MAT 545: Complex GeometryFall 2008Problem Set 1Due on Tuesday, 9/16, at 2:20pm in Math P-131(or by 2pm on 9/9 in Math 3-111)Please write up concise solutions to problems worth 20 points, including one of the 10-pointproblems.Problem 1 (10 pts)Let U ⊂Cnbe a connected open subset.(a) Unique Continuation: If f, g : U −→ C are holomorphic functions and V ⊂ Cnis anonempty open subset such that f|V=g|V, then f =g.(b) Maximum Principle: If f : U −→C is a holomorphic function and maxz∈U|f(z)|=|f(z0)|for some z0∈U, then f is a constant function.(c) Elliptic Regularity: If f : U −→ C is a holomorphic function (and thus f is assumed tobe C1), then f is smooth.Problem 2 (5 pts)Let f(z, w) = sin(w2) − z. Find the Weierstrass polynomialg(z, w) = wd+ a1(z)wd−1+ . . . + ad(z)such that f = g · h near (z, w) = (0, 0 with h(0, 0) 6= 0.Problem 3 (10 pts)Let R be an integral domain, i.e. a commutative ring with identity such that fg 6=0 wheneverf, g ∈R−0.• An element u ∈ R is a unit if u is invertible in R, i.e. uv = 1 for some v ∈R;• An element u ∈ R is irreducible if u is not a unit and u=vw for some v, w ∈ R impliesthat v or w is a unit;• An element u ∈ R is prime if u is not a unit and uz = vw for some v, w, z ∈ R impliesthat either v =z0u or w =z0u for some z0∈R;• R is a principal ideal domain (PID) if every ideal is principal, i.e. of the form pR forsome p∈R;• R is a unique factorization domain (UFD) if for every f ∈ R such that f is not a unitthere exist irreducible elements f1, . . . , fk∈ R such that f = f1. . . fkand f1, . . . , fkare uniquely determined by f up to a permutation and multiplication by units in R;• A polynomial f = a0+ a1x + . . . ∈ R[x] is primitive if only the units in R divide allthe coefficients a0, a1, . . ..Show that:(a) If R is an integral domain and p ∈ R is prime, then R/pR is an integral domain.(b) Any prime element of R is irreducible. If R is UFD, every irreducible element is prime.(c) If R is UFD and f, g ∈ R[x] are primitive, then fg is primitive.(d) If R is UFD, F is the field of fractions of R, and f ∈ R[x] is irreducible, then f is alsoirreducible in F [x].(e) If R is PID, every irreducible element is prime and R is a UFD.(f) If F is a field, then F [x] is a PID.(g) If R is UFD, so is R[x].(h) If R is UFD and f, g ∈ R[x] are relatively prime (have no common divisors other thanunits), there exist relatively prime α, β ∈R[x] and γ ∈ R−0 such thatαf + βg = γ.Problem 4 (5 pts)Find a value of τ such that the tori C/(Z⊕τZ) and C∗/(z ∼2z) are isomorphic as Riemannsurfaces.Problem 5 (5 pts)Show the complex projective space Pnand the total space of the tautological line bundleγ ≡(`, v)∈Pn×Cn+1: v ∈`−→ Pnare complex manifolds. Describe transition maps