Abstract:
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.

Abstract:
Let F be a flat vector bundle over a compact Riemannian manifold M and let f be a Morse function. Let g be a smooth Euclidean metric on F, let g_t=e^{-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 grad(f) 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\left(\frac t\pi\right)+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 a_0 and a_1 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:
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 $grad f$ 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.

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 $G$. 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 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:
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:
Let $M$ be an oriented even-dimensional Riemannian manifold on which a discrete group $\Gamma$ of orientation-preserving isometries acts freely, so that the quotient $X=M/\Gamma$ is compact. We prove a vanishing theorem for a half-kernel of a $\Gamma$-invariant Dirac operator on a $\Gamma$-equivariant Clifford module over $M$, twisted by a sufficiently large power of a $\Gamma$-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 manifold. In particular, if $M$ is an almost complex manifold we prove a vanishing theorem for the half-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. 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 $\spin^c$ 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 $M$ be a complete Riemannian manifold and let $\Omega^*(M)$ denote the space of differential forms on $M$. Let $d:\Omega^*(M) \to \Omega^{*+1}(M)$ be the exterior differential operator and let $\Del=dd^*+d^*d$ be the Laplacian. We establish a sufficient condition for the Schroedinger operator $H=\Del+V(x)$ (where the potential $V(x):\Omega^*(M)\to \Omega^*(M)$ 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 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 $\sigma$-model in superconductivity.

Abstract:
We construct a regularized index of a generalized Dirac operator on a complete Riemannian manifold endowed with a proper action of a unimodular Lie group. We show that the index is preserved by a certain class of non-compact cobordisms and prove a gluing formula for the regularized index. The results of this paper generalize our previous construction of index for compact group action and the recent paper of Mathai and Hochs who studied the case of a Hamiltonian action on a symplectic manifold. As an application of the cobordism invariance of the index we give an affirmative answer to a question of Mathai and Hochs about the independence of the Mathai-Hochs quantization of the metric, connection and other choices.