Reports of the Academy of Sciences of the USSR

1. Volume 171, No. 3

MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR S. P. NOVIKOV, B. Yu. STERNIN

ELLIPTIC OPERATORS AND SUBMANIFOLDS

The present work is a direct continuation of the authors’ work (2). We shall not here redefine the operators \((A^{-1}, SD_F)\), the semigroups \(\mathrm{Op}^{s,t}(X,\xi)\), \(\mathrm{Ell}^{s,t}(X,\xi)\) for a vector bundle \(\xi\) of dimension \(2q\), the “trace” operator \(i_{SD}\), the character of an operator \(\mathrm{Ch}\, A \in H^*(X,Q)[t]/t^{|s,t-q+1}\), the Pontryagin character \(\mathrm{Ph}(n)\in H^*(Y,Q)[t]\), and the restriction

\[ i^*: H^*(X,Q)[t]/t^{l-q+1}\to H^*(Y,Q)[t]/t^{l-q-q'+1}. \]

We introduce here an important special case of the operators \(SD_F\): let \(X\) be a Kähler manifold, \(Y\subset X\) a complex submanifold, and \(n\) the complex normal bundle. Let

\[ F=\sum_j S^j n \subset \sum_j S^j cn; \]

then \(SD_F\) shall be denoted by \(SD_c\).

By analogy with (2), for a (Kähler) bundle \(\xi\) over \(X\) of dimension \(q\), a semigroup of operators \(\mathrm{Op}^{s,t}_c(X,\xi)\) is introduced:

\[ A=(A^{jk}):\quad \sum_{j\le l\,s,t-q} \Gamma^{s-q-j}(X,S^j\xi\otimes E_1) \to \sum_{k\le l\,s,t-q} \Gamma^{t+q+k}(X,S^j\xi\otimes E_2) \]

and the homomorphic trace operator is defined

\[ i_{SD_c}:\mathrm{Op}^{s,t}_c(X,\xi)\to \mathrm{Op}^{s,t}_c(Y,n\oplus i^*\xi), \]

by setting

\[ i_{SD_c}A=(i_{SD_c}A^{jk})\quad\text{and}\quad i_{SD_c}A^{jk}=i_c^*\circ A^{jk}\circ \chi_c. \]

Here it is necessary to explain more precisely what the operators \(i^*\) and \(\chi\) in general are. The restriction with normal derivatives \(i^*\) is the operator which assigns to each smooth section of the bundle \(S^j cn\) on \(Y\) a complex-valued function on \(Y\), depending linearly on this section, according to the following rule: we differentiate the section on \(X\) along some extension to a neighborhood \(U\supset Y\) of the original section of the bundle \(S^j cn\), and then restrict to \(Y\) the result of differentiation. We obtain a linear form

\[ \Gamma(Y,S^j cn)\to \Gamma(Y,1), \]

which may be regarded as a section of the bundle \(S^j cn\) on \(Y\). The operator \(\chi\) is adjoint to \(i^*\). The operators \(i_c^*\) and \(\chi_c\) are obtained analogously, but one must restrict oneself only to complex differentiations.

We now turn to our main result in \(K\)-theory. Recall some standard notation: \(M\eta\) is the Thom complex; \(j:B\to M\eta\) is the inclusion of the base \(B\); \(i_\tau:Mi^*\eta\to M\eta\) for \(i:Y\to X\) is the natural map of Thom complexes. If \(\eta=\eta_X\) is the cotangent bundle, then \(i^*\eta_X=\bar n\oplus \eta_Y\), and we have:

\[ (M\eta_Y,*) \xrightarrow{\,j\,} (Mi^*\eta_X,*) \xrightarrow{\,i_\tau\,} (M\eta_X,*), \]

where \((Mi^*\eta_X,*)\) is the relative Thom complex of the bundle \(\bar n\) over the pair

\[ (M\eta_Y,*)=(B(Y),S(Y)). \]

We define mappings:
\[ S_t: K(X)\to K(X)[t],\qquad \lambda_{-t}: K(X)\to K(X)[t], \]
\[ S_t^R=S_t\circ c: KO(X)\to K(X)[t],\qquad \lambda_{-t}^R=\lambda_{-t}\circ c: KO(X)\to K(X)[t], \]
where
\[ S_t\xi=\sum_{j\geq 0} S^j\xi\,t^j;\qquad \lambda_{-t}\xi=\sum_{j\geq 0}(-1)^j\Lambda^j\xi\,t^j; \]
\[ S_t\xi\circ\lambda_{-t}\xi=1;\quad S_t\xi\oplus\eta=S_t\xi\circ S_t\eta;\quad \lambda_{-t}\xi\oplus\eta=\lambda_{-t}\xi\circ\lambda_{-t}\eta; \]
\[ \delta:\operatorname{Ell}^{s,t}(X,\xi)\to K(M\eta_X,*)[t]/t^{s,t-q+1}; \]
\[ \delta_c=\delta\circ r:\operatorname{Ell}_c^{s,t}(X,\xi)\to K(M\eta_X,*)[t]/t^{s,t-q+1}; \]
\[ 2q=\dim_R \xi,\qquad \delta(A)=\sum_j \alpha(\widetilde{\delta}_j)t^j; \]
\(\widetilde{\delta}_j\) are the elliptic operators indicated in (2) for \(A\);
\(\alpha(\widetilde{\delta}_j)\in K(M\eta_X,*)\) are their Atiyah–Singer elements. Then
\[ \varphi^{-1}\operatorname{ch}\delta(A)=\sum_j\varphi^{-1}\operatorname{ch}\alpha(\widetilde{\delta}_j)t^j=\operatorname{Ch}A,\quad \operatorname{ch}S_t^R\xi=\operatorname{Ph}\xi,\quad \operatorname{ch}S_t\xi=\operatorname{Ch}\xi. \]
Put
\[ i_M^*=j^*\circ i_T^*;\qquad K(M\eta_X,*)[t]/t^{l-q+1}\to K(M\eta_Y,*)[t]/t^{l-q-q'+1} \]
naturally.

The following contravariant theorem (of Riemann–Roch type) holds:

Theorem 1. a) If
\[ A\subset \operatorname{Ell}^{s,t}(X,\xi)\cap i_{SD}^{-1}\operatorname{Ell}^{s,t}(Y,\eta\oplus i^*\xi), \]
then the relation
\[ \lambda_{-t}^R\eta\oplus i^*\xi\circ \delta(i_{SD}A) = i_M^*\bigl(\lambda_{-t}^R\xi\circ\delta(A)\bigr) \]
is satisfied;

b) if
\[ A\in \operatorname{Ell}_c^{s,t}(X,\xi)\cap i_{SD_c}^{-1}\operatorname{Ell}_c^{s,t}(Y,\eta\oplus i^*\xi), \]
then the relation
\[ (\lambda_{-t})\eta\oplus i^*\xi\circ \delta_c(i_{SD_c}A) = i_M^*\bigl(\lambda_{-t}\xi\circ\delta_c(A)\bigr), \]
is satisfied, where
\[ i_M^*=j^*\circ i_T^*,\quad i_T^*,\quad j^* \]
were defined above.

Thus, for \(\operatorname{Ell}^{s,t}(X,\xi)\), the functorial expression is \(\lambda_{-t}^R\xi\circ\delta(A)\) for the case of immersions, moreover contravariantly.

Corollary 1. a)
\[ \operatorname{Ch}(i_{SD}A)=\chi(\eta)\,(i^*\operatorname{Ch}A)\operatorname{Ph}(\eta); \]
b)
\[ \operatorname{Ch}(i_{SD_c}A)=c_q(\eta)\circ i^*(\operatorname{Ch}A)\operatorname{Ch}(\eta). \]

Denote the expression \(\lambda_{-t}^R\xi\circ\delta(A)\) by
\[ R(A)\in K(M\eta_X,*)[t]/t^{l-q+1}. \]
The index formula can be written as follows:
\[ I_a(A)=\left\{\left[\varphi^{-1}\operatorname{ch}\bigl(S_t^R\xi R(A)\bigr)\right]\circ T(c\eta_X),[Y]\right\}_{t=1}. \]
Since
\[ I_a(A^{-1},SD)-I_a(A^{-1})=I_a(i_{SD}A) \]
(see (2)), we have
\[ I_a(A^{-1},SD)-I_a(A^{-1})= \]
\[ =\left\{\left[\varphi^{-1}\operatorname{ch}\bigl(i_M^*R(A)\bigr)\circ S_t^R\eta\oplus i^*\xi\right]\circ T(c\eta_Y),[Y]\right\}_{t=1}. \]
and analogously for \(SD_c\).

Let us now pass to the analytic meaning of the results and to general boundary operators associated with \(Y\). Let
\[ C:\Gamma^s(X,E)\to \Sigma\Gamma^{s_j}(Y,E_j) \]
and
\[ B:\Sigma\Gamma^{t_j}(Y,E_j)\to \Gamma^t(Y,E') \]
be operators with the following properties:

1. The image of \(B\) consists only of sections concentrated on \(Y\); all sections that vanish on any neighborhood \(U\supset Y\) belong to the kernel of \(C\).

2. The operators \(B\) and \(C\) have “symbols”:
   \[ \sigma_B(\tau,\xi):\sum_j\pi_{i^*\eta_X}^*E'_j\to \pi_{i^*\eta_X}^*E', \qquad \sigma_C(\tau,\xi):\pi_{i^*\eta_X}^*E\to \sum_j\pi_{i^*\eta_X}^*E_j. \]

3. \(B\) is a monomorphism and \(C\) is an epimorphism; here \(\tau\) are coordinates in the fibration \(\bar\eta\) on \(Y\), and \(\xi\) are coordinates in the fibration \(\eta_Y\); \(\pi_\eta:B(\eta)\to B\) (for a fibration \(\eta\) with base \(B\)) is the projection.

Let
\[ A:\Gamma^t(X,E')\to\Gamma^s(X,E) \]
be an operator and let \(\sigma_A\) be its symbol,
\[ \sigma_A:\pi_{\eta_X}^*E'\to \pi_{\eta_X}^*E. \]
The symbols are not assumed to be isomorphisms of fibrations.

Lemma 1. The symbol of the operator \(C\circ A\circ B\) is computed by the formula

\[ \sigma_{C\circ A\circ B}(\xi) = \int_{\xi=\mathrm{const}} \sigma_C\circ\sigma_A\circ\sigma_B\,d\tau, \]

where the integration is over the linear mapping depending on the parameter \(\xi\).

An important example: \(C=i^*,\ B=\varkappa\); here we have

\[ \sigma_C(\tau,\xi)[e] = e\otimes \sum_{|m|\le l-q}\tau^m, \]

where \(m=(m_1,\ldots,m_k)\), \(\tau^m=\tau_1^{m_1}\circ\cdots\circ\tau_k^{m_k}\), \(|m|=\sum_j m_j\); \(e\) is a vector of the quotient bundle \(\pi^*_{i^*\eta X}E\);

\[ \sigma_B(\tau,\xi)[P\otimes e']=P(\tau)e', \]

where \(e'\) is a vector of the quotient bundle \(\pi^*_{i^*_{\eta}X}E'\) and \(P\) is a polynomial of degree \(\le l_s,t-q\) on the quotient bundle \(\bar n\). Similarly for \(C=i_c^*,\ B=\varkappa_c\).

Definition 1. A direction \((\tau)\) is called elliptic with respect to the triple of operators \((A,B,C)\) if, for any \(\xi\ne0\), the mapping

\[ \int_{\substack{\xi\ne0\\ \xi=\mathrm{const}}} \sigma_C\circ\sigma_A\circ\sigma_B\,d\tau \]

is an isomorphism at this point.

We obtain important special cases when \(B=\varkappa,\ C=i^*\) or \(B=\varkappa_c,\ C=i_c^*\). In these cases we shall call directions elliptic with respect to the triple \((A,B,C)\), \(A\)-elliptic, respectively, in the real and complex sense.

Definition 2. A submanifold \(Y\subset X\) is called elliptic with respect to the triple \((A,B,C)\) if

\[ \operatorname{codim}_{\mathbf R}Y=\dim_{\mathbf R}\tau \]

and at each point \(y\in Y\) this submanifold is transversal to the elliptic direction \(\tau\).

Let the operator \(A\) be elliptic, and let \(A^{-1}\) be its inverse (mod Comp). Consider the boundary operator

\[ (A^{-1},B,C):\Gamma^s(X,E)\to \Gamma^t(X,E')/\operatorname{Im}B \oplus \sum_j \Gamma^{s_j}(Y,E_j), \]

putting

\[ (A^{-1},B,C)[u]=(A^{-1}u\,[\bmod B]\oplus Cu). \]

It should be noted that operators of the more general type

\[ \begin{pmatrix} A^{-1} & B\\ C & D \end{pmatrix} : \Gamma^s(X,E_1)\oplus \Sigma\Gamma^{t_k}(Y,F'_k) \to \Gamma^t(X,E_2)\oplus \Sigma\Gamma^{s_j}(Y,F_j), \]

where

\[ \begin{pmatrix} A^{-1} & B\\ C & D \end{pmatrix} [u\oplus v] = (Au+Bv\oplus Cu+Dv) \]

and \(A^{-1}\) is elliptic, also fall under our scheme. Put

\[ \sigma=\sigma_D-\int \sigma_C\circ\sigma_A\circ\sigma_B\,d\tau: \pi^*_{\eta Y}(\Sigma F_k)\to \pi^*_{\eta Y}(\Sigma F'_j). \]

The following simple result holds.

Theorem 2.

\[ I_a \begin{pmatrix} A^{-1} & B\\ C & D \end{pmatrix} = I_a(A^{-1})+I_a(D-C\circ A\circ B), \]

where \(\sigma_{D-C\circ A\circ B}=\sigma\); the case of the triple \((A^{-1},B,C)\) corresponds to \(D=0\), and the case of operators of type \((A^{-1},SD_F)\) reduces to operators of the form

\[ \begin{pmatrix} A^{-1} & \varkappa_F\\ i_F^* & 0 \end{pmatrix}. \]

We now return to the operators \(A\subset \mathrm{Op}^{s,t}(X,E)\) or \(A\in \mathrm{Op}^{s,t}_c(X,E)\). Let \(\tau\) be a \(2q'\)-dimensional direction in the real sense at a point \(x\in X\), and let \(E_x\) be the quotient bundle \(E\); \(E_{1x}\) and \(E_{2x}\) are the quotient bundles \(E_1,E_2\) at the points \(x_1\), where \(E_1\) and \(E_2\) are defined by the operator \(A\). We state an assertion useful from the point of view of geometric interpretation.

Lemma 2. The integral

\[ \int_{\xi=\mathrm{const}} \sigma_{i^*}\circ\sigma_A\circ\sigma_{\varkappa}\,d\tau \]

defines a bilinear product

\[ [a,b]\in \operatorname{Hom}_c(E_1,E_2)_x, \]

where

\[ a,b\in \sum_{j\le l-q-q'} S^j c(\tau\oplus E_x); \]

it is symmetric for \(E=0\), Hermitian for \(i_c^*,\varkappa_c\) and Kähler quotient bundles \(E_1,E_2,E\), if the operator \(A\) is naturally related to the Kähler structure. If this product

\[ [a,b]\in \operatorname{Hom}_c(E_1,E_2)_x \]

is nondegenerate in the sense that, for any

\[ a\in \sum_j S^j c(\tau\oplus E_x), \]

we have

\[ \bigcap_{b\in S^1 c(\tau\oplus E_x)} \operatorname{Ker}[a,b]^j=0, \]

then the direction \((\tau)\) is elliptic relative to the operator \(A\) (the triple \(A, i^*, \chi\)) or \((A, i_c^*, \chi_c)\).

Types of operators.

1. Real operators \(\operatorname{Op}^{s,t}_R(X,E) \xrightarrow{c} \operatorname{Op}^{s,t}(X,E)\) correspond to the fact that \(E_1=cE_1'\), \(E_2=cE_2'\) and the operator \(A\) is the complexification of a real one. Then the bilinear product \([a,b]\) indicated in Lemma 2 is real if \(a,b\in \Sigma S^j(\tau\oplus E_x)\). For scalar operators \(E_1'=E_2'=R\) the product \([a,b]\) is scalar and positive definite.

2. Hermitian operators \(A\in \operatorname{Op}^{s,t}_c(X,\xi)\), for which the product \([a,b]\) is Hermitian, \(a,b\in \Sigma S^j(\tau\oplus E)\), although it may also be expressed. The collection of such operators gives the Dolbeault operator (see below) on forms with coefficients in bundles. Here there is also the scalar case \(E=0\), \(E_1=E_2=C\) (more generally, \(E_1\) and \(E_2\) are one-dimensional). In all scalar cases the topological invariants of the operator \(i_{SDc}A\), \(i_{SD}A\) are equal to zero if \(\dim Y<\dim X\).

3. Group operators with coefficients in a bundle (operators associated with a \(G\)-structure in the terminology of \((^1)\). Let us indicate examples:

a) The Hirzebruch operator
\[ E_1=\Lambda^+\mathrm{cn}_X\otimes E,\qquad E_2=\Lambda^-\mathrm{cn}_X\otimes E,\qquad A^{-1}=(d+\delta)\Delta^m\otimes 1, \]
\[ A:\Gamma^{-m-1/2}(X,E_1)\to \Gamma^{m+1/2}(X,E_2),\qquad A\in \operatorname{Op}^{-m-1/2,m+1/2}(X,0). \]
Denote the element \(a(A)\in K(M\eta_X,*)\) by \(h_m(X,E)\), and for the adjoint operator \(A^*\) the element \(a(A^*)\) by
\[ h_m^*(X,E)=-h_m(X,E),\qquad h_m(X,E)=\delta(A)\big|_{t=1}. \]

b) The Euler operator:
\[ E_1=\Lambda^{\mathrm{even}}\mathrm{cn}_X\otimes E,\qquad E_2=\Lambda^{\mathrm{odd}}\mathrm{cn}_X\otimes E,\qquad A^{-1}=(d+\delta)\Delta^m, \]
\[ A:\Gamma^{-m-1/2}(X,E_1)\to \Gamma^{m+1/2}(X,E_2),\qquad a(A)=\chi_m(X,E), \]
\[ a(A^*)=\chi_m^*(X,E)=-\chi_m(X,E). \]
The Euler operator is real if \(E=cE'\). Here \(\chi_m(X,E)=\delta(A)\big|_{t=1}\).

c) The Todd operator
\[ E_1=\Lambda^{\mathrm{even}}\eta_X\otimes E,\qquad E_2=\Lambda^{\mathrm{odd}}\eta_X\otimes E;\qquad A^{-1}=(\partial+\bar\partial)\Box^m\otimes 1; \]
\[ A:\Gamma^{-m-1/2}(X,E_1)\to \Gamma^{m+1/2}(X,E_2);\qquad A\in \operatorname{Op}^{-m-1/2,m+1/2}(X,0); \]
\[ a(A)=t_m(X,E);\qquad a(A^*)=t_m^*(X,E)=-t_m(X,E); \]
\(X,E,E_1,E_2\) are Kähler; \(t_m(X,E)=\delta(A)\big|_{t=1}\).

d) The Dirac operator:
\[ E_1=\Delta^+\eta_X\otimes E,\qquad E_2=\Delta^-\eta_X\otimes E;\qquad A^{-1}=(d+d^*)[d+d^*]^{2m}\otimes 1, \]
where
\[ d:\Gamma(X,\Delta^+\eta_X)\to \Gamma(X,\Delta^-\eta_X) \]
is the ordinary Dirac operator;
\[ A:\Gamma^{-m-1/2}(X,E_1)\to \Gamma^{m+1/2}(X,E_2); \]
\(X,E,E_1,E_2\) are spinor. Put
\[ a(A)=d_m(X,E),\qquad a(A^*)=d_m^*(X,E)=-d_m(X,E). \]

From the preceding results one obtains, respectively, the following formulas for the embedding \(i:Y\subset X\), \(\operatorname{codim}_R Y=2q\):
\[ \text{a) }\quad \lambda_{-}^{R}tn\circ \delta(i_{SD}A) = \sum_{j\le m-q} t^j \left\{ h_{m-q-j}\left(Y,\overline{\Lambda cn}\otimes i^*E\right) + h_{m-q-j}^*\left(Y,\Lambda^+ cn\otimes iE\right) \right\}, \]
\[ \text{b) }\quad \lambda_{-}^{R}tn\circ \delta(i_{SD}A)=0 \quad \text{for the Euler operator, } \dim Y<\dim X, \]
\[ \text{c) }\quad \lambda_{-}tn\circ \delta(i_{SD_c}A) = \sum_{j\le m-q} t^j \left\{ t_{m-q-j}\left(Y,\Lambda^{\mathrm{odd}}\eta\otimes i^*E\right) + t_{m-q-j}^*\left(Y,\Lambda_n^{\mathrm{even}}\otimes i^*E\right) \right\}, \]
\[ \text{d) }\quad \lambda_{-}^{R}tn\circ \delta(i_{SD}A) = \sum_{j\le m-q} \left\{ d_{m-q-j}\left(Y,\Delta^{-}n\otimes i^*E\right) d_{m-q-j}^*\left(Y,\Delta^{+}n\otimes i^*E\right) \right\}t^j, \]
where \(E,Y,X\) are spinor. Note that \(\chi_m(X,E)\), \(h_m(X,E)\), \(t_m(X,E)\), \(d_m(X,E)\) do not depend on \(m\).

It is not difficult to expand these formulas for the cases \(2\dim Y=\dim X\) and \(\operatorname{codim}_R Y=2\). For example, in the first case one must use Corollary 1 of the authors’ paper \((^2)\) and the fact that
\[ \operatorname{ch}^0 A=2^n,\,0,\,(-1)^n,\,1 \]
respectively in cases a), b), c), d). For lack of space we do not present them.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Institute for Problems in Mechanics
Academy of Sciences of the USSR

Received
21 VI 1966

CITED LITERATURE

1. M. F. Atiyah, I. M. Singer, Bull. Am. Math. Soc., 69, No. 3, 422 (1963).
2. S. P. Novikov, B. Yu. Sternin, DAN, 170, No. 6 (1966).
3. B. Yu. Sternin, Tr. Mosk. matem. obshch., 15 (1966).