Siegel varieties are locally symmetric varieties. They are important and interesting in algebraic geometry and number theory. We construct a canonical Hodge bundle on a Siegel variety so that the holomorphic tangent bundle can be embedded into the Hodge bundle; we obtain that the canonical Bergman metric on a Siegel variety is same as the induced Hodge metric and we describe asymptotic behavior of this unique K\"ahler-Einstein metric explicitly; depending on these properties and the uniformitarian of K\"ahler-Einstein manifold, we study extensions of the tangent bundle over any smooth toroidal compactification. We apply these results of Hodge bundles, to study dimension of Siegel cusp modular forms and general type for Siegel varieties

Principal component analysis (PCA) is one of the most commonly used statistical procedures with a wide range of applications. This paper considers both minimax and adaptive estimation of the principal subspace in the high dimensional setting. Under mild technical conditions, we first establish optimal rates of convergence for estimating the principal subspace which sharp with respect to all the parameters, thus providing a complete character ization of the difficulty of the estimation problem in term of the convergence rate. The lower bound is obtained by calculating the local metric entropy an application of Fano's lemma. The rate optimal estimator is constructed using aggregation, which, however, might not be computationally feasible. We then introduce an adaptive procedure for estimating the principal sub space which is fully data driven and can be computed efficiently. It is shown that the estimator attains the optimal rates of convergence simultaneously over a large collection of the parameter spaces. A key idea in our construc tion is a reduction scheme which reduces the sparse PCA problem to a high dimensional multivariate regression problem. This method is potentially useful for other related

We study smooth toroidal compactifications of Siegel varieties thoroughly from the viewpoints of mixed Hodge theory and K\"ahler-Einstein metric. We observe that any cusp of a Siegel space can be identified as a set of certain weight one polarized mixed Hodge structures. We then study the infinity boundary divisors of toroidal compactifications, and obtain a global volume form formula of an arbitrary smooth Siegel variety $\sA_{g,\Gamma}(g>1)$ with a smooth toroidal compactification $\overline{\sA}_{g,\Gamma}$ such that $D_\infty:=\overline{\sA}_{g,\Gamma}\setminus \sA_{g,\Gamma}$ is normal crossing. We use this volume form formula to show that the unique group-invariant K\"ahler-Einstein metric on $\sA_{g,\Gamma}$ endows some restraint combinatorial conditions for all smooth toroidal compactifications of $\sA_{g,\Gamma}.$ Again using the volume form formula, we study the asymptotic behaviour of logarithmical canonical line bundle on any smooth toroidal compactification of $\sA_{g,\Gamma}$ carefully and we obtain that the logarithmical canonical bundle degenerate sharply even though it is big and numerically effective.

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In this paper, we study the transonic shock problem for the full compressible Euler system in a general two-dimensional de Laval nozzle as proposed in Courant and Friedrichs (Supersonic flow and shock waves, Interscience, New York, 1948): given the appropriately large exit pressure pe(x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes pe(x). We solve this problem completely for a general class of de Laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible Euler system. The problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic–elliptic system. One of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted Hölder space with low regularities for a second order quasilinear elliptic equation with a free parameter (the position of the shock curve at one wall of the nozzle) and non-local terms involving the trace on the shock of the first order derivatives of the unknown function.