Papers by M. Verbitsky
Recent papers at arxiv.org.
This page
last updated: 1996.
Three latest papers (in LaTeX 2e):
In the paper
where trianalytic subvarieties were first introduced, we proved that
a hyperkaehler manifold,
if ``generic'' enough, might have only those complex subvarieties
which are trianalytic  i. e., complex analytic with respect to
each induced complex structure. Here we study
the complex analytic deformations of trianalytic
subvarieties. We prove that all deformations of such X
are trianalytic and naturally isomorphic to
X as complex analytic varieties. We show also
that this isomorphism is compatible with the metric
induced from the ambient manifold. Also, we prove that
the Douady space of complex analytic deformations of
X inside its ambient manifold is equipped with a natural
hyperk\"ahler structure. There is some discussion
of singular hyperkaehler varietieis. A sample definition is given;
trianalytic subvarieties and the deformation spaces of holomorphic
bundles (as discussed in my paper ``Hyperholomorphic
bundles'') all have
this ``singular hyperkaehler'' structure. We show that the natural
isomorphism of deformations of the trianalytic subvarieties
(the isomorphism
discussed above) is compatible with the singular hyperkaehler
structure.
 From abstract:

Let M be a compact hyperkaehler manifold.
The hyperkaehler structure equips M with a
set R of complex structures parametrized by
$CP^1$, called "the set of induced complex
structures". It was known previously that
induced complex structures are nonalgebraic, except
may be a countable set. We prove that a
countable set of induced complex structures is
algebraic, and this set is dense in R.
A more general version of this theorem was proven by
Fujiki.
 A note

The paper is 5 pages long, and consists of a rather elementary proof used
in the early version on the paper
about deformations of trianalytic cycles. After this early version
was demolished, I did not use the proof anymore, but it was aesthetically
pleasing anyway. So I put it down as a separate paper. After
I sent it to alggeom I learned that my result was superseded by an
180pages long work of Fujiki (1984). What a mess.
 Published in:
 Math. Res. Lett. 3 763767 (1996)
A paper which I wrote with my friend and classmate (Moscow school #91)
Dima Kaledin
(Ph. D. 1995, MIT). I like it very much and appreciate any feedback.
Here is the abstract:

We study YangMills connections on holomorphic bundles
over complex K\"ahler manifolds of arbitrary dimension, in
the spirit of Hitchin's and Simpson's study of flat
connections. The space of nonHermitian YangMills
(NHYM) connections has dimension twice the space of
Hermitian YangMills connections, and is locally isomorphic
to the complexification of the space of Hermitian
YangMills connections (which is, by Uhlenbeck and Yau,
the same as the space of stable bundles). Further, we study
the NHYM connections over hyperk\"ahler manifolds. We
construct direct and inverse twistor transform from NHYM
bundles on a hyperk\"ahler manifold to holomorphic bundles
over its twistor space. We study the stability and the
modular properties of holomorphic bundles over twistor
spaces, and prove that work of Li and Yau, giving the
notion of stability for bundles over nonK\"ahler manifolds,
can be applied to the twistors. We identify locally the
following two spaces: the space of stable holomorphic
bundles on a twistor space of a hyperk\"ahler manifold and
the space of rational curves in the twistor space of the
``Mukai dual'' hyperk\"ahler manifold.
I wrote a letter
(to A. Beilinson, V. Ginzburg, D. Kazhdan, T. Pantev and Carlos Simpson)
explaining the matters in more detail. Recommended
Hyperkaehler geometry
The articles in Latex 2e format.
We prove Mirror Conjecture for
CalabiYau manifolds equipped with holomorphic symplectic form,
also known as complex manifolds of hyperkaehler type.
We obtain that a complex manifold of hyperkaehler type is mirror dual to
itself. The Mirror Conjecture is stated (following Kontsevich,
ICM talk) as equivalence of certain
algebraic structures related to variations of Hodge structures.
We compute the canonical flat coordinates on the moduli space
of CalabiYau manifolds of hyperkaehler type,
introduced to Mirror Symmetry by Bershadsky, Cecotti, Ooguri and Vafa.
LaTeX 2e, 65 pages.
The articles in Latex 2.09 format.
Let M be a compact simply connected hyperk\"ahler
(or holomorphically symplectic) manifold, \dim H^2(M)=n.
We prove that the Lie group SO(n) acts by automorphisms
on the cohomology ring H^*(M). Under this action,
the space H^2(M) is isomorphic to the fundamental
representation of SO(n). Let A^r be the subring of
H^*(M) generated by H^2(M). We construct an action of the
Lie algebra so(n+2) on the space A, which preserves
A^r. The space A^r is an irreducible representation
of so(n+2). This makes it possible to compute
the ring A^r explicitely.
This article contains
a compression of results from
``Cohomology of hyperkaehler manifolds.''
with most proofs omitted.
We prove that every two points of the connected component of
moduli space of holomorphically symplectic manifolds
can be connected with socalled ``twistor lines'' 
projective lines holomorphically embedded to the moduli space
and corresponding to the hyperk\"ahler structures.
This has interesting implications for the geometry of compact
hyperk\"ahler manifolds and of holomorphic vector
bundles over such manifolds.
This article was written upon demand from GAFA. I am not to be held
accountable for its (relative) lack of content: GAFA wanted to publish a
short version without proofs instead of the longish
Cohomology of hyperkaehler manifolds. Some new results have
found a way here, anyway, and it's generally nicer to read
short papers. Published in GAFA vol. 6 4 pp. 601612 (1996)
In this paper the instanton equation is being generalized
to hyperkaehler manifolds of arbitrary dimension.
The hyperholomorphic bundle is a stable holomorphic bundle
for which this instanton equation can be solved. Its solution
is unique, if exist.
Precise equivalence between such instantons and
stable holomorphic bundles with first two Chern classes invariant
under a natural SU(2)action in cohomology is obtained, thus
generalizing the theorem of Yau and Uhlenbeck.
Deformation properties of hyperholomorphic bundles
are studied. In particular, we show that deformation space of
a hyperholomorphic bundle is again hyperkaehler. As
easy calculations imply, on generic hyperkaehler
manifold all stable bunles are hyperholomorphic.
I wrote an
article, which describes some details
on deformations of hyperholomorphic bundles,
vaguely motivated by Massey products
and homotopical algebra (posted to sci.math.research).
Here is a
paper about what sort of subvarieties a compact holomorphic symplectic
manifold has.
If the complex structure on this manifold is
"generic enough" in its deformation space (we describe the
sufficient conditions explicitly) these subvarieties turn
out to be holomorphically symplectic as well.
Published in Journ. of Alg. Geom. vol. 5 3
pp. 401415 (1996)
Here is a paper,
where I prove that all complex analytic subvarieties of a "generic"
hyperkaehler manifold are also hyperkaehler.
By CalabiYau theorem, all compact holomorphically symplectic
manifolds admit a hyperkaehler structure. In this paper
I prove that under some genericity conditions, all complex analytic
subvarieties of such manifold are also hyperkaehler. Thus we show that
the property of being hyperkaehler is very resilient  not only
deformation spaces of sheaves or bundles over a hyperkaehler manifold
are hyperkaehler, but even complex analytic subvarieties of a hyperkaehler
manifolds are hyperkaehler! Amazing. Published in GAFA, January 95.
Quantum groups.
Here is an article
written with David Kazhdan, where we compute
cohomology of the restricted quantum algebra.
Published in ``Quantum
Deformations of Algebras and Their Representations,''
vol. 7 of Isr. Math. Conf. Proc. (1993)
Here are the Postscript files of the papers above. The old
links to .DVI files were removed,
because our system deletes old .dvi files automatically
anyway.
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