Let $X$ be a projective manifold, $\rho:\tilde X \to X$ its universal covering and $\rho^*: Vect (X) \to Vect(\tilde X)$ the pullback map for the isomorphism classes of vector bundles. This article establishes a connection between the properties of the pullback map $\rho^*$ and the properties of the function theory on $\tilde X$. We prove the following pivotal result: if a universal cover of a projective variety has no nonconstant holomorphic functions then the pullback map $\rho^*$ is almost an imbedding.

A natural Hasse-Schmidt derivation on the exterior algebra of a free module realizes the (small quantum) cohomology ring of the grassmannian $G_k(\mathbb{C}^n)$ as a ring of operators on the exterior algebra of a free module of rank $n$. Classical Pieri's formula can be interpreted as Leibniz's rule enjoyed by special Schubert cycles with respect to the wedge product.

The multifractal spectrum of a Borel measure $\mu$ in $\mathbb{R}^n$ is defined as \[ f_\mu(\alpha) = \dim_H\left\{x:\lim_{r\rightarrow 0} \frac{\log \mu(B(x,r))}{\log r}=\alpha\right\}. \] For self-similar measures under the open set condition the behaviour of this and related functions is well-understood ([CM92],[Ols95],[AP96]); the situation turns out to be very regular and is governed by the so-called ``multifractal formalism''. Recently there has been a lot of interest in understanding how much of the theory carries over to the overlapping case; most of the results obtained, however, apply only to a limited range of $\alpha$ (the ``left half'' of the spectrum). Here we carry out a complete study of the multifractal structure for a class of self-similar measures with overlap which includes the $3$-fold convolution of the Cantor measure. Among other things, we prove that the multifractal formalism fails for many of these measures, but it holds when taking a suitable restriction.

In [St], [WS], Stoll and Wong-Stoll established the Second Main Theorem of meromorphic maps $f: M \rightarrow {\Bbb P}^N({\Bbb C})$ intersecting hyperplanes, under the assumption that $f$ is linear non-degenerate, where $M$ is a $m$-dimensional affine algebraic manifold(the proof actually works for more general category of Stein parabolic manifolds). This paper deals with the degenerate case. Using P. Vojta's method, we show that there exists a finite union of proper linear subspaces of ${\Bbb P}^N({\Bbb C})$, depending only on the given hyperplanes, such that for every (possibly degenerate) meromorphic map $f: M \rightarrow {\Bbb P}^N({\Bbb C})$, if its image is not contained in that union, the inequality of Wong-Stoll's theorem still holds (without the ramification term). We also carefully examine the error terms appearing in the inequality.

The paper considers a relationship between the Chern numbers $K^2_X,\,c_2(X)$ of a smooth minimal surface $X$ of general type and the dimension of the space of infinitesimal deformations of $X$, i.e. $h^1(\Theta_X)$, where $\Theta_X$ is the holomorphic tangent bundle of $X$. We prove that if the ratio of the Chern numbers $\alpha(X) = \frac{c_2(X)}{K^2_X} \leq \frac38$ and $K_X$ is ample then $$h^1(\Theta_X) \leq 9(3c_2-K^2).$$ On the geometric side it is shown that a smooth surface of general type $X$ with $\alpha(X)\leq \frac38$ and $h^1(\Theta_X) \geq 3$ has two distinguished effective divisors $ F$ and $E$ such that $H^1(\Theta_X)$ admits a direct sum decomposition $H^1(\Theta_X) = V_1 \oplus V_0$, where $V_1$ is identified with a subspace of $H^0(\mathcal{O}_X (F))$ while $V_0$ is identified with a subspace of $H^0(\Theta_X \otimes \mathcal{O}_E (E))$. This gives a geometric interpretation of the cohomology classes in $H^1(\Theta_X)$ and allows to bound the dimension of $V_0$ (resp. $V_1$) in terms of geometry of the divisor $E$ (resp. $F$).

The main idea of the paper is to use the natural identification $$H^1(\Theta_X) = Ext^1(\Omega_X, \mathcal{O}_X)$$ where $\Omega_X$ is the holomorphic cotangent bundle of $X$. Then the "universal" extension gives rise to a certain vector bundle whose study constitutes the essential part of the paper.