \arrow CP^1$, where $Tw(X)$ is a twistor space for $X$.

This is very interesting because $Sec(X)$ is a very important space which we are going to study further (e. g, there is a natural plurisubharmonic function on $Sec(X)$, and it would have been very nice to show it is exhausting). Also, every component of the Chow scheme for twistor space $Tw(M)$ for cycles which are flat over $CP^1$ is interpreted as $Sec(M')$ for appropriate hyperkaehler variety $M'$, hence the study of Chow scheme for twistors is basically reduced to the study of $Sec(M)$, for various $M$.

Misha.