Riemann-Roch Theorem

Classical Riemann-Roch theorem is one of the most prominent works. In 1954, F. Hirzebruch generalized Riemann-Roch theorem to high dimension, whose method has profound affection on differential topology and transcendental algebra geometry. Later, in 1958, Grothendieck has regarded the computations of Euler numbers on holomorphic vector bundle in Riemann-Roch-Hirzebruch theorem as a formula of holomorphic mapping, which establishes a relation between some cohomology classes, i.e. Riemann-Roch-Grothendiek theorem. In 1963, M.Atiyah and I.M. Singer generalized Riemann-Roch-Hirzebruch theorem and got Atiyah-Singer index theorem. If we define the operator (?)|- + ((?)|-)^# : ∧^0,* (M)→ ∧^{0, *}(M), we can get the Riemann-Roch-Hirzebruch theorem of compact complex manifolds. There always exists the global Atiyah-Singer index theorem for general elliptic operators on manifolds. However, we need to deal with each kind of common differential operators respectively to get the local index theorem of elliptic differential operators. There are some kinds of method, one common method is the heat equation method. In the proceeding of proving, we use a concrete expression of square of operators whose distinguish feature is that there is no one-order derivative. It plays a crucial role in the proceeding. We also find the one-order derivative has some relation with the choices of connection.Firstly, by using connection, we express the two-order elliptic differential operator L as L = — â–³₀ +b_iâ–½V_Ei^L+ c. Furthermore, we deal with a restricted second order self-adjoint elliptic operator by complex tensor computation. We prove that if the operator L is a restricted second order self-adjoint elliptic operator on complex vector bundle E, then there exists a Hermitian connection â–½^L on complex vector bundle E such that the operator C has Schrodinger expression, i.e. L is a generalized Schrodinger operator. Consequently, we prove that the square of (?)|- + ((?)|-)^# : ∧^0,* (M)→ ∧^{0, *}(M) is a generalized Schrodinger operator.Secondly, on Spin^c(2n) manifold, if we lift a connection â–½^B(= â–½^L + S(B),S(B) is a 1-form defined by odd form B) which preserves metric but isn't the Levi-Civita connection, then we can get a new connection â–½^E, thus we construct a self-adjoint op-erator Vs + C which is a modified operator of Dirac operator Vs and get an asymptotic expansion of the expanded Laplace-Beltrami operator Ajf,2n 02 i 2n = E 0-2 - 7ï¿¡ E ^(0)i2fm^(0)%ymefce,eQe^ + (x < 2).and by using heat equation method, we prove the local index theorem of the Dirac operator Vs:Loc.ind(Vs) = L^ â–  A ($-\\ (Eu â–  â–  â– , E2n).where Ci is the first Chern class on the line bundle Pu of 5pmc(2n)-structure.Lastly, by the imbedding A : U{n) —> Spinc(2n), we got the relation between almost Hermitian manifolds and Spinc(2n) manifolds. We discuss almost Hermitian manifolds from the view of Spinc(2n) manifold and expressed d + d as1 1 2n f—\ nd + d* = -j=Ds + -= ï¿¡ eaA(Ea) - ^ E rf$(^>z^ ZjVfij + v^ej). V2 V2Q=1 ? j>fe=1where A € spin(2n). Furthermore, we proved that the local index theorem of Riemann-Roch operators on K'dhler manifolds is a special case of the local index theorem of Dirac operators on Spinc(2n) manifolds.