Papers by Maxim Braverman
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|Abstract: Let F
be a flat vector bundle over a compact Riemannian manifold M and let f
a Morse function. Let g be a smooth Euclidean metric on F, let gt=e-2tfg
and let r(t) b e the Ray-Singer analytic torsion of F
associated to the metric
gt. Assuming that the vector field
grad(f) satisfies the Morse-Smale transversality conditions, we provide an asymptotic
expansion for log(r(t)) for t
+ of the form:
a0 + a1 t + b log(t/p)+o(1),
where the coefficient b is a half-integer depending only on the Betti numbers of F.
In the case where all the critical values of f are rational, we calculate the coefficients a0 and a1 explicitly in terms of the spectral sequence of a filtration associated to the Morse function.
These results are obtained as an applications of a theorem by Bismut and Zhang.
|Abstract: We generalize
the Novikov inequalities for 1-forms in two different directions: first,
we allow non-isolated critical points (assuming that they are non-degenerate
in the sense of R.Bott), and, secondly, we strengthen the inequalities
by means of twisting by an arbitrary flat bundle.
The proof uses Bismut's modification of the Witten deformation of the de Rham complex; it is based on an explicit estimate on the lower part of the spectrum of the corresponding Laplacian.
In particular, we obtain a new analytic proof of the degenerate Morse inequalities of Bott.
|Abstract: In this paer we give a short exposition of the results of the previous paper and also aply this result to obtain an L2 version of Novikov-Bott inequalities with finite von Neumann algebras.|
|Abstract: We establish an equivariant generalization of the Novikov inequalities which allow to estimate thetopology of the set of critical points of a closed basic invariant form by means of twisted equivariant cohomology of the manifold. We apply these inequalities to study cohomology of the fixed points set of a symplectic torus action. We show that in this case our inequalities are perfect, i.e. they are in fact equalities.|
|Abstract: We give a short exposition of the result of the previous paper and test and apply these inequalities in the case of a finite group. As an application we obtain Novikov type inequalities for a manifold with boundary.|
|Abstract: Let M
be a complete Riemannian manifold and let \Omega*(M) denote the
space of differential forms on M. Let
d: W*(M) W*+1(M)
be the exterior differential operator and let D=dd*+d*d be the Laplacian. We establish a sufficient condition for the Schroedinger operator H=D+V(x) (where the potential V(x):W*(M) W*(M) is a zero order differential operator) to be self-adjoint. Our result generalizes a theorem by Igor Oleinik about self-adjointness of a Schroedinger operator which acts on the space of scalar valued functions.
|Abstract: We introduce
Morse-type inequalities for a holomorphic circle action on a holomorphic
vector bundle over a compact Kaehler manifold. Our inequalities produce
bounds on the multiplicities of weights occurring in the twisted
Dolbeault cohomology in terms of the data of the fixed points and of the
symplectic reduction. This result generalizes both Wu-Zhang extension of
Witten's holomorphic Morse inequalities and Tian-Zhang Morse-type
inequalities for symplectic reduction.
As an application we get a new proof of the Tian-Zhang relative index theorem for symplectic quotients.
|Abstract: We obtain estimates
on the character of the cohomology of an S1-equivariant holomorphic
vector bundle over a Kaehler manifold M in terms of the cohomology
of the Lerman symplectic cuts and the symplectic reduction of M.
In particular, we prove and extend inequalities conjectured by Wu and Zhang.
The proof is based on constructing a flat family of complex spaces Mt such that Mt is isomorphic to M for t0, while M0 is a singular reducible complex space, whose irreducible components are the Lerman symplectic cuts.
|Abstract: Let M
be an oriented even-dimensional Riemannian manifold on which a discrete
of orientation-preserving isometries acts freely,
so that the quotient X=M/G is compact. We prove a vanishing
theorem for a half-kernel of a G
-invariant Dirac operator
on a G
-equivariant Clifford module over M, twisted
by a sufficiently large power of a G
-equivariant line bundle,
whose curvature is non-degenerate at any point of M. This
generalizes our previous vanishing theorems for Dirac operators on a compact
In particular, if M is an almost complex manifold we prove a vanishing theorem for the half-kernel of a spinc Dirac operator, twisted by a line bundle with curvature of a mixed sign. In this case we also relax the assumption of non-degeneracy of the curvature. When M is a complex manifold our results imply analogues of Kodaira and Andreotti-Grauert vanishing theorems for covering manifolds.
As another application, we show that semiclassically the spinc quantization of an almost complex covering manifold gives an "honest" Hilbert space. This generalizes a result of Borthwick and Uribe, who considered quantization of compact manifolds.
Application of our results to homogeneous manifolds of a real semisimple Lie group leads to new proofs of Griffiths-Schmidt and Atiyah-Schmidt vanishing theorems.
|Abstract:Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use the Witten deformation of the Dolbeault complex of L to show, that the cohomology of the sheaf of holomorphic sections of the induced bundle on the Mumford quotient of (X,L) is equal to the G-invariant part on the cohomology of the sheaf of holomorphic sections of L. This result, which was recently proven by C. Teleman by a completely different method, generalizes a theorem of Guillemin and Sternberg, which addressed the global sections. It also shows, that the Morse-type inequalities of Tian and Zhang for symplectic reduction are, in fact, equalities.|
|Abstract: We give a simple proof of the cobordism invariance of the index of an elliptic operator. The proof is based on a study of a Witten-type deformation of an extension of the operator to a complete Riemannian manifold. One of the advantages of our approach is that it allows to treat directly general elliptic operator which are not of Dirac type.|
|Abstract: We show that the index of a transversally elliptic opearator on a compact manifold is invariant under cobordisms. As an application we show that the index of an elliptic operator on an orbifold is invariant under cobordisms of orbifolds.|
|Abstract: We obtain several
essential self-adjointness conditions for a Schroedinger type operator
D*D+V acting in sections of a vector bundle over a manifold M. Here V is
a locally square-integrable bundle map. Our conditions are expressed in
terms of completeness of certain metrics on M; these metrics are naturally
associated to the operator. We do not assume a priori that M is endowed
with a complete Riemannian metric. This allows us to treat e.g. operators
acting on bounded domains in the euclidean space.
For the case when the principal symbol of the operator is scalar, we establish more precise results. The proofs are based onan extension of the Kato inequality which modifies and improves a result of Hess, Schrader and Uhlenbrock.
|Abstract: Let D be
a (generalized) Dirac operator on a non-compact complete Riemannian manifold M acted on by a compact Lie group
G. Let v:M
Lie(G) be an equivariant
map, such that the corresponding vector field on M does not vanish outside
of a compact subset. These data define an element of K-theory of the
transversal cotangent bundle to M. Hence a topological index of the pair (D,v) is defined as an element of the completed ring of characters of
We define an analytic index of (D,v) as an index space of certain deformation of D and we prove that the analytic and topological indexes coincide.
As a main step of the proof, we show that index is an invariant of a certain class of cobordisms, similar to the one considered by Ginzburg, Guillemin and Karshon. In particular, this means that the topological index of Atiyah is also invariant under this class of non-compact cobordisms.
As an application we extend the Atiyah-Segal-Singer equivariant index theorem to our non-compact setting. In particular, we obtain a new proof of this theorem for compact manifolds
|Abstract: We present a short analytic proof of the equality between the analytic and combinatorial torsion. We use the same approach as in the proof given by Burghelea, Friedlander and Kappeler, but avoid using the difficult Mayer-Vietoris type formula for the determinants of elliptic operators. Instead, we provide a direct way of analyzing the behaviour of the determinant of the Witten deformation of the Laplacian. In particular, we show that this determinant can be written as a sum of two terms, one of which has an asymptotic expansion with computable coefficients and the other is very simple (no zeta-function regularization is involved in its definition).|
|Abstract: We study the zeta-regularized determinant of a non self-adjoint elliptic operator on a closed odd-dimensional manifold. We show that, if the spectrum of the operator is symmetric with respect to the imaginary axis, then the determinant is real and its sign is determined by the parity of the number of the eigenvalues of the operator, which lie on the positive part of the imaginary axis. It follows that, for many geometrically defined operators, the phase of the determinant is a topological invariant. In numerous examples, coming from geometry and physics, we calculate the phase of the determinants in purely topological terms. Some of those examples were known in physical literature, but no mathematically rigorous proofs and no general theory were available until now.|
|Abstract: We associate determinant lines to objects of the extended abelian category built out of a von Neumann category with a trace. Using this we suggest constructions of the combinatorial and analytic L2 torsions which, unlike the work of the previous authors requires no additional assumptions. In particular, we do not impose the determinant class condition. Applying a recent result of Burghelea, Friedlander, and Kappeler we obtain a Cheeger-Müller type theorem stating the equality between the combinatorial and the analytic L2 torsions.|
|Abstract: We extend the
Novikov Morse-type inequalities for closed 1-forms in 2 directions. First, we
consider manifolds with boundary. Second, we allow a very degenerate structure
of the critical set of the form, assuming only that the form is non-degenerated
in the sense of Kirwan. In particular, we obtain a generalization of a result of
Floer about the usual Morse inequalities on a manifold with boundary. We also
obtain an equivariant version of our inequalities.
Our proof is based on an application of the Witten deformation technique. The main novelty here is that we consider the neighborhood of the critical set as a manifold with a cylindrical end. This leads to a considerable simplification of the local analysis. In particular, we obtain a new analytic proof of the Morse-Bott inequalities on a closed manifold.
|Abstract: This is a short exposition of the results of the paper Refined Analytic Torsion|
|Abstract: For an acyclic representation of the fundamental group of a compact oriented
odd-dimensional manifold, which is close enough to a unitary representation, we
define a refinement of the Ray-Singer torsion associated to this representation.
This new invariant can be viewed as an analytic counterpart of the refined
combinatorial torsion introduced by Turaev.
The refined analytic torsion is a holomorphic function of the representation of the fundamental group. When the representation is unitary, the absolute value of the refined analytic torsion is equal to the Ray-Singer torsion, while its phase is determined by the eta-invariant. The fact that the Ray-Singer torsion and the eta-invariant can be combined into one holomorphic function allows to use methods of complex analysis to study both invariants. In particular, we extend and improve a result of Farber about the relationship between the Farber-Turaev absolute torsion and the eta-invariant.
|Abstract: We construct a canonical element, called the refined analytic torsion, of the determinant line of the cohomology of a closed oriented odd-dimensional manifold M with coefficients in a flat complex vector bundle E. We compute the Ray-Singer norm of the refined analytic torsion. In particular, if there exists a flat Hermitian metric on E, we show that this norm is equal to 1. We prove a duality theorem, establishing a relationship between the refined analytic torsions corresponding to a flat connection and its dual.|
We show that the refined analytic
torsion is a holomorphic section of the determinant line bundle over the space
of complex representations of the fundamental group of a closed oriented odd
dimensional manifold. Further, we calculate the ratio of the refined analytic
torsion and the Turaev combinatorial torsion.
As an application, we establish a formula relating the eta-invariant and the phase of the Turaev torsion, which extends a theorem of Farber and earlier results of ours. This formula allows to study the spectral flow using methods of combinatorial topology.
|Abstract: We obtain a
vanishing theorem for the kernel of a Dirac operator on a Clifford module
twisted by a sufficiently large power of a line bundle, whose curvature
is non-degenerate at any point of the base manifold. In particular, if
the base manifold is almost complex, we prove a vanishing theorem for the
kernel of a
spinc Dirac operator twisted by a line bundle with
curvature of a mixed sign. In this case we also relax the assumption of
non-degeneracy of the curvature. These results are generalization of a
vanishing theorem of Borthwick and Uribe. As an application we obtain a
new proof of the classical Andreotti-Grauert vanishing theorem for the
cohomology of a compact complex manifold with values in the sheaf of holomorphic
sections of a holomorphic vector bundle, twisted by a large power of a
holomorphic line bundle with curvature of a mixed sign.
As another application we calculate the sign of the index of a signature operator twisted by a large power of a line bundle.
|Abstract: Let M be a compact symplectic manifold on which a compact torus T acts Hamiltonialy with a moment map m.. Suppose there exists a symplectic involution q: M M, such that m.oq =-m.. Assuming that 0 is a regular value of m., we calculate the character of the action of q on the cohomology of M in terms of the character of the action of q on the symplectic reduction m.-1(0)/T of M. This result generalizes a theorem of R. Stanley, who considered the case when M was a toric variety.|
|Abstract: We express the Burghelea-Haller complex Ray-Singer torsion in terms of the square of the refined analytic torsion and the eta-invariant. As an application we obtain new results about the Burghelea-Haller torsion. In particular, we prove a weak version of the Burghelea-Haller conjecture relating their torsion with the square of the Turaev combinatorial torsion.|
|Abstract: We introduce a new canonical trace on odd class logarithmic pseudo-differential operators on an odd dimensional manifold, which vanishes on commutators. When restricted to the algebra of odd class classical pseudo-differential operators our trace coincides with the canonical trace of Kontsevich and Vishik. Using the new trace we construct a new determinant of odd class classical elliptic pseudo-differential operators. This determinant is multiplicative whenever the multiplicative anomaly formula for usual determinants of Kontsevich-Vishik and Okikiolu holds. In particular, it is multiplicative for operators whose leading symbols commute. When restricted to operators of Dirac type our determinant provides a sign refined version of the determinant constructed by Kontsevich and Vishik.|
Deformation of the De Rham Complex and Global Invariants of Manifolds, Ph.D. Thesis. Tel-Aviv University.
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