74 research outputs found
Witten deformation of Ray-Singer analytic torsion
Consider a flat vector bundle F over compact Riemannian manifold M and let f
be a self-indexing Morse function on M. Let g be a smooth Euclidean metric on
F. Set g_t=exp(-2tf)g and let \rho(t) be the Ray-Singer analytic torsion of F
associated to the metric g_t. Assuming that the vector field satisfies
the Morse-Smale transversality conditions, we provide an asymptotic expansion
for log(\rho(t)) for t\to\infty of the form a_0+a_1t+b log(t)+o(1). We present
explicit formulae for coefficients a_0,a_1 and b. In particular, we show that b
is a half integer.Comment: 10 pages. AMS-LaTe
New Proof of the Cobordism Invariance of the Index
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.Comment: Some references are added, some minor changes are made. To appear in
Proc. of AM
Symplectic cutting of Kaehler manifolds
We obtain estimates on the character of the cohomology of an
-equivariant holomorphic vector bundle over a Kaehler manifold in
terms of the cohomology of the Lerman symplectic cuts and the symplectic
reduction of . In particular, we prove and extend inequalities conjectured
by Wu and Zhang.
The proof is based on constructing a flat family of complex spaces such
that is isomorphic to for , while is a singular
reducible complex space, whose irreducible components are the Lerman symplectic
cuts.Comment: 11 pages, LaTeX 2
Holomorphic Morse Inequalities and Symplectic Reduction
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.Comment: LaTeX 2e, 9 page
The spectral Flow of a family of Toeplitz operators
We show that the (graded) spectral flow of a family of Toeplitz operators on
a complete Riemannian manifold is equal to the index of a certain Callias-type
operator. When the dimension of the manifold is even this leads to a
cohomological formula for the spectral flow. As an application, we compute the
spectral flow of a family of Toeplitz operators on a strongly pseudoconvex
domain in . This result is similar to the Boutet de Monvel's computation
of the index of a single Toeplitz operator on a strongly pseudoconvex domain.
Finally, we show that the bulk-boundary correspondence in a tight-binding model
of topological insulators is a special case of our result.
In the appendix, Koen van den Dungen reviewed the main result in the context
of (unbounded) KK-theory.Comment: Minor corrections, some references are adde
Cohomology of the Mumford Quotient
Let be a smooth projective variety acted on by a reductive group . Let
be a positive -equivariant line bundle over . We use the Witten
deformation of the Dolbeault complex of to show, that the cohomology of the
sheaf of holomorphic sections of the induced bundle on the Mumford quotient of
is equal to the -invariant part on the cohomology of the sheaf of
holomorphic sections of . 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.Comment: A mistake in the proof of Theorem 3.1.b is corrected. The definition
of the integration map is slightly changed. To appear in "Quantization of
singular symplectic quotients
Vanishing theorems for the kernel of a Dirac operator
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 \spin^c 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.Comment: A mistake in Theorem 3.13 is corrected. Some othe misprints are
remove
New proof of the Cheeger-Muller Theorem
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).Comment: 13 pages, more details are given in section 5, some misprints are
correcte
Deformation of infinite dimensional differential graded Lie algebras
We introduce a notion of elliptic differential graded Lie algebra. The class
of elliptic algebras contains such examples as the algebra of differential
forms with values in endomorphisms of a flat vector bundle over a compact
manifold, etc.
For elliptic differential graded algebra we construct a complete set of
deformations.
We show that for several deformation problems the existence of a formal power
series solution guarantees the existence of an analytic solution.Comment: 11 pages; AMS-TeX; no figures. A non-correct example was erase
Symmetrized Trace and Symmetrized Determinant of Odd Class Pseudo-Differential Operators
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 up to sign whenever the multiplicative anomaly formula for usual
determinants of Kontsevich-Vishik and Okikiolu holds. When restricted to
operators of Dirac type our determinant provides a sign refined version of the
determinant constructed by Kontsevich and Vishik. We discuss some applications
of the symmetrized determinant to a non-linear -model in
superconductivity.Comment: 21 page
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