Math 524
Homework 3
Let Xbe a compact complex manifold and Ea holomorphic vector bundle
on X. Let cE(t) := Pk
i=1 ci(E)tibe the Chern polynomial of E, where k=
rk E.
Solve at least 3 of the following ex.
When solving ex. 1 and 2, prove the statement in the case when rk E=1
first. Then, in the case of a vector bundle Eof rank >1, use the splitting
theorem for Etogether with the splitting and functoriality properties of the
Chern polynomials.
1. Consider a generic section s∈H0(X, E). Prove that ck(E) is the
Poincare dual of the fundamental class [Z], where Z⊂Xis the locus of
points where the section sis zero.
Recall that in the case when Eis a line bundle,
c1(E) = 1
2πi [¯
∂∂ log |s|2].
2. a) Let Lbe a holomorphic line bundle on X. Compute cE⊗L(t) in
terms of cE(t) and c1(L).
b) Show that c1(E) = c1(VkE), where k= rk E.
3. If Xis a Riemann surface, then integral along Xgives an isomorphism
H2(X;Z)→Z, called the degree map.
Let Xbe a Riemann surface with local coordinate z, let h2dz ⊗d¯zbe
a Hermitian metric on TX. Let dA be the volume form, Kthe Gaussian
curvature of Xand Θ the curvature of the metric connection. Show that
Θ = −2πiKdA and use the Gauss-Bonnet theorem RXKdA = 2πχ(X) to
compute the degree of c1(X) := c1(TX) in terms of the Euler number of X.
(Note: a formula for the Gaussian curvature may be found in [GH], pg.77.
4. Use the Euler sequence from hw. 2 to compute the Chern polynomial
of Pn(defined as the Chern polynomial of its tangent bundle), in terms of
the Poincare dual for the hyperplane class [H].
5. Let Xbe a smooth surface in P4, given as the set of zeroes of a global
section in the vector bundle E→P4. Compute the Chern classes of Xin
terms of the Chern classes of Eand [H]. You will need to use ex.6 in hw. 2.
6. Let n>kbe two positive integers. The Grassmanian G(k, n) is defined
as a set by
G(k, n) = {L⊂Cn|Lis a kdimensionalC−vector subspace of Cn}.
As well, G(k, n) = {P⊂Pn−1|Lis a k−1 dimensional linear subspace of Pn−1}.
a) The Plucker map. Fix the canonical basis {e1, ..., en}in Cn. To any
element L∈G(k, n) given by a basis {v1, ..., vk} ⊂ Cnassociate v1∧...∧vk∈
V:= VkCn. Show that this induces a well defined injective map
µ:G(k, n),→P(V).
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