## Note: Oral Exam Done

So I was able to relax a little for the one or two days leading up to the exam. I felt that I had thoroughly learned information that was pertinent.

At 10:00 AM sharp I present the first Chern class of a line bundle and prove that \(c_1(L)=\Theta\). This takes about twenty minutes and at the end I state the result that \(c_1([D])=\eta_D\).

At first Dr. Shiffman asked me to show this result. But his real question was about showing the Poincare duality between \(\eta_D\) and \((D)\). I wrote that you define \((\eta,\xi)\mapsto \int_M \eta \wedge \xi\) and then using the isomorphisms between the cohomology and homology, we see that this is the intersection number of the corresponding homology classes and use the result of Poincare duality there.

Moving on from this, Dr. Shiffman asked if given two line bundles with the same Chern classes, are the line bundles the same. I strongly answered, "No." He asked to give an example but I didn't know one. He had me write up the long exact sequence, which I knew and wrote up easily. And I understood the obstacle to Chern classes determining line bundles, but didn't know what he was looking for. He helped me define \(\mbox{Pic}_0(M)\) the set of line bundles which go to 0 under the Bockstein map.

Then he asked me when a line bundle associated to a divisor is trivial. I answered correctly. Then he asked about the Chern class of trivial line bundles on a Riemann surface or something similar and I answered correctly, degree must be 0.

Then he gave me the goal of finding a function, on a torus, with degree 0, but not given by a global meromorphic function. I figured you should take two different points and subtract them, I took \(\left(\frac{i}{2}\right)-\left(\frac{1}{2}\right)\). This was true, but when asked how does this show what we want, I wasn't sure. This time I gave the wrong answer, writing the divisor as a quotient. I'll have to figure out why you can't decompose a global section as such, but at the moment of writing this it sort of seems silly to think that it's possible. He told me the answer was related to using the Cauchy integral formula. Integrating around the torus to get 0, and contradicting the simple pole of such a divisor.

I was asked about Hodge theory to compute $H^1(M,\mathcal{O})$ for M a torus. But I had know I idea, trying to think if I can express \(T\) as \(\mathbb{C}^k \times \mathbb{C}\setminus {0}\). Then he asked instead if I can do it for a Kahler manifold M. I was trying to think about \(\mathcal{O}=\Omega^0\). So then he saw this and saw I was trying to use the Dolbeault cohomology groups and helped me in that direction. First notationally. Then asking about the Betti numbers relation to the groups. I gave the correct summation and then he asked for the first Betti number, so I wrote that particular equation. And then he said what about the torus. I said 2. So \(\dim H^{(1,0)}_{\bar{\partial}}(T)+\dim H^{(0,1)}_{\bar{\partial}}(T)=2\) But he asked if there was a relationship between the two groups and I said their dimension are equal so yes that implies each is dimension 1. So \(H^1(T,\mathcal{O})\cong \mathbb{C}\). And then Dr. Shiffman came up to write that \(H^1(T,\mathbb{Z})\cong \mathbb{Z}\times\mathbb{Z}\) and so this leads to an understanding of \(H^1(T,\mathcal{O}^\ast)=Pic(T)\).

Next I explained the Monge-Ampere operator, which was asked by Dr. Spruck. I defined \(dd^c(u)\) for \(u\) plurisubharmonic and then moved on to inductively defining \(dd^c(u_1)\wedge\dots\wedge dd^c(u_n)\wedge T\).

I stated the Bedford-Taylor theorem.

Then there's Poincare-Lelong formula from Griffiths and Harris. And then Dr. Shiffman asked about the one for sections. At first I was a little confused, but he helped me out saying to write \(s=fe\) and then I understood what he was leading me through. \(\partial\bar{\partial}\log|s|^2=\partial\bar{\partial}\log|s|^2\partial\bar{\partial}\log|f|^2+\partial\bar{\partial}\log|e|^2=\mbox{div}(s)-\frac{\sqrt{-1}}{2\pi}\Theta\)

Then he asked about the version of Poincare-Lelong formula from his paper, but I couldn't remember. He discussed it.

End exam at about 11:15 AM. Dr. Shiffman asked me to leave the room and let them discuss for a minute. Soon enough Dr. Shiffman comes out to tell me I passed. Yay!

At 10:00 AM sharp I present the first Chern class of a line bundle and prove that \(c_1(L)=\Theta\). This takes about twenty minutes and at the end I state the result that \(c_1([D])=\eta_D\).

At first Dr. Shiffman asked me to show this result. But his real question was about showing the Poincare duality between \(\eta_D\) and \((D)\). I wrote that you define \((\eta,\xi)\mapsto \int_M \eta \wedge \xi\) and then using the isomorphisms between the cohomology and homology, we see that this is the intersection number of the corresponding homology classes and use the result of Poincare duality there.

Moving on from this, Dr. Shiffman asked if given two line bundles with the same Chern classes, are the line bundles the same. I strongly answered, "No." He asked to give an example but I didn't know one. He had me write up the long exact sequence, which I knew and wrote up easily. And I understood the obstacle to Chern classes determining line bundles, but didn't know what he was looking for. He helped me define \(\mbox{Pic}_0(M)\) the set of line bundles which go to 0 under the Bockstein map.

Then he asked me when a line bundle associated to a divisor is trivial. I answered correctly. Then he asked about the Chern class of trivial line bundles on a Riemann surface or something similar and I answered correctly, degree must be 0.

Then he gave me the goal of finding a function, on a torus, with degree 0, but not given by a global meromorphic function. I figured you should take two different points and subtract them, I took \(\left(\frac{i}{2}\right)-\left(\frac{1}{2}\right)\). This was true, but when asked how does this show what we want, I wasn't sure. This time I gave the wrong answer, writing the divisor as a quotient. I'll have to figure out why you can't decompose a global section as such, but at the moment of writing this it sort of seems silly to think that it's possible. He told me the answer was related to using the Cauchy integral formula. Integrating around the torus to get 0, and contradicting the simple pole of such a divisor.

I was asked about Hodge theory to compute $H^1(M,\mathcal{O})$ for M a torus. But I had know I idea, trying to think if I can express \(T\) as \(\mathbb{C}^k \times \mathbb{C}\setminus {0}\). Then he asked instead if I can do it for a Kahler manifold M. I was trying to think about \(\mathcal{O}=\Omega^0\). So then he saw this and saw I was trying to use the Dolbeault cohomology groups and helped me in that direction. First notationally. Then asking about the Betti numbers relation to the groups. I gave the correct summation and then he asked for the first Betti number, so I wrote that particular equation. And then he said what about the torus. I said 2. So \(\dim H^{(1,0)}_{\bar{\partial}}(T)+\dim H^{(0,1)}_{\bar{\partial}}(T)=2\) But he asked if there was a relationship between the two groups and I said their dimension are equal so yes that implies each is dimension 1. So \(H^1(T,\mathcal{O})\cong \mathbb{C}\). And then Dr. Shiffman came up to write that \(H^1(T,\mathbb{Z})\cong \mathbb{Z}\times\mathbb{Z}\) and so this leads to an understanding of \(H^1(T,\mathcal{O}^\ast)=Pic(T)\).

Next I explained the Monge-Ampere operator, which was asked by Dr. Spruck. I defined \(dd^c(u)\) for \(u\) plurisubharmonic and then moved on to inductively defining \(dd^c(u_1)\wedge\dots\wedge dd^c(u_n)\wedge T\).

I stated the Bedford-Taylor theorem.

Then there's Poincare-Lelong formula from Griffiths and Harris. And then Dr. Shiffman asked about the one for sections. At first I was a little confused, but he helped me out saying to write \(s=fe\) and then I understood what he was leading me through. \(\partial\bar{\partial}\log|s|^2=\partial\bar{\partial}\log|s|^2\partial\bar{\partial}\log|f|^2+\partial\bar{\partial}\log|e|^2=\mbox{div}(s)-\frac{\sqrt{-1}}{2\pi}\Theta\)

Then he asked about the version of Poincare-Lelong formula from his paper, but I couldn't remember. He discussed it.

End exam at about 11:15 AM. Dr. Shiffman asked me to leave the room and let them discuss for a minute. Soon enough Dr. Shiffman comes out to tell me I passed. Yay!

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