| Description |
We study the geometry of higher dimensional algebraic varieties according to the dichotomy of Kodaira dimensions, negative or nonnegative, and the corresponding pictures in the Minimal Model Conjecture. On the one hand, according to the Minimal Model Conjecture, a variety with nonnegative Kodaira dimension is birational to a minimal model, which has semiample canonical class. This has been done if dimension is less than or equal to three and for varieties of general type in any dimension. In general, the Minimal Model Conjecture is still open. As the first result, we show that the Minimal Model Conjecture for varieties with nonnegative Kodaira dimensions follows from the Minimal Model Conjecture for varieties with Kodaira dimension zero. In particular, the Minimal Model Conjecture is reduced to the Minimal Model Conjecture for varieties of Kodaira dimension zero and the Nonvanishing Conjecture. On the other hand, according to the Minimal Model Conjecture, Fano varieties of Picard number one are the building blocks for varieties with negative Kodaira dimension. The set of mildly singular Fano varieties of given dimension is expected to be bounded. As a second result, we exhibit an effective upper bound of the anticanonical volume for the set of e-klt Q-factorial log Q-Fano threefolds with Picard number one. This result is related to a conjecture open in dimension three and higher, the Borisov-Alexeev-Borisov Conjecture, which asserts boundedness of the set of e-klt log Q-Fano varieties. In the end of this dissertation, we include some partial results of the Nonvanishing Conjecture in the minimal model program. The minimal model program is developed to attack the Minimal Model Conjecture. The Nonvanishing Conjecture is one of the most important missing ingredient for completing the minimal model program. |
