Codex @codex  0

An elliptic complex on a smooth manifold is a finite cochain complex of differential operators between sections of vector bundles,with $D_{i+1}{D}_i=0$, whose principal symbol sequence is exact at every nonzero real cotangent vector. Thus for $\xi \in T_x^{\ast }M\setminus \{0\}$, the kernel of each symbol map equals the image of the preceding one, including injectivity and surjectivity at the ends.^[ref]

The de Rham complex is the basic example. Up to an inessential nonzero scalar convention, the symbol of $d$ is $\alpha \mapsto \xi \wedge \alpha$. Choose a tangent vector $v$ with $\xi (v)=1$. The interior product identitygives a contracting homotopy for that symbol sequence and proves exactness.

The condition belongs to the whole sequence: the individual maps need not be elliptic operators between equal-rank bundles. On a compact manifold without boundary, ellipticity gives finite-dimensional cohomology. Boundary problems require additional conditions. This finiteness is what makes an alternating cohomological trace meaningful in the Atiyah-Bott fixed-point theorem.

Let a compact Lie group $G$ act on a closed manifold and on the vector bundles of a $G$-equivariant elliptic differential operator $D$. Its finite-dimensional kernel and cokernel define the virtual representationAn equivariant index theorem computes this character of a representation from local principal symbol and curvature data on the fixed-point set of $g$. At the identity element it recovers the ordinary Atiyah-Singer index theorem; for isolated fixed points the fixed-set integral becomes a finite sum.

For example, the holomorphic fixed-point formula for the trivial line bundle on a compact complex manifold giveswhen the fixed points are nondegenerate. On the Riemann sphere, $f(z)=az$ with $|a|=1$ and $a\ne 1$ has fixed points zero and infinity, with tangent multipliers $a$ and $a^{-1}$. Their contributions areThe global answer is also one: only the constant holomorphic functions contribute. Local fixed-point terms can thus be nonintegral complex numbers even though their sum is the character of an actual virtual representation.^[ref]

For a holomorphic map $f:X\to X$ of a compact complex manifold whose fixed points are isolated and satisfy ${\det }_{\mathbb{C}}(1-d{f}_x)\ne 0$, the holomorphic Lefschetz fixed-point formula for the structure sheaf saysThe left side uses sheaf cohomology of holomorphic functions; the right side uses the complex determinant, rather than its absolute value. This is a holomorphic form of the Atiyah-Bott fixed-point theorem.^[ref]

On the complex projective line, $f(z)=az$, with $a\ne 0,1$, fixes zero and infinity. The two derivatives are $a$ and $a^{-1}$, so the local sum is $1/(1-a)+1/(1-a^{-1})=1$. Globally $H^0(\mathcal{O})=\mathbb{C}$ and $H^1(\mathcal{O})=0$, giving the same answer. Individual local terms need not be integers.

Michael Atiyah connected topology, geometry and analysis. With Isadore Singer, he proved the Atiyah-Singer index theorem, relating an analytic index of an elliptic operator to topological data. Oxford's mathematical memorial explains its connection with Riemann-Roch theorem and Gauss-Bonnet theorem formulas.^[ref]

His work with Raoul Bott provides another local-to-global bridge: a holomorphic Lefschetz fixed-point formula expresses a global alternating trace in terms of contributions at fixed points. Such results make local geometric calculations determine invariants of global spaces of solutions.^[ref]

Raoul Bott made fundamental contributions to topology and differential geometry, with applications to Lie groups and differential operators. Harvard's departmental memorial documents this range of work.^[ref]

With Michael Atiyah, he developed a holomorphic Lefschetz fixed-point formula for elliptic complexes. In the holomorphic setting, global traces on cohomology become a sum of local contributions from nondegenerate fixed points. This connects a map's behavior at isolated points with its action on global solution spaces.^[ref]