Let (X,\Delta) be a log canonical pair with real coefficients. Suppose that K_X+\Delta is PSEF. then we have

• Minimal Model Conjecture (MMC): there exists a minimal model of (X,\Delta)

• Non-vanishing Conjecture (NVC): K_X+\Delta is effective..In this series of talks, we plan to explain the following three results:

I. (NVC) is equivalent to (NVC) for smooth projective varieties.

II. (NVC) implies (MMC)

III. Existence of log canonical flips

In the first talk, we will talk about topic I. If time permits, we will also discuss a recent non-vanishing result when v(K_X)=\dim X-1 or 1.

References:

1. K. Hashizume, On the non-vanishing conjecture and existence of log minimal models

2. V. Lazic, T. Peternell, Abundance for varieties with manydifferential forms; Rationally connected varieties - on a conjecture of Mumford

3. C. Birkar, On existence of log minimal models II,

4. C. Birkar, Existence of log canonical flips and a special LMMP.

5. C. Hacon, C. Xu, Existence of log canonical closures