## ICTP Day 1

So, I’ve put the actual lecture notes below the fold.  Today was a back-to-basics day, to make sure everyone has the same definitions (and I think to give people time to get used to the area before diving into the deep end, and giving me a chance to get used to the 6 hour time difference from home.)  Tu talked about manifolds a bit, Cattani spoke a bit about complex manifolds, line bundles and complex structures, and Trang spoke a bit about algebraic varieties.  Nothing really exciting.  I should note that the lecturer’s notes are, in many cases, on the web already, here, and I’ll be posting brief summaries before the notes each day, and when a lecture series finishes, I’ll be compiling those notes into a single pdf and posting them to my personal website, with links here.  So, here are the notes, please, leave comments with any corrections or comments.

1. Topology, cohomology and sheaf theory 1 – Tu

1.1. Manifolds

A topological space is locally Euclidean if every ${p\in M}$ has a neighborhood ${U}$ and a homeomorphism ${\phi:U\rightarrow V}$, where ${V}$ is an open subset of ${\mathbb{R}^n}$. We call the pair ${(U,\phi)}$ a chart.

Two charts are ${C^\infty}$ compatible if ${\phi\circ\psi^{-1}}$ and ${\psi\circ\phi^{-1}}$ are ${C^\infty}$ functions.

A manifold is a locally Euclidean, Hausdorff, second countable topological space on which there existsa covering by ${C^\infty}$ compatible charts.

The circle is given by ${x^2+y^2=1}$. Taking the upper and lower semicircles, along with the left and right semicircles, provides the desired cover.

1.2. Tangent Space

Let ${r^i}$ be coordinates on an open set in ${\mathbb{R}^n}$ and ${x^i}$ be ${r^i\circ\phi}$.

${f:M\rightarrow\mathbb{R}}$ is ${C^\infty}$ at ${p}$ if there exists ${(U,\phi)}$ with ${p\in U}$ such that ${f\circ\phi^{-1}}$ is ${C^\infty}$ at ${\phi(p)\in \mathbb{R}^n}$.

If ${f\in C^\infty(M)}$ and ${(U,x^1,\ldots,x^n)}$ is a chart containing ${p}$, then we define ${\frac{\partial }{\partial x^i}|_pf=\frac{\partial}{\partial r^i}|_{\phi(p)}f\circ\phi^{-1}}$

${T_pM}$, the tangent space at ${p}$, is the vector space with basis ${\frac{\partial}{\partial x^i}}$ at ${p\in (U,\phi)}$.

Let ${x^i}$ and ${y^j}$ be different sets of coordinates at a point ${p}$. Then ${\frac{\partial}{\partial y^j}=\sum a^i_j\frac{\partial}{\partial x^i}}$, where ${a^i_j=\frac{\partial x^i}{\partial y^j}}$.

The proof is a direct computation.

A vector field is a function ${X:M\rightarrow \coprod_{p\in M} T_pM}$ such that ${X_p\in T_pM}$. We can write it at ${p}$ as ${X=\sum a^i\frac{\partial}{\partial x^i}}$ and we call ${X}$ a ${C^\infty}$ vector field if the ${a^i}$ are all ${C^\infty}$.

1.3. Differential Forms

The cotangent space is ${T^*_pM=\hom(T_pM,\mathbb{R})}$. It has basis ${(dx^i)_p}$ the duals of ${\frac{\partial}{\partial x^i}}$‘s at ${p}$.

A 1-form is a function ${\omega:M\rightarrow \coprod_{p\in M} T^*_pM}$ such that ${\omega_p\in T_p^*M}$. We can write them near ${p}$ as ${\omega=\sum a^i (dx^i)_p}$ and call them ${C^\infty}$ if the ${a^i}$ are.

Let ${V}$ be a real vector space, and let ${\alpha,\beta\in V^*}$, then ${(\alpha\wedge\beta)(v,w)=\alpha(v)\beta(w)-\alpha(w)\beta(v)}$.

It then follows that ${\beta\wedge\alpha=-\alpha\wedge\beta}$ and ${\alpha\wedge\alpha=0}$.

Let ${A_k(V)}$ be the set of alternating functions on ${V}$ with ${k}$ inputs. If ${\alpha\in A_k}$ and ${\beta\in A_\ell}$ then ${\alpha\wedge\beta\in A_{k+\ell}}$ and ${\beta\wedge\alpha=(-1)^{k\ell}\alpha\wedge\beta}$

We define ${A_0(V)=\mathbb{R}}$.

A ${k}$-form on ${M}$ is a map ${\omega:M\rightarrow \coprod_{p\in M} A_k(T^*_pM)}$ such that ${\omega_p\in A_k(T_p^*M)}$. It is called ${C^\infty}$ if the coefficient functions are.

We denote ${\mathcal{A}^i(M)}$ to be the ${C^\infty}$ ${i}$-forms on ${M}$. We have that ${\mathcal{A}^0(M)=C^\infty(M)}$, and ${\mathcal{A}^k(M)=0}$ for ${k}$ greater than the dimension of ${M}$.

1.4. Exterior Derivative

If ${f\in C^\infty(M)}$, define a 1-form ${df\in \mathcal{A}^1(M)}$ on a chart ${(U,x^i)}$ by ${df=\sum \frac{\partial f}{\partial x^i}dx^i}$. More generally, let ${\omega\in \mathcal{A}^k(M)}$ by ${\omega=\sum a_Idx^I}$, then we define ${d\omega=\sum d(a_I)\wedge dx^I}$.

The exterior derivative ${d:\mathcal{A}^k(M)\rightarrow \mathcal{A}^{k+1}(M)}$ satisfies

1. Antiderivation: ${d(\omega\wedge \tau)=(d\omega)\wedge \tau+(-1)^{\deg \tau}\omega\wedge d\tau}$
2. ${d^2=0}$.

1.5. DeRham Cohomology

We define the kernel of ${d}$ to be the closed ${k}$-forms ${Z^k(M)}$ and the image to be the exact ${k}$-forms, ${B^k(M)}$.

${H^k(M)=Z^k(M)/B^k(M)}$.

${H^0(M)}$ is just ${\ker d:C^\infty(M)\rightarrow \mathcal{A}^1(M)}$, because there is no ${\mathcal{A}^{-1}(M)}$. So we end up with the set of locally constant functions on ${M}$, and, because ${M}$ is a manifold, ${H^0(M)}$ is the set of all functions constant on the connected components.

${H^0(M)=\mathbb{R}^r}$ where ${r}$ is the number of connected component of ${M}$.

An element of ${\mathcal{A}^1(\mathbb{R})}$ is just ${f(x)dx}$ where ${f\in C^\infty(\mathbb{R})}$, and element of ${d\mathcal{A}^0(\mathbb{R})}$ is ${dg=g'(x)dx}$. So is every ${C^\infty}$ function equal to ${g'(x)}$ for some ${g\in C^\infty(\mathbb{R})}$? Yes, by the fundamental theorem of calculus, so ${H^k(\mathbb{R})}$ is ${\mathbb{R}}$ for ${k=0}$ and is ${0}$ else.

2. Complex manifolds, Kahler metrics, differential and harmonic forms 1 – Cattani

A complex manifold is a manifold with coordinates holomorphic on ${\mathbb{C}^n}$ rather than ${C^\infty}$ on ${\mathbb{R}^n}$.

What is the difference betwene holomorphic and ${C^\infty}$?

From the PDE point of view, they must satisfy the Cauchy-Riemann equations: ${f(z)=u+iv}$ is holomorphic if and only if ${\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}}$ and ${\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}}$.

The fact that this causes the function ${f}$ to be analytic(holomorphic) is what gives the theory a very algebraic flavor.

We get a map ${S^2\setminus \{N\}\rightarrow \mathbb{C}}$ and another ${S^2\setminus \{S\}\rightarrow \mathbb{C}}$ by stereographic projection. The map from ${S^2\setminus \{N,S\}}$ to itself that takes the image of the first to the image of the second is ${\frac{1}{z}}$ on ${\mathbb{C}^*}$, which is holomorphic. So what are the holomorphic functions? Any holomorphic function gives a bounded entire function by removing a point, and so must be constant! This is in fact true for holomorphic functions on compact connected complex manifolds.

The only global holomorphic functions on a compact connected complex manifold are the constants.

There are no compact complex submanifolds on ${\mathbb{C}^n}$ of dimension greater than 0.

Set ${\mathbb{P}^n=\mathbb{C}^{n+1}\setminus \{0\}/\sim}$ where ${z'\sim z}$ iff there exists ${\lambda\in\mathbb{C}^*}$ such that ${z'=\lambda z}$. We put charts on it by looking at ${U_i=\{[z]\in\mathbb{P}^n|z_i\neq 0\}}$.

When is a compact complex manifold a submanifold of ${\mathbb{P}^n}$?

We will also be interested in

${G(k,n)}$ is the space of ${k}$-dimensional subspaces of ${\mathbb{C}^n}$.

Two manifolds may be ${C^\infty}$ diffeomorphic but not holomorphic.

As a ${C^\infty}$ manifold, a torus is just ${S^1\times S^1}$. However, it can have many complex structures. To get one, view it as ${\mathbb{C}/\Lambda}$ where ${\Lambda}$ is a lattice isomorphic to ${\mathbb{Z}^2}$. There are many different lattices, and so we can get many complex structures this way. It is not hard to see that they are distinct.

The theory of complex manifolds splits into compact and noncompact, we’re only going to look at the compact complex manifolds.

So what do we do? We have no holomorphic functions. A function can be viewed as being a map ${M\rightarrow M\times \mathbb{C}}$ by ${x\mapsto (x,f(x))}$. To get something like a function, we can change ${M\times\mathbb{C}}$ to something that is only locally this product, that is, a line bundle.

A holomorphic line bundle is ${E\stackrel{\pi}{\rightarrow}M}$ where ${M}$ And ${E}$ are complex manifolds and ${\pi}$ is holomorphic such that the fiber over each point is isomorrphic to ${\mathbb{C}}$ and it is locally biholomorphic to ${U_\alpha\times\mathbb{C}}$.

Taking a trivializing open cover ${U_\alpha}$, we get maps ${U_\alpha\cap U_\beta\rightarrow \mathrm{GL}(1,\mathbb{C})}$ (or ${n}$ in the case of more general vector bundles) which we will call ${g_{\alpha\beta}}$, and they will satisfy ${g_{\alpha\beta}g_{\beta\gamma}=g_{\alpha\gamma}}$, these are called the gluing data.

We can get another bundle using ${^tg_{\alpha\beta}^{-1}}$, and these will give the dual bundle, whose fibers are naturally the dual of the original, and call it ${E^*}$. Similarly, we can take ${E,F}$ and put the transition functions in blocks, and we get ${E\oplus F}$. For line bundles, the transition functions are numbers, not matrices, so their product is another invertible function, so we get a line bundle ${E\otimes F}$.

We set ${\mathcal{T}=\{([z],v)|v\in[z]\}\subset \mathbb{P}^n\times\mathbb{C}^{n+1}\}}$ and its projection map to ${\mathbb{P}^n}$ makes it a line bundle with transition maps ${z_i/z_j}$.

So, we replace the notion of a global function with a global section of a line bundle ${E\rightarrow M}$, which is a map ${M\rightarrow E}$ such that ${x\in E_x}$. This means that on each ${U_\alpha}$ we have a function ${f_\alpha}$ and ${f_\alpha=g_{\alpha\beta}f_\beta}$.

Let ${P(x_0,\ldots,z_n)}$ be a homogeneous polynomial of degree ${k}$. Then on ${U_i}$, set ${f_i=P/z_i^k}$, these form a global section of ${(\mathcal{T}^*)^k}$. But, if we remove the dual, there are no global sections.

3. Algebraic varieties and schemes over any scheme. Non singular varieties 1 – Trang

Let ${k}$ be a field and ${k[x_1,\ldots,x_n]}$ the polynomial ring with coefficients in ${k}$. Then we have two objects: polynomials ${P\in k[x_1,\ldots,x_n]}$ and polynomial functions obtained by turning polynomials into maps ${k^n\rightarrow k}$. For finite fields, these are very different, though they are the same for infinite fields.

A subset ${E\subset k^n}$ is called an algebraic set if it is equal to the zero locus of a collection of polynomials.

${k[x_1,\ldots,x_n]}$ is Noetherian.

So any collection of polynomials has zero locus determined by finitely many of them.

3.1. Zariski Topology

Closed sets are the algebraic sets, ${\emptyset}$ and ${k^n}$, and open sets are the complements. This is actually non-Hausdorff.

Let ${k=\bar{k}}$, then if ${P}$ vanishes on an algebraic set ${E=V(I)}$, then some power of ${P}$ is in the ideal ${I}$. In particular, this implies that ${I(E)=\sqrt{I}}$.

Let ${E=\cup E_i}$ where the ${E_i}$ are irreducible, that is, are not the union of two properly contained algebraic sets. Then, ${E_i}$ is irreducible if and only if ${I(E_i)}$ is prime.

If ${f=P|_E}$, we call it an algebraic function, and there are also regular functions.

The set of algebraic functions on ${E}$ is isomorphic to ${k[x_1,\ldots,x_n]/I(E)=A(E)}$.

${\phi}$ is regular at ${x\in E}$ if there exists a Zariski neighborhood of ${x}$ such that ${\phi=f/g}$ on it.

${\phi}$ is regular on ${E}$ if it is regular at each ${x\in E}$.

The set of regular functions on ${E}$ is ring isomorphic to ${A(E)}$.

A morphism of varietes ${\phi:E\rightarrow F}$ is a continuous function which pulls regular functions back to regular functions ${A(F)\rightarrow A(E)}$.

3.2. Projective varieties

${\mathbb{P}^n_k=k^{n+1}\setminus \{0\}/\sim}$ where ${x\sim y}$ if there exsits ${\lambda\in k^*}$ such that ${x=\lambda y}$. Now, we look at ${R=k[x_0,\ldots,x_n]}$ as a graded ring, with gradations the polynomials homogeneous of a given degree ${d}$.

Let ${S}$ be a set of homogeneous polynomials. Then ${V(S)}$ the algebraic projective set of zeroes. These give a Zariski topology on ${\mathbb{P}^n_k}$, and we see that ${I(E)}$ is a homogeneous ideal and we define the coordinate ring to be ${R/I(E)}$.

For projective algebraic sets, a regular function is one that is locally ${f/g}$ with ${\deg f=\deg g}$ and ${g}$ not vanishing on the neighborhood.

An algebraic variety is then a Zariski open subset of a projective variety, and this gives us a category of algebraic varieties.

3.3. Sheaves

Let ${X}$ be a topological space and look at the category of open sets. A presheaf is just a contravariant functor to the category of abelian groups.

A sheaf is a presheaf along with two conditions:

1. ${s\in \mathscr{F}(U)}$ such that its image in ${\mathscr{F}(U_i)}$ is zero for all ${i}$ is zero.
2. If we have ${s_i\in\mathscr{F}(U_i)}$ which agree when restricted to ${U_i\cap U_j}$, then there exists ${s}$ which restricts to each of them.

4. Complex manifolds, Kahler metrics, differential and harmonic forms 2 – Cattani

Example 1.16 and A.1

In mathematics, we often try to reduce problems to linear algebra, because linear algebra is something that we understand very well.

If we set ${g_{\alpha\beta}=D(\phi_\alpha\circ\phi_\beta^{-1})\in \mathrm{GL}(n,\mathbb{R})}$. In fact, we get ${\left\{\left(\begin{array}{cc}A&-B\\ B&A\end{array}\right)\in \mathrm{GL}(2n,\mathbb{R})\right\}}$, and these are the changes of coordinates preserving the matrix ${\left(\begin{array}{cc}0&-I\\ I&0\end{array}\right)}$.

On a complex manifold ${M}$, it is possible to define a linear map ${J_p:T_pM\rightarrow T_pM}$ by ${J_p(\frac{\partial}{\partial x_i}=\frac{\partial }{\partial y_i}}$ and ${J_p(\frac{\partial }{\partial y_i})=-\frac{\partial }{\partial x_i}}$.

Then ${J_p^2=-id}$ and so ${J^2+I=0}$.

Now, take ${[T_pM]_\mathbb{C}=T_pM\otimes_\mathbb{R} \mathbb{C}=T_pM\oplus iT_pM}$. Call ${T'_p}$ and ${T''_p}$, the parts with eigenvalue ${i}$ and ${-i}$.

We can actually, for a complex manifold ${M}$, make ${T_pM}$ into a complex vector space, by defining ${(a+ib)*v=av+bJ_pv}$. And then, this vector space is isomorphic to ${T'_pM}$ by taking ${v}$ to ${v-iJv}$.

So what is the tangent bundle of ${\mathbb{P}^n}$? Let ${L(t)\in\mathbb{P}^n}$ a curve. Fix ${v\in L(p)=L}$. So the curves don’t depend on motion inside the line, thus we end up with ${T(\mathbb{P}^n)=\hom_\mathbb{C}(\mathcal{T},\mathbb{C}^{n+1}/\mathcal{T})}$, and by changing the lines to subspaces, the same is true of Grassmannians.

Now, we write ${[T^*_pM]_\mathbb{C}=T_p^{1,0}(M)\oplus T_p^{0,1}(M)}$ where the first has basis ${dz_i=dx_i+idy_i}$ and the second has basis ${d\bar{z}_i=dx_i-idy_i}$. Then, ${df=\sum_j df(\frac{\partial}{\partial z_j})dz_j+\sum_j df(\frac{\partial}{\partial \bar{z}_j}d\bar{z}_j}$.

We now set ${A^{p,q}(M)}$ to be the space of forms whose terms have ${dz_I\wedge d\bar{z}_{\bar{J}}}$ where ${|I|=p}$ and ${|\bar{J}|=q}$. Then the exterior derivative maps ${d:A^{p,q}(M)\rightarrow A^{p+1,q}(M)\oplus A^{p,q+1}(M)}$. So ${d=\partial+\bar{\partial}}$, both of which square to zero, and also ${\partial\bar{\partial}+\bar{\partial}\partial=0}$.

We then have that ${A^k(M,\mathbb{C})=\oplus_{p+q=k} A^{p,q}(M)}$. The Dolbeault cohomology, then, is ${H^{p,q}_{\bar{\partial}}(M)}$ is the cohomology of ${\bar{\partial}}$.

Inside of ${A^{p,0}(M)}$ is ${\Omega^p(M)}$, which is the collection of the forms with holomorphic, not merely ${C^\infty}$, coefficients.

Now we note that the Dolbealt cohomology is NOT diffeomorphism invariant. It depends on the complex structure of the manifold in question.