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Register a new userSheaves and Duality in the Two-Vertex Graph Riemann-Roch Theorem
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For each graph on two vertices, and each divisor on the graph in the sense of Baker-Norine, we describe a sheaf of vector spaces on a finite category whose zeroth Betti number is the Baker-Norine Graph Riemann-Roch rank of the divisor plus one. We prove duality theorems that generalize the Baker-Norine Graph Riemann-Roch Theorem.
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We extend the Boutet de Monvel Toeplitz index theorem to complex manifold with isolated singularities following the relative $K$-homology theory of Baum, Douglas, and Taylor for manifold with boundary. We apply this index theorem to study the Arveson-Douglas conjecture. Let $ball^m$ be the unit ball in $mathbb{C}^m$, and $I$ an ideal in the polynomial algebra $mathbb{C}[z_1, cdots, z_m]$. We prove that when the zero variety $Z_I$ is a complete intersection space with only isolated singularities and intersects with the unit sphere $mathbb{S}^{2m-1}$ transversely, the representations of $mathbb{C}[z_1, cdots, z_m]$ on the closure of $I$ in $L^2_a(ball^m)$ and also the corresponding quotient space $Q_I$ are essentially normal. Furthermore, we prove an index theorem for Toeplitz operators on $Q_I$ by showing that the representation of $mathbb{C}[z_1, cdots, z_m]$ on the quotient space $Q_I$ gives the fundamental class of the boundary $Z_Icap mathbb{S}^{2m-1}$. In the appendix, we prove with Kai Wang that if $fin L^2_a(ball^m)$ vanishes on $Z_Icap ball ^m$, then $f$ is contained inside the closure of the ideal $I$ in $L^2_a(ball^m)$.
Let G be a torus and M a G-Hamiltonian manifold with Kostant line bundle L and proper moment map. Let P be the weight lattice of G. We consider a parameter k and the multiplicity $m(lambda,k)$ of the quantized representation associated to M and the k-th power of L . We prove that the weighted sum $sum m(lambda,k) f(lambda/k)$ of the value of a test function f on points of the lattice $P/k$ has an asymptotic development in terms of the twisted Duistermaat-Heckman distributions associated to the graded Todd class of M. When M is compact, and f polynomial, the asymptotic series is finite and exact.
The classical Riemann-Roch theorem has been extended by N. Nadirashvili and then M. Gromov and M. Shubin to computing indices of elliptic operators on compact (as well as non-compact) manifolds, when a divisor mandates a finite number of zeros and allows a finite number of poles of solutions. On the other hand, Liouville type theorems count the number of solutions that are allowed to have a pole at infinity. Usually these theorems do not provide the exact dimensions of the spaces of such solutions (only finite-dimensionality, possibly with estimates or asymptotics of the dimension. An important case has been discovered by M. Avellaneda and F. H. Lin and advanced further by J. Moser and M. Struwe. It pertains periodic elliptic operators of divergent type, where, surprisingly, exact dimensions can be computed. This study has been extended by P. Li and Wang and brought to its natural limit for the case of periodic elliptic operators on co-compact abelian coverings by P. Kuchment and Pinchover. Comparison of the results and techniques of Nadirashvili and Gromov and Shubin with those of Kuchment and Pinchover shows significant similarities, as well as appearance of the same combinatorial expressions in the answers. Thus a natural idea was considered that possibly the results could be combined somehow in the case of co-compact abelian coverings, if the infinity is added to the divisor. This work shows that such results indeed can be obtained, although they come out more intricate than a simple-minded expectation would suggest. Namely, the interaction between the finite divisor and the point at infinity is non-trivial.
We produce a Grothendieck transformation from bivariant operational $K$-theory to Chow, with a Riemann-Roch formula that generalizes classical Grothendieck-Verdier-Riemann-Roch. We also produce Grothendieck transformations and Riemann-Roch formulas that generalize the classical Adams-Riemann-Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety $X$ whose equivariant $K$-theory of vector bundles does not surject onto its ordinary $K$-theory, and describe the operational $K$-theory of spherical varieties in terms of fixed-point data. In an appendix, Vezzosi studies operational $K$-theory of derived schemes and constructs a Grothendieck transformation from bivariant algebraic $K$-theory of relatively perfect complexes to bivariant operational $K$-theory.
We calculate the local Riemann-Roch numbers of the zero sections of $T^*S^n$ and $T^*R P^n$, where the local Riemann-Roch numbers are defined by using the $S^1$-bundle structure on their complements associated to the geodesic flows.
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