Sheaf cohomology / 03

Picard group and degree

The Picard group remembers holomorphic line bundles under tensor product; degree is its first numerical shadow.

Sections 6.3 and 6.6-6.8 of the PDF identify holomorphic line bundles with multiplicative Cech cohomology and compute the basic model \(\mathcal O(n)\) on \(\mathbb{CP}^1\).

Picard group

The Picard group of a complex manifold \(X\) is

\[\operatorname{Pic}(X)= \{\text{holomorphic line bundles on }X\}/\cong\]

with group law induced by tensor product. In transition functions,

\[[g_{ij}]+[h_{ij}]=[g_{ij}h_{ij}],\qquad -[g_{ij}]=[g_{ij}^{-1}].\]

The classification by cocycles gives

\[\operatorname{Pic}(X)\cong H^1(X,\mathcal O_X^*).\]

This is not a slogan. It says exactly that a line bundle is a 1-cocycle with values in invertible holomorphic functions, modulo gauge changes.

Degree on compact Riemann surfaces

If \(L\cong\mathcal O(D)\), define

\[\deg L=\deg D.\]

This is well-defined because principal divisors have degree zero. It satisfies

\[\deg(L\otimes M)=\deg L+\deg M, \qquad \deg(L^\vee)=-\deg L.\]

Why principal divisors have degree zero

For a meromorphic function \(f:X\to\mathbb{CP}^1\) on a compact Riemann surface, the number of preimages of a generic value, counted with multiplicity, is constant. Counting preimages of \(0\) and \(\infty\) gives the same number:

\[\sum_{f(p)=0}\operatorname{ord}_p(f)= \sum_{f(p)=\infty}-\operatorname{ord}_p(f).\]

Therefore \(\deg((f))=0\).

The bundles \(\mathcal O(n)\) on \(\mathbb{CP}^1\)

Let \(\mathcal O(1)\) be the hyperplane bundle. Its powers are

\[\mathcal O(n)=\mathcal O(1)^{\otimes n}.\]

On the two standard charts \(U_0,U_\infty\), the section convention of Article 01 gives compatibility by \(z^{-n}\) on \(\mathbb C^*\); the reciprocal frame convention uses \(z^n\). The global sections are homogeneous polynomials of degree \(n\):

\[a_0Z_0^n+a_1Z_0^{n-1}Z_1+\cdots+a_n Z_1^n.\]

Thus

\[h^0(\mathbb{CP}^1,\mathcal O(n))= \begin{cases} n+1,& n\ge0,\\ 0,& n<0. \end{cases}\]

Example 1: \(\mathcal O(2)\) and quadratic sections

A section of \(\mathcal O(2)\) is a quadratic homogeneous form

\[aZ_0^2+bZ_0Z_1+cZ_1^2.\]

On \(U_0\) this becomes \(a+bz+cz^2\). Its zero divisor has degree two, counting multiplicity. For instance \(Z_1^2\) has divisor \(2[0]\), while \(Z_0Z_1\) has divisor \([0]+[\infty]\).

Example 2: no global sections of \(\mathcal O(-1)\)

A holomorphic section of \(\mathcal O(-1)\) would have an effective zero divisor whose degree is \(\deg\mathcal O(-1)=-1\). Effective divisors have nonnegative degree, so no nonzero holomorphic section can exist. Therefore \(H^0(\mathbb{CP}^1,\mathcal O(-1))=0\).

Degree does not classify everything

On \(\mathbb{CP}^1\), degree classifies line bundles:

\[\operatorname{Pic}(\mathbb{CP}^1)\cong\mathbb Z, \qquad \mathcal O(n)\leftrightarrow n.\]

On higher genus Riemann surfaces, degree is not enough. The kernel of the degree map,

\[\operatorname{Pic}^0(X)=\ker(\deg:\operatorname{Pic}(X)\to\mathbb Z),\]

is the Jacobian variety. It contains many nontrivial degree-zero bundles.

Example 3: degree zero on an elliptic curve

Let \(E\) be an elliptic curve. The line bundle \(\mathcal O([p]-[0])\) has degree zero. It is trivial precisely when \([p]-[0]\) is a principal divisor, which happens exactly for \(p=0\) in the elliptic curve group law. Varying \(p\) gives the standard isomorphism \(E\cong\operatorname{Pic}^0(E)\), \(p\mapsto\mathcal O([p]-[0])\); the calculation above identifies its kernel.

Example 4: canonical degree

The canonical bundle \(K_X\) has degree

\[\deg K_X=2g-2.\]

For \(\mathbb{CP}^1\), this gives \(\deg K=-2\). For an elliptic curve, \(\deg K=0\) and in fact \(K_E\) is trivial because \(dz\) descends from \(\mathbb C\) to \(E=\mathbb C/\Lambda\).

The Picard group as a refinement of degree

Degree is numerical, while the Picard group is geometric. A divisor first gives a line bundle, then degree forgets most of that line bundle. Riemann-Roch depends on both layers: degree gives the leading dimension count, and the special contribution comes from the residual line bundle \(K-D\).