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SBU MAT 545 - Problem set 7

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MAT 545: Complex GeometryFall 2008Problem Set 7Due on Monday, 12/15, at 2:20pm in Math P-131(or by 2pm on 12/15 in Math 3-111)Please write up clear and concise solutions to problems worth 20 pts.Problem 1 (10 pts)(a) Let Xa⊂P3be a smooth hypersurface of degree a≥1. Show thatdimCH0¯∂(Xa; KXa) =(0, if a≤3;1, if a=4.Determine the Hodge diamonds for Xawith a≤4.(b) Let Ya⊂ P4be a smooth hypersurface of degree a ≥ 1. Determine the Hodge diamonds for Yawith a≤5.Note: the quartic surface X4⊂ P3is a K3 surface; the quintic Y5⊂ P4is a Calabi-Yau 3-fold,popular in string theory.Problem 2 (5 pts)Let u : P1−→ Pnbe a holomorphic map of degree d (thus, u∗[P1] = d[P1] ∈ H2(Pn)). If d ≤ n, showthat u(P1) is contained in some linearly embedded Pdin Pn.Note: this is a special case of the Castenuovo bound. It implies for example that every degree 2(rational) curve in P3is in fact contained in some hyperplane P2⊂P3. This makes it possible to useclassical Schubert calculus (homology intersections on G(k, n)) to determine the number of suchconics in P3that pass through a points and 8−2a lines in general position.Problem 3 (5 pts)Let Σ be a compact connected Riemann surface (complex one-dimensional manifold). Show thatΣ can be holomorphically embedded into PNfor some N.Problem 4 (5 pts)Let M be a complex manifold of dimension at least 2 and x ∈ M. Show that the sheaf IxofO-modules is not isomorphic to the sheaf of holomorphic sections of any line bundle L−→M.Note: Recall that for any open subset U ⊂M,Ix(U) =f ∈O(U): f(x)=0 if x∈U;this is a module over the ring O(U).Problem 5 (10 pts)Let Γ be a complete lattice in C2(i.e. the Z-span of 4 R-linearly independent vectors v1, . . . , v4∈C2).Thus, M ≡C2/Γ is diffeomorphic to (S1)4.(a) Show that the Kahler structure (complex structure and symplectic form) on C4induce a Kahlerstructure on M. Describe a basis for H2(M; Z).(b) Find a lattice Γ so that H1,1(M; Z) = {0} and thus M is not projective (cannot be embeddedinto PNfor any N).Problem 6 (10 pts)(a) Let C ⊂P3be a complete intersection of bi-degree (a, b) (so C = s−1(0), where s is a holomorphicsection of the bundle O(a)⊕O(b)−→P3which is transverse to the zero set). Determine the degreeof C in P3and the genus of C.(b) If C ⊂ P3is a smooth rational curve of degree 3 (thus, C ≈ P1and [C] = 3[P1] ∈ H2(P3)) andC is not contained in any hyperplane P2of P3, then C is not a complete intersection in P3. Showthat such a curve C actually

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