Manin’s Conjecture predicts the asymptotic formula for the counting function of rational points over number fields or global function fields. Our results for (non-algebraic) schemes and rigid analytic spaces were previously only known in dimensions ≤3 and ≤2, respectively, and our results for formal schemes and Berkovich spaces are completely new. This not only recovers previous results for complex varieties, complex algebraic spaces, and complex analytic spaces, but also greatly extends the scope of the relative minimal model program with scaling to a broader class of geometric spaces, including formal schemes, rigid analytic spaces, and Berkovich spaces, all in equal characteristic zero.
In this talk, I will discuss recent joint work with Shiji Lyu that establishes the relative minimal model program with scaling for excellent schemes, excellent algebraic spaces, and analytic spaces simultaneously in equal characteristic zero. Their result is now fundamental to our understanding of the birational classification of quasi-projective varieties and has numerous applications. In 2010, Birkar, Cascini, Hacon, and McKernan proved a relative version of the minimal model program for projective morphisms of complex quasi-projective varieties, called the relative minimal model program with scaling.
This talk is based on joint work with Sho Ejiri. In particular, we obtain the decomposition theorem for Frobenius splitting varieties with nef tangent bundle. In this talk, I will introduce the result that every ordinary variety with numerically flat tangent bundle is an etale quotient of an ordinary Abelian variety. The "flat” part of their theorem is a smooth projective variety with numerically flat tangent bundle. A positive characteristic analog of the above decomposition theorem was proved by Kanemitsu and Watanabe. The theorem states that, up to an etale cover, such a variety has a smooth fibration admitting a smooth algebraic fiber space over an abelian variety whose fibers are Fano varieties, so one can say that such a variety decomposes into the "positive” part and the "flat” part. Demailly, Peternell and Schneider established a decomposition theorem for a smooth projective complex variety with nef tangent bundle. If you drive, the most convenient public parking is in the pay lot whose entrance is on 34th Street between Market and Chestnut Streets.The positivity condition imposed on the tangent bundle of a smooth projective variety is known to restrict the geometric structure of the variety. Coming from the airport by train (about 15 minutes): the University City Rail Station is the second stop after you leave the airport. We are about a 15 minute walk from the main 30th Street Station and 5 minutes from the University City Rail Station at 32nd and Spruce (=South Street & Convention Avenue). Note 33rd Street runs one way north while Walnut runs one way west. The building is at 209 South 33rd Street (the Southeast corner of 33rd. The Mathematics Department Office is located on the fourth (top) floor of David Rittenhouse Laboratory ("DRL").
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