A conic bundle is a contraction $X\to Z$ between normal varieties of relative dimension $1$ such that $-K_X$ is relatively ample. We prove a conjecture of Shokurov that predicts that if $X\to Z$ is a conic bundle such that X has canonical singularities and Z is $\mathbb {Q}$ -Gorenstein, then Z is always $\frac {1}{2}$ -lc, and the multiplicities of the fibres over codimension $1$ points are bounded from above by $2$ . Both values $\frac {1}{2}$ and $2$ are sharp. This is achieved by solving a more general conjecture of Shokurov on singularities of bases of lc-trivial fibrations of relative dimension $1$ with canonical singularities.

Let $\Gamma $ be a finite set, and $X\ni x$ a fixed kawamata log terminal germ. For any lc germ $(X\ni x,B:=\sum _{i} b_iB_i)$ , such that $b_i\in \Gamma $ , Nakamura’s conjecture, which is equivalent to the ascending chain condition conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor E over $X\ni x$ , such that $a(E,X,B)=\mathrm {mld}(X\ni x,B)$ , and $a(E,X,0)$ is bounded from above. We extend Nakamura’s conjecture to the setting that $X\ni x$ is not necessarily fixed and $\Gamma $ satisfies the descending chain condition, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of $a(E,X,0)$ for any such E.

In this article we will mainly introduce the basic ideas of Witten deformation, which were first introduced by Witten on [W], and some applications of it. The first part of this article mainly focuses on deformation of Dirac operators and some important analytic facts about the deformed Dirac operators. In the second part of this article some applications of Witten deformation will be given (mainly referring to [WZ]), to be more specific, an analytic proof of Poincaré-Hopf index theorem and Real Morse nequilities will be given. Also we will use Witten deformation to prove that the Thom Smale complex is quasiisomorphism to the de-Rham complex (Witten suggested that Thom Smale complex can be recovered from his deformation in [W] and his suggestion was first realized by Helffer and Sjöstrand [HS], the proof in this article is given by Bismut and Zhang in [BZ1]). And in the last part an analytic proof of Atiyah vanishing theorem (referring to [Z1], [Z2]) via Witten deformation will be given.