arXiv subjects

Maxim Braverman

Publications and source records attributed to Maxim Braverman.

51 records · Page 3Linked to original sources

arXiv2001

Index theorem for equivariant Dirac operators on non-compact manifolds

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.

math-ph↗

arXiv2001

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.

math.SP↗

arXiv2000

Cohomology of the Mumford Quotient

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.

math.SG↗

arXiv1998

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.

math.DG↗

arXiv1998

Vanishing theorems on covering manifolds

Let $M$ be an oriented even-dimensional Riemannian manifold on which a discrete group $Γ$ of orientation-preserving isometries acts freely, so that the quotient $X=M/Γ$ is compact. We prove a vanishing theorem for a half-kernel of a $Γ$-invariant Dirac operator on a $Γ$-equivariant Clifford module over $M$, twisted by a sufficiently large power of a $Γ$-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.

math.DG↗

arXiv1997

Symplectic cutting of Kaehler manifolds

We obtain estimates on the character of the cohomology of an $S^1$-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 $M_t$ such that $M_t$ is isomorphic to $M$ for $t\not=0$, while $M_0$ is a singular reducible complex space, whose irreducible components are the Lerman symplectic cuts.

alg-geom↗

arXiv1997

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.

dg-ga↗

arXiv1997

Tempered currents and the cohomology of the remote fiber of a real polynomial map

Let $p:R^n\to R$ be a polynomial map. Consider the complex $S'Ω^*(\RR^n)$ of tempered currents on $R^n$ with the twisted differential $d_p=d-dp$ where $d$ is the usual exterior differential and $dp$ stands for the exterior multiplication by $dp$. Let $t\in R$ and let $F_t=p^{-1}(t)$. In this paper we prove that the reduced cohomology $\tilda H^k(F_t;C)$ of $F_t$ is isomorphic to $H^{k+1}(S'Ω^*(\RR^n),d_p)$ in the case when $p$ is homogeneous and $t$ is any positive real number. We conjecture that this isomorphism holds for any polynomial $p$, for $t$ large enough (we call the $F_t$ for $t >> 0$ the remote fiber of $p$) and we prove this conjecture for polynomials that satisfy certain technical condition. The result is analogous to that of A. Dimca and M. Saito, who give a similar (algebraic) way to compute the reduced cohomology of the generic fiber of a complex polynomial.

alg-geom↗

arXiv1996

On self-adjointness of a Schroedinger operator

Let $M$ be a complete Riemannian manifold and let $Ω^*(M)$ denote the space of differential forms on $M$. Let $d:Ω^*(M) \to Ω^{*+1}(M)$ 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 $V(x):Ω^*(M)\to Ω^*(M)$ 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.

funct-an↗

arXiv1996

Novikov inequalities with symmetry

We establish an equivariant generalization of the Novikov inequalities which allow to estimate the topology of the set of critical points of a closed basic invariant 1-form by means of twisted equivariant cohomology of the manifold. We test and apply our inequalities in the case of a finite group. As an application we obtain Novikov type inequalities for a manifold with boundary.

dg-ga↗

arXiv1995

The Novikov-Bott inequalities

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. We also obtain an $L^2$ version of these inequalities with finite von Neumann algebras. The proof of the main theorem 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.

dg-ga↗

arXiv1995

Novikov type inequalities for differential forms with non-isolated zeros

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.

dg-ga↗

arXiv1995

Witten deformation of the analytic torsion and the spectral sequence of a filtration

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 ρ(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(ρ(t)) for t\to +\infty of the form a_0+a_1t+b\log\left(\frac tπ\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.

dg-ga↗

arXiv1994

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.

hep-th↗

arXiv1994

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 ρ(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(ρ(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.

dg-ga↗