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Let $(G,X)$ be a Shimura pair of Hodge type such that $G$ is theMumford--Tate group of some elements of $X$. We assume that for each simplefactor $G_0$ of $G^{\ad}$ there exists a simple factor of $G_{0\dbR}$ which iscompact. Let $N\Ge 3$. We show that for many compact open subgroups $K$ of$G(\dbA_f)$, the Shimura variety $\Sh(G,X)/K$ has a projective integral model$\scrN$ over $\dbZ[{1\over N}]$ which is a finite scheme over a certain Mumfordmoduli scheme $\scrA_{g,1,N}$. Equivalently, we show that if $A$ is an abelianvariety over a number field and if the Mumford--Tate group of $A_{\dbC}$ is$G$, then $A$ has potentially good reduction everywhere. The last resultrepresents significant progress towards the proof of a conjecture of Morita. If$\scrN$ is smooth over $\dbZ[{1\over N}]$, then it is a N\'eron model of itsgeneric fibre. In this way one gets in arbitrary mixed characteristic, the veryfirst examples of general nature of projective N\'eron models whose genericfibres are not finite schemes over abelian varieties.Comment: 24 pages, final version accepted for publication in Crell

Projective integral models of Shimura varieties of Hodge type with compact factors