Problem 1 (yau-contest 2019)
Section: Geometry and Topology

Let $\mathrm{Conf}_n$ be the following submanifold of $\mathbb{C}^n$:
$$\mathrm{Conf}_n=\{(z_1,z_2,\cdots,z_n)\in\mathbb{C}^n \mid z_i\ne z_j\text{ for any } i\ne j\}.$$
For every pair $(i,j)$ with $i\ne j$, we define the complex valued 1-form
$$\omega_{ij}:=\frac{dz_i-dz_j}{z_i-z_j}.$$
(a) Show that for any $i\ne j$, $\omega_{ij}$ represents a non-zero de Rham cohomology class in $H^1(\mathrm{Conf}_n,\mathbb{C})$.
(b) Show that for any pair-wise distinct indices $i,j,k$,
$$\omega_{ij}\wedge\omega_{jk}+\omega_{jk}\wedge\omega_{ki}+\omega_{ki}\wedge\omega_{ij}=0.$$

Problem 2 (yau-contest 2019)
Section: Geometry and Topology

Let $M$ be a compact oriented manifold of (real) dimension $4$. Consider the following symmetric bilinear form on $H^2(M)$
$$H^2(M)\times H^2(M)\to \mathbb{R}, \qquad ([\alpha],[\beta])\mapsto \int_M \alpha\wedge \beta.$$
Let $\tau(M)$ be the signature of this bilinear form, i.e. the number of positive eigenvalues minus the number of negative eigenvalues. Compute $\tau(M)$ for $M=S^4$, $\mathbb{CP}^2$ and $S^2\times S^2$.

Problem 3 (yau-contest 2019)
Section: Geometry and Topology

Let $X=\mathbb{R}^4/\sim$, where
$$\begin{aligned}
(x_1,x_2,x_3,x_4)&\sim (x_1,x_2+1,x_3,x_4),\\
(x_1,x_2,x_3,x_4)&\sim (x_1,x_2,x_3,x_4+1),\\
(x_1,x_2,x_3,x_4)&\sim (x_1+1,x_2,x_3,x_4),\\
(x_1,x_2,x_3,x_4)&\sim (x_1,x_2+x_4,x_3+1,x_4).
\end{aligned}$$
Compute $H_1(X,\mathbb{Z})$.

Problem 4 (yau-contest 2019)
Section: Geometry and Topology

Let $E$ be a vector bundle over a smooth manifold $M$. Let $\nabla^E$ be a connection on $E$ and $R^E\in \Omega^2(M,\mathrm{End}(E))$ be its curvature tensor. For any polynomial $f(x)=a_0+a_1x+a_2x^2+\cdots+a_n x^n$, we denote
$$f(R^E)=a_0+a_1 R^E+a_2 (R^E)^2+\cdots+a_n (R^E)^n\in \Omega^*(M,\mathrm{End}(E)).$$
Here $(R^E)^k\in \Omega^{2k}(M,\mathrm{End}(E))$ is the $k$-th wedge product on forms combined with matrix multiplications on $\mathrm{End}(E)$.
(a) Show that the differential form $\operatorname{tr}\big[f(R^E)\big]\in \Omega^*(M)$ is closed
$$d\,\operatorname{tr}\big[f(R^E)\big]=0.$$
Here $\operatorname{tr}$ is the trace on $\mathrm{End}(E)$.
(b) Let $\nabla^E,\,\tilde{\nabla}^E$ be two connections on $E$ and $R^E,\,\tilde{R}^E$ be the corresponding curvature tensors. Show that there exists a differential form $\omega\in\Omega^*(M)$ such that
$$\operatorname{tr}\big[f(R^E)\big]-\operatorname{tr}\big[f(\tilde{R}^E)\big]=d\omega.$$

Problem 5 (yau-contest 2019)
Section: Geometry and Topology

(a) Let $u$ be a smooth function over a Riemannian manifold $(M,g)$. Prove the following Bochner's formula
$$\tfrac12\,\Delta\lVert\nabla u\rVert^2=\lVert\nabla\nabla u\rVert^2+\mathrm{Ric}(\nabla u,\nabla u)+g(\nabla\Delta u,\nabla u),$$
where $\Delta$ is the Laplacian and $\lVert\bullet\rVert^2=g(\bullet,\bullet)$.
(b) Let $(S^2,g)$ be the standard unit sphere and $E$ be a constant. Show that the only smooth positive solution to
$$\Delta\ln f+E f^2=1$$
is $f=\dfrac{1}{A+\phi}$ where $A$ is a constant and $\phi$ is some first eigenfunction of $S^2$.